Bernoulli Articles (Project Euclid)
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A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables
http://projecteuclid.org/euclid.bj/1274821072
<strong>Michael V. Boutsikas</strong>, <strong>Eutichia Vaggelatou</strong><p><strong>Source: </strong>Bernoulli, Volume 16, Number 2, 301--330.</p><p><strong>Abstract:</strong><br/>
Let X 1 , X 2 , …, X n be a sequence of independent or locally dependent random variables taking values in ℤ + . In this paper, we derive sharp bounds, via a new probabilistic method, for the total variation distance between the distribution of the sum ∑ i =1 n X i and an appropriate Poisson or compound Poisson distribution. These bounds include a factor which depends on the smoothness of the approximating Poisson or compound Poisson distribution. This “smoothness factor” is of order O( σ −2 ), according to a heuristic argument, where σ 2 denotes the variance of the approximating distribution. In this way, we offer sharp error estimates for a large range of values of the parameters. Finally, specific examples concerning appearances of rare runs in sequences of Bernoulli trials are presented by way of illustration.
</p>projecteuclid.org/euclid.bj/1274821072_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTOn predictive density estimation for location families under integrated absolute error losshttp://projecteuclid.org/euclid.bj/1495505090<strong>Tatsuya Kubokawa</strong>, <strong>Éric Marchand</strong>, <strong>William E. Strawderman</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3197--3212.</p><p><strong>Abstract:</strong><br/>
This paper is concerned with estimating a predictive density under integrated absolute error ($L_{1}$) loss. Based on a spherically symmetric observable $X\sim p_{X}(\Vert x-\mu\Vert^{2})$, $x,\mu \in \mathbb{R}^{d}$, we seek to estimate the (unimodal) density of $Y\sim q_{Y}(\Vert y-\mu \Vert^{2})$, $y\in \mathbb{R}^{d}$. We focus on the benchmark (and maximum likelihood for unimodal $p$) plug-in density estimator $q_{Y}(\Vert y-X\Vert^{2})$ and, for $d\geq 4$, we establish its inadmissibility, as well as provide plug-in density improvements, as measured by the frequentist risk taken with respect to $X$. Sharper results are obtained for the subclass of scale mixtures of normal distributions which include the normal case. The findings rely on the duality between the predictive density estimation problem with a point estimation problem of estimating $\mu$ under a loss which is a concave function of $\Vert \hat{\mu}-\mu\Vert^{2}$, Stein estimation results and techniques applicable to such losses, and further properties specific to scale mixtures of normal distributions. Finally, (i) we address univariate implications for cases where there exist parametric restrictions on $\mu$, and (ii) we show quite generally for logconcave $q_{Y}$ that improvements on the benchmark mle can always be found among the scale expanded predictive densities $\frac{1}{c}q_{Y}(\frac{(y-x)^{2}}{c^{2}})$, with $c-1$ positive but not too large.
</p>projecteuclid.org/euclid.bj/1495505090_20170522220508Mon, 22 May 2017 22:05 EDTTransportation and concentration inequalities for bifurcating Markov chainshttp://projecteuclid.org/euclid.bj/1495505091<strong>S. Valère Bitseki Penda</strong>, <strong>Mikael Escobar-Bach</strong>, <strong>Arnaud Guillin</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3213--3242.</p><p><strong>Abstract:</strong><br/>
We investigate the transportation inequality for bifurcating Markov chains which are a class of processes indexed by a regular binary tree. Fitting well models like cell growth when each individual gives birth to exactly two offsprings, we use transportation inequalities to provide useful concentration inequalities. We also study deviation inequalities for the empirical means under relaxed assumptions on the Wasserstein contraction for the Markov kernels. Applications to bifurcating nonlinear autoregressive processes are considered for point-wise estimates of the non-linear autoregressive function.
</p>projecteuclid.org/euclid.bj/1495505091_20170522220508Mon, 22 May 2017 22:05 EDTPólya urn schemes with infinitely many colorshttp://projecteuclid.org/euclid.bj/1495505092<strong>Antar Bandyopadhyay</strong>, <strong>Debleena Thacker</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3243--3267.</p><p><strong>Abstract:</strong><br/>
In this work, we introduce a class of balanced urn schemes with infinitely many colors indexed by ${\mathbb{Z} }^{d}$, where the replacement schemes are given by the transition matrices associated with bounded increment random walks. We show that the color of the $n$th selected ball follows a Gaussian distribution on ${\mathbb{R} }^{d}$ after ${\mathcal{O} }(\log n)$ centering and ${\mathcal{O} }(\sqrt{\log n})$ scaling irrespective of whether the underlying walk is null recurrent or transient. We also provide finer asymptotic similar to local limit theorems for the expected configuration of the urn. The proofs are based on a novel representation of the color of the $n$th selected ball as “slowed down” version of the underlying random walk.
</p>projecteuclid.org/euclid.bj/1495505092_20170522220508Mon, 22 May 2017 22:05 EDTApproximate local limit theorems with effective rate and application to random walks in random sceneryhttp://projecteuclid.org/euclid.bj/1495505093<strong>Rita Giuliano</strong>, <strong>Michel Weber</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3268--3310.</p><p><strong>Abstract:</strong><br/>
We show that the Bernoulli part extraction method can be used to obtain approximate forms of the local limit theorem for sums of independent lattice valued random variables, with effective error term. That is with explicit parameters and universal constants. We also show that our estimates allow us to recover Gnedenko and provide a version with effective bounds of Gamkrelidze’s local limit theorem. We further establish by this method a local limit theorem with effective remainder for random walks in random scenery.
</p>projecteuclid.org/euclid.bj/1495505093_20170522220508Mon, 22 May 2017 22:05 EDTOn Stein’s method for products of normal random variables and zero bias couplingshttp://projecteuclid.org/euclid.bj/1495505094<strong>Robert E. Gaunt</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3311--3345.</p><p><strong>Abstract:</strong><br/>
In this paper, we extend Stein’s method to the distribution of the product of $n$ independent mean zero normal random variables. A Stein equation is obtained for this class of distributions, which reduces to the classical normal Stein equation in the case $n=1$. This Stein equation motivates a generalisation of the zero bias transformation. We establish properties of this new transformation, and illustrate how they may be used together with the Stein equation to assess distributional distances for statistics that are asymptotically distributed as the product of independent central normal random variables. We end by proving some product normal approximation theorems.
</p>projecteuclid.org/euclid.bj/1495505094_20170522220508Mon, 22 May 2017 22:05 EDTWeak convergence of empirical copula processes indexed by functionshttp://projecteuclid.org/euclid.bj/1495505095<strong>Dragan Radulović</strong>, <strong>Marten Wegkamp</strong>, <strong>Yue Zhao</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3346--3384.</p><p><strong>Abstract:</strong><br/>
Weak convergence of the empirical copula process indexed by a class of functions is established. Two scenarios are considered in which either some smoothness of these functions or smoothness of the underlying copula function is required.
A novel integration by parts formula for multivariate, right-continuous functions of bounded variation, which is perhaps of independent interest, is proved. It is a key ingredient in proving weak convergence of a general empirical process indexed by functions of bounded variation.
</p>projecteuclid.org/euclid.bj/1495505095_20170522220508Mon, 22 May 2017 22:05 EDTSieve maximum likelihood estimation for a general class of accelerated hazards models with bundled parametershttp://projecteuclid.org/euclid.bj/1495505096<strong>Xingqiu Zhao</strong>, <strong>Yuanshan Wu</strong>, <strong>Guosheng Yin</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3385--3411.</p><p><strong>Abstract:</strong><br/>
In semiparametric hazard regression, nonparametric components may involve unknown regression parameters. Such intertwining effects make model estimation and inference much more difficult than the case in which the parametric and nonparametric components can be separated out. We study the sieve maximum likelihood estimation for a general class of hazard regression models, which include the proportional hazards model, the accelerated failure time model, and the accelerated hazards model. Coupled with the cubic B-spline, we propose semiparametric efficient estimators for the parameters that are bundled inside the nonparametric component. We overcome the challenges due to intertwining effects of the bundled parameters, and establish the consistency and asymptotic normality properties of the estimators. We carry out simulation studies to examine the finite-sample properties of the proposed method, and demonstrate its efficiency gain over the conventional estimating equation approach. For illustration, we apply our proposed method to a study of bone marrow transplantation for patients with acute leukemia.
</p>projecteuclid.org/euclid.bj/1495505096_20170522220508Mon, 22 May 2017 22:05 EDTIntegrated empirical processes in $L^{p}$ with applications to estimate probability metricshttp://projecteuclid.org/euclid.bj/1495505097<strong>Javier Cárcamo</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3412--3436.</p><p><strong>Abstract:</strong><br/>
We discuss the convergence in distribution of the $r$-fold (reverse) integrated empirical process in the space $L^{p}$, for $1\le p\le\infty$. In the case $1\le p<\infty$, we find the necessary and sufficient condition on a positive random variable $X$ so that this process converges weakly in $L^{p}$. This condition defines a Lorentz space and can be also characterized in terms of several integrability conditions related to the process $\{(X-t)^{r}_{+}:t\ge0\}$. For $p=\infty$, we obtain an integrability requirement on $X$ guaranteeing the convergence of the integrated empirical process. In particular, these results imply a limit theorem for the stop-loss distance between the empirical and the true distribution. As an application, we derive the asymptotic distribution of an estimator of the Zolotarev distance between two probability distributions. The connections of the involved processes with equilibrium distributions and stochastic integrals with respect to the Brownian bridge are also briefly explained.
</p>projecteuclid.org/euclid.bj/1495505097_20170522220508Mon, 22 May 2017 22:05 EDTEfficiency and bootstrap in the promotion time cure modelhttp://projecteuclid.org/euclid.bj/1495505098<strong>François Portier</strong>, <strong>Anouar El Ghouch</strong>, <strong>Ingrid Van Keilegom</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3437--3468.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider a semiparametric promotion time cure model and study the asymptotic properties of its nonparametric maximum likelihood estimator (NPMLE). First, by relying on a profile likelihood approach, we show that the NPMLE may be computed by a single maximization over a set whose dimension equals the dimension of the covariates plus one. Next, using $Z$-estimation theory for semiparametric models, we derive the asymptotics of both the parametric and nonparametric components of the model and show their efficiency. We also express the asymptotic variance of the estimator of the parametric component. Since the variance is difficult to estimate, we develop a weighted bootstrap procedure that allows for a consistent approximation of the asymptotic law of the estimators. As in the Cox model, it turns out that suitable tools are the martingale theory for counting processes and the infinite dimensional $Z$-estimation theory. Finally, by means of simulations, we show the accuracy of the bootstrap approximation.
</p>projecteuclid.org/euclid.bj/1495505098_20170522220508Mon, 22 May 2017 22:05 EDTNon-central limit theorems for random fields subordinated to gamma-correlated random fieldshttp://projecteuclid.org/euclid.bj/1495505099<strong>Nikolai Leonenko</strong>, <strong>M. Dolores Ruiz-Medina</strong>, <strong>Murad S. Taqqu</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3469--3507.</p><p><strong>Abstract:</strong><br/>
A reduction theorem is proved for functionals of Gamma-correlated random fields with long-range dependence in $d$-dimensional space. As a particular case, integrals of non-linear functions of chi-squared random fields, with Laguerre rank being equal to one and two, are studied. When the Laguerre rank is equal to one, the characteristic function of the limit random variable, given by a Rosenblatt-type distribution, is obtained. When the Laguerre rank is equal to two, a multiple Wiener–Itô stochastic integral representation of the limit distribution is derived and an infinite series representation, in terms of independent random variables, is obtained for the limit.
</p>projecteuclid.org/euclid.bj/1495505099_20170522220508Mon, 22 May 2017 22:05 EDTAsymptotic expansions and hazard rates for compound and first-passage distributionshttp://projecteuclid.org/euclid.bj/1495505100<strong>Ronald W. Butler</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3508--3536.</p><p><strong>Abstract:</strong><br/>
A general theory which provides asymptotic tail expansions for density, survival, and hazard rate functions is developed for both absolutely continuous and integer-valued distributions. The expansions make use of Tauberian theorems which apply to moment generating functions (MGFs) with boundary singularities that are of gamma-type or log-type. Standard Tauberian theorems from Feller [ An Introduction to Probability Theory and Its Applications II (1971) Wiley] can provide a limited theory but these theorems do not suffice in providing a complete theory as they are not capable of explaining tail behaviour for compound distributions and other complicated distributions which arise in stochastic modelling settings. Obtaining such a complete theory for absolutely continuous distributions requires introducing new “Ikehara” conditions based upon Tauberian theorems whose development and application have been largely confined to analytic number theory. For integer-valued distributions, a complete theory is developed by applying Darboux’s theorem used in analytic combinatorics. Characterizations of asymptotic hazard rates for both absolutely continuous and integer-valued distributions are developed in conjunction with these expansions. The main applications include the ruin distribution in the Cramér–Lundberg and Sparre Andersen models, more general classes of compound distributions, and first-passage distributions in finite-state semi-Markov processes. Such first-passage distributions are shown to have exponential-like/geometric-like tails which mimic the behaviour of first-passage distributions in Markov processes even though the holding-time MGFs involved with such semi-Markov processes are typically not rational.
</p>projecteuclid.org/euclid.bj/1495505100_20170522220508Mon, 22 May 2017 22:05 EDTEfficient Bayesian estimation and uncertainty quantification in ordinary differential equation modelshttp://projecteuclid.org/euclid.bj/1495505101<strong>Prithwish Bhaumik</strong>, <strong>Subhashis Ghosal</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3537--3570.</p><p><strong>Abstract:</strong><br/>
Often the regression function is specified by a system of ordinary differential equations (ODEs) involving some unknown parameters. Typically analytical solution of the ODEs is not available, and hence likelihood evaluation at many parameter values by numerical solution of equations may be computationally prohibitive. Bhaumik and Ghosal ( Electron. J. Stat. 9 (2015) 3124–3154) considered a Bayesian two-step approach by embedding the model in a larger nonparametric regression model, where a prior is put through a random series based on B-spline basis functions. A posterior on the parameter is induced from the regression function by minimizing an integrated weighted squared distance between the derivative of the regression function and the derivative suggested by the ODEs. Although this approach is computationally fast, the Bayes estimator is not asymptotically efficient. In this paper, we suggest a modification of the two-step method by directly considering the distance between the function in the nonparametric model and that obtained from a four stage Runge–Kutta (RK$4$) method. We also study the asymptotic behavior of the posterior distribution of $\mathbf{\theta}$ based on an approximate likelihood obtained from an RK$4$ numerical solution of the ODEs. We establish a Bernstein–von Mises theorem for both methods which assures that Bayesian uncertainty quantification matches with the frequentist one and the Bayes estimator is asymptotically efficient.
</p>projecteuclid.org/euclid.bj/1495505101_20170522220508Mon, 22 May 2017 22:05 EDTFractional Brownian motion satisfies two-way crossinghttp://projecteuclid.org/euclid.bj/1495505102<strong>Rémi Peyre</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3571--3597.</p><p><strong>Abstract:</strong><br/>
We prove the following result: For $(Z_{t})_{t\in\mathbf{R}}$ a fractional Brownian motion with arbitrary Hurst parameter, for any stopping time $\tau$, there exist arbitrarily small $\varepsilon>0$ such that $Z_{\tau+\varepsilon}<Z_{\tau}$, with asymptotic behaviour when $\varepsilon\searrow0$ satisfying a bound of iterated logarithm type. As a consequence, fractional Brownian motion satisfies the “two-way crossing” property, which has important applications in financial mathematics.
</p>projecteuclid.org/euclid.bj/1495505102_20170522220508Mon, 22 May 2017 22:05 EDTAdaptive estimation for bifurcating Markov chainshttp://projecteuclid.org/euclid.bj/1495505103<strong>S. Valère Bitseki Penda</strong>, <strong>Marc Hoffmann</strong>, <strong>Adélaïde Olivier</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3598--3637.</p><p><strong>Abstract:</strong><br/>
In a first part, we prove Bernstein-type deviation inequalities for bifurcating Markov chains (BMC) under a geometric ergodicity assumption, completing former results of Guyon and Bitseki Penda, Djellout and Guillin. These preliminary results are the key ingredient to implement nonparametric wavelet thresholding estimation procedures: in a second part, we construct nonparametric estimators of the transition density of a BMC, of its mean transition density and of the corresponding invariant density, and show smoothness adaptation over various multivariate Besov classes under $L^{p}$-loss error, for $1\leq p<\infty$. We prove that our estimators are (nearly) optimal in a minimax sense. As an application, we obtain new results for the estimation of the splitting size-dependent rate of growth-fragmentation models and we extend the statistical study of bifurcating autoregressive processes.
</p>projecteuclid.org/euclid.bj/1495505103_20170522220508Mon, 22 May 2017 22:05 EDTA proof of the Shepp–Olkin entropy concavity conjecturehttp://projecteuclid.org/euclid.bj/1495505104<strong>Erwan Hillion</strong>, <strong>Oliver Johnson</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3638--3649.</p><p><strong>Abstract:</strong><br/>
We prove the Shepp–Olkin conjecture, which states that the entropy of the sum of independent Bernoulli random variables is concave in the parameters of the individual random variables. Our proof refines an argument previously presented by the same authors, which resolved the conjecture in the monotonic case (where all the parameters are simultaneously increasing). In fact, we show that the monotonic case is the worst case, using a careful analysis of concavity properties of the derivatives of the probability mass function. We propose a generalization of Shepp and Olkin’s original conjecture, to consider Rényi and Tsallis entropies.
</p>projecteuclid.org/euclid.bj/1495505104_20170522220508Mon, 22 May 2017 22:05 EDTThe failure of the profile likelihood method for a large class of semi-parametric modelshttp://projecteuclid.org/euclid.bj/1495505105<strong>Eric Beutner</strong>, <strong>Laurent Bordes</strong>, <strong>Laurent Doyen</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3650--3684.</p><p><strong>Abstract:</strong><br/>
We consider a semi-parametric model for recurrent events. The model consists of an unknown hazard rate function, the infinite-dimensional parameter of the model, and a parametrically specified effective age function. We will present a condition on the family of effective age functions under which the profile likelihood function evaluated at the parameter vector $\mathbf{{\theta}}$, say, exceeds the profile likelihood function evaluated at the parameter vector $\tilde{\boldsymbol {\theta}}$, say, with probability $p$. From this we derive a condition under which profile likelihood inference for the finite-dimensional parameter of the model leads to inconsistent estimates. Examples will be presented. In particular, we will provide an example where the profile likelihood function is monotone with probability one regardless of the true data generating process. We also discuss the relation of our results to other semi-parametric models like the accelerated failure time model and Cox’s proportional hazards model.
</p>projecteuclid.org/euclid.bj/1495505105_20170522220508Mon, 22 May 2017 22:05 EDTSome monotonicity properties of parametric and nonparametric Bayesian banditshttp://projecteuclid.org/euclid.bj/1495505106<strong>Yaming Yu</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3685--3710.</p><p><strong>Abstract:</strong><br/>
One of two independent stochastic processes (arms) is to be selected at each of $n$ stages. The selection is sequential and depends on past observations as well as the prior information. The objective is to maximize the expected future-discounted sum of the $n$ observations. We study structural properties of this classical bandit problem, in particular how the maximum expected payoff and the optimal strategy vary with the priors, in two settings: (a) observations from each arm have an exponential family distribution and different arms are assigned independent conjugate priors; (b) observations from each arm have a nonparametric distribution and different arms are assigned independent Dirichlet process priors. In both settings, we derive results of the following type: (i) for a particular arm and a fixed prior weight, the maximum expected payoff increases as the prior mean yield increases; (ii) for a fixed prior mean yield, the maximum expected payoff increases as the prior weight decreases. Specializing to the one-armed bandit, the second result captures the intuition that, given the same immediate payoff, the less one knows about an arm, the more desirable it becomes because there remains more information to be gained when selecting that arm. In the parametric case, our results extend those of ( Ann. Statist. 20 (1992) 1625–1636) concerning Bernoulli and normal bandits (see also (In Time Series and Related Topics (2006) pp. 284–294 IMS)). In the nonparametric case, we extend those of ( Ann. Statist. 13 (1985) 1523–1534). A key tool in the derivation is stochastic orders.
</p>projecteuclid.org/euclid.bj/1495505106_20170522220508Mon, 22 May 2017 22:05 EDTAccelerated Gibbs sampling of normal distributions using matrix splittings and polynomialshttp://projecteuclid.org/euclid.bj/1495505107<strong>Colin Fox</strong>, <strong>Albert Parker</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3711--3743.</p><p><strong>Abstract:</strong><br/>
Standard Gibbs sampling applied to a multivariate normal distribution with a specified precision matrix is equivalent in fundamental ways to the Gauss–Seidel iterative solution of linear equations in the precision matrix. Specifically, the iteration operators, the conditions under which convergence occurs, and geometric convergence factors (and rates) are identical. These results hold for arbitrary matrix splittings from classical iterative methods in numerical linear algebra giving easy access to mature results in that field, including existing convergence results for antithetic-variable Gibbs sampling, REGS sampling, and generalizations. Hence, efficient deterministic stationary relaxation schemes lead to efficient generalizations of Gibbs sampling. The technique of polynomial acceleration that significantly improves the convergence rate of an iterative solver derived from a symmetric matrix splitting may be applied to accelerate the equivalent generalized Gibbs sampler. Identicality of error polynomials guarantees convergence of the inhomogeneous Markov chain, while equality of convergence factors ensures that the optimal solver leads to the optimal sampler. Numerical examples are presented, including a Chebyshev accelerated SSOR Gibbs sampler applied to a stylized demonstration of low-level Bayesian image reconstruction in a large 3-dimensional linear inverse problem.
</p>projecteuclid.org/euclid.bj/1495505107_20170522220508Mon, 22 May 2017 22:05 EDTSimulation of hitting times for Bessel processes with non-integer dimensionhttp://projecteuclid.org/euclid.bj/1495505108<strong>Madalina Deaconu</strong>, <strong>Samuel Herrmann</strong>. <p><strong>Source: </strong>Bernoulli, Volume 23, Number 4B, 3744--3771.</p><p><strong>Abstract:</strong><br/>
In this paper, we complete and improve the study of the simulation of the hitting times of some given boundaries for Bessel processes. These problems are of great interest in many application fields as finance and neurosciences. In a previous work ( Ann. Appl. Probab. 23 (2013) 2259–2289), the authors introduced a new method for the simulation of hitting times for Bessel processes with integer dimension. The method, called walk on moving spheres algorithm (WoMS), was based mainly on the explicit formula for the distribution of the hitting time and on the connection between the Bessel process and the Euclidean norm of the Brownian motion. This method does not apply anymore for a non-integer dimension. In this paper we consider the simulation of the hitting time of Bessel processes with non-integer dimension $\delta\geq1$ and provide a new algorithm by using the additivity property of the laws of squared Bessel processes. We split each simulation step of the algorithm in two parts: one is using the integer dimension case and the other one considers hitting time of a Bessel process starting from zero.
</p>projecteuclid.org/euclid.bj/1495505108_20170522220508Mon, 22 May 2017 22:05 EDTDirichlet curves, convex order and Cauchy distributionhttp://projecteuclid.org/euclid.bj/1501142434<strong>Gérard Letac</strong>, <strong>Mauro Piccioni</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 1--29.</p><p><strong>Abstract:</strong><br/>
If $\alpha$ is a probability on $\mathbb{R}^{d}$ and $t>0$, the Dirichlet random probability $P_{t}\sim\mathcal{D}(t\alpha)$ is such that for any measurable partition $(A_{0},\ldots,A_{k})$ of $\mathbb{R}^{d}$ the random variable $(P_{t}(A_{0}),\ldots,P_{t}(A_{k}))$ is Dirichlet distributed with parameters $(t\alpha(A_{0}),\ldots,t\alpha(A_{k}))$. If $\int_{\mathbb{R}^{d}}\log(1+\Vert x\Vert )\alpha(dx)<\infty$ the random variable $\int_{\mathbb{R}^{d}}xP_{t}(dx)$ of $\mathbb{R}^{d}$ does exist: let $\mu(t\alpha)$ be its distribution. The Dirichlet curve associated to the probability $\alpha$ is the map $t\mapsto\mu(t\alpha)$. It has simple properties like $\lim_{t\searrow0}\mu(t\alpha)=\alpha$ and $\lim_{t\rightarrow\infty}\mu(t\alpha)=\delta_{m}$ when $m=\int_{\mathbb{R}^{d}}x\alpha(dx)$ exists. The present paper shows that if $m$ exists and if $\psi$ is a convex function on $\mathbb{R}^{d}$ then $t\mapsto\int_{\mathbb{R}^{d}}\psi(x)\mu(t\alpha)(dx)$ is a decreasing function, which means that $t\mapsto\mu(t\alpha)$ is decreasing according to the Strassen convex order of probabilities. The second aim of the paper is to prove a group of results around the following question: if $\mu(t\alpha)=\mu(s\alpha)$ for some $0\leq s<t$, can we claim that $\mu$ is Cauchy distributed in $\mathbb{R}^{d}?$
</p>projecteuclid.org/euclid.bj/1501142434_20170727040115Thu, 27 Jul 2017 04:01 EDTRandom tessellations associated with max-stable random fieldshttp://projecteuclid.org/euclid.bj/1501142435<strong>Clément Dombry</strong>, <strong>Zakhar Kabluchko</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 30--52.</p><p><strong>Abstract:</strong><br/>
With any max-stable random process $\eta$ on $\mathcal{X}=\mathbb{Z}^{d}$ or $\mathbb{R}^{d}$, we associate a random tessellation of the parameter space $\mathcal{X}$. The construction relies on the Poisson point process representation of the max-stable process $\eta$ which is seen as the pointwise maximum of a random collection of functions $\Phi=\{\phi_{i},i\geq1\}$. The tessellation is constructed as follows: two points $x,y\in\mathcal{X}$ are in the same cell if and only if there exists a function $\phi\in\Phi$ that realizes the maximum $\eta$ at both points $x$ and $y$, that is, $\phi(x)=\eta(x)$ and $\phi(y)=\eta(y)$. We characterize the distribution of cells in terms of coverage and inclusion probabilities. Most interesting is the stationary case where the asymptotic properties of the cells are strongly related to the ergodic and mixing properties of the max-stable process $\eta$ and to its conservative/dissipative and positive/null decompositions.
</p>projecteuclid.org/euclid.bj/1501142435_20170727040115Thu, 27 Jul 2017 04:01 EDTProper scoring rules and Bregman divergencehttp://projecteuclid.org/euclid.bj/1501142436<strong>Evgeni Y. Ovcharov</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 53--79.</p><p><strong>Abstract:</strong><br/>
Proper scoring rules measure the quality of probabilistic forecasts. They induce dissimilarity measures of probability distributions known as Bregman divergences. We survey the literature on both entities and present their mathematical properties in a unified theoretical framework. Score and Bregman divergences are developed as a single concept. We formalize the proper affine scoring rules and present a motivating example from robust estimation. And lastly, we develop the elements of the regularity theory of entropy functions and describe under what conditions a general convex function may be identified as the entropy function of a proper scoring rule and whether this association is unique.
</p>projecteuclid.org/euclid.bj/1501142436_20170727040115Thu, 27 Jul 2017 04:01 EDTThe logarithmic law of sample covariance matrices near singularityhttp://projecteuclid.org/euclid.bj/1501142437<strong>Xuejun Wang</strong>, <strong>Xiao Han</strong>, <strong>Guangming Pan</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 80--114.</p><p><strong>Abstract:</strong><br/>
Let $B=(b_{jk})_{p\times n}=(Y_{1},Y_{2},\ldots,Y_{n})$ be a collection of independent real random variables with mean zero and variance one. Suppose that $\Sigma$ is a $p$ by $p$ population covariance matrix. Let $X_{k}=\Sigma^{1/2}Y_{k}$ for $k=1,2,\ldots,n$ and $\hat{\Sigma}_{1}=\frac{1}{n}\sum_{k=1}^{n}X_{k}X_{k}^{T}$. Under the moment condition $\mathop{\mathrm{sup}}_{p,n}\max_{1\leq j\leq p,1\leq k\leq n}\mathbb{E}b_{jk}^{4}<\infty$, we prove that the log determinant of the sample covariance matrix $\hat{\Sigma}_{1}$ satisfies
\[\frac{\log\operatorname{det}\hat{\Sigma}_{1}-\sum_{k=1}^{p}\log(1-\frac{k}{n})-\log\det\Sigma}{\sqrt{-2\log(1-\frac{p}{n})}}\xrightarrow[\qquad]{d}N(0,1),\] when $p/n\rightarrow1$ and $p<n$. For $p=n$, we prove that
\[\frac{\log\det\hat{\Sigma}_{1}+n\log n-\log(n-1)!-\log\det\Sigma}{\sqrt{2\log n}}\xrightarrow[\qquad]{d}N(0,1).\]
</p>projecteuclid.org/euclid.bj/1501142437_20170727040115Thu, 27 Jul 2017 04:01 EDTMaximum entropy distribution of order statistics with given marginalshttp://projecteuclid.org/euclid.bj/1501142438<strong>Cristina Butucea</strong>, <strong>Jean-François Delmas</strong>, <strong>Anne Dutfoy</strong>, <strong>Richard Fischer</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 115--155.</p><p><strong>Abstract:</strong><br/>
We consider distributions of ordered random vectors with given one-dimensional marginal distributions. We give an elementary necessary and sufficient condition for the existence of such a distribution with finite entropy. In this case, we give explicitly the density of the unique distribution which achieves the maximal entropy and compute the value of its entropy. This density is the unique one which has a product form on its support and the given one-dimensional marginals. The proof relies on the study of copulas with given one-dimensional marginal distributions for its order statistics.
</p>projecteuclid.org/euclid.bj/1501142438_20170727040115Thu, 27 Jul 2017 04:01 EDTMultiple collisions in systems of competing Brownian particleshttp://projecteuclid.org/euclid.bj/1501142439<strong>Cameron Bruggeman</strong>, <strong>Andrey Sarantsev</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 156--201.</p><p><strong>Abstract:</strong><br/>
Consider a finite system of competing Brownian particles on the real line. Each particle moves as a Brownian motion, with drift and diffusion coefficients depending only on its current rank relative to the other particles. We find a sufficient condition for a.s. absence of a total collision (when all particles collide) and of other types of collisions, say of the three lowest-ranked particles. This continues the work of Ichiba, Karatzas and Shkolnikov [ Probab. Theory Related Fields 156 (2013) 229–248] and Sarantsev (2016).
</p>projecteuclid.org/euclid.bj/1501142439_20170727040115Thu, 27 Jul 2017 04:01 EDTRate of convergence for Hilbert space valued processeshttp://projecteuclid.org/euclid.bj/1501142440<strong>Moritz Jirak</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 202--230.</p><p><strong>Abstract:</strong><br/>
Consider a stationary, linear Hilbert space valued process. We establish Berry–Esseen type results with optimal convergence rates under sharp dependence conditions on the underlying coefficient sequence of the linear operators. The case of non-linear Bernoulli-shift sequences is also considered. If the sequence is $m$-dependent, the optimal rate $(n/m)^{1/2}$ is reached. If the sequence is weakly geometrically dependent, the rate $(n/\log n)^{1/2}$ is obtained.
</p>projecteuclid.org/euclid.bj/1501142440_20170727040115Thu, 27 Jul 2017 04:01 EDTPosterior concentration rates for empirical Bayes procedures with applications to Dirichlet process mixtureshttp://projecteuclid.org/euclid.bj/1501142441<strong>Sophie Donnet</strong>, <strong>Vincent Rivoirard</strong>, <strong>Judith Rousseau</strong>, <strong>Catia Scricciolo</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 231--256.</p><p><strong>Abstract:</strong><br/>
We provide conditions on the statistical model and the prior probability law to derive contraction rates of posterior distributions corresponding to data-dependent priors in an empirical Bayes approach for selecting prior hyper-parameter values. We aim at giving conditions in the same spirit as those in the seminal article of Ghosal and van der Vaart [ Ann. Statist. 35 (2007) 192–223]. We then apply the result to specific statistical settings: density estimation using Dirichlet process mixtures of Gaussian densities with base measure depending on data-driven chosen hyper-parameter values and intensity function estimation of counting processes obeying the Aalen model. In the former setting, we also derive recovery rates for the related inverse problem of density deconvolution. In the latter, a simulation study for inhomogeneous Poisson processes illustrates the results.
</p>projecteuclid.org/euclid.bj/1501142441_20170727040115Thu, 27 Jul 2017 04:01 EDTA note on the convex infimum convolution inequalityhttp://projecteuclid.org/euclid.bj/1501142442<strong>Naomi Feldheim</strong>, <strong>Arnaud Marsiglietti</strong>, <strong>Piotr Nayar</strong>, <strong>Jing Wang</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 257--270.</p><p><strong>Abstract:</strong><br/>
We characterize the symmetric real random variables which satisfy the one dimensional convex infimum convolution inequality of Maurey. We deduce Talagrand’s two-level concentration for random vector $(X_{1},\ldots,X_{n})$, where $X_{i}$’s are independent real random variables whose tails satisfy certain exponential type decay condition.
</p>projecteuclid.org/euclid.bj/1501142442_20170727040115Thu, 27 Jul 2017 04:01 EDTSparse oracle inequalities for variable selection via regularized quantizationhttp://projecteuclid.org/euclid.bj/1501142443<strong>Clément Levrard</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 271--296.</p><p><strong>Abstract:</strong><br/>
We give oracle inequalities on procedures which combines quantization and variable selection via a weighted Lasso $k$-means type algorithm. The results are derived for a general family of weights, which can be tuned to size the influence of the variables in different ways. Moreover, these theoretical guarantees are proved to adapt the corresponding sparsity of the optimal codebooks, suggesting that these procedures might be of particular interest in high dimensional settings. Even if there is no sparsity assumption on the optimal codebooks, our procedure is proved to be close to a sparse approximation of the optimal codebooks, as has been done for the Generalized Linear Models in regression. If the optimal codebooks have a sparse support, we also show that this support can be asymptotically recovered, providing an asymptotic consistency rate. These results are illustrated with Gaussian mixture models in arbitrary dimension with sparsity assumptions on the means, which are standard distributions in model-based clustering.
</p>projecteuclid.org/euclid.bj/1501142443_20170727040115Thu, 27 Jul 2017 04:01 EDTThe $\lambda$-invariant measures of subcritical Bienaymé–Galton–Watson processeshttp://projecteuclid.org/euclid.bj/1501142444<strong>Pascal Maillard</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 297--315.</p><p><strong>Abstract:</strong><br/>
A $\lambda$-invariant measure of a sub-Markov chain is a left eigenvector of its transition matrix of eigenvalue $\lambda$. In this article, we give an explicit integral representation of the $\lambda$-invariant measures of subcritical Bienaymé–Galton–Watson processes killed upon extinction, that is, upon hitting the origin. In particular, this characterizes all quasi-stationary distributions of these processes. Our formula extends the Kesten–Spitzer formula for the (1-)invariant measures of such a process and can be interpreted as the identification of its minimal $\lambda$-Martin entrance boundary for all $\lambda$. In the particular case of quasi-stationary distributions, we also present an equivalent characterization in terms of semi-stable subordinators.
Unlike Kesten and Spitzer’s arguments, our proofs are elementary and do not rely on Martin boundary theory.
</p>projecteuclid.org/euclid.bj/1501142444_20170727040115Thu, 27 Jul 2017 04:01 EDTHitting probabilities for the Greenwood model and relations to near constancy oscillationhttp://projecteuclid.org/euclid.bj/1501142445<strong>M. Möhle</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 316--332.</p><p><strong>Abstract:</strong><br/>
We derive some properties of the Greenwood epidemic Galton–Watson branching model. Formulas for the probability $h(i,j)$ that the associated Markov chain $X$ hits state $j$ when started from state $i\ge j$ are obtained. For $j\ge1$, it follows that $h(i,j)$ slightly oscillates with varying $i$ and has infinitely many accumulation points. In particular, $h(i,j)$ does not converge as $i\to\infty$. It is shown that there exists a Markov chain $Y$ which is Siegmund dual to the chain $X$. The hitting probabilities of the dual Markov chain $Y$ are investigated.
</p>projecteuclid.org/euclid.bj/1501142445_20170727040115Thu, 27 Jul 2017 04:01 EDTFunctional inequalities for Gaussian convolutions of compactly supported measures: Explicit bounds and dimension dependencehttp://projecteuclid.org/euclid.bj/1501142446<strong>Jean-Baptiste Bardet</strong>, <strong>Nathaël Gozlan</strong>, <strong>Florent Malrieu</strong>, <strong>Pierre-André Zitt</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 333--353.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to establish various functional inequalities for the convolution of a compactly supported measure and a standard Gaussian distribution on $\mathbb{R}^{d}$. We especially focus on getting good dependence of the constants on the dimension. We prove that the Poincaré inequality holds with a dimension-free bound. For the logarithmic Sobolev inequality, we improve the best known results (Zimmermann, JFA 2013) by getting a bound that grows linearly with the dimension. We also establish transport-entropy inequalities for various transport costs.
</p>projecteuclid.org/euclid.bj/1501142446_20170727040115Thu, 27 Jul 2017 04:01 EDTLarge deviations for stochastic heat equation with rough dependence in spacehttp://projecteuclid.org/euclid.bj/1501142447<strong>Yaozhong Hu</strong>, <strong>David Nualart</strong>, <strong>Tusheng Zhang</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 354--385.</p><p><strong>Abstract:</strong><br/>
In this paper, we establish a large deviation principle for the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter $H\in (\frac{1}{4},\frac{1}{2})$ in the space variable.
</p>projecteuclid.org/euclid.bj/1501142447_20170727040115Thu, 27 Jul 2017 04:01 EDTThe maximum likelihood threshold of a graphhttp://projecteuclid.org/euclid.bj/1501142448<strong>Elizabeth Gross</strong>, <strong>Seth Sullivant</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 386--407.</p><p><strong>Abstract:</strong><br/>
The maximum likelihood threshold of a graph is the smallest number of data points that guarantees that maximum likelihood estimates exist almost surely in the Gaussian graphical model associated to the graph. We show that this graph parameter is connected to the theory of combinatorial rigidity. In particular, if the edge set of a graph $G$ is an independent set in the $(n-1)$-dimensional generic rigidity matroid, then the maximum likelihood threshold of $G$ is less than or equal to $n$. This connection allows us to prove many results about the maximum likelihood threshold. We conclude by showing that these methods give exact bounds on the number of observations needed for the score matching estimator to exist with probability one.
</p>projecteuclid.org/euclid.bj/1501142448_20170727040115Thu, 27 Jul 2017 04:01 EDTUniform measure density condition and game regularity for tug-of-war gameshttp://projecteuclid.org/euclid.bj/1501142449<strong>Joonas Heino</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 408--432.</p><p><strong>Abstract:</strong><br/>
We show that a uniform measure density condition implies game regularity for all $2<p<\infty$ in a stochastic game called “tug-of-war with noise”. The proof utilizes suitable choices of strategies combined with estimates for the associated stopping times and density estimates for the sum of independent and identically distributed random vectors.
</p>projecteuclid.org/euclid.bj/1501142449_20170727040115Thu, 27 Jul 2017 04:01 EDTThe van den Berg–Kesten–Reimer operator and inequality for infinite spaceshttp://projecteuclid.org/euclid.bj/1501142450<strong>Richard Arratia</strong>, <strong>Skip Garibaldi</strong>, <strong>Alfred W. Hales</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 433--448.</p><p><strong>Abstract:</strong><br/>
We remove the hypothesis “$S$ is finite” from the BKR inequality for product measures on $S^{d}$, which raises some issues related to descriptive set theory. We also discuss the extension of the BKR operator and inequality, from 2 events to 2 or more events, and we remove, in one sense, the hypothesis that $d$ be finite.
</p>projecteuclid.org/euclid.bj/1501142450_20170727040115Thu, 27 Jul 2017 04:01 EDTJackknife empirical likelihood goodness-of-fit tests for U-statistics based general estimating equationshttp://projecteuclid.org/euclid.bj/1501142451<strong>Hanxiang Peng</strong>, <strong>Fei Tan</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 449--464.</p><p><strong>Abstract:</strong><br/>
Motivated by applications to goodness of fit U-statistic testing, the jackknife empirical likelihood (JEL) for vector U-statistics is justified with two approaches and the Wilks theorems are proved. This generalizes empirical likelihood (EL) for general estimating equations (GEE’s) to U-statistics based GEE’s. The results are extended to allow for the use of estimated constraints and for the number of constraints to grow with the sample size. It is demonstrated that the JEL can be used to construct EL tests for moment based distribution characteristics (e.g., skewness, coefficient of variation) with less computational burden and more flexibility than the usual EL. This can be done in the U-statistic representation approach and the vector U-statistic approach which were illustrated with several examples including JEL tests for Pearson’s correlation, Goodman–Kruskal’s Gamma, overdisperson, U-quantiles, variance components, and the simplicial depth function. The JEL tests are asymptotically distribution free. Simulations were run to exhibit power improvement of the JEL tests with incorporation of side information.
</p>projecteuclid.org/euclid.bj/1501142451_20170727040115Thu, 27 Jul 2017 04:01 EDTPower of the spacing test for least-angle regressionhttp://projecteuclid.org/euclid.bj/1501142452<strong>Jean-Marc Azaïs</strong>, <strong>Yohann De Castro</strong>, <strong>Stéphane Mourareau</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 465--492.</p><p><strong>Abstract:</strong><br/>
Recent advances in Post-Selection Inference have shown that conditional testing is relevant and tractable in high-dimensions. In the Gaussian linear model, further works have derived unconditional test statistics such as the Kac–Rice Pivot for general penalized problems. In order to test the global null, a prominent offspring of this breakthrough is the Spacing test that accounts the relative separation between the first two knots of the celebrated least-angle regression (LARS) algorithm. However, no results have been shown regarding the distribution of these test statistics under the alternative. For the first time, this paper addresses this important issue for the Spacing test and shows that it is unconditionally unbiased. Furthermore, we provide the first extension of the Spacing test to the frame of unknown noise variance.
More precisely, we investigate the power of the Spacing test for LARS and prove that it is unbiased: its power is always greater or equal to the significance level $\alpha$. In particular, we describe the power of this test under various scenarii: we prove that its rejection region is optimal when the predictors are orthogonal; as the level $\alpha$ goes to zero, we show that the probability of getting a true positive is much greater than $\alpha$; and we give a detailed description of its power in the case of two predictors. Moreover, we numerically investigate a comparison between the Spacing test for LARS, the Pearson’s chi-squared test (goodness of fit) and a numerical testing procedure based on the maximal correlation.
When the noise variance is unknown, our analysis unleashes a new test statistic that can be computed in cubic time in the population size and which we refer to as the $t$-Spacing test for LARS. The $t$-Spacing test involves the first two knots of the LARS algorithm and we give its distribution under the null hypothesis. Interestingly, numerical experiments witness that the $t$-Spacing test for LARS enjoys the same aforementioned properties as the Spacing test.
</p>projecteuclid.org/euclid.bj/1501142452_20170727040115Thu, 27 Jul 2017 04:01 EDTInference in Ising modelshttp://projecteuclid.org/euclid.bj/1501142453<strong>Bhaswar B. Bhattacharya</strong>, <strong>Sumit Mukherjee</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 493--525.</p><p><strong>Abstract:</strong><br/>
The Ising spin glass is a one-parameter exponential family model for binary data with quadratic sufficient statistic. In this paper, we show that given a single realization from this model, the maximum pseudolikelihood estimate (MPLE) of the natural parameter is $\sqrt{a_{N}}$-consistent at a point whenever the log-partition function has order $a_{N}$ in a neighborhood of that point. This gives consistency rates of the MPLE for ferromagnetic Ising models on general weighted graphs in all regimes, extending the results of Chatterjee ( Ann. Statist. 35 (2007) 1931–1946) where only $\sqrt{N}$-consistency of the MPLE was shown. It is also shown that consistent testing, and hence estimation, is impossible in the high temperature phase in ferromagnetic Ising models on a converging sequence of simple graphs, which include the Curie–Weiss model. In this regime, the sufficient statistic is distributed as a weighted sum of independent $\chi^{2}_{1}$ random variables, and the asymptotic power of the most powerful test is determined. We also illustrate applications of our results on synthetic and real-world network data.
</p>projecteuclid.org/euclid.bj/1501142453_20170727040115Thu, 27 Jul 2017 04:01 EDTA MOSUM procedure for the estimation of multiple random change pointshttp://projecteuclid.org/euclid.bj/1501142454<strong>Birte Eichinger</strong>, <strong>Claudia Kirch</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 526--564.</p><p><strong>Abstract:</strong><br/>
In this work, we investigate statistical properties of change point estimators based on moving sum statistics. We extend results for testing in a classical situation with multiple deterministic change points by allowing for random exogenous change points that arise in Hidden Markov or regime switching models among others. To this end, we consider a multiple mean change model with possible time series errors and prove that the number and location of change points are estimated consistently by this procedure. Additionally, we derive rates of convergence for the estimation of the location of the change points and show that these rates are strict by deriving the limit distribution of properly scaled estimators. Because the small sample behavior depends crucially on how the asymptotic (long-run) variance of the error sequence is estimated, we propose to use moving sum type estimators for the (long-run) variance and derive their asymptotic properties. While they do not estimate the variance consistently at every point in time, they can still be used to consistently estimate the number and location of the changes. In fact, this inconsistency can even lead to more precise estimators for the change points. Finally, some simulations illustrate the behavior of the estimators in small samples showing that its performance is very good compared to existing methods.
</p>projecteuclid.org/euclid.bj/1501142454_20170727040115Thu, 27 Jul 2017 04:01 EDTOn conditional moments of high-dimensional random vectors given lower-dimensional projectionshttp://projecteuclid.org/euclid.bj/1501142455<strong>Lukas Steinberger</strong>, <strong>Hannes Leeb</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 565--591.</p><p><strong>Abstract:</strong><br/>
One of the most widely used properties of the multivariate Gaussian distribution, besides its tail behavior, is the fact that conditional means are linear and that conditional variances are constant. We here show that this property is also shared, in an approximate sense, by a large class of non-Gaussian distributions. We allow for several conditioning variables and we provide explicit non-asymptotic results, whereby we extend earlier findings of Hall and Li ( Ann. Statist. 21 (1993) 867–889) and Leeb ( Ann. Statist. 41 (2013) 464–483).
</p>projecteuclid.org/euclid.bj/1501142455_20170727040115Thu, 27 Jul 2017 04:01 EDTLocal block bootstrap for inhomogeneous Poisson marked point processeshttp://projecteuclid.org/euclid.bj/1501142456<strong>William Garner</strong>, <strong>Dimitris N. Politis</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 592--615.</p><p><strong>Abstract:</strong><br/>
The asymptotic theory for the sample mean of a marked point process in $d$ dimensions is established, allowing for the possibility that the underlying Poisson point process is inhomogeneous. A novel local block bootstrap method for resampling inhomogeneous Poisson marked point processes is introduced, and its consistency is proven for the sample mean and related statistics. Finite-sample simulations are carried out to complement the asymptotic results, and demonstrate the feasibility of the proposed methodology.
</p>projecteuclid.org/euclid.bj/1501142456_20170727040115Thu, 27 Jul 2017 04:01 EDTChange-point estimators with true identification propertyhttp://projecteuclid.org/euclid.bj/1501142457<strong>Chi Tim Ng</strong>, <strong>Woojoo Lee</strong>, <strong>Youngjo Lee</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 616--660.</p><p><strong>Abstract:</strong><br/>
The change-point problem is reformulated as a penalized likelihood estimation problem. A new non-convex penalty function is introduced to allow consistent estimation of the number of change points, and their locations and sizes. Penalized likelihood methods based on LASSO and SCAD penalties may not satisfy such a property. The asymptotic properties for the local solutions are established and numerical studies are conducted to highlight their performance. An application to copy number variation is discussed.
</p>projecteuclid.org/euclid.bj/1501142457_20170727040115Thu, 27 Jul 2017 04:01 EDTDesigns from good Hadamard matriceshttp://projecteuclid.org/euclid.bj/1501142458<strong>Chenlu Shi</strong>, <strong>Boxin Tang</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 661--671.</p><p><strong>Abstract:</strong><br/>
Hadamard matrices are very useful mathematical objects for the construction of various statistical designs. Some Hadamard matrices are better than others in terms of the qualities of designs they produce. In this paper, we provide a theoretical investigation into such good Hadamard matrices and discuss their applications in the construction of nonregular factorial designs and supersaturated designs.
</p>projecteuclid.org/euclid.bj/1501142458_20170727040115Thu, 27 Jul 2017 04:01 EDTCurvature and transport inequalities for Markov chains in discrete spaceshttp://projecteuclid.org/euclid.bj/1501142459<strong>Max Fathi</strong>, <strong>Yan Shu</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 672--698.</p><p><strong>Abstract:</strong><br/>
We study various transport-information inequalities under three different notions of Ricci curvature in the discrete setting: the curvature-dimension condition of Bakry and Émery (In Séminaire de Probabilités, XIX, 1983/84 (1985) 177–206 Springer), the exponential curvature-dimension condition of Bauer et al. ( Li-Yau Inequality on Graphs (2013)) and the coarse Ricci curvature of Ollivier ( J. Funct. Anal. 256 (2009) 810–864). We prove that under a curvature-dimension condition or coarse Ricci curvature condition, an $L_{1}$ transport-information inequality holds; while under an exponential curvature-dimension condition, some weak-transport information inequalities hold. As an application, we establish a Bonnet–Myers theorem under the curvature-dimension condition $\operatorname{CD}(\kappa,\infty)$ of Bakry and Émery (In Séminaire de Probabilités, XIX, 1983/84 (1985) 177–206 Springer).
</p>projecteuclid.org/euclid.bj/1501142459_20170727040115Thu, 27 Jul 2017 04:01 EDTOptimal adaptive inference in random design binary regressionhttp://projecteuclid.org/euclid.bj/1501142460<strong>Rajarshi Mukherjee</strong>, <strong>Subhabrata Sen</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 699--739.</p><p><strong>Abstract:</strong><br/>
We construct confidence sets for the regression function in nonparametric binary regression with an unknown design density – a nuisance parameter in the problem. These confidence sets are adaptive in $L^{2}$ loss over a continuous class of Sobolev type spaces. Adaptation holds in the smoothness of the regression function, over the maximal parameter spaces where adaptation is possible, provided the design density is smooth enough. We identify two key regimes – one where adaptation is possible, and one where some critical regions must be removed. We address related questions about goodness of fit testing and adaptive estimation of relevant infinite dimensional parameters.
</p>projecteuclid.org/euclid.bj/1501142460_20170727040115Thu, 27 Jul 2017 04:01 EDTTesting for instability in covariance structureshttp://projecteuclid.org/euclid.bj/1501142461<strong>Chihwa Kao</strong>, <strong>Lorenzo Trapani</strong>, <strong>Giovanni Urga</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 740--771.</p><p><strong>Abstract:</strong><br/>
We propose a test for the stability over time of the covariance matrix of multivariate time series. The analysis is extended to the eigensystem to ascertain changes due to instability in the eigenvalues and/or eigenvectors. Using strong Invariance Principles and Law of Large Numbers, we normalise the CUSUM-type statistics to calculate their supremum over the whole sample. The power properties of the test versus alternative hypotheses, including also the case of breaks close to the beginning/end of sample are investigated theoretically and via simulation. We extend our theory to test for the stability of the covariance matrix of a multivariate regression model. The testing procedures are illustrated by studying the stability of the principal components of the term structure of 18 US interest rates.
</p>projecteuclid.org/euclid.bj/1501142461_20170727040115Thu, 27 Jul 2017 04:01 EDTSecond and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in timehttp://projecteuclid.org/euclid.bj/1501142462<strong>ZhiQiang Gao</strong>, <strong>Quansheng Liu</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 1, 772--800.</p><p><strong>Abstract:</strong><br/>
Consider a branching random walk in which the offspring distribution and the moving law both depend on an independent and identically distributed random environment indexed by the time. For the normalised counting measure of the number of particles of generation $n$ in a given region, we give the second and third orders asymptotic expansions of the central limit theorem under rather weak assumptions on the moments of the underlying branching and moving laws. The obtained results and the developed approaches shed light on higher order expansions. In the proofs, the Edgeworth expansion of central limit theorems for sums of independent random variables, truncating arguments and martingale approximation play key roles. In particular, we introduce a new martingale, show its rate of convergence, as well as the rates of convergence of some known martingales, which are of independent interest.
</p>projecteuclid.org/euclid.bj/1501142462_20170727040115Thu, 27 Jul 2017 04:01 EDTThe minimum of a branching random walk outside the boundary casehttps://projecteuclid.org/euclid.bj/1505980879<strong>Julien Barral</strong>, <strong>Yueyun Hu</strong>, <strong>Thomas Madaule</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 801--841.</p><p><strong>Abstract:</strong><br/>
This paper is a complement to the studies on the minimum of a real-valued branching random walk. In the boundary case [ Electron. J. Probab. 10 (2005) 609–631], Aïdékon in a seminal paper [ Ann. Probab. 41 (2013) 1362–1426] obtained the convergence in law of the minimum after a suitable renormalization. We study here the situation when the log-generating function of the branching random walk explodes at some positive point and it cannot be reduced to the boundary case. In the associated thermodynamics framework, this corresponds to a first-order phase transition, while the boundary case corresponds to a second-order phase transition.
</p>projecteuclid.org/euclid.bj/1505980879_20170921040135Thu, 21 Sep 2017 04:01 EDTUniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplershttps://projecteuclid.org/euclid.bj/1505980880<strong>Christophe Andrieu</strong>, <strong>Anthony Lee</strong>, <strong>Matti Vihola</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 842--872.</p><p><strong>Abstract:</strong><br/>
We establish quantitative bounds for rates of convergence and asymptotic variances for iterated conditional sequential Monte Carlo (i-cSMC) Markov chains and associated particle Gibbs samplers [ J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 (2010) 269–342]. Our main findings are that the essential boundedness of potential functions associated with the i-cSMC algorithm provide necessary and sufficient conditions for the uniform ergodicity of the i-cSMC Markov chain, as well as quantitative bounds on its (uniformly geometric) rate of convergence. Furthermore, we show that the i-cSMC Markov chain cannot even be geometrically ergodic if this essential boundedness does not hold in many applications of interest. Our sufficiency and quantitative bounds rely on a novel non-asymptotic analysis of the expectation of a standard normalizing constant estimate with respect to a “doubly conditional” SMC algorithm. In addition, our results for i-cSMC imply that the rate of convergence can be improved arbitrarily by increasing $N$, the number of particles in the algorithm, and that in the presence of mixing assumptions, the rate of convergence can be kept constant by increasing $N$ linearly with the time horizon. We translate the sufficiency of the boundedness condition for i-cSMC into sufficient conditions for the particle Gibbs Markov chain to be geometrically ergodic and quantitative bounds on its geometric rate of convergence, which imply convergence of properties of the particle Gibbs Markov chain to those of its corresponding Gibbs sampler. These results complement recently discovered, and related, conditions for the particle marginal Metropolis–Hastings (PMMH) Markov chain.
</p>projecteuclid.org/euclid.bj/1505980880_20170921040135Thu, 21 Sep 2017 04:01 EDTDomains of attraction on countable alphabetshttps://projecteuclid.org/euclid.bj/1505980881<strong>Zhiyi Zhang</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 873--894.</p><p><strong>Abstract:</strong><br/>
For each probability distribution on a countable alphabet, a sequence of positive functionals are developed as tail indices. By and only by the asymptotic behavior of these indices, domains of attraction for all probability distributions on the alphabet are defined. The three main domains of attraction are shown to contain distributions with thick tails, thin tails and no tails respectively, resembling in parallel the three main domains of attraction, Fréchet, Gumbel and Weibull families, for continuous random variables on the real line. In addition to the probabilistic merits associated with the domains, the tail indices are partially motivated by the fact that there exists an unbiased estimator for every index in the sequence, which is therefore statistically observable, provided that the sample is sufficiently large.
</p>projecteuclid.org/euclid.bj/1505980881_20170921040135Thu, 21 Sep 2017 04:01 EDTWavelet estimation for operator fractional Brownian motionhttps://projecteuclid.org/euclid.bj/1505980882<strong>Patrice Abry</strong>, <strong>Gustavo Didier</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 895--928.</p><p><strong>Abstract:</strong><br/>
Operator fractional Brownian motion (OFBM) is the natural vector-valued extension of the univariate fractional Brownian motion. Instead of a scalar parameter, the law of an OFBM scales according to a Hurst matrix that affects every component of the process. In this paper, we develop the wavelet analysis of OFBM, as well as a new estimator for the Hurst matrix of bivariate OFBM. For OFBM, the univariate-inspired approach of analyzing the entry-wise behavior of the wavelet spectrum as a function of the (wavelet) scales is fraught with difficulties stemming from mixtures of power laws. Instead we consider the evolution along scales of the eigenstructure of the wavelet spectrum. This is shown to yield consistent and asymptotically normal estimators of the Hurst eigenvalues, and also of the eigenvectors under assumptions. A simulation study is included to demonstrate the good performance of the estimators under finite sample sizes.
</p>projecteuclid.org/euclid.bj/1505980882_20170921040135Thu, 21 Sep 2017 04:01 EDTAsymptotics for the maximum sample likelihood estimator under informative selection from a finite populationhttps://projecteuclid.org/euclid.bj/1505980883<strong>Daniel Bonnéry</strong>, <strong>F. Jay Breidt</strong>, <strong>François Coquet</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 929--955.</p><p><strong>Abstract:</strong><br/>
Inference for the parametric distribution of a response given covariates is considered under informative selection of a sample from a finite population. Under this selection, the conditional distribution of a response in the sample, given the covariates and given that it was selected for observation, is not the same as the conditional distribution of the response in the finite population, given only the covariates. It is instead a weighted version of the conditional distribution of interest. Inference must be modified to account for this informative selection. An established approach in this context is maximum “sample likelihood”, developing a weight function that reflects the informative sampling design, then treating the observations as if they were independently distributed according to the weighted distribution. While the sample likelihood methodology has been widely applied, its theoretical foundation has been less developed. A precise asymptotic description of a wide range of informative selection mechanisms is proposed. Under this framework, consistency and asymptotic normality of the maximum sample likelihood estimators are established. The theory allows for the possibility of nuisance parameters that describe the selection mechanism. The proposed regularity conditions are verifiable for various sample schemes, motivated by real problems in surveys. Simulation results for these examples illustrate the quality of the asymptotic approximations, and demonstrate a practical approach to variance estimation that combines aspects of model-based information theory and design-based variance estimation.
</p>projecteuclid.org/euclid.bj/1505980883_20170921040135Thu, 21 Sep 2017 04:01 EDTHörmander-type theorem for Itô processes and related backward SPDEshttps://projecteuclid.org/euclid.bj/1505980884<strong>Jinniao Qiu</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 956--970.</p><p><strong>Abstract:</strong><br/>
A Hörmander-type theorem is established for Itô processes and related backward stochastic partial differential equations (BSPDEs). A short self-contained proof is also provided for the $L^{2}$-theory of linear, possibly degenerate BSPDEs, in which new gradient estimates are obtained.
</p>projecteuclid.org/euclid.bj/1505980884_20170921040135Thu, 21 Sep 2017 04:01 EDTAn upper bound on the convergence rate of a second functional in optimal sequence alignmenthttps://projecteuclid.org/euclid.bj/1505980885<strong>Raphael Hauser</strong>, <strong>Heinrich Matzinger</strong>, <strong>Ionel Popescu</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 971--992.</p><p><strong>Abstract:</strong><br/>
Consider finite sequences $X_{[1,n]}=X_{1},\ldots,X_{n}$ and $Y_{[1,n]}=Y_{1},\ldots,Y_{n}$ of length $n$, consisting of i.i.d. samples of random letters from a finite alphabet, and let $S$ and $T$ be chosen i.i.d. randomly from the unit ball in the space of symmetric scoring functions over this alphabet augmented by a gap symbol. We prove a probabilistic upper bound of linear order in $(\ln(n))^{1/4}n^{3/4}$ for the deviation of the score relative to $T$ of optimal alignments with gaps of $X_{[1,n]}$ and $Y_{[1,n]}$ relative to $S$. It remains an open problem to prove a lower bound. Our result contributes to the understanding of the microstructure of optimal alignments relative to one given scoring function, extending a theory begun in ( J. Stat. Phys. 153 (2013) 512–529).
</p>projecteuclid.org/euclid.bj/1505980885_20170921040135Thu, 21 Sep 2017 04:01 EDTMixing time and cutoff for a random walk on the ring of integers mod $n$https://projecteuclid.org/euclid.bj/1505980886<strong>Michael Bate</strong>, <strong>Stephen Connor</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 993--1009.</p><p><strong>Abstract:</strong><br/>
We analyse a random walk on the ring of integers mod $n$, which at each time point can make an additive ‘step’ or a multiplicative ‘jump’. When the probability of making a jump tends to zero as an appropriate power of $n$, we prove the existence of a total variation pre-cutoff for this walk. In addition, we show that the process obtained by subsampling our walk at jump times exhibits a true cutoff, with mixing time dependent on whether the step distribution has zero mean.
</p>projecteuclid.org/euclid.bj/1505980886_20170921040135Thu, 21 Sep 2017 04:01 EDTExponential mixing properties for time inhomogeneous diffusion processes with killinghttps://projecteuclid.org/euclid.bj/1505980887<strong>Pierre Del Moral</strong>, <strong>Denis Villemonais</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1010--1032.</p><p><strong>Abstract:</strong><br/>
We consider an elliptic and time-inhomogeneous diffusion process with time-periodic coefficients evolving in a bounded domain of $\mathbb{R}^{d}$ with a smooth boundary. The process is killed when it hits the boundary of the domain (hard killing) or after an exponential time (soft killing) associated with some bounded rate function. The branching particle interpretation of the non absorbed diffusion again behaves as a set of interacting particles evolving in an absorbing medium. Between absorption times, the particles evolve independently one from each other according to the diffusion evolution operator; when a particle is absorbed, another selected particle splits into two offsprings. This article is concerned with the stability properties of these non absorbed processes. Under some classical ellipticity properties on the diffusion process and some mild regularity properties of the hard obstacle boundaries, we prove an uniform exponential strong mixing property of the process conditioned to not be killed. We also provide uniform estimates w.r.t. the time horizon for the interacting particle interpretation of these non-absorbed processes, yielding what seems to be the first result of this type for this class of diffusion processes evolving in soft and hard obstacles, both in homogeneous and non-homogeneous time settings.
</p>projecteuclid.org/euclid.bj/1505980887_20170921040135Thu, 21 Sep 2017 04:01 EDTFunctional central limit theorems in $L^{2}(0,1)$ for logarithmic combinatorial assemblieshttps://projecteuclid.org/euclid.bj/1505980888<strong>Koji Tsukuda</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1033--1052.</p><p><strong>Abstract:</strong><br/>
Functional central limit theorems in $L^{2}(0,1)$ for logarithmic combinatorial assemblies are presented. The random elements argued in this paper are viewed as elements taking values in $L^{2}(0,1)$ whereas the Skorokhod space is argued as a framework of weak convergences in functional central limit theorems for random combinatorial structures in the literature. It enables us to treat other standardized random processes which converge weakly to a corresponding Gaussian process with additional assumptions.
</p>projecteuclid.org/euclid.bj/1505980888_20170921040135Thu, 21 Sep 2017 04:01 EDTInference for a two-component mixture of symmetric distributions under log-concavityhttps://projecteuclid.org/euclid.bj/1505980889<strong>Fadoua Balabdaoui</strong>, <strong>Charles R. Doss</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1053--1071.</p><p><strong>Abstract:</strong><br/>
In this article, we revisit the problem of estimating the unknown zero-symmetric distribution in a two-component location mixture model, considered in previous works, now under the assumption that the zero-symmetric distribution has a log-concave density. When consistent estimators for the shift locations and mixing probability are used, we show that the nonparametric log-concave Maximum Likelihood estimator (MLE) of both the mixed density and that of the unknown zero-symmetric component are consistent in the Hellinger distance. In case the estimators for the shift locations and mixing probability are $\sqrt{n}$-consistent, we establish that these MLE’s converge to the truth at the rate $n^{-2/5}$ in the $L_{1}$ distance. To estimate the shift locations and mixing probability, we use the estimators proposed by ( Ann. Statist. 35 (2007) 224–251). The unknown zero-symmetric density is efficiently computed using the R package logcondens.mode.
</p>projecteuclid.org/euclid.bj/1505980889_20170921040135Thu, 21 Sep 2017 04:01 EDTOn matrix estimation under monotonicity constraintshttps://projecteuclid.org/euclid.bj/1505980890<strong>Sabyasachi Chatterjee</strong>, <strong>Adityanand Guntuboyina</strong>, <strong>Bodhisattva Sen</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1072--1100.</p><p><strong>Abstract:</strong><br/>
We consider the problem of estimating an unknown $n_{1}\times n_{2}$ matrix $\mathbf{\theta}^{*}$ from noisy observations under the constraint that $\mathbf{\theta}^{*}$ is nondecreasing in both rows and columns. We consider the least squares estimator (LSE) in this setting and study its risk properties. We show that the worst case risk of the LSE is $n^{-1/2}$, up to multiplicative logarithmic factors, where $n=n_{1}n_{2}$ and that the LSE is minimax rate optimal (up to logarithmic factors). We further prove that for some special $\mathbf{\theta}^{*}$, the risk of the LSE could be much smaller than $n^{-1/2}$; in fact, it could even be parametric, that is, $n^{-1}$ up to logarithmic factors. Such parametric rates occur when the number of “rectangular” blocks of $\mathbf{\theta}^{*}$ is bounded from above by a constant. We also derive an interesting adaptation property of the LSE which we term variable adaptation – the LSE adapts to the “intrinsic dimension” of the problem and performs as well as the oracle estimator when estimating a matrix that is constant along each row/column. Our proofs, which borrow ideas from empirical process theory, approximation theory and convex geometry, are of independent interest.
</p>projecteuclid.org/euclid.bj/1505980890_20170921040135Thu, 21 Sep 2017 04:01 EDTEfficient estimation for generalized partially linear single-index modelshttps://projecteuclid.org/euclid.bj/1505980891<strong>Li Wang</strong>, <strong>Guanqun Cao</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1101--1127.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the estimation for generalized partially linear single-index models, where the systematic component in the model has a flexible semi-parametric form with a general link function. We propose an efficient and practical approach to estimate the single-index link function, single-index coefficients as well as the coefficients in the linear component of the model. The estimation procedure is developed by applying quasi-likelihood and polynomial spline smoothing. We derive large sample properties of the estimators and show the convergence rate of each component of the model. Asymptotic normality and semiparametric efficiency are established for the coefficients in both the single-index and linear components. By making use of spline basis approximation and Fisher score iteration, our approach has numerical advantages in terms of computing efficiency and stability in practice. Both simulated and real data examples are used to illustrate our proposed methodology.
</p>projecteuclid.org/euclid.bj/1505980891_20170921040135Thu, 21 Sep 2017 04:01 EDTCritical points of multidimensional random Fourier series: Central limitshttps://projecteuclid.org/euclid.bj/1505980892<strong>Liviu I. Nicolaescu</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1128--1170.</p><p><strong>Abstract:</strong><br/>
We investigate certain families $X^{\hbar}$, $0<\hbar\ll1$ of stationary Gaussian random smooth functions on the $m$-dimensional torus $\mathbb{T}^{m}:=\mathbb{R}^{m}/\mathbb{Z}^{m}$ approaching the white noise as $\hbar\to0$. We show that there exists universal constants $c_{1},c_{2}>0$ such that for any cube $B\subset\mathbb{R}^{m}$ of size $r\leq1/2$, the number of critical points of $X^{\hbar}$ in the region $B\bmod\mathbb{Z}^{m}\subset\mathbb{T}^{m}$ has mean $\sim c_{1}\operatorname{vol}(B)\hbar^{-m}$, variance $\sim c_{2}\operatorname{vol}(B)\hbar^{-m}$, and satisfies a central limit theorem as $\hbar\searrow0$.
</p>projecteuclid.org/euclid.bj/1505980892_20170921040135Thu, 21 Sep 2017 04:01 EDTDeterminantal point process models on the spherehttps://projecteuclid.org/euclid.bj/1505980893<strong>Jesper Møller</strong>, <strong>Morten Nielsen</strong>, <strong>Emilio Porcu</strong>, <strong>Ege Rubak</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1171--1201.</p><p><strong>Abstract:</strong><br/>
We consider determinantal point processes on the $d$-dimensional unit sphere $\mathbb{S}^{d}$. These are finite point processes exhibiting repulsiveness and with moment properties determined by a certain determinant whose entries are specified by a so-called kernel which we assume is a complex covariance function defined on $\mathbb{S}^{d}\times\mathbb{S}^{d}$. We review the appealing properties of such processes, including their specific moment properties, density expressions and simulation procedures. Particularly, we characterize and construct isotropic DPPs models on $\mathbb{S}^{d}$, where it becomes essential to specify the eigenvalues and eigenfunctions in a spectral representation for the kernel, and we figure out how repulsive isotropic DPPs can be. Moreover, we discuss the shortcomings of adapting existing models for isotropic covariance functions and consider strategies for developing new models, including a useful spectral approach.
</p>projecteuclid.org/euclid.bj/1505980893_20170921040135Thu, 21 Sep 2017 04:01 EDTBaxter’s inequality for finite predictor coefficients of multivariate long-memory stationary processeshttps://projecteuclid.org/euclid.bj/1505980894<strong>Akihiko Inoue</strong>, <strong>Yukio Kasahara</strong>, <strong>Mohsen Pourahmadi</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1202--1232.</p><p><strong>Abstract:</strong><br/>
For a multivariate stationary process, we develop explicit representations for the finite predictor coefficient matrices, the finite prediction error covariance matrices and the partial autocorrelation function (PACF) in terms of the Fourier coefficients of its phase function in the spectral domain. The derivation is based on a novel alternating projection technique and the use of the forward and backward innovations corresponding to predictions based on the infinite past and future, respectively. We show that such representations are ideal for studying the rates of convergence of the finite predictor coefficients, prediction error covariances, and the PACF as well as for proving a multivariate version of Baxter’s inequality for a multivariate FARIMA process with a common fractional differencing order for all components of the process.
</p>projecteuclid.org/euclid.bj/1505980894_20170921040135Thu, 21 Sep 2017 04:01 EDTSmooth backfitting for additive modeling with small errors-in-variables, with an application to additive functional regression for multiple predictor functionshttps://projecteuclid.org/euclid.bj/1505980895<strong>Kyunghee Han</strong>, <strong>Hans-Georg Müller</strong>, <strong>Byeong U. Park</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1233--1265.</p><p><strong>Abstract:</strong><br/>
We study smooth backfitting when there are errors-in-variables, which is motivated by functional additive models for a functional regression model with a scalar response and multiple functional predictors that are additive in the functional principal components of the predictor processes. The development of a new smooth backfitting technique for the estimation of the additive component functions in functional additive models with multiple functional predictors requires to address the difficulty that the eigenfunctions and therefore the functional principal components of the predictor processes, which are the arguments of the proposed additive model, are unknown and need to be estimated from the data. The available estimated functional principal components contain an error that is small for large samples but nevertheless affects the estimation of the additive component functions. This error-in-variables situation requires to develop new asymptotic theory for smooth backfitting. Our analysis also pertains to general situations where one encounters errors in the predictors for an additive model, when the errors become smaller asymptotically. We also study the finite sample properties of the proposed method for the application in functional additive regression through a simulation study and a real data example.
</p>projecteuclid.org/euclid.bj/1505980895_20170921040135Thu, 21 Sep 2017 04:01 EDTBump detection in heterogeneous Gaussian regressionhttps://projecteuclid.org/euclid.bj/1505980896<strong>Farida Enikeeva</strong>, <strong>Axel Munk</strong>, <strong>Frank Werner</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1266--1306.</p><p><strong>Abstract:</strong><br/>
We analyze the effect of a heterogeneous variance on bump detection in a Gaussian regression model. To this end, we allow for a simultaneous bump in the variance and specify its impact on the difficulty to detect the null signal against a single bump with known signal strength. This is done by calculating lower and upper bounds, both based on the likelihood ratio.
Lower and upper bounds together lead to explicit characterizations of the detection boundary in several subregimes depending on the asymptotic behavior of the bump heights in mean and variance. In particular, we explicitly identify those regimes, where the additional information about a simultaneous bump in variance eases the detection problem for the signal. This effect is made explicit in the constant and/or the rate, appearing in the detection boundary.
We also discuss the case of an unknown bump height and provide an adaptive test and some upper bounds in that case.
</p>projecteuclid.org/euclid.bj/1505980896_20170921040135Thu, 21 Sep 2017 04:01 EDTQuenched invariance principles for the discrete Fourier transforms of a stationary processhttps://projecteuclid.org/euclid.bj/1505980897<strong>David Barrera</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1307--1350.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the asymptotic behavior of the normalized cadlag functions generated by the discrete Fourier transforms of a stationary centered square-integrable process, started at a point.
We prove that the quenched invariance principle holds for averaged frequencies under no assumption other than ergodicity, and that this result holds also for almost every fixed frequency under a certain generalization of the Hannan condition and a certain rotated form of the Maxwell and Woodroofe condition which, under a condition of weak dependence that we specify, is guaranteed for a.e. frequency. If the process is in particular weakly mixing, our results describe the asymptotic distributions of the normalized discrete Fourier transforms at every frequency other than $0$ and $\pi$ under the generalized Hannan condition.
We prove also that under a certain regularity hypothesis the conditional centering is irrelevant for averaged frequencies, and that the same holds for a given fixed frequency under the rotated Maxwell and Woodroofe condition but not necessarily under the generalized Hannan condition. In particular, this implies that the hypothesis of regularity is not sufficient for functional convergence without random centering at a.e. fixed frequency.
The proofs are based on martingale approximations and combine results from Ergodic theory of recent and classical origin with approximation results by contemporary authors and with some facts from Harmonic Analysis and Functional Analysis.
</p>projecteuclid.org/euclid.bj/1505980897_20170921040135Thu, 21 Sep 2017 04:01 EDTThe eigenvalues of the sample covariance matrix of a multivariate heavy-tailed stochastic volatility modelhttps://projecteuclid.org/euclid.bj/1505980898<strong>Anja Janssen</strong>, <strong>Thomas Mikosch</strong>, <strong>Mohsen Rezapour</strong>, <strong>Xiaolei Xie</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1351--1393.</p><p><strong>Abstract:</strong><br/>
We consider a multivariate heavy-tailed stochastic volatility model and analyze the large-sample behavior of its sample covariance matrix. We study the limiting behavior of its entries in the infinite-variance case and derive results for the ordered eigenvalues and corresponding eigenvectors. Essentially, we consider two different cases where the tail behavior either stems from the i.i.d. innovations of the process or from its volatility sequence. In both cases, we make use of a large deviations technique for regularly varying time series to derive multivariate $\alpha$-stable limit distributions of the sample covariance matrix. For the case of heavy-tailed innovations, we show that the limiting behavior resembles that of completely independent observations. In contrast to this, for a heavy-tailed volatility sequence the possible limiting behavior is more diverse and allows for dependencies in the limiting distributions which are determined by the structure of the underlying volatility sequence.
</p>projecteuclid.org/euclid.bj/1505980898_20170921040135Thu, 21 Sep 2017 04:01 EDTAmerican options with asymmetric information and reflected BSDEhttps://projecteuclid.org/euclid.bj/1505980899<strong>Neda Esmaeeli</strong>, <strong>Peter Imkeller</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1394--1426.</p><p><strong>Abstract:</strong><br/>
We consider an American contingent claim on a financial market where the buyer has additional information. Both agents (seller and buyer) observe the same prices, while the information available to them may differ due to some extra exogenous knowledge the buyer has. The buyer’s information flow is modeled by an initial enlargement of the reference filtration. It seems natural to investigate the value of the American contingent claim with asymmetric information. We provide a representation for the cost of the additional information relying on some results on reflected backward stochastic differential equations (RBSDE). This is done by using an interpretation of prices of American contingent claims with extra information for the buyer by solutions of appropriate RBSDE.
</p>projecteuclid.org/euclid.bj/1505980899_20170921040135Thu, 21 Sep 2017 04:01 EDTMaximum likelihood estimation for the Fréchet distribution based on block maxima extracted from a time serieshttps://projecteuclid.org/euclid.bj/1505980900<strong>Axel Bücher</strong>, <strong>Johan Segers</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1427--1462.</p><p><strong>Abstract:</strong><br/>
The block maxima method in extreme-value analysis proceeds by fitting an extreme-value distribution to a sample of block maxima extracted from an observed stretch of a time series. The method is usually validated under two simplifying assumptions: the block maxima should be distributed exactly according to an extreme-value distribution and the sample of block maxima should be independent. Both assumptions are only approximately true. The present paper validates that the simplifying assumptions can in fact be safely made.
For general triangular arrays of block maxima attracted to the Fréchet distribution, consistency and asymptotic normality is established for the maximum likelihood estimator of the parameters of the limiting Fréchet distribution. The results are specialized to the common setting of block maxima extracted from a strictly stationary time series. The case where the underlying random variables are independent and identically distributed is further worked out in detail. The results are illustrated by theoretical examples and Monte Carlo simulations.
</p>projecteuclid.org/euclid.bj/1505980900_20170921040135Thu, 21 Sep 2017 04:01 EDTCharacterization of the convergence in total variation and extension of the Fourth Moment Theorem to invariant measures of diffusionshttps://projecteuclid.org/euclid.bj/1505980901<strong>Seiichiro Kusuoka</strong>, <strong>Ciprian A. Tudor</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1463--1496.</p><p><strong>Abstract:</strong><br/>
We give necessary and sufficient conditions to characterize the convergence in distribution of a sequence of arbitrary random variables to a probability distribution which is the invariant measure of a diffusion process. This class of target distributions includes the most known continuous probability distributions. Precisely speaking, we characterize the convergence in total variation to target distributions which are not Gaussian or Gamma distributed, in terms of the Malliavin calculus and of the coefficients of the associated diffusion process. We also prove that, among the distributions whose associated squared diffusion coefficient is a polynomial of second degree (with some restrictions on its coefficients), the only possible limits of sequences of multiple integrals are the Gaussian and the Gamma laws.
</p>projecteuclid.org/euclid.bj/1505980901_20170921040135Thu, 21 Sep 2017 04:01 EDTExact and fast simulation of max-stable processes on a compact set using the normalized spectral representationhttps://projecteuclid.org/euclid.bj/1505980902<strong>Marco Oesting</strong>, <strong>Martin Schlather</strong>, <strong>Chen Zhou</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1497--1530.</p><p><strong>Abstract:</strong><br/>
The efficiency of simulation algorithms for max-stable processes relies on the choice of the spectral representation: different choices result in different sequences of finite approximations to the process. We propose a constructive approach yielding a normalized spectral representation that solves an optimization problem related to the efficiency of simulating max-stable processes. The simulation algorithm based on the normalized spectral representation can be regarded as max-importance sampling. Compared to other simulation algorithms hitherto, our approach has at least two advantages. First, it allows the exact simulation of a comprising class of max-stable processes. Second, the algorithm has a stopping time with finite expectation. In practice, our approach has the potential of considerably reducing the simulation time of max-stable processes.
</p>projecteuclid.org/euclid.bj/1505980902_20170921040135Thu, 21 Sep 2017 04:01 EDTAsymptotic analysis of covariance parameter estimation for Gaussian processes in the misspecified casehttps://projecteuclid.org/euclid.bj/1505980903<strong>François Bachoc</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1531--1575.</p><p><strong>Abstract:</strong><br/>
In parametric estimation of covariance function of Gaussian processes, it is often the case that the true covariance function does not belong to the parametric set used for estimation. This situation is called the misspecified case. In this case, it has been shown that, for irregular spatial sampling of observation points, Cross Validation can yield smaller prediction errors than Maximum Likelihood. Motivated by this observation, we provide a general asymptotic analysis of the misspecified case, for independent and uniformly distributed observation points. We prove that the Maximum Likelihood estimator asymptotically minimizes a Kullback–Leibler divergence, within the misspecified parametric set, while Cross Validation asymptotically minimizes the integrated square prediction error. In Monte Carlo simulations, we show that the covariance parameters estimated by Maximum Likelihood and Cross Validation, and the corresponding Kullback–Leibler divergences and integrated square prediction errors, can be strongly contrasting. On a more technical level, we provide new increasing-domain asymptotic results for independent and uniformly distributed observation points.
</p>projecteuclid.org/euclid.bj/1505980903_20170921040135Thu, 21 Sep 2017 04:01 EDTOn branching process with rare neutral mutationhttps://projecteuclid.org/euclid.bj/1505980904<strong>Airam Blancas</strong>, <strong>Víctor Rivero</strong>. <p><strong>Source: </strong>Bernoulli, Volume 24, Number 2, 1576--1612.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the genealogical structure of a Galton–Watson process with neutral mutations. Namely, we extend in two directions the asymptotic results obtained in Bertoin [ Stochastic Process. Appl. 120 (2010) 678–697]. In the critical case, we construct the version of the model in Bertoin [ Stochastic Process. Appl. 120 (2010) 678–697], conditioned not to be extinct. We establish a version of the limit theorems in Bertoin [ Stochastic Process. Appl. 120 (2010) 678–697], when the reproduction law has an infinite variance and it is in the domain of attraction of an $\alpha$-stable distribution, both for the unconditioned process and for the process conditioned to nonextinction. In the latter case, we obtain the convergence (after re-normalization) of the allelic sub-populations towards a tree indexed CSBP with immigration.
</p>projecteuclid.org/euclid.bj/1505980904_20170921040135Thu, 21 Sep 2017 04:01 EDT