Bernoulli Articles (Project Euclid)

A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables

Thu, 05 Aug 2010 15:41 EDT

Michael V. Boutsikas, Eutichia Vaggelatou

Source: Bernoulli, Volume 16, Number 2, 301--330.

Abstract:
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.

On branching process with rare neutral mutation

Thu, 21 Sep 2017 04:01 EDT

Airam Blancas, Víctor Rivero.

Source: Bernoulli, Volume 24, Number 2, 1576--1612.

Abstract:
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.

Regularity of BSDEs with a convex constraint on the gains-process

Thu, 01 Feb 2018 22:01 EST

Bruno Bouchard, Romuald Elie, Ludovic Moreau.

Source: Bernoulli, Volume 24, Number 3, 1613--1635.

Abstract:
We consider the minimal super-solution of a backward stochastic differential equation with constraint on the gains-process. The terminal condition is given by a function of the terminal value of a forward stochastic differential equation. Under boundedness assumptions on the coefficients, we show that the first component of the solution is Lipschitz in space and $\frac{1}{2}$-Hölder in time with respect to the initial data of the forward process. Its path is continuous before the time horizon at which its left-limit is given by a face-lifted version of its natural boundary condition. This first component is actually equal to its own face-lift. We only use probabilistic arguments. In particular, our results can be extended to certain non-Markovian settings.

Optimal scaling of the independence sampler: Theory and practice

Thu, 01 Feb 2018 22:01 EST

Clement Lee, Peter Neal.

Source: Bernoulli, Volume 24, Number 3, 1636--1652.

Abstract:
The independence sampler is one of the most commonly used MCMC algorithms usually as a component of a Metropolis-within-Gibbs algorithm. The common focus for the independence sampler is on the choice of proposal distribution to obtain an as high as possible acceptance rate. In this paper, we have a somewhat different focus concentrating on the use of the independence sampler for updating augmented data in a Bayesian framework where a natural proposal distribution for the independence sampler exists. Thus, we concentrate on the proportion of the augmented data to update to optimise the independence sampler. Generic guidelines for optimising the independence sampler are obtained for independent and identically distributed product densities mirroring findings for the random walk Metropolis algorithm. The generic guidelines are shown to be informative beyond the narrow confines of idealised product densities in two epidemic examples.

On the weak approximation of a skew diffusion by an Euler-type scheme

Thu, 01 Feb 2018 22:01 EST

Noufel Frikha.

Source: Bernoulli, Volume 24, Number 3, 1653--1691.

Abstract:
We study the weak approximation error of a skew diffusion with bounded measurable drift and Hölder diffusion coefficient by an Euler-type scheme, which consists of iteratively simulating skew Brownian motions with constant drift. We first establish two sided Gaussian bounds for the density of this approximation scheme. Then, a bound for the difference between the densities of the skew diffusion and its Euler approximation is obtained. Notably, the weak approximation error is shown to be of order $h^{\eta/2}$, where $h$ is the time step of the scheme, $\eta$ being the Hölder exponent of the diffusion coefficient.

Parametrized measure models

Thu, 01 Feb 2018 22:01 EST

Nihat Ay, Jürgen Jost, Hông Vân Lê, Lorenz Schwachhöfer.

Source: Bernoulli, Volume 24, Number 3, 1692--1725.

Abstract:
We develop a new and general notion of parametric measure models and statistical models on an arbitrary sample space $\Omega$ which does not assume that all measures of the model have the same null sets. This is given by a differentiable map from the parameter manifold $M$ into the set of finite measures or probability measures on $\Omega$, respectively, which is differentiable when regarded as a map into the Banach space of all signed measures on $\Omega$. Furthermore, we also give a rigorous definition of roots of measures and give a natural characterization of the Fisher metric and the Amari–Chentsov tensor as the pullback of tensors defined on the space of roots of measures. We show that many features such as the preservation of this tensor under sufficient statistics and the monotonicity formula hold even in this very general set-up.

Unbiased Monte Carlo: Posterior estimation for intractable/infinite-dimensional models

Thu, 01 Feb 2018 22:01 EST

Sergios Agapiou, Gareth O. Roberts, Sebastian J. Vollmer.

Source: Bernoulli, Volume 24, Number 3, 1726--1786.

Abstract:
We provide a general methodology for unbiased estimation for intractable stochastic models. We consider situations where the target distribution can be written as an appropriate limit of distributions, and where conventional approaches require truncation of such a representation leading to a systematic bias. For example, the target distribution might be representable as the $L^{2}$-limit of a basis expansion in a suitable Hilbert space; or alternatively the distribution of interest might be representable as the weak limit of a sequence of random variables, as in MCMC. Our main motivation comes from infinite-dimensional models which can be parameterised in terms of a series expansion of basis functions (such as that given by a Karhunen–Loeve expansion). We introduce and analyse schemes for direct unbiased estimation along such an expansion. However, a substantial component of our paper is devoted to the study of MCMC schemes which, due to their infinite dimensionality, cannot be directly implemented, but which can be effectively estimated unbiasedly. For all our methods we give theory to justify the numerical stability for robust Monte Carlo implementation, and in some cases we illustrate using simulations. Interestingly the computational efficiency of our methods is usually comparable to simpler methods which are biased. Crucial to the effectiveness of our proposed methodology is the construction of appropriate couplings, many of which resonate strongly with the Monte Carlo constructions used in the coupling from the past algorithm.

On Gaussian comparison inequality and its application to spectral analysis of large random matrices

Thu, 01 Feb 2018 22:01 EST

Fang Han, Sheng Xu, Wen-Xin Zhou.

Source: Bernoulli, Volume 24, Number 3, 1787--1833.

Abstract:
Recently, Chernozhukov, Chetverikov, and Kato ( Ann. Statist. 42 (2014) 1564–1597) developed a new Gaussian comparison inequality for approximating the suprema of empirical processes. This paper exploits this technique to devise sharp inference on spectra of large random matrices. In particular, we show that two long-standing problems in random matrix theory can be solved: (i) simple bootstrap inference on sample eigenvalues when true eigenvalues are tied; (ii) conducting two-sample Roy’s covariance test in high dimensions. To establish the asymptotic results, a generalized $\varepsilon$-net argument regarding the matrix rescaled spectral norm and several new empirical process bounds are developed and of independent interest.

Extrema of rescaled locally stationary Gaussian fields on manifolds

Thu, 01 Feb 2018 22:01 EST

Wanli Qiao, Wolfgang Polonik.

Source: Bernoulli, Volume 24, Number 3, 1834--1859.

Abstract:
Given a class of centered Gaussian random fields $\{X_{h}(s),s\in\mathbb{R}^{n},h\in(0,1]\}$, define the rescaled fields $\{Z_{h}(t)=X_{h}(h^{-1}t),t\in\mathcal{M}\}$, where $\mathcal{M}$ is a compact Riemannian manifold. Under the assumption that the fields $Z_{h}(t)$ satisfy a local stationary condition, we study the limit behavior of the extreme values of these rescaled Gaussian random fields, as $h$ tends to zero. Our main result can be considered as a generalization of a classical result of Bickel and Rosenblatt ( Ann. Statist. 1 (1973) 1071–1095), and also of results by Mikhaleva and Piterbarg ( Theory Probab. Appl. 41 (1997) 367–379).

Strong consistency of multivariate spectral variance estimators in Markov chain Monte Carlo

Thu, 01 Feb 2018 22:01 EST

Dootika Vats, James M. Flegal, Galin L. Jones.

Source: Bernoulli, Volume 24, Number 3, 1860--1909.

Abstract:
Markov chain Monte Carlo (MCMC) algorithms are used to estimate features of interest of a distribution. The Monte Carlo error in estimation has an asymptotic normal distribution whose multivariate nature has so far been ignored in the MCMC community. We present a class of multivariate spectral variance estimators for the asymptotic covariance matrix in the Markov chain central limit theorem and provide conditions for strong consistency. We examine the finite sample properties of the multivariate spectral variance estimators and its eigenvalues in the context of a vector autoregressive process of order 1.

Finite sample properties of the mean occupancy counts and probabilities

Thu, 01 Feb 2018 22:01 EST

Geoffrey Decrouez, Michael Grabchak, Quentin Paris.

Source: Bernoulli, Volume 24, Number 3, 1910--1941.

Abstract:
For a probability distribution $P$ on an at most countable alphabet $\mathcal{A}$, this article gives finite sample bounds for the expected occupancy counts $\mathbb{E}K_{n,r}$ and probabilities $\mathbb{E}M_{n,r}$. Both upper and lower bounds are given in terms of the counting function $\nu$ of $P$. Special attention is given to the case where $\nu$ is bounded by a regularly varying function. In this case, it is shown that our general results lead to an optimal-rate control of the expected occupancy counts and probabilities with explicit constants. Our results are also put in perspective with Turing’s formula and recent concentration bounds to deduce bounds in probability. At the end of the paper, we discuss an extension of the occupancy problem to arbitrary distributions in a metric space.

Tree formulas, mean first passage times and Kemeny’s constant of a Markov chain

Thu, 01 Feb 2018 22:01 EST

Jim Pitman, Wenpin Tang.

Source: Bernoulli, Volume 24, Number 3, 1942--1972.

Abstract:
This paper offers some probabilistic and combinatorial insights into tree formulas for the Green function and hitting probabilities of Markov chains on a finite state space. These tree formulas are closely related to loop-erased random walks by Wilson’s algorithm for random spanning trees, and to mixing times by the Markov chain tree theorem . Let $m_{ij}$ be the mean first passage time from $i$ to $j$ for an irreducible chain with finite state space $S$ and transition matrix $(p_{ij};i,j\in S)$. It is well known that $m_{jj}=1/\pi_{j}=\Sigma^{(1)}/\Sigma_{j}$, where $\pi$ is the stationary distribution for the chain, $\Sigma_{j}$ is the tree sum, over $n^{n-2}$ trees $\mathbf{t}$ spanning $S$ with root $j$ and edges $i\rightarrow k$ directed towards $j$, of the tree product $\prod_{i\rightarrow k\in\mathbf{t}}p_{ik}$, and $\Sigma^{(1)}:=\sum_{j\in S}\Sigma_{j}$. Chebotarev and Agaev ( Linear Algebra Appl. 356 (2002) 253–274) derived further results from Kirchhoff’s matrix tree theorem . We deduce that for $i\ne j$, $m_{ij}=\Sigma_{ij}/\Sigma_{j}$, where $\Sigma_{ij}$ is the sum over the same set of $n^{n-2}$ spanning trees of the same tree product as for $\Sigma_{j}$, except that in each product the factor $p_{kj}$ is omitted where $k=k(i,j,\mathbf{t})$ is the last state before $j$ in the path from $i$ to $j$ in $\mathbf{t}$. It follows that Kemeny’s constant $\sum_{j\in S}m_{ij}/m_{jj}$ equals $\Sigma^{(2)}/\Sigma^{(1)}$, where $\Sigma^{(r)}$ is the sum, over all forests $\mathbf{f}$ labeled by $S$ with $r$ directed trees, of the product of $p_{ij}$ over edges $i\rightarrow j$ of $\mathbf{f}$. We show that these results can be derived without appeal to the matrix tree theorem. A list of relevant literature is also reviewed.

When do wireless network signals appear Poisson?

Thu, 01 Feb 2018 22:01 EST

H. Paul Keeler, Nathan Ross, Aihua Xia.

Source: Bernoulli, Volume 24, Number 3, 1973--1994.

Abstract:
We consider the point process of signal strengths from transmitters in a wireless network observed from a fixed position under models with general signal path loss and random propagation effects. We show via coupling arguments that under general conditions this point process of signal strengths can be well-approximated by an inhomogeneous Poisson or a Cox point processes on the positive real line. We also provide some bounds on the total variation distance between the laws of these point processes and both Poisson and Cox point processes. Under appropriate conditions, these results support the use of a spatial Poisson point process for the underlying positioning of transmitters in models of wireless networks, even if in reality the positioning does not appear Poisson. We apply the results to a number of models with popular choices for positioning of transmitters, path loss functions, and distributions of propagation effects.

Strong convergence of the symmetrized Milstein scheme for some CEV-like SDEs

Thu, 01 Feb 2018 22:01 EST

Mireille Bossy, Héctor Olivero.

Source: Bernoulli, Volume 24, Number 3, 1995--2042.

Abstract:
In this paper, we study the rate of convergence of a symmetrized version of the Milstein scheme applied to the solution of the one dimensional SDE \[X_{t}=x_{0}+\int_{0}^{t}{b(X_{s})\,ds}+\int_{0}^{t}{\sigma\vert X_{s}\vert^{\alpha}\,dW_{s}},\qquad x_{0}>0,\sigma>0,\alpha\in[\frac{1}{2},1).\] Assuming $b(0)/\sigma^{2}$ big enough, and $b$ smooth, we prove a strong rate of convergence of order one, recovering the classical result of Milstein for SDEs with smooth diffusion coefficient. In contrast with other recent results, our proof does not relies on Lamperti transformation, and it can be applied to a wide class of drift functions. On the downside, our hypothesis on the critical parameter value $b(0)/\sigma^{2}$ is more restrictive than others available in the literature. Some numerical experiments and comparison with various other schemes complement our theoretical analysis that also applies for the simple projected Milstein scheme with same convergence rate.

Deviation of polynomials from their expectations and isoperimetry

Thu, 01 Feb 2018 22:01 EST

Lavrentin M. Arutyunyan, Egor D. Kosov.

Source: Bernoulli, Volume 24, Number 3, 2043--2063.

Abstract:
The article is divided into two parts. In the first part, we study the deviation of a polynomial from its mathematical expectation. This deviation can be estimated from above by Carbery–Wright inequality, so we investigate estimates of the deviation from below. We obtain such type estimates in two different cases: for Gaussian measures and a polynomial of an arbitrary degree and for an arbitrary log-concave measure but only for polynomials of the second degree. In the second part, we deal with the isoperimetric inequality and the Poincaré inequality for probability measures on the real line that are images of the uniform distributions on convex compact sets in $\mathbb{R}^{n}$ under polynomial mappings.

On parameter estimation of hidden telegraph process

Thu, 01 Feb 2018 22:01 EST

Rafail Z. Khasminskii, Yury A. Kutoyants.

Source: Bernoulli, Volume 24, Number 3, 2064--2090.

Abstract:
The problem of parameter estimation is considered for the two-state telegraph process, observed in the white Gaussian observation noise. An online one-step Maximum Likelihood Estimator process is constructed, using a preliminary Method of Moments Estimator. The obtained estimation procedure is shown to be asymptotically normal and asymptotically efficient in the large sample regime.

A general approach to posterior contraction in nonparametric inverse problems

Thu, 01 Feb 2018 22:01 EST

Bartek Knapik, Jean-Bernard Salomond.

Source: Bernoulli, Volume 24, Number 3, 2091--2121.

Abstract:
In this paper, we propose a general method to derive an upper bound for the contraction rate of the posterior distribution for nonparametric inverse problems. We present a general theorem that allows us to derive contraction rates for the parameter of interest from contraction rates of the related direct problem of estimating transformed parameter of interest. An interesting aspect of this approach is that it allows us to derive contraction rates for priors that are not related to the singular value decomposition of the operator. We apply our result to several examples of linear inverse problems, both in the white noise sequence model and the nonparametric regression model, using priors based on the singular value decomposition of the operator, location-mixture priors and splines prior, and recover minimax adaptive contraction rates.

Bayesian non-parametric inference for $\Lambda$-coalescents: Posterior consistency and a parametric method

Thu, 01 Feb 2018 22:01 EST

Jere Koskela, Paul A. Jenkins, Dario Spanò.

Source: Bernoulli, Volume 24, Number 3, 2122--2153.

Abstract:
We investigate Bayesian non-parametric inference of the $\Lambda$-measure of $\Lambda$-coalescent processes with recurrent mutation, parametrised by probability measures on the unit interval. We give verifiable criteria on the prior for posterior consistency when observations form a time series, and prove that any non-trivial prior is inconsistent when all observations are contemporaneous. We then show that the likelihood given a data set of size $n\in \mathbb{N}$ is constant across $\Lambda$-measures whose leading $n-2$ moments agree, and focus on inferring truncated sequences of moments. We provide a large class of functionals which can be extremised using finite computation given a credible region of posterior truncated moment sequences, and a pseudo-marginal Metropolis–Hastings algorithm for sampling the posterior. Finally, we compare the efficiency of the exact and noisy pseudo-marginal algorithms with and without delayed acceptance acceleration using a simulation study.

The function-indexed sequential empirical process under long-range dependence

Thu, 01 Feb 2018 22:01 EST

Jannis Buchsteiner.

Source: Bernoulli, Volume 24, Number 3, 2154--2175.

Abstract:
Let $(\boldsymbol{X}_{j})_{j\geq1}$ be a multivariate long-range dependent Gaussian process. We study the asymptotic behavior of the corresponding sequential empirical process indexed by a class of functions. If some entropy condition is satisfied we have weak convergence to a linear combination of Hermite processes.

On optimality of empirical risk minimization in linear aggregation

Thu, 01 Feb 2018 22:01 EST

Adrien Saumard.

Source: Bernoulli, Volume 24, Number 3, 2176--2203.

Abstract:
In the first part of this paper, we show that the small-ball condition, recently introduced by ( J. ACM 62 (2015) Art. 21, 25), may behave poorly for important classes of localized functions such as wavelets, piecewise polynomials or for trigonometric polynomials, in particular leading to suboptimal estimates of the rate of convergence of ERM for the linear aggregation problem. In a second part, we recover optimal rates of convergence for the excess risk of ERM when the dictionary is made of trigonometric functions. Considering the bounded case, we derive the concentration of the excess risk around a single point, which is an information far more precise than the rate of convergence. In the general setting of a $L_{2}$ noise, we finally refine the small ball argument by rightly selecting the directions we are looking at, in such a way that we obtain optimal rates of aggregation for the Fourier dictionary.

Dynamics of an adaptive randomly reinforced urn

Thu, 01 Feb 2018 22:01 EST

Giacomo Aletti, Andrea Ghiglietti, Anand N. Vidyashankar.

Source: Bernoulli, Volume 24, Number 3, 2204--2255.

Abstract:
Adaptive randomly reinforced urn (ARRU) is a two-color urn model where the updating process is defined by a sequence of non-negative random vectors $\{(D_{1,n},D_{2,n});n\geq1\}$ and randomly evolving thresholds which utilize accruing statistical information for the updates. Let $m_{1}=E[D_{1,n}]$ and $m_{2}=E[D_{2,n}]$. In this paper, we undertake a detailed study of the dynamics of the ARRU model. First, for the case $m_{1}\neq m_{2}$, we establish $L_{1}$ bounds on the increments of the urn proportion, that is, the proportion of ball colors in the urn, at fixed and increasing times under very weak assumptions on the random threshold sequences. As a consequence, we deduce weak consistency of the evolving urn proportions. Second, under slightly stronger conditions, we establish the strong consistency of the urn proportions for all finite values of $m_{1}$ and $m_{2}$. Specifically, we show that when $m_{1}=m_{2}$, the proportion converges to a non-degenerate random variable. Third, we establish the asymptotic distribution, after appropriate centering and scaling, for the proportion of sampled ball colors and urn proportions for the case $m_{1}=m_{2}$. In the process, we resolve the issue concerning the asymptotic distribution of the proportion of sampled ball colors for a randomly reinforced urn (RRU). To address the technical issues, we establish results on the harmonic moments of the total number of balls in the urn at different times under very weak conditions, which is of independent interest.

Equilibrium of the interface of the grass-bushes-trees process

Thu, 01 Feb 2018 22:01 EST

Enrique Andjel, Thomas Mountford, Daniel Valesin.

Source: Bernoulli, Volume 24, Number 3, 2256--2277.

Abstract:
We consider the grass-bushes-trees process, which is a two-type contact process in which one of the types is dominant. Individuals of the dominant type can give birth on empty sites and sites occupied by non-dominant individuals, whereas non-dominant individuals can only give birth at empty sites. We study the shifted version of this process so that it is ‘seen from the rightmost dominant individual’ (which is well defined if the process occurs in an appropriate subset of the configuration space); we call this shifted process the grass-bushes-trees interface (GBTI) process. The set of stationary distributions of the GBTI process is fully characterized, and precise conditions for convergence to these distributions are given.

Schwarz type model comparison for LAQ models

Thu, 01 Feb 2018 22:01 EST

Shoichi Eguchi, Hiroki Masuda.

Source: Bernoulli, Volume 24, Number 3, 2278--2327.

Abstract:
For model-comparison purpose, we study asymptotic behavior of the marginal quasi-log likelihood associated with a family of locally asymptotically quadratic (LAQ) statistical experiments. Our result entails a far-reaching extension of applicable scope of the classical approximate Bayesian model comparison due to Schwarz, with frequentist-view theoretical foundation. In particular, the proposed statistics can deal with both ergodic and non-ergodic stochastic process models, where the corresponding $M$-estimator may of multi-scaling type and the asymptotic quasi-information matrix may be random. We also deduce the consistency of the multistage optimal-model selection where we select an optimal sub-model structure step by step, so that computational cost can be much reduced. Focusing on some diffusion type models, we illustrate the proposed method by the Gaussian quasi-likelihood for diffusion-type models in details, together with several numerical experiments.

M-estimators of location for functional data

Thu, 01 Feb 2018 22:01 EST

Beatriz Sinova, Gil González-Rodríguez, Stefan Van Aelst.

Source: Bernoulli, Volume 24, Number 3, 2328--2357.

Abstract:
M-estimators of location are widely used robust estimators of the center of univariate or multivariate real-valued data. This paper aims to study M-estimates of location in the framework of functional data analysis. To this end, recent developments for robust nonparametric density estimation by means of M-estimators are considered. These results can also be applied in the context of functional data analysis and allow to state conditions for the existence and uniqueness of location M-estimates in this setting. Properties of these functional M-estimators are investigated. In particular, their consistency is shown and robustness is studied by means of their breakdown point and their influence function. The finite-sample performance of the M-estimators is explored by simulation. The M-estimators are also empirically compared to trimmed means for functional data.

On the local semicircular law for Wigner ensembles

Thu, 01 Feb 2018 22:01 EST

Friedrich Götze, Alexey Naumov, Alexander Tikhomirov, Dmitry Timushev.

Source: Bernoulli, Volume 24, Number 3, 2358--2400.

Abstract:
We consider a random symmetric matrix $\mathbf{X}=[X_{jk}]_{j,k=1}^{n}$ with upper triangular entries being i.i.d. random variables with mean zero and unit variance. We additionally suppose that $\mathbb{E}|X_{11}|^{4+\delta}=:\mu_{4+\delta}<\infty$ for some $\delta>0$. The aim of this paper is to significantly extend a recent result of the authors Götze, Naumov and Tikhomirov (2015) and show that with high probability the typical distance between the Stieltjes transform of the empirical spectral distribution (ESD) of the matrix $n^{-\frac{1}{2}}\mathbf{X}$ and Wigner’s semicircle law is of order $(nv)^{-1}\log n$, where $v$ denotes the distance to the real line in the complex plane. We apply this result to the rate of convergence of the ESD to the distribution function of the semicircle law as well as to rigidity of eigenvalues and eigenvector delocalization significantly extending a recent result by Götze, Naumov and Tikhomirov (2015). The result on delocalization is optimal by comparison with GOE ensembles. Furthermore the techniques of this paper provide a new shorter proof for the optimal $O(n^{-1})$ rate of convergence of the expected ESD to the semicircle law.

On the Poisson equation for Metropolis–Hastings chains

Thu, 01 Feb 2018 22:01 EST

Aleksandar Mijatović, Jure Vogrinc.

Source: Bernoulli, Volume 24, Number 3, 2401--2428.

Abstract:
This paper defines an approximation scheme for a solution of the Poisson equation of a geometrically ergodic Metropolis–Hastings chain $\Phi$. The scheme is based on the idea of weak approximation and gives rise to a natural sequence of control variates for the ergodic average $S_{k}(F)=(1/k)\sum_{i=1}^{k}F(\Phi_{i})$, where $F$ is the force function in the Poisson equation. The main results show that the sequence of the asymptotic variances (in the CLTs for the control-variate estimators) converges to zero and give a rate of this convergence. Numerical examples in the case of a double-well potential are discussed.

Adaptive confidence sets for matrix completion

Mon, 26 Mar 2018 04:00 EDT

Alexandra Carpentier, Olga Klopp, Matthias Löffler, Richard Nickl.

Source: Bernoulli, Volume 24, Number 4A, 2429--2460.

Abstract:
In the present paper, we study the problem of existence of honest and adaptive confidence sets for matrix completion. We consider two statistical models: the trace regression model and the Bernoulli model. In the trace regression model, we show that honest confidence sets that adapt to the unknown rank of the matrix exist even when the error variance is unknown. Contrary to this, we prove that in the Bernoulli model, honest and adaptive confidence sets exist only when the error variance is known a priori. In the course of our proofs, we obtain bounds for the minimax rates of certain composite hypothesis testing problems arising in low rank inference.

Evolution of the Wasserstein distance between the marginals of two Markov processes

Mon, 26 Mar 2018 04:00 EDT

Aurélien Alfonsi, Jacopo Corbetta, Benjamin Jourdain.

Source: Bernoulli, Volume 24, Number 4A, 2461--2498.

Abstract:
In this paper, we are interested in the time derivative of the Wasserstein distance between the marginals of two Markov processes. As recalled in the introduction, the Kantorovich duality leads to a natural candidate for this derivative. Up to the sign, it is the sum of the integrals with respect to each of the two marginals of the corresponding generator applied to the corresponding Kantorovich potential. For pure jump processes with bounded intensity of jumps, we prove that the evolution of the Wasserstein distance is actually given by this candidate. In dimension one, we show that this remains true for Piecewise Deterministic Markov Processes. We apply the formula to estimate the exponential decrease rate of the Wasserstein distance between the marginals of two birth and death processes with the same generator in terms of the Wasserstein curvature.

The sharp constant for the Burkholder–Davis–Gundy inequality and non-smooth pasting

Mon, 26 Mar 2018 04:00 EDT

Walter Schachermayer, Florian Stebegg.

Source: Bernoulli, Volume 24, Number 4A, 2499--2530.

Abstract:
We revisit the celebrated family of BDG-inequalities introduced by Burkholder, Gundy ( Acta Math. 124 (1970) 249–304) and Davis ( Israel J. Math. 8 (1970) 187–190) for continuous martingales. For the inequalities $\mathbb{E}[\tau^{\frac{p}{2}}]\leq C_{p}\mathbb{E}[(B^{*}(\tau))^{p}]$ with $0<p<2$ we propose a connection of the optimal constant $C_{p}$ with an ordinary integro-differential equation which gives rise to a numerical method of finding this constant. Based on numerical evidence, we are able to calculate, for $p=1$, the explicit value of the optimal constant $C_{1}$, namely $C_{1}=1.27267\ldots$ . In the course of our analysis, we find a remarkable appearance of “non-smooth pasting” for a solution of a related ordinary integro-differential equation.

The $M/G/\infty$ estimation problem revisited

Mon, 26 Mar 2018 04:00 EDT

Alexander Goldenshluger.

Source: Bernoulli, Volume 24, Number 4A, 2531--2568.

Abstract:
The subject of this paper is the $M/G/\infty$ estimation problem : the goal is to estimate the service time distribution $G$ of the $M/G/\infty$ queue from the arrival–departure observations without identification of customers. We develop estimators of $G$ and derive exact non-asymptotic expressions for their mean squared errors. The problem of estimating the service time expectation is addressed as well. We present some numerical results on comparison of different estimators of the service time distribution.

Special weak Dirichlet processes and BSDEs driven by a random measure

Mon, 26 Mar 2018 04:00 EDT

Elena Bandini, Francesco Russo.

Source: Bernoulli, Volume 24, Number 4A, 2569--2609.

Abstract:
This paper considers a forward BSDE driven by a random measure, when the underlying forward process $X$ is a special semimartingale, or even more generally, a special weak Dirichlet process. Given a solution $(Y,Z,U)$, generally $Y$ appears to be of the type $u(t,X_{t})$ where $u$ is a deterministic function. In this paper, we identify $Z$ and $U$ in terms of $u$ applying stochastic calculus with respect to weak Dirichlet processes.

Perturbation theory for Markov chains via Wasserstein distance

Mon, 26 Mar 2018 04:00 EDT

Daniel Rudolf, Nikolaus Schweizer.

Source: Bernoulli, Volume 24, Number 4A, 2610--2639.

Abstract:
Perturbation theory for Markov chains addresses the question of how small differences in the transition probabilities of Markov chains are reflected in differences between their distributions. We prove powerful and flexible bounds on the distance of the $n$th step distributions of two Markov chains when one of them satisfies a Wasserstein ergodicity condition. Our work is motivated by the recent interest in approximate Markov chain Monte Carlo (MCMC) methods in the analysis of big data sets. By using an approach based on Lyapunov functions, we provide estimates for geometrically ergodic Markov chains under weak assumptions. In an autoregressive model, our bounds cannot be improved in general. We illustrate our theory by showing quantitative estimates for approximate versions of two prominent MCMC algorithms, the Metropolis–Hastings and stochastic Langevin algorithms.

Gaussian approximation for high dimensional vector under physical dependence

Mon, 26 Mar 2018 04:00 EDT

Xianyang Zhang, Guang Cheng.

Source: Bernoulli, Volume 24, Number 4A, 2640--2675.

Abstract:
We develop a Gaussian approximation result for the maximum of a sum of weakly dependent vectors, where the data dimension is allowed to be exponentially larger than sample size. Our result is established under the physical/functional dependence framework. This work can be viewed as a substantive extension of Chernozhukov et al. ( Ann. Statist. 41 (2013) 2786–2819) to time series based on a variant of Stein’s method developed therein.

Equivalence classes of staged trees

Mon, 26 Mar 2018 04:00 EDT

Christiane Görgen, Jim Q. Smith.

Source: Bernoulli, Volume 24, Number 4A, 2676--2692.

Abstract:
In this paper, we give a complete characterization of the statistical equivalence classes of CEGs and of staged trees. We are able to show that all graphical representations of the same model share a common polynomial description. Then, simple transformations on that polynomial enable us to traverse the corresponding class of graphs. We illustrate our results with a real analysis of the implicit dependence relationships within a previously studied dataset.

Max-linear models on directed acyclic graphs

Mon, 26 Mar 2018 04:00 EDT

Nadine Gissibl, Claudia Klüppelberg.

Source: Bernoulli, Volume 24, Number 4A, 2693--2720.

Abstract:
We consider a new recursive structural equation model where all variables can be written as max-linear function of their parental node variables and independent noise variables. The model is max-linear in terms of the noise variables, and its causal structure is represented by a directed acyclic graph. We detail the relation between the weights of the recursive structural equation model and the coefficients in its max-linear representation. In particular, we characterize all max-linear models which are generated by a recursive structural equation model, and show that its max-linear coefficient matrix is the solution of a fixed point equation. We also find the minimum directed acyclic graph representing the recursive structural equations of the variables. The model structure introduces a natural order between the node variables and the max-linear coefficients. This yields representations of the vector components, which are based on the minimum number of node and noise variables.

Coalescence of Euclidean geodesics on the Poisson–Delaunay triangulation

Mon, 26 Mar 2018 04:00 EDT

David Coupier, Christian Hirsch.

Source: Bernoulli, Volume 24, Number 4A, 2721--2751.

Abstract:
Let us consider Euclidean first-passage percolation on the Poisson–Delaunay triangulation. We prove almost sure coalescence of any two semi-infinite geodesics with the same asymptotic direction. The proof is based on an argument of Burton–Keane type and makes use of the concentration property for shortest-path lengths in the considered graphs. Moreover, by considering the specific example of the relative neighborhood graph, we illustrate that our approach extends to further well-known graphs in computational geometry. As an application, we show that the expected number of semi-infinite geodesics starting at a given vertex and leaving a disk of a certain radius grows at most sublinearly in the radius.

Sticky processes, local and true martingales

Mon, 26 Mar 2018 04:00 EDT

Miklós Rásonyi, Hasanjan Sayit.

Source: Bernoulli, Volume 24, Number 4A, 2752--2775.

Abstract:
We prove that for a so-called sticky process $S$ there exists an equivalent probability $Q$ and a $Q$-martingale $\tilde{S}$ that is arbitrarily close to $S$ in $L^{p}(Q)$ norm. For continuous $S$, $\tilde{S}$ can be chosen arbitrarily close to $S$ in supremum norm. In the case where $S$ is a local martingale we may choose $Q$ arbitrarily close to the original probability in the total variation norm. We provide examples to illustrate the power of our results and present an application in mathematical finance.

A new approach to estimator selection

Mon, 26 Mar 2018 04:00 EDT

O.V. Lepski.

Source: Bernoulli, Volume 24, Number 4A, 2776--2810.

Abstract:
In the framework of an abstract statistical model, we discuss how to use the solution of one estimation problem ( Problem A ) in order to construct an estimator in another, completely different, Problem B . As a solution of Problem A we understand a data-driven selection from a given family of estimators $\mathbf{A}(\mathfrak{H})=\{\widehat{A}_{\mathfrak{h}},\mathfrak{h}\in\mathfrak{H}\}$ and establishing for the selected estimator so-called oracle inequality. If $\hat{\mathfrak{h}}\in\mathfrak{H}$ is the selected parameter and $\mathbf{B}(\mathfrak{H})=\{\widehat{B}_{\mathfrak{h}},\mathfrak{h}\in\mathfrak{H}\}$ is an estimator’s collection built in Problem B , we suggest to use the estimator $\widehat{B}_{\hat{\mathfrak{h}}}$. We present very general selection rule led to selector $\hat{\mathfrak{h}}$ and find conditions under which the estimator $\widehat{B}_{\hat{\mathfrak{h}}}$ is reasonable. Our approach is illustrated by several examples related to adaptive estimation.

Concentration and moderate deviations for Poisson polytopes and polyhedra

Mon, 26 Mar 2018 04:00 EDT

Julian Grote, Christoph Thäle.

Source: Bernoulli, Volume 24, Number 4A, 2811--2841.

Abstract:
The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the $d$-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation probabilities are derived and the relative error in the central limit theorem on a logarithmic scale is investigated for a large class of key geometric characteristics. This includes the number of lower-dimensional faces and the intrinsic volumes of the random polytopes. Furthermore, moderate deviation principles for the spatial empirical measures induced by these functionals are also established using the method of cumulants. The results are applied to a class of zero cells associated with Poisson hyperplane mosaics. As a special case, this comprises the typical Poisson–Voronoi cell conditioned on having large inradius.

Large deviations for locally monotone stochastic partial differential equations driven by Lévy noise

Mon, 26 Mar 2018 04:00 EDT

Jie Xiong, Jianliang Zhai.

Source: Bernoulli, Volume 24, Number 4A, 2842--2874.

Abstract:
We establish a large deviation principle for a type of stochastic partial differential equations (SPDEs) with locally monotone coefficients driven by Lévy noise. The weak convergence method plays an important role.

Large deviations and applications for Markovian Hawkes processes with a large initial intensity

Mon, 26 Mar 2018 04:00 EDT

Xuefeng Gao, Lingjiong Zhu.

Source: Bernoulli, Volume 24, Number 4A, 2875--2905.

Abstract:
Hawkes process is a class of simple point processes that is self-exciting and has clustering effect. The intensity of this point process depends on its entire past history. It has wide applications in finance, insurance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we study linear Hawkes process with an exponential kernel in the asymptotic regime where the initial intensity of the Hawkes process is large. We establish large deviations for Hawkes processes in this regime as well as the regime when both the initial intensity and the time are large. We illustrate the strength of our results by discussing the applications to insurance and queueing systems.

Concentration inequalities for separately convex functions

Mon, 26 Mar 2018 04:00 EDT

Antoine Marchina.

Source: Bernoulli, Volume 24, Number 4A, 2906--2933.

Abstract:
We provide new comparison inequalities for separately convex functions of independent random variables. Our method is based on the decomposition in Doob martingale. However, we only impose that the martingale increments are stochastically bounded. For this purpose, building on the results of Bentkus ( Lith. Math. J. 48 (2008) 237–255; Lith. Math. J. 48 (2008) 137–157; Bounds for the stop loss premium for unbounded risks under the variance constraints (2010) Preprint), we establish comparison inequalities for random variables stochastically dominated from below and from above. We illustrate our main results by showing how they can be used to derive deviation or moment inequalities for functions which are both separately convex and separately Lipschitz, for weighted empirical distribution functions, for suprema of randomized empirical processes and for chaos of order two.

Nonparametric volatility estimation in scalar diffusions: Optimality across observation frequencies

Mon, 26 Mar 2018 04:00 EDT

Jakub Chorowski.

Source: Bernoulli, Volume 24, Number 4A, 2934--2990.

Abstract:
The nonparametric volatility estimation problem of a scalar diffusion process observed at equidistant time points is addressed. Using the spectral representation of the volatility in terms of the invariant density and an eigenpair of the infinitesimal generator the first known estimator that attains the minimax optimal convergence rates for both high and low-frequency observations is constructed. The proofs are based on a posteriori error bounds for generalized eigenvalue problems as well as the path properties of scalar diffusions and stochastic analysis. The finite sample performance is illustrated by a numerical example.

Simultaneous quantile inference for non-stationary long-memory time series

Mon, 26 Mar 2018 04:00 EDT

Weichi Wu, Zhou Zhou.

Source: Bernoulli, Volume 24, Number 4A, 2991--3012.

Abstract:
We consider the simultaneous or functional inference of time-varying quantile curves for a class of non-stationary long-memory time series. New uniform Bahadur representations and Gaussian approximation schemes are established for a broad class of non-stationary long-memory linear processes. Furthermore, an asymptotic distribution theory is developed for the maxima of a class of non-stationary long-memory Gaussian processes. Using the latter theoretical results, simultaneous confidence bands for the aforementioned quantile curves with asymptotically correct coverage probabilities are constructed.

Simultaneous nonparametric regression analysis of sparse longitudinal data

Mon, 26 Mar 2018 04:00 EDT

Hongyuan Cao, Weidong Liu, Zhou Zhou.

Source: Bernoulli, Volume 24, Number 4A, 3013--3038.

Abstract:
Longitudinal data arise frequently in many scientific inquiries. To capture the dynamic relationship between longitudinal covariates and response, varying coefficient models have been proposed with point-wise inference procedures. This paper considers the challenging problem of asymptotically accurate simultaneous inference of varying coefficient models for sparse and irregularly observed longitudinal data via the local linear kernel method. The error and covariate processes are modeled as very general classes of non-Gaussian and non-stationary processes and are allowed to be statistically dependent. Simultaneous confidence bands (SCBs) with asymptotically correct coverage probabilities are constructed to assess the overall pattern and magnitude of the dynamic association between the response and covariates. A simulation based method is proposed to overcome the problem of slow convergence of the asymptotic results. Simulation studies demonstrate that the proposed inference procedure performs well in realistic settings and is favored over the existing point-wise and Bonferroni methods. A longitudinal dataset from the Chicago Health and Aging Project is used to illustrate our methodology.

Nested particle filters for online parameter estimation in discrete-time state-space Markov models

Mon, 26 Mar 2018 04:00 EDT

Dan Crisan, Joaquín Míguez.

Source: Bernoulli, Volume 24, Number 4A, 3039--3086.

Abstract:
We address the problem of approximating the posterior probability distribution of the fixed parameters of a state-space dynamical system using a sequential Monte Carlo method. The proposed approach relies on a nested structure that employs two layers of particle filters to approximate the posterior probability measure of the static parameters and the dynamic state variables of the system of interest, in a vein similar to the recent “sequential Monte Carlo square” (SMC$^{2}$) algorithm. However, unlike the SMC$^{2}$ scheme, the proposed technique operates in a purely recursive manner. In particular, the computational complexity of the recursive steps of the method introduced herein is constant over time. We analyse the approximation of integrals of real bounded functions with respect to the posterior distribution of the system parameters computed via the proposed scheme. As a result, we prove, under regularity assumptions, that the approximation errors vanish asymptotically in $L_{p}$ ($p\ge1$) with convergence rate proportional to $\frac{1}{\sqrt{N}}+\frac{1}{\sqrt{M}}$, where $N$ is the number of Monte Carlo samples in the parameter space and $N\times M$ is the number of samples in the state space. This result also holds for the approximation of the joint posterior distribution of the parameters and the state variables. We discuss the relationship between the SMC$^{2}$ algorithm and the new recursive method and present a simple example in order to illustrate some of the theoretical findings with computer simulations.

Applications of distance correlation to time series

Mon, 26 Mar 2018 04:00 EDT

Richard A. Davis, Muneya Matsui, Thomas Mikosch, Phyllis Wan.

Source: Bernoulli, Volume 24, Number 4A, 3087--3116.

Abstract:
The use of empirical characteristic functions for inference problems, including estimation in some special parametric settings and testing for goodness of fit, has a long history dating back to the 70s. More recently, there has been renewed interest in using empirical characteristic functions in other inference settings. The distance covariance and correlation, developed by Székely et al. ( Ann. Statist. 35 (2007) 2769–2794) and Székely and Rizzo ( Ann. Appl. Stat. 3 (2009) 1236–1265) for measuring dependence and testing independence between two random vectors, are perhaps the best known illustrations of this. We apply these ideas to stationary univariate and multivariate time series to measure lagged auto- and cross-dependence in a time series. Assuming strong mixing, we establish the relevant asymptotic theory for the sample auto- and cross-distance correlation functions. We also apply the auto-distance correlation function (ADCF) to the residuals of an autoregressive processes as a test of goodness of fit. Under the null that an autoregressive model is true, the limit distribution of the empirical ADCF can differ markedly from the corresponding one based on an i.i.d. sequence. We illustrate the use of the empirical auto- and cross-distance correlation functions for testing dependence and cross-dependence of time series in a variety of contexts.

On limit theory for Lévy semi-stationary processes

Mon, 26 Mar 2018 04:00 EDT

Andreas Basse-O’Connor, Claudio Heinrich, Mark Podolskij.

Source: Bernoulli, Volume 24, Number 4A, 3117--3146.

Abstract:
In this paper, we present some limit theorems for power variation of Lévy semi-stationary processes in the setting of infill asymptotics. Lévy semi-stationary processes, which are a one-dimensional analogue of ambit fields, are moving average type processes with a multiplicative random component, which is usually referred to as volatility or intermittency. From the mathematical point of view this work extends the asymptotic theory investigated in ( Power variation for a class of stationary increments Lévy driven moving averages . Preprint), where the authors derived the limit theory for $k$th order increments of stationary increments Lévy driven moving averages. The asymptotic results turn out to heavily depend on the interplay between the given order of the increments, the considered power $p>0$, the Blumenthal–Getoor index $\beta\in(0,2)$ of the driving pure jump Lévy process $L$ and the behaviour of the kernel function $g$ at $0$ determined by the power $\alpha$. In this paper, we will study the first order asymptotic theory for Lévy semi-stationary processes with a random volatility/intermittency component and present some statistical applications of the probabilistic results.

Wide consensus aggregation in the Wasserstein space. Application to location-scatter families

Mon, 26 Mar 2018 04:00 EDT

Pedro C. Álvarez-Esteban, Eustasio del Barrio, Juan A. Cuesta-Albertos, Carlos Matrán.

Source: Bernoulli, Volume 24, Number 4A, 3147--3179.

Abstract:
We introduce a general theory for a consensus-based combination of estimations of probability measures. Potential applications include parallelized or distributed sampling schemes as well as variations on aggregation from resampling techniques like boosting or bagging. Taking into account the possibility of very discrepant estimations, instead of a full consensus we consider a “wide consensus” procedure. The approach is based on the consideration of trimmed barycenters in the Wasserstein space of probability measures. We provide general existence and consistency results as well as suitable properties of these robustified Fréchet means. In order to get quick applicability, we also include characterizations of barycenters of probabilities that belong to (non necessarily elliptical) location and scatter families. For these families, we provide an iterative algorithm for the effective computation of trimmed barycenters, based on a consistent algorithm for computing barycenters, guarantying applicability in a wide setting of statistical problems.

Posteriors, conjugacy, and exponential families for completely random measures

Wed, 18 Apr 2018 04:06 EDT

Tamara Broderick, Ashia C. Wilson, Michael I. Jordan.

Source: Bernoulli, Volume 24, Number 4B, 3181--3221.

Abstract:
We demonstrate how to calculate posteriors for general Bayesian nonparametric priors and likelihoods based on completely random measures (CRMs). We further show how to represent Bayesian nonparametric priors as a sequence of finite draws using a size-biasing approach – and how to represent full Bayesian nonparametric models via finite marginals. Motivated by conjugate priors based on exponential family representations of likelihoods, we introduce a notion of exponential families for CRMs, which we call exponential CRMs. This construction allows us to specify automatic Bayesian nonparametric conjugate priors for exponential CRM likelihoods. We demonstrate that our exponential CRMs allow particularly straightforward recipes for size-biased and marginal representations of Bayesian nonparametric models. Along the way, we prove that the gamma process is a conjugate prior for the Poisson likelihood process and the beta prime process is a conjugate prior for a process we call the odds Bernoulli process. We deliver a size-biased representation of the gamma process and a marginal representation of the gamma process coupled with a Poisson likelihood process.

Applications of pathwise Burkholder–Davis–Gundy inequalities

Wed, 18 Apr 2018 04:06 EDT

Pietro Siorpaes.

Source: Bernoulli, Volume 24, Number 4B, 3222--3245.

Abstract:
In this paper, after generalizing the pathwise Burkholder–Davis–Gundy (BDG) inequalities from discrete time to cadlag semimartingales, we present several applications of the pathwise inequalities. In particular we show that they allow to extend the classical BDG inequalities 1 to the Bessel process of order $\alpha\geq1$ 2 to the case of a random exponent $p$ 3 to martingales stopped at a time $\tau$ which belongs to a well studied class of random times

Entropy production in nonlinear recombination models

Wed, 18 Apr 2018 04:06 EDT

Pietro Caputo, Alistair Sinclair.

Source: Bernoulli, Volume 24, Number 4B, 3246--3282.

Abstract:
We study the convergence to equilibrium of a class of nonlinear recombination models. In analogy with Boltzmann’s H-theorem from kinetic theory, and in contrast with previous analysis of these models, convergence is measured in terms of relative entropy. The problem is formulated within a general framework that we refer to as Reversible Quadratic Systems. Our main result is a tight quantitative estimate for the entropy production functional. Along the way, we establish some new entropy inequalities generalizing Shearer’s and related inequalities.

Bounded size biased couplings, log concave distributions and concentration of measure for occupancy models

Wed, 18 Apr 2018 04:06 EDT

Jay Bartroff, Larry Goldstein, Ümit Işlak.

Source: Bernoulli, Volume 24, Number 4B, 3283--3317.

Abstract:
Threshold-type counts based on multivariate occupancy models with log concave marginals admit bounded size biased couplings under weak conditions, leading to new concentration of measure results for random graphs, germ-grain models in stochastic geometry and multinomial allocation models. The results obtained compare favorably with classical methods, including the use of McDiarmid’s inequality, negative association, and self bounding functions.

Parametric inference for nonsynchronously observed diffusion processes in the presence of market microstructure noise

Wed, 18 Apr 2018 04:06 EDT

Teppei Ogihara.

Source: Bernoulli, Volume 24, Number 4B, 3318--3383.

Abstract:
We study parametric inference for diffusion processes when observations occur nonsynchronously and are contaminated by market microstructure noise. We construct a quasi-likelihood function and study asymptotic mixed normality of maximum-likelihood- and Bayes-type estimators based on it. We also prove the local asymptotic normality of the model and asymptotic efficiency of our estimator when the diffusion coefficients are deterministic and noise follows a normal distribution. We conjecture that our estimator is asymptotically efficient even when the latent process is a general diffusion process. An estimator for the quadratic covariation of the latent process is also constructed. Some numerical examples show that this estimator performs better compared to existing estimators of the quadratic covariation.

The Gamma Stein equation and noncentral de Jong theorems

Wed, 18 Apr 2018 04:06 EDT

Christian Döbler, Giovanni Peccati.

Source: Bernoulli, Volume 24, Number 4B, 3384--3421.

Abstract:
We study the Stein equation associated with the one-dimensional Gamma distribution, and provide novel bounds, allowing one to effectively deal with test functions supported by the whole real line. We apply our estimates to derive new quantitative results involving random variables that are non-linear functionals of random fields, namely: (i) a non-central quantitative de Jong theorem for sequences of degenerate $U$-statistics satisfying minimal uniform integrability conditions, significantly extending previous findings by de Jong ( J. Multivariate Anal. 34 (1990) 275–289), Nourdin, Peccati and Reinert ( Ann. Probab. 38 (2010) 1947–1985) and Döbler and Peccati ( Electron. J. Probab. 22 (2017) no. 2), (ii) a new Gamma approximation bound on the Poisson space, refining previous estimates by Peccati and Thäle ( ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013) 525–560) and (iii) new Gamma bounds on a Gaussian space, strengthening estimates by Nourdin and Peccati ( Probab. Theory Related Fields 145 (2009) 75–118). As a by-product of our analysis, we also deduce a new inequality for Gamma approximations via exchangeable pairs, that is of independent interest.

Expected number and height distribution of critical points of smooth isotropic Gaussian random fields

Wed, 18 Apr 2018 04:06 EDT

Dan Cheng, Armin Schwartzman.

Source: Bernoulli, Volume 24, Number 4B, 3422--3446.

Abstract:
We obtain formulae for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension. The results hold in general in the sense that there are no restrictions on the covariance function of the field except for smoothness and isotropy. The results are based on a characterization of the distribution of the Hessian of the Gaussian field by means of the family of Gaussian orthogonally invariant (GOI) matrices, of which the Gaussian orthogonal ensemble (GOE) is a special case. The obtained formulae depend on the covariance function only through a single parameter (Euclidean space) or two parameters (spheres), and include the special boundary case of random Laplacian eigenfunctions.

A unified matrix model including both CCA and F matrices in multivariate analysis: The largest eigenvalue and its applications

Wed, 18 Apr 2018 04:06 EDT

Xiao Han, Guangming Pan, Qing Yang.

Source: Bernoulli, Volume 24, Number 4B, 3447--3468.

Abstract:
Let $\mathbf{Z}_{M_{1}\times N}=\mathbf{T}^{\frac{1}{2}}\mathbf{X}$ where $(\mathbf{T}^{\frac{1}{2}})^{2}=\mathbf{T}$ is a positive definite matrix and $\mathbf{X}$ consists of independent random variables with mean zero and variance one. This paper proposes a unified matrix model \[\mathbf{\Omega}=(\mathbf{Z}\mathbf{U}_{2}\mathbf{U}_{2}^{T}\mathbf{Z}^{T})^{-1}\mathbf{Z}\mathbf{U}_{1}\mathbf{U}_{1}^{T}\mathbf{Z}^{T},\] where $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ are isometric with dimensions $N\times N_{1}$ and $N\times(N-N_{2})$ respectively such that $\mathbf{U}_{1}^{T}\mathbf{U}_{1}=\mathbf{I}_{N_{1}}$, $\mathbf{U}_{2}^{T}\mathbf{U}_{2}=\mathbf{I}_{N-N_{2}}$ and $\mathbf{U}_{1}^{T}\mathbf{U}_{2}=0$. Moreover, $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ (random or non-random) are independent of $\mathbf{Z}_{M_{1}\times N}$ and with probability tending to one, $\operatorname{rank}(\mathbf{U}_{1})=N_{1}$ and $\operatorname{rank}(\mathbf{U}_{2})=N-N_{2}$. We establish the asymptotic Tracy–Widom distribution for its largest eigenvalue under moment assumptions on $\mathbf{X}$ when $N_{1},N_{2}$ and $M_{1}$ are comparable. The asymptotic distributions of the maximum eigenvalues of the matrices used in Canonical Correlation Analysis (CCA) and of F matrices (including centered and non-centered versions) can be both obtained from that of $\mathbf{\Omega}$ by selecting appropriate matrices $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$. Moreover, via appropriate matrices $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$, this matrix $\mathbf{\Omega}$ can be applied to some multivariate testing problems that cannot be done by both types of matrices. To see this, we explore two more applications. One is in the MANOVA approach for testing the equivalence of several high-dimensional mean vectors, where $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ are chosen to be two nonrandom matrices. The other one is in the multivariate linear model for testing the unknown parameter matrix, where $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ are random. For each application, theoretical results are developed and various numerical studies are conducted to investigate the empirical performance.

Statistical inference for the doubly stochastic self-exciting process

Wed, 18 Apr 2018 04:06 EDT

Simon Clinet, Yoann Potiron.

Source: Bernoulli, Volume 24, Number 4B, 3469--3493.

Abstract:
We introduce and show the existence of a Hawkes self-exciting point process with exponentially-decreasing kernel and where parameters are time-varying. The quantity of interest is defined as the integrated parameter $T^{-1}\int_{0}^{T}\theta_{t}^{*}\,dt$, where $\theta_{t}^{*}$ is the time-varying parameter, and we consider the high-frequency asymptotics. To estimate it naïvely, we chop the data into several blocks, compute the maximum likelihood estimator (MLE) on each block, and take the average of the local estimates. The asymptotic bias explodes asymptotically, thus we provide a non-naïve estimator which is constructed as the naïve one when applying a first-order bias reduction to the local MLE. We show the associated central limit theorem. Monte Carlo simulations show the importance of the bias correction and that the method performs well in finite sample, whereas the empirical study discusses the implementation in practice and documents the stochastic behavior of the parameters.

Small deviations of a Galton–Watson process with immigration

Wed, 18 Apr 2018 04:06 EDT

Nadia Sidorova.

Source: Bernoulli, Volume 24, Number 4B, 3494--3521.

Abstract:
We consider a Galton–Watson process with immigration $(\mathcal{Z}_{n})$, with offspring probabilities $(p_{i})$ and immigration probabilities $(q_{i})$. In the case when $p_{0}=0$, $p_{1}\neq0$, $q_{0}=0$ (that is, when $\operatorname{essinf}(\mathcal{Z}_{n})$ grows linearly in $n$), we establish the asymptotics of the left tail $\mathbb{P}\{\mathcal{W}<\varepsilon\}$, as $\varepsilon\downarrow0$, of the martingale limit $\mathcal{W}$ of the process $(\mathcal{Z}_{n})$. Further, we consider the first generation $\mathcal{K}$ such that $\mathcal{Z}_{\mathcal{K}}\operatorname{essinf}(\mathcal{Z}_{\mathcal{K}})$ and study the asymptotic behaviour of $\mathcal{K}$ conditionally on $\{\mathcal{W}<\varepsilon\}$, as $\varepsilon\downarrow 0$. We find the growth scale and the fluctuations of $\mathcal{K}$ and compare the results with those for standard Galton–Watson processes.

Testing for simultaneous jumps in case of asynchronous observations

Wed, 18 Apr 2018 04:06 EDT

Ole Martin, Mathias Vetter.

Source: Bernoulli, Volume 24, Number 4B, 3522--3567.

Abstract:
This paper proposes a novel test for simultaneous jumps in a bivariate Itô semimartingale when observation times are asynchronous and irregular. Inference is built on a realized correlation coefficient for the squared jumps of the two processes which is estimated using bivariate power variations of Hayashi–Yoshida type without an additional synchronization step. An associated central limit theorem is shown whose asymptotic distribution is assessed using a bootstrap procedure. Simulations show that the test works remarkably well in comparison with the much simpler case of regular observations.

Statistical estimation of the Oscillating Brownian Motion

Wed, 18 Apr 2018 04:06 EDT

Antoine Lejay, Paolo Pigato.

Source: Bernoulli, Volume 24, Number 4B, 3568--3602.

Abstract:
We study the asymptotic behavior of estimators of a two-valued, discontinuous diffusion coefficient in a Stochastic Differential Equation, called an Oscillating Brownian Motion. Using the relation of the latter process with the Skew Brownian Motion, we propose two natural consistent estimators, which are variants of the integrated volatility estimator and take the occupation times into account. We show the stable convergence of the renormalized errors’ estimations toward some Gaussian mixture, possibly corrected by a term that depends on the local time. These limits stem from the lack of ergodicity as well as the behavior of the local time at zero of the process. We test both estimators on simulated processes, finding a complete agreement with the theoretical predictions.

Correlated continuous time random walks and fractional Pearson diffusions

Wed, 18 Apr 2018 04:06 EDT

N.N. Leonenko, I. Papić, A. Sikorskii, N. Šuvak.

Source: Bernoulli, Volume 24, Number 4B, 3603--3627.

Abstract:
Continuous time random walks have random waiting times between particle jumps. We define the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions (fPDs). The jumps in these CTRWs are obtained from Markov chains through the Bernoulli urn-scheme model and Wright–Fisher model. The jumps are correlated so that the limiting processes are not Lévy but diffusion processes with non-independent increments. The waiting times are selected from the domain of attraction of a stable law.

Detecting Markov random fields hidden in white noise

Wed, 18 Apr 2018 04:06 EDT

Ery Arias-Castro, Sébastien Bubeck, Gábor Lugosi, Nicolas Verzelen.

Source: Bernoulli, Volume 24, Number 4B, 3628--3656.

Abstract:
Motivated by change point problems in time series and the detection of textured objects in images, we consider the problem of detecting a piece of a Gaussian Markov random field hidden in white Gaussian noise. We derive minimax lower bounds and propose near-optimal tests.

Large volatility matrix estimation with factor-based diffusion model for high-frequency financial data

Wed, 18 Apr 2018 04:06 EDT

Donggyu Kim, Yi Liu, Yazhen Wang.

Source: Bernoulli, Volume 24, Number 4B, 3657--3682.

Abstract:
Large volatility matrices are involved in many finance practices, and estimating large volatility matrices based on high-frequency financial data encounters the “curse of dimensionality”. It is a common approach to impose a sparsity assumption on the large volatility matrices to produce consistent volatility matrix estimators. However, due to the existence of common factors, assets are highly correlated with each other, and it is not reasonable to assume the volatility matrices are sparse in financial applications. This paper incorporates factor influence in the asset pricing model and investigates large volatility matrix estimation under the factor price model together with some sparsity assumption. We propose to model asset prices by assuming that asset prices are governed by common factors and that the assets with similar characteristics share the same association with the factors. We then impose some reasonable sparsity condition on the part of the volatility matrices after accounting for the factor contribution. Under the proposed factor-based model and sparsity assumption, we develop an estimation scheme called “blocking and regularizing”. Asymptotic properties of the proposed estimator are studied, and its finite sample performance is tested via extensive numerical studies to support theoretical results.

Adaptive estimation of high-dimensional signal-to-noise ratios

Wed, 18 Apr 2018 04:06 EDT

Nicolas Verzelen, Elisabeth Gassiat.

Source: Bernoulli, Volume 24, Number 4B, 3683--3710.

Abstract:
We consider the equivalent problems of estimating the residual variance, the proportion of explained variance $\eta$ and the signal strength in a high-dimensional linear regression model with Gaussian random design. Our aim is to understand the impact of not knowing the sparsity of the vector of regression coefficients and not knowing the distribution of the design on minimax estimation rates of $\eta$. Depending on the sparsity $k$ of the vector regression coefficients, optimal estimators of $\eta$ either rely on estimating the vector of regression coefficients or are based on $U$-type statistics. In the important situation where $k$ is unknown, we build an adaptive procedure whose convergence rate simultaneously achieves the minimax risk over all $k$ up to a logarithmic loss which we prove to be non avoidable. Finally, the knowledge of the design distribution is shown to play a critical role. When the distribution of the design is unknown, consistent estimation of explained variance is indeed possible in much narrower regimes than for known design distribution.

Efficient strategy for the Markov chain Monte Carlo in high-dimension with heavy-tailed target probability distribution

Wed, 18 Apr 2018 04:06 EDT

Kengo Kamatani.

Source: Bernoulli, Volume 24, Number 4B, 3711--3750.

Abstract:
The purpose of this paper is to introduce a new Markov chain Monte Carlo method and to express its effectiveness by simulation and high-dimensional asymptotic theory. The key fact is that our algorithm has a reversible proposal kernel, which is designed to have a heavy-tailed invariant probability distribution. A high-dimensional asymptotic theory is studied for a class of heavy-tailed target probability distributions. When the number of dimensions of the state space passes to infinity, we will show that our algorithm has a much higher convergence rate than the pre-conditioned Crank–Nicolson (pCN) algorithm and the random-walk Metropolis algorithm.

The class of multivariate max-id copulas with $\ell_{1}$-norm symmetric exponent measure

Wed, 18 Apr 2018 04:06 EDT

Christian Genest, Johanna G. Nešlehová, Louis-Paul Rivest.

Source: Bernoulli, Volume 24, Number 4B, 3751--3790.

Abstract:
Members of the well-known family of bivariate Galambos copulas can be expressed in a closed form in terms of the univariate Fréchet distribution. This formula extends to any dimension and can be used to define a whole new class of tractable multivariate copulas that are generated by suitable univariate distributions. This paper gives necessary and sufficient conditions on the underlying univariate distribution which ensure that the resulting copula exists. It is also shown that these new copulas are in fact dependence structures of certain max-id distributions with $\ell_{1}$-norm symmetric exponent measure. The basic dependence properties of this new class of multivariate exchangeable copulas is investigated, and an efficient algorithm is provided for generating observations from distributions in this class.

Optimal estimation of a large-dimensional covariance matrix under Stein’s loss

Wed, 18 Apr 2018 04:06 EDT

Olivier Ledoit, Michael Wolf.

Source: Bernoulli, Volume 24, Number 4B, 3791--3832.

Abstract:
This paper introduces a new method for deriving covariance matrix estimators that are decision-theoretically optimal within a class of nonlinear shrinkage estimators. The key is to employ large-dimensional asymptotics: the matrix dimension and the sample size go to infinity together, with their ratio converging to a finite, nonzero limit. As the main focus, we apply this method to Stein’s loss. Compared to the estimator of Stein (Estimation of a covariance matrix (1975); J. Math. Sci. 34 (1986) 1373–1403), ours has five theoretical advantages: (1) it asymptotically minimizes the loss itself, instead of an estimator of the expected loss; (2) it does not necessitate post-processing via an ad hoc algorithm (called “isotonization”) to restore the positivity or the ordering of the covariance matrix eigenvalues; (3) it does not ignore any terms in the function to be minimized; (4) it does not require normality; and (5) it is not limited to applications where the sample size exceeds the dimension. In addition to these theoretical advantages, our estimator also improves upon Stein’s estimator in terms of finite-sample performance, as evidenced via extensive Monte Carlo simulations. To further demonstrate the effectiveness of our method, we show that some previously suggested estimators of the covariance matrix and its inverse are decision-theoretically optimal in the large-dimensional asymptotic limit with respect to the Frobenius loss function.

Covariance estimation via sparse Kronecker structures

Wed, 18 Apr 2018 04:06 EDT

Chenlei Leng, Guangming Pan.

Source: Bernoulli, Volume 24, Number 4B, 3833--3863.

Abstract:
The problem of estimating covariance matrices is central to statistical analysis and is extensively addressed when data are vectors. This paper studies a novel Kronecker-structured approach for estimating such matrices when data are matrices and arrays. Focusing on matrix-variate data, we present simple approaches to estimate the row and the column correlation matrices, formulated separately via convex optimization. We also discuss simple thresholding estimators motivated by the recent development in the literature. Non-asymptotic results show that the proposed method greatly outperforms methods that ignore the matrix structure of the data. In particular, our framework allows the dimensionality of data to be arbitrary order even for fixed sample size, and works for flexible distributions beyond normality. Simulations and data analysis further confirm the competitiveness of the method. An extension to general array-data is also outlined.

Robust dimension-free Gram operator estimates

Wed, 18 Apr 2018 04:06 EDT

Ilaria Giulini.

Source: Bernoulli, Volume 24, Number 4B, 3864--3923.

Abstract:
In this paper, we investigate the question of estimating the Gram operator by a robust estimator from an i.i.d. sample in a separable Hilbert space and we present uniform bounds that hold under weak moment assumptions. The approach consists in first obtaining non-asymptotic dimension-free bounds in finite-dimensional spaces using some PAC-Bayesian inequalities related to Gaussian perturbations of the parameter and then in generalizing the results in a separable Hilbert space. We show both from a theoretical point of view and with the help of some simulations that such a robust estimator improves the behavior of the classical empirical one in the case of heavy tail data distributions.

Uniform dimension results for a family of Markov processes

Wed, 18 Apr 2018 04:06 EDT

Xiaobin Sun, Yimin Xiao, Lihu Xu, Jianliang Zhai.

Source: Bernoulli, Volume 24, Number 4B, 3924--3951.

Abstract:
In this paper, we prove uniform Hausdorff and packing dimension results for the images of a large family of Markov processes. The main tools are the two covering principles in Xiao (In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2 (2004) 261–338 Amer. Math. Soc.). As applications, uniform Hausdorff and packing dimension results for certain classes of Lévy processes, stable jump diffusions and non-symmetric stable-type processes are obtained.