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.

Probability approximation of point processes with Papangelou conditional intensity

Tue, 09 May 2017 04:00 EDT

Giovanni Luca Torrisi.

Source: Bernoulli, Volume 23, Number 4A, 2210--2256.

Abstract:
We give general bounds in the Gaussian and Poisson approximations of innovations (or Skorohod integrals) defined on the space of point processes with Papangelou conditional intensity. We apply the general results to Gibbs point processes with pair potential and determinantal point processes. In particular, we provide explicit error bounds and quantitative limit theorems for stationary, inhibitory and finite range Gibbs point processes with pair potential and $\beta$-Ginibre point processes.

The geometric foundations of Hamiltonian Monte Carlo

Tue, 09 May 2017 04:00 EDT

Michael Betancourt, Simon Byrne, Sam Livingstone, Mark Girolami.

Source: Bernoulli, Volume 23, Number 4A, 2257--2298.

Abstract:
Although Hamiltonian Monte Carlo has proven an empirical success, the lack of a rigorous theoretical understanding of the algorithm has in many ways impeded both principled developments of the method and use of the algorithm in practice. In this paper, we develop the formal foundations of the algorithm through the construction of measures on smooth manifolds, and demonstrate how the theory naturally identifies efficient implementations and motivates promising generalizations.

Large-sample approximations for variance-covariance matrices of high-dimensional time series

Tue, 09 May 2017 04:00 EDT

Ansgar Steland, Rainer von Sachs.

Source: Bernoulli, Volume 23, Number 4A, 2299--2329.

Abstract:
Distributional approximations of (bi-) linear functions of sample variance-covariance matrices play a critical role to analyze vector time series, as they are needed for various purposes, especially to draw inference on the dependence structure in terms of second moments and to analyze projections onto lower dimensional spaces as those generated by principal components. This particularly applies to the high-dimensional case, where the dimension $d$ is allowed to grow with the sample size $n$ and may even be larger than $n$. We establish large-sample approximations for such bilinear forms related to the sample variance-covariance matrix of a high-dimensional vector time series in terms of strong approximations by Brownian motions and the uniform (in the dimension) consistent estimation of their covariances. The results cover weakly dependent as well as many long-range dependent linear processes and are valid for uniformly $\ell_{1}$-bounded projection vectors, which arise, either naturally or by construction, in many statistical problems extensively studied for high-dimensional series. Among those problems are sparse financial portfolio selection, sparse principal components, the LASSO, shrinkage estimation and change-point analysis for high-dimensional time series, which matter for the analysis of big data and are discussed in greater detail.

Laws of the iterated logarithm for symmetric jump processes

Tue, 09 May 2017 04:00 EDT

Panki Kim, Takashi Kumagai, Jian Wang.

Source: Bernoulli, Volume 23, Number 4A, 2330--2379.

Abstract:
Based on two-sided heat kernel estimates for a class of symmetric jump processes on metric measure spaces, the laws of the iterated logarithm (LILs) for sample paths, local times and ranges are established. In particular, the LILs are obtained for $\beta$-stable-like processes on $\alpha$-sets with $\beta>0$.

Cutting down $\mathbf{p}$-trees and inhomogeneous continuum random trees

Tue, 09 May 2017 04:00 EDT

Nicolas Broutin, Minmin Wang.

Source: Bernoulli, Volume 23, Number 4A, 2380--2433.

Abstract:
We study a fragmentation of the $\mathbf{p}$-trees of Camarri and Pitman. We give exact correspondences between the $\mathbf{p}$-trees and trees which encode the fragmentation. We then use these results to study the fragmentation of the inhomogeneous continuum random trees (scaling limits of $\mathbf{p}$-trees) and give distributional correspondences between the initial tree and the tree encoding the fragmentation. The theorems for the inhomogeneous continuum random tree extend previous results by Bertoin and Miermont about the cut tree of the Brownian continuum random tree.

A new rejection sampling method without using hat function

Tue, 09 May 2017 04:00 EDT

Hongsheng Dai.

Source: Bernoulli, Volume 23, Number 4A, 2434--2465.

Abstract:
This paper proposes a new exact simulation method, which simulates a realisation from a proposal density and then uses exact simulation of a Langevin diffusion to check whether the proposal should be accepted or rejected. Comparing to the existing coupling from the past method, the new method does not require constructing fast coalescence Markov chains. Comparing to the existing rejection sampling method, the new method does not require the proposal density function to bound the target density function. The new method is much more efficient than existing methods for certain problems. An application on exact simulation of the posterior of finite mixture models is presented.

Spectral analysis of high-dimensional sample covariance matrices with missing observations

Tue, 09 May 2017 04:00 EDT

Kamil Jurczak, Angelika Rohde.

Source: Bernoulli, Volume 23, Number 4A, 2466--2532.

Abstract:
We study high-dimensional sample covariance matrices based on independent random vectors with missing coordinates. The presence of missing observations is common in modern applications such as climate studies or gene expression micro-arrays. A weak approximation on the spectral distribution in the “large dimension $d$ and large sample size $n$” asymptotics is derived for possibly different observation probabilities in the coordinates. The spectral distribution turns out to be strongly influenced by the missingness mechanism. In the null case under the missing at random scenario where each component is observed with the same probability $p$, the limiting spectral distribution is a Marčenko–Pastur law shifted by $(1-p)/p$ to the left. As $d/n\rightarrow y\in(0,1)$, the almost sure convergence of the extremal eigenvalues to the respective boundary points of the support of the limiting spectral distribution is proved, which are explicitly given in terms of $y$ and $p$. Eventually, the sample covariance matrix is positive definite if $p$ is larger than \[1-(1-\sqrt{y})^{2},\] whereas this is not true any longer if $p$ is smaller than this quantity.

Convergence rates of Laplace-transform based estimators

Tue, 09 May 2017 04:00 EDT

Arnoud V. den Boer, Michel Mandjes.

Source: Bernoulli, Volume 23, Number 4A, 2533--2557.

Abstract:
This paper considers the problem of estimating probabilities of the form $\mathbb{P}(Y\leq w)$, for a given value of $w$, in the situation that a sample of i.i.d. observations $X_{1},\ldots,X_{n}$ of $X$ is available, and where we explicitly know a functional relation between the Laplace transforms of the non-negative random variables $X$ and $Y$. A plug-in estimator is constructed by calculating the Laplace transform of the empirical distribution of the sample $X_{1},\ldots,X_{n}$, applying the functional relation to it, and then (if possible) inverting the resulting Laplace transform and evaluating it in $w$. We show, under mild regularity conditions, that the resulting estimator is weakly consistent and has expected absolute estimation error $O(n^{-1/2}\log(n+1))$. We illustrate our results by two examples: in the first we estimate the distribution of the workload in an M/G/1 queue from observations of the input in fixed time intervals, and in the second we identify the distribution of the increments when observing a compound Poisson process at equidistant points in time (usually referred to as “decompounding”).

Randomized pivots for means of short and long memory linear processes

Tue, 09 May 2017 04:00 EDT

Miklós Csörgő, Masoud M. Nasari, Mohamedou Ould-Haye.

Source: Bernoulli, Volume 23, Number 4A, 2558--2586.

Abstract:
In this paper, we introduce randomized pivots for the means of short and long memory linear processes. We show that, under the same conditions, these pivots converge in distribution to the same limit as that of their classical non-randomized counterparts. We also present numerical results that indicate that these randomized pivots significantly outperform their classical counterparts and as a result they lead to a more accurate inference about the population mean.

Strongly degenerate time inhomogeneous SDEs: Densities and support properties. Application to Hodgkin–Huxley type systems

Tue, 09 May 2017 04:00 EDT

R. Höpfner, E. Löcherbach, M. Thieullen.

Source: Bernoulli, Volume 23, Number 4A, 2587--2616.

Abstract:
In this paper, we study the existence of densities for strongly degenerate stochastic differential equations (SDEs) whose coefficients depend on time and are not globally Lipschitz. In these models, neither local ellipticity nor the strong Hörmander condition is satisfied. In this general setting, we show that continuous transition densities indeed exist in all neighborhoods of points where the weak Hörmander condition is satisfied. We also exhibit regions where these densities remain positive. We then apply these results to stochastic Hodgkin–Huxley models with periodic input as a first step towards the study of ergodicity properties of such systems in the sense of Meyn and Tweedie ( Adv. in Appl. Probab. 25 (1993) 487–517; Adv. in Appl. Probab. 25 (1993) 518–548).

Extended generalised variances, with applications

Tue, 09 May 2017 04:00 EDT

Luc Pronzato, Henry P. Wynn, Anatoly A. Zhigljavsky.

Source: Bernoulli, Volume 23, Number 4A, 2617--2642.

Abstract:
We consider a measure $\psi_{k}$ of dispersion which extends the notion of Wilk’s generalised variance for a $d$-dimensional distribution, and is based on the mean squared volume of simplices of dimension $k\leq d$ formed by $k+1$ independent copies. We show how $\psi_{k}$ can be expressed in terms of the eigenvalues of the covariance matrix of the distribution, also when a $n$-point sample is used for its estimation, and prove its concavity when raised at a suitable power. Some properties of dispersion-maximising distributions are derived, including a necessary and sufficient condition for optimality. Finally, we show how this measure of dispersion can be used for the design of optimal experiments, with equivalence to $A$ and $D$-optimal design for $k=1$ and $k=d$, respectively. Simple illustrative examples are presented.

Multilevel Richardson–Romberg extrapolation

Tue, 09 May 2017 04:00 EDT

Vincent Lemaire, Gilles Pagès.

Source: Bernoulli, Volume 23, Number 4A, 2643--2692.

Abstract:
We propose and analyze a Multilevel Richardson–Romberg (ML2R) estimator which combines the higher order bias cancellation of the Multistep Richardson–Romberg method introduced in [ Monte Carlo Methods Appl. 13 (2007) 37–70] and the variance control resulting from Multilevel Monte Carlo (MLMC) paradigm (see [ Ann. Appl. Probab. 24 (2014) 1585–1620, In Large-Scale Scientific Computing (2001) 58–67 Berlin]). Thus, in standard frameworks like discretization schemes of diffusion processes, the root mean squared error (RMSE) $\varepsilon>0$ can be achieved with our ML2R estimator with a global complexity of $\varepsilon^{-2}\log(1/\varepsilon)$ instead of $\varepsilon^{-2}(\log(1/\varepsilon))^{2}$ with the standard MLMC method, at least when the weak error $\mathbf{E}[Y_{h}]-\mathbf{E}[Y_{0}]$ of the biased implemented estimator $Y_{h}$ can be expanded at any order in $h$ and $\Vert Y_{h}-Y_{0}\Vert_{2}=O(h^{\frac{1}{2}})$. The ML2R estimator is then halfway between a regular MLMC and a virtual unbiased Monte Carlo. When the strong error $\Vert Y_{h}-Y_{0}\Vert_{2}=O(h^{\frac{\beta}{2}})$, $\beta<1$, the gain of ML2R over MLMC becomes even more striking. We carry out numerical simulations to compare these estimators in two settings: vanilla and path-dependent option pricing by Monte Carlo simulation and the less classical Nested Monte Carlo simulation.

Efficiency transfer for regression models with responses missing at random

Tue, 09 May 2017 04:00 EDT

Ursula U. Müller, Anton Schick.

Source: Bernoulli, Volume 23, Number 4A, 2693--2719.

Abstract:
We consider independent observations on a random pair $(X,Y)$, where the response $Y$ is allowed to be missing at random but the covariate vector $X$ is always observed. We demonstrate that characteristics of the conditional distribution of $Y$ given $X$ can be estimated efficiently using complete case analysis, that is, one can simply omit incomplete cases and work with an appropriate efficient estimator which remains efficient. This means in particular that we do not have to use imputation or work with inverse probability weights. Those approaches will never be better (asymptotically) than the above complete case method. This efficiency transfer is a general result and holds true for all regression models for which the distribution of $Y$ given $X$ and the marginal distribution of $X$ do not share common parameters. We apply it to the general homoscedastic semiparametric regression model. This includes models where the conditional expectation is modeled by a complex semiparametric regression function, as well as all basic models such as linear regression and nonparametric regression. We discuss estimation of various functionals of the conditional distribution, for example, of regression parameters and of the error distribution.

Inference under biased sampling and right censoring for a change point in the hazard function

Tue, 09 May 2017 04:00 EDT

Yassir Rabhi, Masoud Asgharian.

Source: Bernoulli, Volume 23, Number 4A, 2720--2745.

Abstract:
Length-biased survival data commonly arise in cross-sectional surveys and prevalent cohort studies on disease duration. Ignoring biased sampling leads to bias in estimating the hazard-of-failure and the survival-time in the population. We address estimating the location of a possible change-point of an otherwise smooth hazard function when the collected data form a biased sample from the target population and the data are subject to informative censoring. We provide two estimation methodologies, for the location and size of the change-point, adapted to two scenarios of the truncation distribution: known and unknown. While the estimators in the first case show gain in efficiency as compared to those in the second case, the latter is more robust to the form of the truncation distribution. In both cases, the change-point estimators can achieve the rate $\mathcal{O}_{p}(1/n)$. We study the asymptotic properties of the estimates and devise interval-estimators for the location and size of the change, paving the way towards making statistical inference about whether or not a change-point exists. Several simulated examples are discussed to assess the finite sample behavior of the estimators. The proposed methods are then applied to analyze a set of survival data collected on elderly Canadian citizen (aged 65$+$) suffering from dementia.

A generalized divergence for statistical inference

Tue, 09 May 2017 04:00 EDT

Abhik Ghosh, Ian R. Harris, Avijit Maji, Ayanendranath Basu, Leandro Pardo.

Source: Bernoulli, Volume 23, Number 4A, 2746--2783.

Abstract:
The power divergence (PD) and the density power divergence (DPD) families have proven to be useful tools in the area of robust inference. In this paper, we consider a superfamily of divergences which contains both of these families as special cases. The role of this superfamily is studied in several statistical applications, and desirable properties are identified and discussed. In many cases, it is observed that the most preferred minimum divergence estimator within the above collection lies outside the class of minimum PD or minimum DPD estimators, indicating that this superfamily has real utility, rather than just being a routine generalization. The limitation of the usual first order influence function as an effective descriptor of the robustness of the estimator is also demonstrated in this connection.

Conditional convex orders and measurable martingale couplings

Tue, 09 May 2017 04:00 EDT

Lasse Leskelä, Matti Vihola.

Source: Bernoulli, Volume 23, Number 4A, 2784--2807.

Abstract:
Strassen’s classical martingale coupling theorem states that two random vectors are ordered in the convex (resp. increasing convex) stochastic order if and only if they admit a martingale (resp. submartingale) coupling. By analysing topological properties of spaces of probability measures equipped with a Wasserstein metric and applying a measurable selection theorem, we prove a conditional version of this result for random vectors conditioned on a random element taking values in a general measurable space. We provide an analogue of the conditional martingale coupling theorem in the language of probability kernels, and discuss how it can be applied in the analysis of pseudo-marginal Markov chain Monte Carlo algorithms. We also illustrate how our results imply the existence of a measurable minimiser in the context of martingale optimal transport.

Sharp thresholds for Gibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction

Tue, 09 May 2017 04:00 EDT

Benedikt Jahnel, Christof Külske.

Source: Bernoulli, Volume 23, Number 4A, 2808--2827.

Abstract:
We investigate the Gibbs properties of the fuzzy Potts model on the $d$-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernández, den Hollander and Martínez [ J. Stat. Phys. 156 (2014) 203–220] for their study of the Gibbs–non-Gibbs transitions of a dynamical Kac–Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac–Potts model with class size unequal two. On the way to this result, we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments.

On Stein operators for discrete approximations

Tue, 09 May 2017 04:00 EDT

Neelesh S. Upadhye, Vydas Čekanavičius, P. Vellaisamy.

Source: Bernoulli, Volume 23, Number 4A, 2828--2859.

Abstract:
In this paper, a new method based on probability generating functions is used to obtain multiple Stein operators for various random variables closely related to Poisson, binomial and negative binomial distributions. Also, the Stein operators for certain compound distributions, where the random summand satisfies Panjer’s recurrence relation, are derived. A well-known perturbation approach for Stein’s method is used to obtain total variation bounds for the distributions mentioned above. The importance of such approximations is illustrated, for example, by the binomial convoluted with Poisson approximation to sums of independent and dependent indicator random variables.

Information criteria for multivariate CARMA processes

Tue, 09 May 2017 04:00 EDT

Vicky Fasen, Sebastian Kimmig.

Source: Bernoulli, Volume 23, Number 4A, 2860--2886.

Abstract:
Multivariate continuous-time ARMA$(p,q)$ ($\operatorname{MCARMA} (p,q)$) processes are the continuous-time analog of the well-known vector ARMA$(p,q)$ processes. They have attracted interest over the last years. Methods to estimate the parameters of an MCARMA process require an identifiable parametrization such as the Echelon form with a fixed Kronecker index, which is in the one-dimensional case the degree $p$ of the autoregressive polynomial. Thus, the Kronecker index has to be known in advance before parameter estimation can be done. When this is not the case, information criteria can be used to estimate the Kronecker index and the degrees $(p,q)$, respectively. In this paper, we investigate information criteria for MCARMA processes based on quasi maximum likelihood estimation. Therefore, we first derive the asymptotic properties of quasi maximum likelihood estimators for MCARMA processes in a misspecified parameter space. Then, we present necessary and sufficient conditions for information criteria to be strongly and weakly consistent, respectively. In particular, we study the well-known Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) as special cases.

From trees to seeds: On the inference of the seed from large trees in the uniform attachment model

Tue, 09 May 2017 04:00 EDT

Sébastien Bubeck, Ronen Eldan, Elchanan Mossel, Miklós Z. Rácz.

Source: Bernoulli, Volume 23, Number 4A, 2887--2916.

Abstract:
We study the influence of the seed in random trees grown according to the uniform attachment model, also known as uniform random recursive trees. We show that different seeds lead to different distributions of limiting trees from a total variation point of view. To do this, we construct statistics that measure, in a certain well-defined sense, global “balancedness” properties of such trees. Our paper follows recent results on the same question for the preferential attachment model.

Guided proposals for simulating multi-dimensional diffusion bridges

Tue, 09 May 2017 04:00 EDT

Moritz Schauer, Frank van der Meulen, Harry van Zanten.

Source: Bernoulli, Volume 23, Number 4A, 2917--2950.

Abstract:
A Monte Carlo method for simulating a multi-dimensional diffusion process conditioned on hitting a fixed point at a fixed future time is developed. Proposals for such diffusion bridges are obtained by superimposing an additional guiding term to the drift of the process under consideration. The guiding term is derived via approximation of the target process by a simpler diffusion processes with known transition densities. Acceptance of a proposal can be determined by computing the likelihood ratio between the proposal and the target bridge, which is derived in closed form. We show under general conditions that the likelihood ratio is well defined and show that a class of proposals with guiding term obtained from linear approximations fall under these conditions.

Convergence of sequential quasi-Monte Carlo smoothing algorithms

Mon, 22 May 2017 22:05 EDT

Mathieu Gerber, Nicolas Chopin.

Source: Bernoulli, Volume 23, Number 4B, 2951--2987.

Abstract:
Gerber and Chopin [ J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 (2015) 509–579] recently introduced Sequential quasi-Monte Carlo (SQMC) algorithms as an efficient way to perform filtering in state–space models. The basic idea is to replace random variables with low-discrepancy point sets, so as to obtain faster convergence than with standard particle filtering. Gerber and Chopin (2015) describe briefly several ways to extend SQMC to smoothing, but do not provide supporting theory for this extension. We discuss more thoroughly how smoothing may be performed within SQMC, and derive convergence results for the so-obtained smoothing algorithms. We consider in particular SQMC equivalents of forward smoothing and forward filtering backward sampling, which are the most well-known smoothing techniques. As a preliminary step, we provide a generalization of the classical result of Hlawka and Mück [ Computing (Arch. Elektron. Rechnen) 9 (1972) 127–138] on the transformation of QMC point sets into low discrepancy point sets with respect to non uniform distributions. As a corollary of the latter, we note that we can slightly weaken the assumptions to prove the consistency of SQMC.

Baxter’s inequality and sieve bootstrap for random fields

Mon, 22 May 2017 22:05 EDT

Marco Meyer, Carsten Jentsch, Jens-Peter Kreiss.

Source: Bernoulli, Volume 23, Number 4B, 2988--3020.

Abstract:
The concept of the autoregressive (AR) sieve bootstrap is investigated for the case of spatial processes in $\mathbb{Z}^{2}$. This procedure fits AR models of increasing order to the given data and, via resampling of the residuals, generates bootstrap replicates of the sample. The paper explores the range of validity of this resampling procedure and provides a general check criterion which allows to decide whether the AR sieve bootstrap asymptotically works for a specific statistic of interest or not. The criterion may be applied to a large class of stationary spatial processes. As another major contribution of this paper, a weighted Baxter-inequality for spatial processes is provided. This result yields a rate of convergence for the finite predictor coefficients, i.e. the coefficients of finite-order AR model fits, towards the autoregressive coefficients which are inherent to the underlying process under mild conditions. The developed check criterion is applied to some particularly interesting statistics like sample autocorrelations and standardized sample variograms. A simulation study shows that the procedure performs very well compared to normal approximations as well as block bootstrap methods in finite samples.

Testing the maximal rank of the volatility process for continuous diffusions observed with noise

Mon, 22 May 2017 22:05 EDT

Tobias Fissler, Mark Podolskij.

Source: Bernoulli, Volume 23, Number 4B, 3021--3066.

Abstract:
In this paper, we present a test for the maximal rank of the volatility process in continuous diffusion models observed with noise. Such models are typically applied in mathematical finance, where latent price processes are corrupted by microstructure noise at ultra high frequencies. Using high frequency observations, we construct a test statistic for the maximal rank of the time varying stochastic volatility process. Our methodology is based upon a combination of a matrix perturbation approach and pre-averaging. We will show the asymptotic mixed normality of the test statistic and obtain a consistent testing procedure. We complement the paper with a simulation and an empirical study showing the performances on finite samples.

Distribution of linear statistics of singular values of the product of random matrices

Mon, 22 May 2017 22:05 EDT

Friedrich Götze, Alexey Naumov, Alexander Tikhomirov.

Source: Bernoulli, Volume 23, Number 4B, 3067--3113.

Abstract:
In this paper we consider the product of two independent random matrices ${\mathbf{X}}^{(1)}$ and ${\mathbf{X}}^{(2)}$. Assume that $X_{jk}^{(q)},1\le j,k\le n,q=1,2$, are i.i.d. random variables with $\mathbb{E}X_{jk}^{(q)}=0,\operatorname{Var}X_{jk}^{(q)}=1$. Denote by $s_{1}({\mathbf{W}}),\ldots,s_{n}({\mathbf{W}})$ the singular values of ${\mathbf{W}}:=\frac{1}{n}{\mathbf{X}}^{(1)}\mathbf{X}^{(2)}$. We prove the central limit theorem for linear statistics of the squared singular values $s_{1}^{2}({\mathbf{W}}),\ldots,s_{n}^{2}({\mathbf{W}})$ showing that the limiting variance depends on $\kappa_{4}:=\mathbb{E}(X_{11}^{(1)})^{4}-3$.

Studentized U -quantile processes under dependence with applications to change-point analysis

Mon, 22 May 2017 22:05 EDT

Daniel Vogel, Martin Wendler.

Source: Bernoulli, Volume 23, Number 4B, 3114--3144.

Abstract:
Many popular robust estimators are $U$-quantiles, most notably the Hodges–Lehmann location estimator and the $Q_{n}$ scale estimator. We prove a functional central limit theorem for the $U$-quantile process without any moment assumptions and under weak short-range dependence conditions. We further devise an estimator for the long-run variance and show its consistency, from which the convergence of the studentized version of the $U$-quantile process to a standard Brownian motion follows. This result can be used to construct CUSUM-type change-point tests based on $U$-quantiles, which do not rely on bootstrapping procedures. We demonstrate this approach in detail with the example of the Hodges–Lehmann estimator for robustly detecting changes in the central location. A simulation study confirms the very good efficiency and robustness properties of the test. Two real-life data sets are analyzed.

“Building” exact confidence nets

Mon, 22 May 2017 22:05 EDT

Andrew F. Francis, Milan Stehlík, Henry P. Wynn.

Source: Bernoulli, Volume 23, Number 4B, 3145--3165.

Abstract:
Confidence nets, that is, collections of confidence intervals that fill out the parameter space and whose exact parameter coverage can be computed, are familiar in nonparametric statistics. Here, the distributional assumptions are based on invariance under the action of a finite reflection group. Exact confidence nets are exhibited for a single parameter, based on the root system of the group. The main result is a formula for the generating function of the coverage interval probabilities. The proof makes use of the theory of “buildings” and the Chevalley factorization theorem for the length distribution on Cayley graphs of finite reflection groups.

Eigen structure of a new class of covariance and inverse covariance matrices

Mon, 22 May 2017 22:05 EDT

Heather Battey.

Source: Bernoulli, Volume 23, Number 4B, 3166--3177.

Abstract:
There is a one to one mapping between a $p$ dimensional strictly positive definite covariance matrix $\Sigma$ and its matrix logarithm $L$. We exploit this relationship to study the structure induced on $\Sigma$ through a sparsity constraint on $L$. Consider $L$ as a random matrix generated through a basis expansion, with the support of the basis coefficients taken as a simple random sample of size $s=s^{*}$ from the index set $[p(p+1)/2]=\{1,\ldots,p(p+1)/2\}$. We find that the expected number of non-unit eigenvalues of $\Sigma$, denoted $\mathbb{E}[|\mathcal{A}|]$, is approximated with near perfect accuracy by the solution of the equation \[\frac{4p+p(p-1)}{2(p+1)}[\log (\frac{p}{p-d})-\frac{d}{2p(p-d)}]-s^{*}=0.\] Furthermore, the corresponding eigenvectors are shown to possess only ${p-|\mathcal{A}^{c}|}$ non-zero entries. We use this result to elucidate the precise structure induced on $\Sigma$ and $\Sigma^{-1}$. We demonstrate that a positive definite symmetric matrix whose matrix logarithm is sparse is significantly less sparse in the original domain. This finding has important implications in high dimensional statistics where it is important to exploit structure in order to construct consistent estimators in non-trivial norms. An estimator exploiting the structure of the proposed class is presented.

Influence functions for penalized M-estimators

Mon, 22 May 2017 22:05 EDT

Marco Avella-Medina.

Source: Bernoulli, Volume 23, Number 4B, 3178--3196.

Abstract:
We study the local robustness properties of general nondifferentiable penalized M-estimators via the influence function. More precisely, we propose a framework that allows us to define rigorously the influence function as the limiting influence function of a sequence of approximating estimators. We show that it can be used to characterize the robustness properties of a wide range of sparse estimators and we derive its form for general penalized M-estimators including lasso and adaptive lasso type estimators. We prove that our influence function is equivalent to a derivative in the sense of distribution theory.

On predictive density estimation for location families under integrated absolute error loss

Mon, 22 May 2017 22:05 EDT

Tatsuya Kubokawa, Éric Marchand, William E. Strawderman.

Source: Bernoulli, Volume 23, Number 4B, 3197--3212.

Abstract:
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.

Transportation and concentration inequalities for bifurcating Markov chains

Mon, 22 May 2017 22:05 EDT

S. Valère Bitseki Penda, Mikael Escobar-Bach, Arnaud Guillin.

Source: Bernoulli, Volume 23, Number 4B, 3213--3242.

Abstract:
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.

Pólya urn schemes with infinitely many colors

Mon, 22 May 2017 22:05 EDT

Antar Bandyopadhyay, Debleena Thacker.

Source: Bernoulli, Volume 23, Number 4B, 3243--3267.

Abstract:
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.

Approximate local limit theorems with effective rate and application to random walks in random scenery

Mon, 22 May 2017 22:05 EDT

Rita Giuliano, Michel Weber.

Source: Bernoulli, Volume 23, Number 4B, 3268--3310.

Abstract:
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.

On Stein’s method for products of normal random variables and zero bias couplings

Mon, 22 May 2017 22:05 EDT

Robert E. Gaunt.

Source: Bernoulli, Volume 23, Number 4B, 3311--3345.

Abstract:
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.

Weak convergence of empirical copula processes indexed by functions

Mon, 22 May 2017 22:05 EDT

Dragan Radulović, Marten Wegkamp, Yue Zhao.

Source: Bernoulli, Volume 23, Number 4B, 3346--3384.

Abstract:
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.

Sieve maximum likelihood estimation for a general class of accelerated hazards models with bundled parameters

Mon, 22 May 2017 22:05 EDT

Xingqiu Zhao, Yuanshan Wu, Guosheng Yin.

Source: Bernoulli, Volume 23, Number 4B, 3385--3411.

Abstract:
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.

Integrated empirical processes in $L^{p}$ with applications to estimate probability metrics

Mon, 22 May 2017 22:05 EDT

Javier Cárcamo.

Source: Bernoulli, Volume 23, Number 4B, 3412--3436.

Abstract:
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.

Efficiency and bootstrap in the promotion time cure model

Mon, 22 May 2017 22:05 EDT

François Portier, Anouar El Ghouch, Ingrid Van Keilegom.

Source: Bernoulli, Volume 23, Number 4B, 3437--3468.

Abstract:
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.

Non-central limit theorems for random fields subordinated to gamma-correlated random fields

Mon, 22 May 2017 22:05 EDT

Nikolai Leonenko, M. Dolores Ruiz-Medina, Murad S. Taqqu.

Source: Bernoulli, Volume 23, Number 4B, 3469--3507.

Abstract:
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.

Asymptotic expansions and hazard rates for compound and first-passage distributions

Mon, 22 May 2017 22:05 EDT

Ronald W. Butler.

Source: Bernoulli, Volume 23, Number 4B, 3508--3536.

Abstract:
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.

Efficient Bayesian estimation and uncertainty quantification in ordinary differential equation models

Mon, 22 May 2017 22:05 EDT

Prithwish Bhaumik, Subhashis Ghosal.

Source: Bernoulli, Volume 23, Number 4B, 3537--3570.

Abstract:
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.

Fractional Brownian motion satisfies two-way crossing

Mon, 22 May 2017 22:05 EDT

Rémi Peyre.

Source: Bernoulli, Volume 23, Number 4B, 3571--3597.

Abstract:
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.

Adaptive estimation for bifurcating Markov chains

Mon, 22 May 2017 22:05 EDT

S. Valère Bitseki Penda, Marc Hoffmann, Adélaïde Olivier.

Source: Bernoulli, Volume 23, Number 4B, 3598--3637.

Abstract:
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.

A proof of the Shepp–Olkin entropy concavity conjecture

Mon, 22 May 2017 22:05 EDT

Erwan Hillion, Oliver Johnson.

Source: Bernoulli, Volume 23, Number 4B, 3638--3649.

Abstract:
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.

The failure of the profile likelihood method for a large class of semi-parametric models

Mon, 22 May 2017 22:05 EDT

Eric Beutner, Laurent Bordes, Laurent Doyen.

Source: Bernoulli, Volume 23, Number 4B, 3650--3684.

Abstract:
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.

Some monotonicity properties of parametric and nonparametric Bayesian bandits

Mon, 22 May 2017 22:05 EDT

Yaming Yu.

Source: Bernoulli, Volume 23, Number 4B, 3685--3710.

Abstract:
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.

Accelerated Gibbs sampling of normal distributions using matrix splittings and polynomials

Mon, 22 May 2017 22:05 EDT

Colin Fox, Albert Parker.

Source: Bernoulli, Volume 23, Number 4B, 3711--3743.

Abstract:
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.

Simulation of hitting times for Bessel processes with non-integer dimension

Mon, 22 May 2017 22:05 EDT

Madalina Deaconu, Samuel Herrmann.

Source: Bernoulli, Volume 23, Number 4B, 3744--3771.

Abstract:
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.

Dirichlet curves, convex order and Cauchy distribution

Thu, 27 Jul 2017 04:01 EDT

Gérard Letac, Mauro Piccioni.

Source: Bernoulli, Volume 24, Number 1, 1--29.

Abstract:
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 $\mathbb{R}^{d}?$

Random tessellations associated with max-stable random fields

Thu, 27 Jul 2017 04:01 EDT

Clément Dombry, Zakhar Kabluchko.

Source: Bernoulli, Volume 24, Number 1, 30--52.

Abstract:
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.

Proper scoring rules and Bregman divergence

Thu, 27 Jul 2017 04:01 EDT

Evgeni Y. Ovcharov.

Source: Bernoulli, Volume 24, Number 1, 53--79.

Abstract:
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.

The logarithmic law of sample covariance matrices near singularity

Thu, 27 Jul 2017 04:01 EDT

Xuejun Wang, Xiao Han, Guangming Pan.

Source: Bernoulli, Volume 24, Number 1, 80--114.

Abstract:
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 \[\frac{\log\det\hat{\Sigma}_{1}+n\log n-\log(n-1)!-\log\det\Sigma}{\sqrt{2\log n}}\xrightarrow[\qquad]{d}N(0,1).\]

Maximum entropy distribution of order statistics with given marginals

Thu, 27 Jul 2017 04:01 EDT

Cristina Butucea, Jean-François Delmas, Anne Dutfoy, Richard Fischer.

Source: Bernoulli, Volume 24, Number 1, 115--155.

Abstract:
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.

Multiple collisions in systems of competing Brownian particles

Thu, 27 Jul 2017 04:01 EDT

Cameron Bruggeman, Andrey Sarantsev.

Source: Bernoulli, Volume 24, Number 1, 156--201.

Abstract:
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).

Rate of convergence for Hilbert space valued processes

Thu, 27 Jul 2017 04:01 EDT

Moritz Jirak.

Source: Bernoulli, Volume 24, Number 1, 202--230.

Abstract:
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.

Posterior concentration rates for empirical Bayes procedures with applications to Dirichlet process mixtures

Thu, 27 Jul 2017 04:01 EDT

Sophie Donnet, Vincent Rivoirard, Judith Rousseau, Catia Scricciolo.

Source: Bernoulli, Volume 24, Number 1, 231--256.

Abstract:
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.

A note on the convex infimum convolution inequality

Thu, 27 Jul 2017 04:01 EDT

Naomi Feldheim, Arnaud Marsiglietti, Piotr Nayar, Jing Wang.

Source: Bernoulli, Volume 24, Number 1, 257--270.

Abstract:
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.

Sparse oracle inequalities for variable selection via regularized quantization

Thu, 27 Jul 2017 04:01 EDT

Clément Levrard.

Source: Bernoulli, Volume 24, Number 1, 271--296.

Abstract:
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.

The $\lambda$-invariant measures of subcritical Bienaymé–Galton–Watson processes

Thu, 27 Jul 2017 04:01 EDT

Pascal Maillard.

Source: Bernoulli, Volume 24, Number 1, 297--315.

Abstract:
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.

Hitting probabilities for the Greenwood model and relations to near constancy oscillation

Thu, 27 Jul 2017 04:01 EDT

M. Möhle.

Source: Bernoulli, Volume 24, Number 1, 316--332.

Abstract:
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.

Functional inequalities for Gaussian convolutions of compactly supported measures: Explicit bounds and dimension dependence

Thu, 27 Jul 2017 04:01 EDT

Jean-Baptiste Bardet, Nathaël Gozlan, Florent Malrieu, Pierre-André Zitt.

Source: Bernoulli, Volume 24, Number 1, 333--353.

Abstract:
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.

Large deviations for stochastic heat equation with rough dependence in space

Thu, 27 Jul 2017 04:01 EDT

Yaozhong Hu, David Nualart, Tusheng Zhang.

Source: Bernoulli, Volume 24, Number 1, 354--385.

Abstract:
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.

The maximum likelihood threshold of a graph

Thu, 27 Jul 2017 04:01 EDT

Elizabeth Gross, Seth Sullivant.

Source: Bernoulli, Volume 24, Number 1, 386--407.

Abstract:
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.

Uniform measure density condition and game regularity for tug-of-war games

Thu, 27 Jul 2017 04:01 EDT

Joonas Heino.

Source: Bernoulli, Volume 24, Number 1, 408--432.

Abstract:
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.

The van den Berg–Kesten–Reimer operator and inequality for infinite spaces

Thu, 27 Jul 2017 04:01 EDT

Richard Arratia, Skip Garibaldi, Alfred W. Hales.

Source: Bernoulli, Volume 24, Number 1, 433--448.

Abstract:
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.

Jackknife empirical likelihood goodness-of-fit tests for U-statistics based general estimating equations

Thu, 27 Jul 2017 04:01 EDT

Hanxiang Peng, Fei Tan.

Source: Bernoulli, Volume 24, Number 1, 449--464.

Abstract:
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.

Power of the spacing test for least-angle regression

Thu, 27 Jul 2017 04:01 EDT

Jean-Marc Azaïs, Yohann De Castro, Stéphane Mourareau.

Source: Bernoulli, Volume 24, Number 1, 465--492.

Abstract:
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.

Inference in Ising models

Thu, 27 Jul 2017 04:01 EDT

Bhaswar B. Bhattacharya, Sumit Mukherjee.

Source: Bernoulli, Volume 24, Number 1, 493--525.

Abstract:
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.

A MOSUM procedure for the estimation of multiple random change points

Thu, 27 Jul 2017 04:01 EDT

Birte Eichinger, Claudia Kirch.

Source: Bernoulli, Volume 24, Number 1, 526--564.

Abstract:
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.

On conditional moments of high-dimensional random vectors given lower-dimensional projections

Thu, 27 Jul 2017 04:01 EDT

Lukas Steinberger, Hannes Leeb.

Source: Bernoulli, Volume 24, Number 1, 565--591.

Abstract:
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).

Local block bootstrap for inhomogeneous Poisson marked point processes

Thu, 27 Jul 2017 04:01 EDT

William Garner, Dimitris N. Politis.

Source: Bernoulli, Volume 24, Number 1, 592--615.

Abstract:
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.

Change-point estimators with true identification property

Thu, 27 Jul 2017 04:01 EDT

Chi Tim Ng, Woojoo Lee, Youngjo Lee.

Source: Bernoulli, Volume 24, Number 1, 616--660.

Abstract:
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.

Designs from good Hadamard matrices

Thu, 27 Jul 2017 04:01 EDT

Chenlu Shi, Boxin Tang.

Source: Bernoulli, Volume 24, Number 1, 661--671.

Abstract:
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.

Curvature and transport inequalities for Markov chains in discrete spaces

Thu, 27 Jul 2017 04:01 EDT

Max Fathi, Yan Shu.

Source: Bernoulli, Volume 24, Number 1, 672--698.

Abstract:
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).

Optimal adaptive inference in random design binary regression

Thu, 27 Jul 2017 04:01 EDT

Rajarshi Mukherjee, Subhabrata Sen.

Source: Bernoulli, Volume 24, Number 1, 699--739.

Abstract:
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.

Testing for instability in covariance structures

Thu, 27 Jul 2017 04:01 EDT

Chihwa Kao, Lorenzo Trapani, Giovanni Urga.

Source: Bernoulli, Volume 24, Number 1, 740--771.

Abstract:
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.

Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time

Thu, 27 Jul 2017 04:01 EDT

ZhiQiang Gao, Quansheng Liu.

Source: Bernoulli, Volume 24, Number 1, 772--800.

Abstract:
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.