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The paper addresses asymptotic estimation of normal means under sparsity. The primary focus is estimation of multivariate normal means where we obtain exact asymptotic minimax error under global-local shrinkage prior. |
We consider a binary sequence generated by thresholding a hidden continuous sequence. The hidden variables are assumed to have a compound symmetry covariance structure with a single parameter characterizing the common correlation. |
Over the last two decades, many exciting variable selection methods have been developed for finding a small group of covariates that are associated with the response from a large pool. Can the discoveries from these data mining approaches be spurious due to high dimensionality and limited sample size? |
We develop a framework for the operationalization of models and parameters by combining de Finetti's representation theorem with a conditional form of Sanov's theorem. This synthesis, the tilted de Finetti theorem, shows that conditioning exchangeable sequences on empirical moment constraints yields predictive... |
This paper is concerned with adaptive kernel estimation of the Lévy density N(x) for bounded-variation pure-jump Lévy processes. The sample path is observed at n discrete instants in the "high frequency" context (\Delta = \Delta(n) tends to zero while n\Delta tends to infinity). |
Variable screening is a fast dimension reduction technique for assisting high dimensional feature selection. As a preselection method, it selects a moderate size subset of candidate variables for further refining via feature selection to produce the final model. |
In this paper, we study modulus of continuity and rate of convergence of series of conditionally sub-Gaussian random fields. This framework includes both classical series representations of Gaussian fields and LePage series representations of stable fields. |
We study the problem of estimating the mode and maximum of an unknown regression function in the presence of noise. We adopt the Bayesian approach by using tensor-product B-splines and endowing the coefficients with Gaussian priors. |
In this paper, we propose a new threshold-kernel jump-detection method for jump-diffusion processes, which iteratively applies thresholding and kernel methods in an approximately optimal way to achieve improved finite-sample performance. We use the expected number of jump misclassifications as the objective function t... |
Recently, the binary expansion testing framework was introduced to test the independence of two continuous random variables by utilizing symmetry statistics that are complete sufficient statistics for dependence. We develop a new test based on an ensemble approach that uses the sum of squared symmetry statistics and d... |
Asymptotic properties, both consistency and weak convergence, of estimators arising in a general class of dynamic recurrent event models are presented. The class of models take into account the impact of interventions after each event occurrence, the impact of accumulating event occurrences, the induced informative an... |
In this paper we propose a new approach to study the properties of the Partial Least Squares (PLS) estimator. This approach relies on the link between PLS and discrete orthogonal polynomials. |
We propose a new concept of codivergence, which quantifies the similarity between two probability measures $P_1, P_2$ relative to a reference probability measure $P_0$. In the neighborhood of the reference measure $P_0$, a codivergence behaves like an inner product between the measures $P_1 - P_0$ and $P_2 - P_0$. |
Tensor clustering, which seeks to extract underlying cluster structures from noisy tensor observations, has gained increasing attention. One extensively studied model for tensor clustering is the tensor block model, which postulates the existence of clustering structures along each mode and has found broad application... |
Classical inequality curves and inequality measures are defined for distributions with finite mean value. Moreover, their empirical counterparts are not resistant to outliers. |
Efficient automatic protein classification is of central importance in genomic annotation. As an independent way to check the reliability of the classification, we propose a statistical approach to test if two sets of protein domain sequences coming from two families of the Pfam database are significantly different. |
In this paper, we consider the problem of testing the equality of two multivariate distributions based on geometric graphs constructed using the interpoint distances between the observations. These include the tests based on the minimum spanning tree and the $K$-nearest neighbor (NN) graphs, among others. |
Gaussian processes (GP) are widely used as a metamodel for emulating time-consuming computer codes. We focus on problems involving categorical inputs, with a potentially large number L of levels (typically several tens), partitioned in G << L groups of various sizes. |
New types of designs called nested space-filling designs have been proposed for conducting multiple computer experiments with different levels of accuracy. In this article, we develop several approaches to constructing such designs. |
We consider component-wise estimation of order restricted location/scale parameters $\theta_1$ and $\theta_2$ ($\theta_1\leq \theta_2$) of a general bivariate distribution under the squared error loss function. To find improvements over the best location/scale equivariant estimators (BLEE/BSEE) of $\theta_1$ and $\the... |
Given a sequence \xi_1, \xi_2,... of X-valued, exchangeable random elements, let q(\xi^(n)) and p_m(\xi^(n)) stand for posterior and predictive distribution, respectively, given \xi^(n) = (\xi_1,..., \xi_n). We provide an upper bound for limsup b_n d_[[X]](q(\xi^(n)), \delta_\empiricn) and limsup b_n d_[X^m](p_m(\xi^(... |
We prove an L2 recovery bound for a family of sparse estimators defined as minimizers of some empirical loss functions -- which include hinge loss and logistic loss. More precisely, we achieve an upper-bound for coefficients estimation scaling as (k*/n)\log(p/k*): n,p is the size of the design matrix and k* the dimens... |
Under scenario of high frequency data, consistent estimator of realized Laplace transform of volatility is proposed by \citet{TT2012a} and related central limit theorem has been well established. In this paper, we investigate the asymptotic tail behaviour of the empirical realized Laplace transform of volatility (ERLT... |
We consider the empirical versions of geometric quantile and halfspace depth, and study their extremal behaviour as a function of the sample size. The objective of this study is to establish connection between the rates of convergence and tail behaviour of the corresponding underlying distributions. |
The Metropolis algorithm is one of the Markov chain Monte Carlo (MCMC) methods that realize sampling from the target probability distribution. In this paper, we are concerned with the sampling from the distribution in non-identifiable cases that involve models with Fisher information matrices that may fail to be inver... |
In this paper, we study the asymptotic behavior of the extreme eigenvalues and eigenvectors of the spiked covariance matrices, in the supercritical regime. Specifically, we derive the joint distribution of the extreme eigenvalues and the generalized components of their associated eigenvectors in this regime. |
We prove oracle inequalities for a penalized log-likelihood criterion that hold even if the data are not independent and not stationary, based on a martingale approach. The assumptions are checked for various contexts: density estimation with independent and identically distributed (i.i.d) data, hidden Markov models, ... |
Standardness is a popular assumption in the literature on set estimation. It also appears in statistical approaches to topological data analysis, where it is common to assume that the data were sampled from a probability measure that satisfies the standard assumption. |
In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension $n\times p$ and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here each of the principal components $\mathbf{v}_1,\dots,\mathbf{v}_r$ has at most $s_0$ non-zero entries. |
We consider selection of random predictors for high-dimensional regression problem with binary response for a general loss function. Important special case is when the binary model is semiparametric and the response function is misspecified under parametric model fit. |
Given a large matrix containing independent data entries, we consider the problem of detecting a submatrix inside the data matrix that contains larger-than-usual values. Different from previous literature, we do not have exact information about the dimension of the potential elevated submatrix. |
In this paper, we establish optimal rates of adaptive estimation of a vector in the multi-reference alignment model, a problem with important applications in fields such as signal processing, image processing, and computer vision, among others. We describe how this model can be viewed as a multivariate Gaussian mixtur... |
This paper is concerned with the study of the embedding circulant matrix method to simulate stationary complex-valued Gaussian sequences. The method is, in particular, shown to be well-suited to generate circularly-symmetric stationary Gaussian processes. |
In this paper, we give sufficient conditions to establish central limit theorems for boundary estimates of Poisson point processes. The considered estimates are obtained by smoothing some bias corrected extreme values of the point process. |
This paper describes an estimator of the additive components of a nonparametric additive model with a known link function. When the additive components are twice continuously differentiable, the estimator is asymptotically normally distributed with a rate of convergence in probability of n^{-2/5}. |
Analysing statistical properties of neural networks is a central topic in statistics and machine learning. However, most results in the literature focus on the properties of the neural network minimizing the training error. |
Often when we deal with `Big Data', the true effects we are interested in are Rare and Weak (RW). Researchers measure a large number of features, hoping to find perhaps only a small fraction of them to be relevant to the research in question; the effect sizes of the relevant features are individually small so the ... |
In this article, we consider the estimation of unknown parameters of Weibull distribution when the lifetime data are observed in the presence of progressively type-I hybrid censoring scheme. The Newton-Raphson algorithm, Expectation-Maximization (EM) algorithm and Stochastic EM (SEM) algorithm are utilized to derive t... |
Differential entropy and log determinant of the covariance matrix of a multivariate Gaussian distribution have many applications in coding, communications, signal processing and statistical inference. In this paper we consider in the high dimensional setting optimal estimation of the differential entropy and the log-d... |
This paper proposes a novel non-parametric multidimensional convex regression estimator which is designed to be robust to adversarial perturbations in the empirical measure. We minimize over convex functions the maximum (over Wasserstein perturbations of the empirical measure) of the absolute regression errors. |
In quantitative finance, we often model asset prices as a noisy Ito semimartingale. As this model is not identifiable, approximating by a time-changed Levy process can be useful for generative modelling. |
For a joint probability density function f(x) of a random vector X the mixed partial derivatives of log f(x) can be interpreted as limiting cumulants in an infinitesimally small open neighborhood around x. Moreover, setting them to zero everywhere gives independence and conditional independence conditions. |
Principal component analysis continues to be a powerful tool in dimension reduction of high dimensional data. We assume a variance-diverging model and use the high-dimension, low-sample-size asymptotics to show that even though the principal component directions are not consistent, the sample and prediction principal ... |
Lomax distribution has been widely used in economics, business and actuarial sciences. Due to its importance, we consider the statistical inference of this model under joint type-II censoring scenario. |
Traditionally, assessing the accuracy of inference based on regression quantiles has relied on the Bahadur representation. This provides an error of order $n^{-1/4}$ in normal approximations, and suggests that inference based on regression quantiles may not be as reliable as that based on other (smoother) approaches, ... |
We formalize the problem of detecting a community in a network into testing whether in a given (random) graph there is a subgraph that is unusually dense. We observe an undirected and unweighted graph on N nodes. |
This paper takes a different look on the problem of testing the mutual independence of the components of a high-dimensional vector. Instead of testing if all pairwise associations (e.g. all pairwise Kendall's $\tau$) between the components vanish, we are interested in the (null)-hypothesis that all pairwise associ... |
Covariate shift is a common transfer learning scenario where the marginal distributions of input variables vary between source and target data while the conditional distribution of the output variable remains consistent. The existing notions describing differences between marginal distributions face limitations in han... |
This paper deals with multivariate Gaussian models for which the covariance matrix is a Kronecker product of two matrices. We consider maximum likelihood estimation of the model parameters, in particular of the covariance matrix. |
We consider a model selection estimator of the covariance of a random process. Using the Unbiased Risk Estimation (URE) method, we build an estimator of the risk which allows to select an estimator in a collection of model. |
We propose Rényi information generating function and discuss its properties. A connection between the Rényi information generating function and the diversity index is proposed for discrete type random variables. |
For reconstruction of low-rank matrices from undersampled measurements, we develop an iterative algorithm based on least-squares estimation. While the algorithm can be used for any low-rank matrix, it is also capable of exploiting a-priori knowledge of matrix structure. |
Estimating the maximum mean finds a variety of applications in practice. In this paper, we study estimation of the maximum mean using an upper confidence bound (UCB) approach where the sampling budget is adaptively allocated to one of the systems. |
We determine under which conditions three bivariate copulas are compatible, viz. they are the bivariate marginals of the same trivariate copula, and, then, construct the class of these copulas. In particular, the upper and lower bounds for this class of trivariate copulas are determined. |
In this paper, we construct the wavelet eigenvalue regression methodology in high dimensions. We assume that possibly non-Gaussian, finite-variance $p$-variate measurements are made of a low-dimensional $r$-variate ($r \ll p$) fractional stochastic process with non-canonical scaling coordinates and in the presence of ... |
We investigate the problem of estimating a function $f$ based on observations from its noisy convolution when the noise exhibits long-range dependence. We construct an adaptive estimator based on the kernel method, derive minimax lower bound for the $L^2$-risk when $f$ belongs to Sobolev space and show that such estim... |
Robustness to outliers is often a desirable property of statistical estimators. Indeed many well known estimators offer very good optimal performance in theory but are unusable in applied contexts because of their sensitivity to outliers. |
We introduce tests for the goodness of fit of point patterns via methods from topological data analysis. More precisely, the persistent Betti numbers give rise to a bivariate functional summary statistic for observed point patterns that is asymptotically Gaussian in large observation windows. |
Model selection is often performed by empirical risk minimization. The quality of selection in a given situation can be assessed by risk bounds, which require assumptions both on the margin and the tails of the losses used. |
In this study, simultaneous predictive distributions for independent Poisson observables were considered and the performance of predictive distributions was evaluated using the Kullback-Leibler (K-L) loss. This study proposes a class of empirical Bayesian predictive distributions that dominate the Bayesian predictive ... |
Conditional local independence is an asymmetric independence relation among continuous time stochastic processes. It describes whether the evolution of one process is directly influenced by another process given the histories of additional processes, and it is important for the description and learning of causal relat... |
We develop the concept of exponential stochastic inequality (ESI), a novel notation that simultaneously captures high-probability and in-expectation statements. It is especially well suited to succinctly state, prove, and reason about excess-risk and generalization bounds in statistical learning, specifically, but not... |
Correlated random fields are a common way to model dependence struc- tures in high-dimensional data, especially for data collected in imaging. One important parameter characterizing the degree of dependence is the asymp- totic variance which adds up all autocovariances in the temporal and spatial domain. |
Consider a truncated circular unitary matrix which is a $p_n$ by $p_n$ submatrix of an $n$ by $n$ circular unitary matrix by deleting the last $n-p_n$ columns and rows. Jiang and Qi (2017) proved that the maximum absolute value of the eigenvalues (known as spectral radius) of the truncated matrix, after properly norma... |
We study the estimation of the high-dimensional covariance matrix andits eigenvalues under dynamic volatility models. Data under such modelshave nonlinear dependency both cross-sectionally and temporally. |
We construct an infinitely exchangeable process on the set $\cate$ of subsets of the power set of the natural numbers $\mathbb{N}$ via a Poisson point process with mean measure $\Lambda$ on the power set of $\mathbb{N}$. Each $E\in\cate$ has a least monotone cover in $\catf$, the collection of monotone subsets of $\ca... |
Nowadays, several data analysis problems require for complexity reduction, mainly meaning that they target at removing the non-influential covariates from the model and at delivering a sparse model. When categorical covariates are present, with their levels being dummy coded, the number of parameters included in the m... |
We consider the classical problems of estimating the mean of an $n$-dimensional normally (with identity covariance matrix) or Poisson distributed vector under the squared loss. In a Bayesian setting the optimal estimator is given by the prior-dependent conditional mean. |
We consider a problem of estimating a sparse group of sparse normal mean vectors. The proposed approach is based on penalized likelihood estimation with complexity penalties on the number of nonzero mean vectors and the numbers of their "significant" components, and can be performed by a computationally fast a... |
In this paper we study Bayesian networks from a commutative algebra perspective. We characterize a class of toric Bayesian nets, and provide the first example of a Bayesian net which is proved non-toric under any linear change of variables. |
In this article, we derive a new generalization of Chebyshev inequality for random vectors. We demonstrate that the new generalization is much less conservative than the classical generalization. |
In this article, we introduce and study a one sided tempered stable first order autoregressive model called TAR(1). Under the assumption of stationarity of the model, the marginal probability density function of the error term is found. |
In this paper, we consider the tail probabilities of extremals of $\beta$-Jacobi ensemble which plays an important role in multivariate analysis. The key steps in constructing estimators rely on the rate functions of large deviations. |
An important issue in social network analysis refers to the development of algorithms for estimating optimal parameters of a social network model, using data available from the network itself. This entails solving an optimization problem. |
Under mild assumptions on the kernel, we obtain the best known error rates in a regularized learning scenario taking place in the corresponding reproducing kernel Hilbert space (RKHS). The main novelty in the analysis is a proof that one can use a regularization term that grows significantly slower than the standard q... |
Most of the modern literature on robust mean estimation focuses on designing estimators which obtain optimal sub-Gaussian concentration bounds under minimal moment assumptions and sometimes also assuming contamination. This work looks at robustness in terms of Gaussian and bootstrap approximations, mainly in the regim... |
In this paper we examine some relative orderings of upper and lower records. It is shown that if m > n, the mth upper record ages faster than the nth upper record, where the data sets come from a sequence of independent and identically distributed observations from a continuous distribution. |
In the present paper we propose a new estimator of entropy based on smooth estimators of quantile density. The consistency and asymptotic distribution of the proposed estimates are obtained. |
This work delves into presenting a probabilistic method for analyzing linear process data with weakly dependent innovations, focusing on detecting change-points in the mean and estimating its spectral density. We develop a test for identifying change-points in the mean of data coming from such a model, aiming to detec... |
We propose new concepts of statistical depth, multivariate quantiles, ranks and signs, based on canonical transportation maps between a distribution of interest on $R^d$ and a reference distribution on the $d$-dimensional unit ball. The new depth concept, called Monge-Kantorovich depth, specializes to halfspace depth ... |
Although the Lasso has been extensively studied, the relationship between its prediction performance and the correlations of the covariates is not fully understood. In this paper, we give new insights into this relationship in the context of multiple linear regression. |
We consider estimation of a multivariate normal mean vector under sum of squared error loss. We propose a new class of smooth estimators parameterized by \alpha dominating the James-Stein estimator. |
The aim of this work is to estimate a quadratic functional of a unknown Wigner function from noisy tomographic data. The Wigner function can be seen as the representation of the quantum state of a light beam. |
This paper considers fixed effects (FE) estimation for linear panel data models under possible model misspecification when both the number of individuals, $n$, and the number of time periods, $T$, are large. We first clarify the probability limit of the FE estimator and argue that this probability limit can be regarde... |
The paper search for the minimum of the entropy of a two- dimensional distribution in the Fréchet class, the class of distributions with given marginals. The main result for discrete distributions is an algorithm for building the minimizing distribution, which is given by the maximum distribution function of the Fréch... |
The difficulties of estimating and representing the distributions of functional data mean that principal component methods play a substantially greater role in functional data analysis than in more conventional finite-dimensional settings. Local maxima and minima in principal component functions are of direct importan... |
Stein Variational Gradient Descent (SVGD), a popular sampling algorithm, is often described as the kernelized gradient flow for the Kullback-Leibler divergence in the geometry of optimal transport. We introduce a new perspective on SVGD that instead views SVGD as the (kernelized) gradient flow of the chi-squared diver... |
In 1991 Stute introduced a class of estimators called conditional U-statistics. They can be seen as a generalization of the Nadaraya-Watson estimator, and their strong pointwise consistency to the general regression function has been obtained in the same paper by Stute. |
Mixture of experts (MoE) models are widely applied for conditional probability density estimation problems. We demonstrate the richness of the class of MoE models by proving denseness results in Lebesgue spaces, when inputs and outputs variables are both compactly supported. |
We derive a Bernstein von-Mises theorem in the context of misspecified, non-i.i.d., hierarchical models parametrized by a finite-dimensional parameter of interest. We apply our results to hierarchical models containing non-linear operators, including the squared integral operator, and PDE-constrained inverse problems.... |
We consider the recovery of an unknown function $f$ from a noisy observation of the solution $u_f$ to a partial differential equation that can be written in the form $\mathcal{L} u_f=c(f,u_f)$, for a differential operator $\mathcal{L}$ that is rich enough to recover $f$ from $\mathcal{L} u_f$. Examples include the tim... |
We study minimax lower bounds for function estimation problems on large graph when the target function is smoothly varying over the graph. We derive minimax rates in the context of regression and classification problems on graphs that satisfy an asymptotic shape assumption and with a smoothness condition on the target... |
We introduce closed-form transition density expansions for multivariate affine jump-diffusion processes. The expansions rely on a general approximation theory which we develop in weighted Hilbert spaces for random variables which possess all polynomial moments. |
It is well known that the finite-sample null distribution of the Jarque-Bera Lagrange Multiplier (LM) test for normality and its adjusted version (ALM) introduced by Urzua differ considerably from their asymptotic chi^2(2) limit. Here, we present results from Monte Carlo simulations using 10^7 replications which yield... |
In this paper we present a general convex optimization approach for solving high-dimensional multiple response tensor regression problems under low-dimensional structural assumptions. We consider using convex and weakly decomposable regularizers assuming that the underlying tensor lies in an unknown low-dimensional su... |
Prediction sets based on full conformal prediction have seen an increasing interest in statistical learning due to their universal marginal coverage guarantees. However, practitioners have refrained from using it in applications for two reasons: Firstly, it comes at very high computational costs, exceeding even that o... |
Causal models are important tools to understand complex phenomena and predict the outcome of controlled experiments, also known as interventions. In this work, we present statistical rates of estimation for linear cyclic causal models under the assumption of homoscedastic Gaussian noise by analyzing both the LLC estim... |
In this paper new tests for the independence of two high-dimensional vectors are investigated. We consider the case where the dimension of the vectors increases with the sample size and propose multivariate analysis of variance-type statistics for the hypothesis of a block diagonal covariance matrix. |
A novel and comprehensive methodology designed to tackle the challenges posed by extreme values in the context of random censorship is introduced. The main focus is on the analysis of integrals based on the product-limit estimator of normalized upper order statistics, called extreme Kaplan--Meier integrals. |
This work concerns testing the number of parameters in one hidden layer multilayer perceptron (MLP). For this purpose we assume that we have identifiable models, up to a finite group of transformations on the weights, this is for example the case when the number of hidden units is know. |
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