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In this paper, we characterize the asymptotic and large scale behavior of the eigenvalues of wavelet random matrices 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 ... |
This paper proposes new tests of conditional independence of two random variables given a single-index involving an unknown finite-dimensional parameter. The tests employ Rosenblatt transforms and are shown to be distribution-free while retaining computational convenience. |
In the present paper, we propose and analyze a novel method for estimating a univariate regression function of bounded variation. The underpinning idea is to combine two classical tools in nonparametric statistics, namely isotonic regression and the estimation of additive models. |
Consider n individuals who, by popular vote, choose among q >= 2 alternatives, one of which is "better" than the others. Assume that each individual votes independently at random, and that the probability of voting for the better alternative is larger than the probability of voting for any other. |
The $T$-test is probably the most popular statistical test; it is routinely recommended by the textbooks. The applicability of the test relies upon the validity of normal or Student's approximation to the distribution of Student's statistic $\,t_n$. |
We study the optimal sample complexity of neighbourhood selection in linear structural equation models, and compare this to best subset selection (BSS) for linear models under general design. We show by example that -- even when the structure is \emph{unknown} -- the existence of underlying structure can reduce the sa... |
Correlations between asset returns are important in many financial applications. In recent years, multivariate volatility models have been used to describe the time-varying feature of the correlations. |
Discussion of ``EQUI-energy sampler'' by Kou, Zhou and Wong [<a href="https://arxiv.org/abs/math.ST/0507080" data-arxiv-id="math.ST/0507080" class="link-https">math.ST/0507080</a>] |
Estimating the number of unseen species is an important problem in many scientific endeavors. Its most popular formulation, introduced by Fisher, uses $n$ samples to predict the number $U$ of hitherto unseen species that would be observed if $t\cdot n$ new samples were collected. |
This paper deals with the identification of the multivariate fractional Brownian motion, a recently developed extension of the fractional Brownian motion to the multivariate case. This process is a $p$-multivariate self-similar Gaussian process parameterized by $p$ different Hurst exponents $H_i$, $p$ scaling coeffici... |
We consider a dynamical system with small noise for which the drift is parametrized by a finite dimensional parameter. For this model we consider minimum distance estimation from continuous time observations under $l^p$-penalty imposed on the parameters in the spirit of the Lasso approach with the aim of simultaneous ... |
The persistent Betti numbers are used in topological data analysis to infer the scales at which topological features appear and disappear in the filtration of a topological space. Most commonly by means of the corresponding barcode or persistence diagram. |
This paper considers the problem of testing for the presence of a continuous part in a semimartingale sampled at high frequency. We provide two tests, one where the null hypothesis is that a continuous component is present, the other where the continuous component is absent, and the model is then driven by a pure jump... |
Asymptotic theory of tail index estimation has been studied extensively in the frequentist literature on extreme values, but rarely in the Bayesian context. We investigate whether popular Bayesian kernel mixture models are able to support heavy tailed distributions and consistently estimate the tail index. |
The density function of the limiting spectral distribution of general sample covariance matrices is usually unknown. We propose to use kernel estimators which are proved to be consistent. |
Bagging is a commonly used ensemble technique in statistics and machine learning to improve the performance of prediction procedures. In this paper, we study the prediction risk of variants of bagged predictors under the proportional asymptotics regime, in which the ratio of the number of features to the number of obs... |
We study an epidemic type change in innovations of a first order autoregressive process $ y_{n,k} = \varphi_n y_{n,k-1} + \epsilon_{k} + a_{n,k}$, where $\phi_n$ is either a constant in $(-1,1)$ or a sequence in $(0,1)$, converging to 1. For $k$ inside some unknown interval $\mathbb{I}_n^\ast=(k^\ast,k^\ast+\ell^\ast]... |
We provide the lacking theory for a test of normality based on the empirical moment generating function. |
This note proves a weak type of the sharpness principle as conjectured by Gneiting, Balabdaoui, and Raftery in 2007 in connection with probabilistic forecasting subject to calibration constraints. A strong version of such a principle still awaits a proper formulation. |
We analyze the performance of spectral clustering for community extraction in stochastic block models. We show that, under mild conditions, spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of the maximum expected degree is as small as $\... |
The stochastic block model (SBM) is a popular framework for studying community detection in networks. This model is limited by the assumption that all nodes in the same community are statistically equivalent and have equal expected degrees. |
We study polynomial time algorithms for estimating the mean of a heavy-tailed multivariate random vector. We assume only that the random vector $X$ has finite mean and covariance. |
Notion of median in one dimension is a foundational element in nonparametric statistics. It has been extended to multi-dimensional cases both in location and in regression via notions of data depth. |
In this paper, we provide non-asymptotic upper bounds on the error of sampling from a target density using three schemes of discretized Langevin diffusions. The first scheme is the Langevin Monte Carlo (LMC) algorithm, the Euler discretization of the Langevin diffusion. |
This paper considers a multivariate spatial random field, with each component having univariate marginal distributions of the skew-Gaussian type. We assume that the field is defined spatially on the unit sphere embedded in $\mathbb{R}^3$, allowing for modeling data available over large portions of planet Earth. |
We consider the problem of frequency estimation of the periodic signal multiplied by a stationary Gaussian process (Ornstein-Uhlenbeck) and observed in the presence of the white Gaussian noise. We show the consistency and asymptotic normality of the maximum likelihood estimator in the asymptotics of small noise in obs... |
We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of random-coefficient AR(1) process driven by i.i.d. innovations in the domain of normal attraction of an $\alpha$-stable distribution, $0< \alpha \le 2$, as both $N$ and the time scale $n$ tend to infinity, possibly at a different r... |
This article is a review of theoretical advances in the research field of algebraic geometry and Bayesian statistics in the last two decades. Many statistical models and learning machines which contain hierarchical structures or latent variables are called nonidentifiable, because the map from a parameter to a statist... |
We analyze the effect of a heterogeneous variance on bump detection in a Gaussian regression model. To this end we allow for a simultaneous bump in the variance and specify its impact on the difficulty to detect the null signal against a single bump with known signal strength. |
We study the distribution of hard-, soft-, and adaptive soft-thresholding estimators within a linear regression model where the number of parameters k can depend on sample size n and may diverge with n. In addition to the case of known error-variance, we define and study versions of the estimators when the error-varia... |
Combining test statistics from independent trials or experiments is a popular method of meta-analysis. However, there is very limited theoretical understanding of the power of the combined test, especially in high-dimensional models considering composite hypotheses tests. |
We consider the nonparametric estimation of a periodic function that is observed in additive Gaussian white noise after convolution with a ``boxcar,'' the indicator function of an interval. This is an idealized model for the problem of recovery of noisy signals and images observed with ``motion blur. |
Inference for partially observed Markov process models has been a longstanding methodological challenge with many scientific and engineering applications. Iterated filtering algorithms maximize the likelihood function for partially observed Markov process models by solving a recursive sequence of filtering problems. |
In this paper, we propose the multivariate range Value-at-Risk (MRVaR) and the multivariate range covariance (MRCov) as two risk measures and explore their desirable properties in risk management. In particular, we explain that such range-based risk measures are appropriate for risk management of regulation and invest... |
The controlled branching process is a generalization of the classical Bienaymé-Galton-Watson branching process. It is a useful model for describing the evolution of populations in which the population size at each generation needs to be controlled. |
Given any finite set F of (n - 1)-dimensional subspaces of R^n we give examples of nongaussian probability measures in R^n whose marginal distribution in each subspace from F is gaussian. However, if F is an infinite family of such (n - 1)-dimensional subspaces then such a nongaussian probability measure in R^n does n... |
It was shown in [6] that the Wasserstein distance is equivalent to the Mean Optimal Sub-Pattern Assignment (MOSPA) measure for empirical probability density functions. A more recent paper [7], extends on it by drawing new connections between the MOSPA concept, which is getting a foothold in the multi-target tracking c... |
The covariance function and the variogram play very important roles in modelling and in prediction of spatial and spatio-temporal data. The assumption of second order stationarity, in space and time, is often made in the analysis of spatial data and the spatio-temporal data. |
We consider the problem of estimating the tail index $\alpha$ of a distribution satisfying a $(\alpha, \beta)$ second-order Pareto-type condition, where \beta is the second-order coefficient. When $\beta$ is available, it was previously proved that $\alpha$ can be estimated with the oracle rate $n^{-\beta/(2\beta+1)}$... |
Motivated by questions about dense (non-sparse) signals in high-dimensional data analysis, we study the unconditional out-of-sample prediction error (predictive risk) associated with three popular linear estimators for high-dimensional linear models: ridge regression estimators, scalar multiples of the ordinary least s... |
A test based on tapering is proposed for use in testing a global linear hypothesis under a functional linear model. The test statistic is constructed as a weighted sum of squared linear combinations of Fourier coefficients, a tapered quadratic form, in which higher Fourier frequencies are down-weighted so as to emphas... |
Given $p$-dimensional Gaussian vectors $X_i \stackrel{iid}{\sim} N(0, \Sigma)$, $1 \leq i \leq n$, where $p \geq n$, we are interested in testing a null hypothesis where $\Sigma = I_p$ against an alternative hypothesis where all eigenvalues of $\Sigma$ are $1$, except for $r$ of them are larger than $1$ (i.e., spiked e... |
This paper advances the general theory of continuous sparse regularisation on measures with the Beurling-LASSO (BLASSO). This TV-regularized convex program on the space of measures allows to recover a sparse measure using a noisy observation from an appropriate measurement operator. |
This paper studies one-sided hypothesis testing under random sampling without replacement. That is, when $n+1$ binary random variables $X_1,\ldots, X_{n+1}$ are subject to a permutation invariant distribution and $n$ binary random variables $X_1,\ldots, X_{n}$ are observed, we have proposed randomized tests with a ran... |
Factorial clustering methods have been developed in recent years thanks to the improving of computational power. These methods perform a linear transformation of data and a clustering on transformed data optimizing a common criterion. |
For a regression model, we consider the risk of the maximum likelihood estimator with respect to $\alpha$-divergence, which includes the special cases of Kullback-Leibler divergence, Hellinger distance and $\chi^2$ divergence. The asymptotic expansion of the risk with respect to the sample size $n$ is given up to the ... |
In dealing with veracity of data analytics, fuzzy methods are more and more relying on probabilistic and statistical techniques to underpin their applicability. Conversely, standard statistical models usually disregard to take into account the inherent fuzziness of choices and this issue is particularly worthy of note... |
We formulate the notion of minimax estimation under storage or communication constraints, and prove an extension to Pinsker's theorem for nonparametric estimation over Sobolev ellipsoids. Placing limits on the number of bits used to encode any estimator, we give tight lower and upper bounds on the excess risk due ... |
In numerous applications data are observed at random times and an estimated graph of the spectral density may be relevant for characterizing and explaining phenomena. By using a wavelet analysis, one derives a nonparametric estimator of the spectral density of a Gaussian process with stationary increments (or a statio... |
For the class of stationary Gaussian long memory processes, we study some properties of the least-squares predictor of X_{n+1} based on (X_n, ..., X_1). The predictor is obtained by projecting X_{n+1} onto the finite past and the coefficients of the predictor are estimated on the same realisation. |
The aim of this paper is to extend Azzalini's method. This extension is done in two stages: consider two dependent and non-identically distributed random variables say $X_1$ and $X_2$; model the dependence between $X_1$ and $X_2$ by a copula. |
We develop a class of tests for semiparametric vector autoregressive (VAR) models with unspecified innovation densities, based on the recent measure-transportation-based concepts of multivariate {\it center-outward ranks} and {\it signs}. We show that these concepts, combined with Le Cam's asymptotic theory of sta... |
In this paper, novel closed-form point estimators of the beta distribution are proposed and investigated. The first estimators are a modified version of Pearson's method of moments. |
We study the rate of convergence of posterior distributions in density estimation problems for log-densities in periodic Sobolev classes characterized by a smoothness parameter p. The posterior expected density provides a nonparametric estimation procedure attaining the optimal minimax rate of convergence under Hellin... |
Network-valued time series are currently a common form of network data. However, the study of the aggregate behavior of network sequences generated from network-valued stochastic processes is relatively rare. |
This paper deals with a new Bayesian approach to the standard one-sample $z$- and $t$- tests. More specifically, let $x_1,\ldots,x_n$ be an independent random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. |
Tensors have broad applications in neuroimaging, data mining, digital marketing, etc. CANDECOMP/PARAFAC (CP) tensor decomposition can effectively reduce the number of parameters to gain dimensionality-reduction and thus plays a key role in tensor regression. |
The Stochastic Block Model (Holland et al., 1983) is a mixture model for heterogeneous network data. Unlike the usual statistical framework, new nodes give additional information about the previous ones in this model. |
The present paper obtains a complete description of the limit distributions of sample covariances in N x n panel data when N and n jointly increase, possibly at different rate. The panel is formed by N independent samples of length n from random-coefficient AR(1) process with the tail distribution function of the rand... |
The asymptotic decision theory by Le Cam and Hajek has been given a lucid perspective by the Ibragimov-Hasminskii theory on convergence of the likelihood random field. Their scheme has been applied to stochastic processes by Kutoyants, and today this plot is called the IHK program. |
Quasi-Monte Carlo (QMC) methods for estimating integrals are attractive since the resulting estimators typically converge at a faster rate than pseudo-random Monte Carlo. However, they can be difficult to set up on arbitrary posterior densities within the Bayesian framework, in particular for inverse problems. |
In this paper, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are multivariate continuous-time ARMA processes. Thereby, the sampling distance is fixed and the driving Lévy process has at least a finite fourth mome... |
This paper considers estimation and inference in semiparametric econometric models. Standard procedures estimate the model based on an independence restriction that induces a minimum distance between a joint cumulative distribution function and the product of the marginal cumulative distribution functions. |
Consider a Gaussian nonparametric regression problem having both an unknown mean function and unknown variance function. This article presents a class of difference-based kernel estimators for the variance function. |
We define data transformations that leave certain classes of distributions invariant, while acting in a specific manner upon the parameters of the said distributions. It is shown that under such transformations the maximum likelihood estimators behave in exactly the same way as the parameters being estimated. |
Given observations from a stationary time series, permutation tests allow one to construct exactly level $\alpha$ tests under the null hypothesis of an i.i.d. (or, more generally, exchangeable) distribution. On the other hand, when the null hypothesis of interest is that the underlying process is an uncorrelated seque... |
We study simple binary hypothesis testing under both local differential privacy (LDP) and communication constraints. We qualify our results as either minimax optimal or instance optimal: the former hold for the set of distribution pairs with prescribed Hellinger divergence and total variation distance, whereas the lat... |
This paper considers non-negative integer-valued autoregressive processes where the autoregression parameter is close to unity. We consider the asymptotics of this `near unit root' situation. |
We systematically investigate the preservation of differential privacy in functional data analysis, beginning with functional mean estimation and extending to varying coefficient model estimation. Our work introduces a distributed learning framework involving multiple servers, each responsible for collecting several s... |
Depth is a concept that measures the `centrality' of a point in a given data cloud or in a given probability distribution. Every depth defines a family of so-called trimmed regions. |
Big data can easily be contaminated by outliers or contain variables with heavy-tailed distributions, which makes many conventional methods inadequate. To address this challenge, we propose the adaptive Huber regression for robust estimation and inference. |
We consider the problem of estimating the unknown response function in the Gaussian white noise model. We first utilize the recently developed Bayesian maximum a posteriori "testimation" procedure of Abramovich et al. (2007) for recovering an unknown high-dimensional Gaussian mean vector. |
This paper aims at developing a quasi-Bayesian analysis of the nonparametric instrumental variables model, with a focus on the asymptotic properties of quasi-posterior distributions. In this paper, instead of assuming a distributional assumption on the data generating process, we consider a quasi-likelihood induced fr... |
There is growing interest in developing statistical estimators that achieve exponential concentration around a population target even when the data distribution has heavier than exponential tails. More recent activity has focused on extending such ideas beyond Euclidean spaces to Hilbert spaces and Riemannian manifold... |
Two goodness-of-fit tests for copulas are being investigated. The first one deals with the case of elliptical copulas and the second one deals with independent copulas. |
We propose a study of a distribution registration model for general deformation functions. In this framework, we provide estimators of the deformations as well as a goodness of fit test of the model. |
Log-linear models are a well-established method for describing statistical dependencies among a set of n random variables. The observed frequencies of the n-tuples are explained by a joint probability such that its logarithm is a sum of functions, where each function depends on as few variables as possible. |
This paper presents a randomization-based framework for estimating causal effects under interference between units, motivated by challenges that arise in analyzing experiments on social networks. The framework integrates three components: |
A large dimensional characterization of robust M-estimators of covariance (or scatter) is provided under the assumption that the dataset comprises independent (essentially Gaussian) legitimate samples as well as arbitrary deterministic samples, referred to as outliers. Building upon recent random matrix advances in th... |
We investigate eigenvectors of rank-one deformations of random matrices $\boldsymbol B = \boldsymbol A + \theta \boldsymbol {uu}^*$ in which $\boldsymbol A \in \mathbb R^{N \times N}$ is a Wigner real symmetric random matrix, $\theta \in \mathbb R^+$, and $\boldsymbol u$ is uniformly distributed on the unit sphere. It... |
We introduce a new one-parameter family of divergence measures, called bounded Bhattacharyya distance (BBD) measures, for quantifying the dissimilarity between probability distributions. These measures are bounded, symmetric and positive semi-definite and do not require absolute continuity. |
Zuckermann [10] considers the problem of optimal control of a finite dam assuming that the input process is Wiener with positive drift term \mu \geq 0. Lam and Lou [7] treat the case where the input is a Wiener process with a reflecting boundary at its infimum, with a positive drift term, using the long-run Average an... |
In this paper we use an iterative algorithm for solving Fredholm equations of the first kind. The basic algorithm is known and is based on an EM algorithm when involved functions are non-negative and integrable. |
One fundamental goal of high-dimensional statistics is to detect or recover planted structure (such as a low-rank matrix) hidden in noisy data. A growing body of work studies low-degree polynomials as a restricted model of computation for such problems: it has been demonstrated in various settings that low-degree poly... |
In this paper, we are concerned with the independence test for $k$ high-dimensional sub-vectors of a normal vector, with fixed positive integer $k$. A natural high-dimensional extension of the classical sample correlation matrix, namely block correlation matrix, is raised for this purpose. |
This paper introduces studies on exponentaited generalized Weibull Gompertz distribution EGWGD which generalizes a lot of distributions. Several properties of the EGWGD such as reversed (hazard) function, moments, maximum likelihood estimation, mean residual (past) lifetime, MTTF, MTTR, MTBF, maintainability, availabi... |
In this article we prove a generalization of the Ejsmont characterization of the multivariate normal distribution. Based on it, we propose a new test for independence and normality. |
Let $X_{1}=(W_{1},Y_{1}),\ldots,X_{n}=(W_{n},Y_{n})$ be $n$ pairs of independent random variables. We assume that, for each $i\in\{1,\ldots,n\}$, the conditional distribution of $Y_{i}$ given $W_{i}$ belongs to a one-parameter exponential family with parameter ${\boldsymbol{\gamma}}^{\star}(W_{i})\in{\mathbb{R}}$, or ... |
For exponentially distributed lifetimes, we consider the prediction of future order statistics based on having observed the first $m$ order statistics. We focus on the previously less explored aspects of predicting: |
The paper "An efficient sampling scheme for the eigenvalues of dual Wishart matrices", by I. ~Santamaría and V. |
This paper considers the problem of estimating a mean pattern in the setting of Grenander's pattern theory. Shape variability in a data set of curves or images is modeled by the random action of elements in a compact Lie group on an infinite dimensional space. |
An asymmetric Dirichlet kernel version of the Gasser-Müller estimator is introduced for regression surfaces on the simplex, extending the univariate analog proposed by Chen [Statist. Sinica, 10(1) (2000), pp. 73-91]. Its asymptotic properties are investigated under the condition that the design points are known and fi... |
The Multi-Reference Alignment (MRA) problem aims at the recovery of an unknown signal from repeated observations under the latent action of a group of cyclic isometries, in the presence of additive noise of high intensity $\sigma$. It is a more tractable version of the celebrated cryo EM model. |
We address the statistical issue of determining the maximal spaces (maxisets) where model selection procedures attain a given rate of convergence. By considering first general dictionaries, then orthonormal bases, we characterize these maxisets in terms of approximation spaces. |
Given an arbitrary long but finite sequence of observations from a finite set, we construct a simple process that approximates the sequence, in the sense that with high probability the empirical frequency, as well as the empirical one-step transitions along a realization from the approximating process, are close to tha... |
Our research proposes a novel method for reducing the dimensionality of functional data, specifically for the case where the response is a scalar and the predictor is a random function. Our method utilizes distance covariance, and has several advantages over existing methods. |
We study a broad class of asymmetric copulas introduced by Liebscher (2008) as a combination of multiple - usually symmetric - copulas. The main thrust of the paper is to provide new theoretical properties including exact tail dependence expressions and stability properties. |
Autoregressive and moving-average (ARMA) models with stable Paretian errors is one of the most studied models for time series with infinite variance. Estimation methods for these models have been studied by many researchers but the problem of diagnostic checking fitted models has not been addressed. |
The Frisch--Waugh--Lovell Theorem states the equivalence of the coefficients from the full and partial regressions. I further show the equivalence between various standard errors. |
Stochastic simulators such as Monte-Carlo estimators are widely used in science and engineering to study physical systems through their probabilistic representation. Global sensitivity analysis aims to identify the input parameters which have the most important impact on the output. |
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