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In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators of persistent homology. |
Estimation procedures based on recursive algorithms are interesting and powerful techniques that are able to deal rapidly with (very) large samples of high dimensional data. The collected data may be contaminated by noise so that robust location indicators, such as the geometric median, may be preferred to the mean. |
We contemplate an experimental situation in a $2^k$-factorial experiment with acute resource crunch so that we need to conduct just a saturated design [SD] - with the understanding that precision of the estimates cannot be estimated from the data. It is known beforehand which effect(s)/interaction(s) are likely to be ... |
This paper investigates the fundamental limits for detecting a high-dimensional sparse matrix contaminated by white Gaussian noise from both the statistical and computational perspectives. We consider $p\times p$ matrices whose rows and columns are individually $k$-sparse. |
The paper is concerned with asymptotic properties of the principal components analysis of functional data. The currently available results assume the existence of the fourth moment. |
In the paper, we suggest three tests on the validity of a factor model which can be applied for both small dimensional and large dimensional data. Both the exact and asymptotic distributions of the resulting test statistics are derived under classical and high-dimensional asymptotic regimes. |
A common problem in Phase II clinical trials is the comparison of dose response curves corresponding to different treatment groups. If the effect of the dose level is described by parametric regression models and the treatments differ in the administration frequency (but not in the sort of drug) a reasonable assumptio... |
Standard statistical techniques often require transforming data to have mean $0$ and standard deviation $1$. Typically, this process of "standardization" or "normalization" is applied across subjects when each subject produces a single number. |
Thurstone's latent-normal model, introduced a century ago to describe human preferences in psychometrics (1927), remains a cornerstone for modeling random rankings. Yet when the underlying normals differ in distribution, the joint law of ranks $R_{i}:=\sum_{j=1}^{n}\mathbf{1}_{X_{j}\leq X_{i}}$ is virtually unexpl... |
We design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of their final time marginals. In contrast to traditional genealogical tree based models, these new particle algorithms can ... |
We focus on one-sided, mixture-based stopping rules for the problem of sequential testing a simple null hypothesis against a composite alternative. For the latter, we consider two cases---either a discrete alternative or a continuous alternative that can be embedded into an exponential family. |
We tackle the inverse problem of reconstructing an unknown finite measure $\mu$ from a noisy observation of a generalized moment of $\mu$ defined as the integral of a continuous and bounded operator $\Phi$ with respect to $\mu$. When only a quadratic approximation $\Phi_m$ of the operator is known, we introduce the $L... |
Averaging provides an alternative to bandwidth selection for density kernel estimation. We propose a procedure to combine linearly several kernel estimators of a density obtained from different, possibly data-driven, bandwidths. |
We study the problem of nonparametric estimation of a multivariate function $g:\mathbb {R}^d\to\mathbb{R}$ that can be represented as a composition of two unknown smooth functions $f:\mathbb{R}\to\mathbb{R}$ and $G:\mathbb{R}^d\to \mathbb{R}$. We suppose that $f$ and $G$ belong to known smoothness classes of functions... |
Autoregressive models are a class of time series models that are important in both applied and theoretical statistics. Typically, inferential devices such as confidence sets and hypothesis tests for time series models require nuanced asymptotic arguments and constructions. |
The analysis of the joint cumulative distribution function (CDF) with bivariate event time data is a challenging problem both theoretically and numerically. This paper develops a tensor spline-based sieve maximum likelihood estimation method to estimate the joint CDF with bivariate current status data. |
We consider the problem of inferring the conditional independence graph (CIG) of a multivariate stationary dicrete-time Gaussian random process based on a finite length observation. Using information-theoretic methods, we derive a lower bound on the error probability of any learning scheme for the underlying process C... |
We study full Bayesian procedures for sparse linear regression when errors have a symmetric but otherwise unknown distribution. The unknown error distribution is endowed with a symmetrized Dirichlet process mixture of Gaussians. |
The Gaussian sequence model is a canonical model in nonparametric estimation. In this study, we introduce a multivariate version of the Gaussian sequence model and investigate adaptive estimation over the multivariate Sobolev ellipsoids, where adaptation is not only to unknown smoothness but also to arbitrary quadrati... |
We consider a time-ordered sequence of networks stemming from stochastic block models where nodes gradually change their memberships over time, and no network at any single time point contains sufficient signal strength to recover its community structure. To estimate the time-varying community structure, we develop KD... |
For a multinormal distribution with a $p$-dimensional mean vector ${\mbtheta}$ and an arbitrary unknown dispersion matrix ${\mbSigma}$, Rao ([9], [10]) proposed two tests for the problem of testing $ H_{0}:{\mbtheta}_{1} = {\bf 0}, {\mbtheta}_{2} = {\bf 0}, {\mbSigma}~ \hbox{unspecified},~\hbox{versus}~H_{1}:{\mbtheta}... |
The pair correlation function is a fundamental spatial point process characteristic that, given the intensity function, determines second order moments of the point process. Non-parametric estimation of the pair correlation function is a typical initial step of a statistical analysis of a spatial point pattern. |
Our goal is to establish large deviations and concentration inequalities for the maximum likelihood estimator of the drift parameter of the Ornstein-Uhlenbeck process without tears. We propose a new strategy to establish large deviation results which allows us, via a suitable transformation, to circumvent the classica... |
A finite set is "hidden" if its elements are not directly enumerable or if its size cannot be ascertained via a deterministic query. In public health, epidemiology, demography, ecology and intelligence analysis, researchers have developed a wide variety of indirect statistical approaches, under different model... |
In this paper, we investigate several properties of the weighted varextropy measure and obtain it for specific distribution functions, such as the equilibrium and weighted distributions. We also obtain bounds for the weighted varextropy, as well as for weighted residual varextropy and weighted past varextropy. |
We provide new characterizations of two-points and some related distributions. We use properties of independence and/or identity of the distributions of suitable linear forms of random variables. |
The purpose of this note is to show that in a widely cited paper by Yakir [Ann. Statist. 25 (1997) 2117--2126, doi:<a href="https://doi.org/10.1214/aos/1069362390" data-doi="10.1214/aos/1069362390" class="link-https link-external" rel="external noopener nofollow">https://doi.org/10.1214/aos/1069362390</a>], the proof t... |
For point patterns observed in natura, spatial heterogeneity is more the rule than the exception. In numerous applications, this can be mathematically handled by the flexible class of log Gaussian Cox processes (LGCPs); in brief, a LGCP is a Cox process driven by an underlying log Gaussian random field (log GRF). |
We provide a new algorithm for the treatment of the noisy inversion of the Radon transform using an appropriate thresholding technique adapted to a well-chosen new localized basis. We establish minimax results and prove their optimality. |
In this paper, we observe a sparse mean vector through Gaussian noise and we aim at estimating some additive functional of the mean in the minimax sense. More precisely, we generalize the results of (Collier et al., 2017, 2019) to a very large class of functionals. |
We formulate simple equivalent conditions for the validity of Bayes' formula for conditional densities. We show that for any random variables X and Y (with values in arbitrary measurable spaces), the following are equivalent: <br>1. |
M-type smoothing splines are a broad class of spline estimators that include the popular least-squares smoothing spline but also spline estimators that are less susceptible to outlying observations and model-misspecification. However, available asymptotic theory only covers smoothing spline estimators based on smooth ... |
We consider non-linear Bayesian inverse problems of determining the parameter $f$. For the posterior distribution with a class of Gaussian process priors, we study the statistical performance of variational Bayesian inference to the posterior with variational sets consisting of Gaussian measures or a mean-field family... |
Tests for proportional hazards assumption concerning specified covariates or groups of covariates are proposed. The class of alternatives is wide: log-hazard rates under different values of covariates may cross, approach, go away. |
This paper studies the problem of recovering the hidden vertex correspondence between two edge-correlated random graphs. We focus on the Gaussian model where the two graphs are complete graphs with correlated Gaussian weights and the Erdős-Rényi model where the two graphs are subsampled from a common parent Erdős-Rény... |
In multiple change-point problems, different data segments often follow different distributions, for which the changes may occur in the mean, scale or the entire distribution from one segment to another. Without the need to know the number of change-points in advance, we propose a nonparametric maximum likelihood appr... |
We establish a general, non-asymptotic error analysis framework for understanding the effects of incremental approximations made by practical approaches for Bayesian sequential learning (BSL) on their long-term inference performance. Our setting covers inverse problems, state estimation, and parameter-state estimation... |
We consider estimation of covariance matrices and their inverses (a.k.a. precision matrices) for high-dimensional stationary and locally stationary time series. In the latter case the covariance matrices evolve smoothly in time, thus forming a covariance matrix function. |
Covariance matrix estimation and principal component analysis (PCA) are two cornerstones of multivariate analysis. Classic textbook solutions perform poorly when the dimension of the data is of a magnitude similar to the sample size, or even larger. |
This work is concerned with the estimation of the intensity parameter of a stationary determinantal point process. We consider the standard estimator, corresponding to the number of observed points per unit volume and a recently introduced median-based estimator more robust to outliers. |
Kernel density estimation is a well known method involving a smoothing parameter (the bandwidth) that needs to be tuned by the user. Although this method has been widely used the bandwidth selection remains a challenging issue in terms of balancing algorithmic performance and statistical relevance. |
E-values have gained attention as potential alternatives to p-values as measures of uncertainty, significance and evidence. In brief, e-values are realized by random variables with expectation at most one under the null; examples include betting scores, (point null) Bayes factors, likelihood ratios and stopped superma... |
In this paper, we propose a cone projected power iteration algorithm to recover the first principal eigenvector from a noisy positive semidefinite matrix. When the true principal eigenvector is assumed to belong to a convex cone, the proposed algorithm is fast and has a tractable error. |
We derive limiting distributions of symmetrized estimators of scatter, where instead of all $n(n-1)/2$ pairs of the $n$ observations we only consider $nd$ suitably chosen pairs, $1 \le d < \lfloor n/2\rfloor$. It turns out that the resulting estimators are asymptotically equivalent to the original one whenever $d =... |
Online learning is an inferential paradigm in which parameters are updated incrementally from sequentially available data, in contrast to batch learning, where the entire dataset is processed at once. In this paper, we assume that mini-batches from the full dataset become available sequentially. |
We consider deconvolution from repeated observations with unknown error distribution. So far, this model has mostly been studied under the additional assumption that the errors are symmetric. |
To go beyond standard first-order asymptotics for Cox regression, we develop parametric bootstrap and second-order methods. In general, computation of $P$-values beyond first order requires more model specification than is required for the likelihood function. |
In this paper, we investigate the spectral properties of the sample canonical correlation (SCC) matrix under the alternative hypothesis to provide a more comprehensive description of the association between two sets of variables. Our research involves establishing the relationship between the eigenvalues of the SCC ma... |
Linear regression is perhaps one of the most popular statistical concepts, which permeates almost every scientific field of study. Due to the technical simplicity and wide applicability of linear regression, attention is almost always quickly directed to the algorithmic or computational side of linear regression. |
A Condorcet voting scheme chooses a winning candidate as one who defeats all others in pairwise majority rule. We provide a review which includes the rigorous mathematical treatment for calculating the limiting probability of a Condorcet winner for any number of candidates and value of $n$ odd or even and with arbitra... |
As a starting point we prove a functional central limit theorem for estimators of the invariant measure of a geometrically ergodic Harris-recurrent Markov chain in a multi-scale space. This allows to construct confidence bands for the invariant density with optimal (up to undersmoothing) $L^{\infty}$-diameter by using... |
Maximum likelihood estimators for time-dependent mean functions within Gaussian processes are provided in the context of continuous observations. We find the widest possible class of mean functions for which the likelihood function can be written explicitly. |
We consider forecasting a single time series using a large number of predictors in the presence of a possible nonlinear forecast function. Assuming that the predictors affect the response through the latent factors, we propose to first conduct factor analysis and then apply sufficient dimension reduction on the estima... |
We propose a nonparametric parameter estimation of confidence intervals when the underlying has large or infinite variance. We explain the method by a simple numerical example and provide an application to estimate the coupling strength in neuronal networks. |
We consider parameter estimation under sparse linear regression -- an extensively studied problem in high-dimensional statistics and compressed sensing. While the minimax framework has been one of the most fundamental approaches for studying statistical optimality in this problem, we identify two important issues that... |
We study the problem of estimating the joint probability mass function (pmf) over two random variables. In particular, the estimation is based on the observation of $m$ samples containing both variables and $n$ samples missing one fixed variable. |
The bias of the sample means of the arms in multi-armed bandits is an important issue in adaptive data analysis that has recently received considerable attention in the literature. Existing results relate in precise ways the sign and magnitude of the bias to various sources of data adaptivity, but do not apply to the ... |
Optimal transport and Wasserstein distances are flourishing in many scientific fields as a means for comparing and connecting random structures. Here we pioneer the use of an optimal transport distance between Lévy measures to solve a statistical problem. |
This paper provides some extended results on estimating parameter matrix of several regression models when the covariate or response possesses weaker moment condition. We study the $M$-estimator of Fan et al. (Ann Stat 49(3):1239--1266, 2021) for matrix completion model with $(1+\epsilon)$-th moment noise. |
We revisit the estimation of the extreme value index for randomly censored data from a heavy tailed distribution. We introduce a new class of estimators which encompasses earlier proposals given in Worms and Worms (2014) and Beirlant et al. (2018), which were shown to have good bias properties compared with the pseudo... |
In a recent article (Proc. Natl. |
A smooth test to simultaneously compare $K$ copulas, where $K \geq 2$ is proposed. The $K$ observed populations can be paired, and the test statistic is constructed based on the differences between moment sequences, called copula coefficients. |
In this paper, we build upon the asymptotic theory for GARCH processes, considering the general class of augmented GARCH($p$, $q$) processes. Our contribution is to complement the well-known univariate asymptotics by providing a joint (bivariate) functional central limit theorem of the sample quantile and the r-th abs... |
A novel self-normalization procedure for CUSUM-based change detection in the mean of a locally stationary time series is introduced. Classical self-normalization relies on the factorization of a constant long-run variance and a stochastic factor. |
A half century ago, Huber evaluated the minimax asymptotic variance in scalar location estimation, $ \min_\psi \max_{F \in {\cal F}_\epsilon} V(\psi, F) = \frac{1}{I(F_\epsilon^*)} $, where $V(\psi,F)$ denotes the asymptotic variance of the $(M)$-estimator for location with score function $\psi$, and $I(F_\epsilon^*)$ ... |
We explore the arguments for maximizing the `evidence' as an algorithm for model selection. We show, using a new definition of model complexity which we term `flexibility', that maximizing the evidence should appeal to both Bayesian and Frequentist statisticians. |
For the conditional mean function of panel count model with time-varying coefficients, we propose to use local kernel regression method for estimation. Partial log-likelihood with local polynomial is formed for estimation. |
This paper develops bootstrap methods to construct uniform confidence bands for nonparametric spectral estimation of Lévy densities under high-frequency observations. We assume that we observe $n$ discrete observations at frequency $1/\Delta > 0$, and work with the high-frequency setup where $\Delta = \Delta_{n} \t... |
Naive Bayes classifiers have proven to be useful in many prediction problems with complete training data. Here we consider the situation where a naive Bayes classifier is trained with data where the response is right censored. |
This paper studies the problem of testing the null assumption of no-change in the mean of chronologically ordered independent observations on a random variable $X$ {\it versus} the at most one change in the mean alternative hypothesis. The approach taken is via a Darling-Erdős type self-normalized maximal deviation be... |
Variational inference (VI) is a popular method for approximating intractable posterior distributions in Bayesian inference and probabilistic machine learning. In this paper, we introduce a general framework for quantifying the statistical accuracy of mean-field variational inference (MFVI) for posterior approximation ... |
We show that one can perform causal inference in a natural way for continuous-time scenarios using tools from stochastic analysis. This provides new alternatives to the positivity condition for inverse probability weighting. |
We propose statistical tests to discriminate between the finite and infinite activity of jumps in a semimartingale discretely observed at high frequency. The two statistics allow for a symmetric treatment of the problem: we can either take the null hypothesis to be finite activity, or infinite activity. |
We study the problem of testing the null hypothesis that X and Y are conditionally independent given Z, where each of X, Y and Z may be functional random variables. This generalises testing the significance of X in a regression model of scalar response Y on functional regressors X and Z. We show however that even in t... |
This work provides a unified analysis of the properties of the sample covariance matrix $\Sigma_n$ over the class of $p\times p$ population covariance matrices $\Sigma$ of reduced effective rank $r_e(\Sigma)$. This class includes scaled factor models and covariance matrices with decaying spectrum. |
We consider the diffusion $(R_t^r)_{t\ge0}$ generated by the equation $dR_t^r=dt+\mu R_t^r dB_t$ with $R_0^r\triangleq r\ge0$ fixed, and where $\mu\neq0$ is given, and $(B_t)_{t\ge0}$ is standard Brownian motion. We assume that $(R_t^r)_{t\ge0}$ is stopped at $\mathcal{S}_A^r\triangleq\inf\{t\ge0\colon R_t^r=A\}$ with... |
Many theoretical results for the lasso require the samples to be iid. Recent work has provided guarantees for the lasso assuming that the time series is generated by a sparse Vector Auto-Regressive (VAR) model with Gaussian innovations. |
The presence of a sparse "truth" has been a constant assumption in the theoretical analysis of sparse PCA and is often implicit in its methodological development. This naturally raises questions about the properties of sparse PCA methods and how they depend on the assumption of sparsity. |
We consider the estimation of the multi-period optimal portfolio obtained by maximizing an exponential utility. Employing Jeffreys' non-informative prior and the conjugate informative prior, we derive stochastic representations for the optimal portfolio weights at each time point of portfolio reallocation. |
The analysis of animal movement has gained attention recently. New continuous-time models and statistical methods have been developed to estimate some sets related to their movements, such as the home-range and the core-area among others, when the information of the trajectory is provided by a GPS. |
Case-cohort design, an outcome-dependent sampling design for censored survival data, is increasingly used in biomedical research. The development of asymptotic theory for a case-cohort design in the current literature primarily relies on counting process stochastic integrals. |
A variety of indices aim to quantify the impact of input variables on a response, typically the output from a complex computer code or black-box model. Most commonly used, the Sobol' index typically measures the influence of some inputs from an explained variance perspective. |
The aim of this paper is to introduce an adaptive penalized estimator for identifying the true reduced parametric model under the sparsity assumption. In particular, we deal with the framework where the unpenalized estimator of the structural parameters needs simultaneously multiple rates of convergence (i.e. the so-c... |
In this article we define new FrÈchet features for random cumulative distribution functions using contrast. These contrasts allow to construct Wasserstein costs and our new features minimize the average costs as the FrÈchet mean minimizes the mean square Wasserstein$_2$ distance. |
We consider a non-linear Bayesian data assimilation model for the periodic two-dimensional Navier-Stokes equations with initial condition modelled by a Gaussian process prior. We show that if the system is updated with sufficiently many discrete noisy measurements of the velocity field, then the posterior distribution... |
The local Hurst exponent, a measure employed to detect the presence of dependence in a time series, may also be used to investigate the source of intraday variation observed in the returns in foreign exchange markets. Given that changes in the local Hurst exponent may be due to either a time-varying range, or standard... |
We consider different types of predictive intervals and ask whether they are elicitable, i.e. are unique minimizers of a loss or scoring function in expectation. The equal-tailed interval is elicitable, with a rich class of suitable loss functions, though subject to translation invariance, or positive homogeneity and ... |
We study the problem of testing the existence of a dense subhypergraph. The null hypothesis is an Erdos-Renyi uniform random hypergraph and the alternative hypothesis is a uniform random hypergraph that contains a dense subhypergraph. |
In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the ... |
We develop a technique for establishing lower bounds on the sample complexity of Least Squares (or, Empirical Risk Minimization) for large classes of functions. As an application, we settle an open problem regarding optimality of Least Squares in estimating a convex set from noisy support function measurements in dime... |
Suppose two networks are observed for the same set of nodes, where each network is assumed to be generated from a weighted stochastic block model. This paper considers the problem of testing whether the community memberships of the two networks are the same. |
Let $\alpha_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)$ be the predictive distributions of a sequence $(X_1,X_2,\ldots)$ of $p$-dimensional random vectors. Suppose $$\alpha_n= \mathcal{N} _p (M_n,Q_n)$$ where $M_n=\frac{1}{n}\sum_{i=1}^nX_i$ and $Q_n=\frac{1}{n}\sum_{i=1}^n(X_i-M_n)(X_i-M_n)^t$. |
Various graphical models are widely used in reliability to provide a qualitative description of domain experts hypotheses about how a system might fail. Here we argue that the semantics developed within standard causal Bayesian networks are not rich enough to fully capture the intervention calculus needed for this dom... |
This paper is concerned with various Mallows ranking models. We study the statistical properties of the MLE of Mallows' $\phi$ model. |
Recent developments in Reinforcement learning have significantly enhanced sequential decision-making in uncertain environments. Despite their strong performance guarantees, most existing work has focused primarily on improving the operational accuracy of learned control policies and the convergence rates of learning a... |
We study the clustering task under anisotropic Gaussian Mixture Models where the covariance matrices from different clusters are unknown and are not necessarily the identical matrix. We characterize the dependence of signal-to-noise ratios on the cluster centers and covariance matrices and obtain the minimax lower bou... |
We study a new privacy model where users belong to certain sensitive groups and we would like to conduct statistical inference on whether there is significant differences in outcomes between the various groups. In particular we do not consider the outcome of users to be sensitive, rather only the membership to certain... |
We consider the setup of nonparametric {\em blind regression} for estimating the entries of a large $m \times n$ matrix, when provided with a small, random fraction of noisy measurements. We assume that all rows $u \in [m]$ and columns $i \in [n]$ of the matrix are associated to latent features $x_{\text{row}}(u)$ and... |
Conjunctive Bayesian networks (CBNs) are graphical models that describe the accumulation of events which are constrained in the order of their occurrence. A CBN is given by a partial order on a (finite) set of events. |
In reliability and life testing studies, the topic of estimating hazard rate has received great attention in recent years since an estimate of hazard rate is a quite useful tool for making decisions. Some works have included nonparametric approaches while some have considered parametric structural models for complete ... |
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