Multivariate kernel density estimations have received much spate of interest. In addition to conventional methods of (non-)classical associated-kernels for (un)bounded densities and bandwidth selections, the multiple extended-beta kernel (MEBK) estimators with Bayesian adaptive bandwidths are invested to gain a deeper and better insight into the estimation of multivariate density functions. Being unimodal, the univariate extended-beta smoother has an adaptable compact support which is suitable for each dataset, always limited. The support of the density MBEK estimator can be known or estimated by extreme values. Thus, asymptotical properties for the (non-)normalized estimators are established. Explicit and general choices of bandwidths using the flexible Bayesian adaptive method are provided. Behavioural analyses, specifically undertaken on the sensitive edges of the estimator support, are studied and compared to Gaussian and gamma kernel estimators. Finally, simulation studies and three applications on original and usual real-data sets of the proposed method yielded very interesting advantages with respect to its flexibility as well as its universality.
Multivariate associated kernel estimators, which depend on both target point and bandwidth matrix, are appropriate for partially or totally bounded distributions and generalize the classical ones as Gaussian. Previous studies on multivariate associated kernels have been restricted to product of univariate associated kernels, also considered having diagonal bandwidth matrices. However, it is shown in classical cases that for certain forms of target density such as multimodal, the use of full bandwidth matrices offers the potential for significantly improved density estimation. In this paper, ge…
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. The purpose of this paper is to compare a recently developped bandwidth selection method for kernel density estimation to those which are commonly used by now (at least those which are implemented in the R-package). This new method is called Penalized Comparison to Overfitting (PCO). It has been pro…
The problem of fast computation of multivariate kernel density estimation (KDE) is still an open research problem. In our view, the existing solutions do not resolve this matter in a satisfactory way. One of the most elegant and efficient approach utilizes the fast Fourier transform. Unfortunately, the existing FFT-based solution suffers from a serious limitation, as it can accurately operate only with the constrained (i.e., diagonal) multivariate bandwidth matrices. In this paper we describe the problem and give a satisfactory solution. The proposed solution may be successfully used also in o…
The performance of multivariate kernel density estimation (KDE) depends strongly on the choice of bandwidth matrix. The high computational cost required for its estimation provides a big motivation to develop fast and accurate methods. One of such methods is based on the Fast Fourier Transform. However, the currently available implementation works very well only for the univariate KDE and it's multivariate extension suffers from a very serious limitation as it can accurately operate only with diagonal bandwidth matrices. A more general solution is presented where the above mentioned limitation…
The paper deals with the density estimation on Rd under sup- norm loss. We provide with fully data-driven estimation procedure and establish for it so called sup-norm oracle inequality. The pro- posed estimator allows to take into account not only approximation properties of the underlying density but eventual independence struc- ture as well. Our results contain, as a particular case, the complete solution of the bandwidth selection problem in multivariate density model. Usefulness of the developed approach is illustrated by appli- cation to adaptive estimation over anisotropic Nikolskii clas…
Dhirubhai Ambani Institute of Technology (DAIICT)（迪鲁巴伊·安巴尼信息技术学院）
The article derives a novel Gram-Charlier A (GCA) Series based Extended Rule-of-Thumb (ExROT) for bandwidth selection in Kernel Density Estimation (KDE). There are existing various bandwidth selection rules achieving minimization of the Asymptotic Mean Integrated Square Error (AMISE) between the estimated probability density function (PDF) and the actual PDF. The rules differ in a way to estimate the integration of the squared second order derivative of an unknown PDF $(f(\cdot))$, identified as the roughness $R(f''(\cdot))$. The simplest Rule-of-Thumb (ROT) estimates $R(f''(\cdot))$ with an a…
It is shown that, for kernel-based classification with univariate distributions and two populations, optimal bandwidth choice has a dichotomous character. If the two densities cross at just one point, where their curvatures have the same signs, then minimum Bayes risk is achieved using bandwidths which are an order of magnitude larger than those which minimize pointwise estimation error. On the other hand, if the curvature signs are different, or if there are multiple crossing points, then bandwidths of conventional size are generally appropriate. The range of different modes of behavior is na…
Matrix variate beta (MVB) distributions are used in different fields of hypothesis testing, multivariate correlation analysis, zero regression, canonical correlation analysis and etc. In this approach a unified methodology is proposed to generate matrix variate distributions by combining the kernel of MVB distributions of different types with an unknown Borel measurable function of trace operator over matrix space, called generator component. The latter component is a principal element of these newly defined generator type matrix variate distributions. The matrix variate Kummer beta distributi…
We consider the solution X = (Xt) t$\ge$0 of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density $μ$. We assume that a continuous record of observations X T = (Xt) 0$\le$t$\le$T is available. In the case without jumps, Reiss and Dalalyan (2007) and Strauch (2018) have found convergence rates of invariant density estimators, under respectively isotropic and anisotropic H{ö}lder smoothness constraints, which are considerably faster than those known from standard multivariate density estimation. We extend the previous wor…
Statistical inference for sparse covariance matrices is crucial to reveal dependence structure of large multivariate data sets, but lacks scalable and theoretically supported Bayesian methods. In this paper, we propose beta-mixture shrinkage prior, computationally more efficient than the spike and slab prior, for sparse covariance matrices and establish its minimax optimality in high-dimensional settings. The proposed prior consists of beta-mixture shrinkage and gamma priors for off-diagonal and diagonal entries, respectively. To ensure positive definiteness of the resulting covariance matrix,…
Kernel density estimation and kernel regression are powerful but computationally expensive techniques: a direct evaluation of kernel density estimates at $M$ evaluation points given $N$ input sample points requires a quadratic $\mathcal{O}(MN)$ operations, which is prohibitive for large scale problems. For this reason, approximate methods such as binning with Fast Fourier Transform or the Fast Gauss Transform have been proposed to speed up kernel density estimation. Among these fast methods, the Fast Sum Updating approach is an attractive alternative, as it is an exact method and its speed is …
This paper investigates the size performance of Wald tests for CAViaR models (Engle and Manganelli, 2004). We find that the usual estimation strategy on test statistics yields inaccuracies. Indeed, we show that existing density estimation methods cannot adapt to the time-variation in the conditional probability densities of CAViaR models. Consequently, we develop a method called adaptive random bandwidth which can approximate time-varying conditional probability densities robustly for inference testing on CAViaR models based on the asymptotic normality of the model parameter estimator. This pr…
There exist several endeavors proposing a new family of extended distributions using the beta-generating technique. This is a well-known mechanism in developing flexible distributions, by embedding the cumulative distribution function (cdf) of a baseline distribution within the beta distribution that acts as a generator. Univariate beta-generated distributions offer many fruitful and tractable properties and have applications in hydrology, biology and environmental sciences amongst other fields. In the univariate cases, this extension works well, however, for multivariate cases, the beta distr…
Important information concerning a multivariate data set, such as clusters and modal regions, is contained in the derivatives of the probability density function. Despite this importance, nonparametric estimation of higher order derivatives of the density functions have received only relatively scant attention. Kernel estimators of density functions are widely used as they exhibit excellent theoretical and practical properties, though their generalization to density derivatives has progressed more slowly due to the mathematical intractabilities encountered in the crucial problem of bandwidth (…
In this paper, we study convex optimization methods for computing the trace norm regularized least squares estimate in multivariate linear regression. The so-called factor estimation and selection (FES) method, recently proposed by Yuan et al. [22], conducts parameter estimation and factor selection simultaneously and have been shown to enjoy nice properties in both large and finite samples. To compute the estimates, however, can be very challenging in practice because of the high dimensionality and the trace norm constraint. In this paper, we explore a variant of Nesterov's smooth method [20]…
Hyogo University of Health Sciences（兵库医疗大学）；Chiba University（千叶大学）
This study proposes multivariate kernel density estimation by stagewise minimization algorithm based on $U$-divergence and a simple dictionary. The dictionary consists of an appropriate scalar bandwidth matrix and a part of the original data. The resulting estimator brings us data-adaptive weighting parameters and bandwidth matrices, and realizes a sparse representation of kernel density estimation. We develop the non-asymptotic error bound of estimator obtained via the proposed stagewise minimization algorithm. It is confirmed from simulation studies that the proposed estimator performs compe…
This paper studies nonparametric estimation of parameters of multivariate Hawkes processes. We consider the Bayesian setting and derive posterior concentration rates. First rates are derived for L1-metrics for stochastic intensities of the Hawkes process. We then deduce rates for the L1-norm of interactions functions of the process. Our results are exemplified by using priors based on piecewise constant functions, with regular or random partitions and priors based on mixtures of Betas distributions. Numerical illustrations are then proposed with in mind applications for inferring functional co…
Given a sample from a multivariate distribution $F$, the uniform random variates generated independently and rearranged in the order specified by the componentwise ranks of the original sample look like a sample from the copula of $F$. This idea can be regarded as a variant on Baker's [J. Multivariate Anal. 99 (2008) 2312--2327] copula construction and leads to the definition of the empirical beta copula. The latter turns out to be a particular case of the empirical Bernstein copula, the degrees of all Bernstein polynomials being equal to the sample size. Necessary and sufficient conditions ar…
We provide finite sample properties of sparse multivariate ARCH processes, where the linear representation of ARCH models allows for an ordinary least squares estimation. Under the restricted strong convexity of the unpenalized loss function, regularity conditions on the penalty function, strict stationary and beta-mixing process, we prove non-asymptotic error bounds on the regularized ARCH estimator. Based on the primal-dual witness method of Loh and Wainwright (2017), we establish variable selection consistency, including the case when the penalty function is non-convex. These theoretical re…
A Local Orthogonal Polynomial Expansion (LOrPE) of the empirical density function is proposed as a novel method to estimate the underlying density. The estimate is constructed by matching localized expectation values of orthogonal polynomials to the values observed in the sample. LOrPE is related to several existing methods, and generalizes straightforwardly to multivariate settings. By manner of construction, it is similar to Local Likelihood Density Estimation (LLDE). In the limit of small bandwidths, LOrPE functions as Kernel Density Estimation (KDE) with high-order (effective) kernels inhe…
The Dirichlet distribution, also known as multivariate beta, is the most used to analyse frequencies or proportions data. Maximum likelihood is widespread for estimation of Dirichlet's parameters. However, for small sample sizes, the maximum likelihood estimator may shows a significant bias. In this paper, Dirchlet's parameters estimation is obtained through modified score functions aiming at mean and median bias reduction of the maximum likelihood estimator, respectively. A simulation study and an application compare the adjusted score approaches with maximum likelihood.
Two key ingredients to carry out inference on the copula of multivariate observations are the empirical copula process and an appropriate resampling scheme for the latter. Among the existing techniques used for i.i.d. observations, the multiplier bootstrap of Rémillard and Scaillet (J. Multivariate Anal. 100 (2009) 377-386) frequently appears to lead to inference procedures with the best finite-sample properties. Bücher and Ruppert (J. Multivariate Anal. 116 (2013) 208-229) recently proposed an extension of this technique to strictly stationary strongly mixing observations by adapting the depe…
In this work, we investigate the statistical computation of the Boltzmann entropy of statistical samples. For this purpose, we use both histogram and kernel function to estimate the probability density function of statistical samples. We find that, due to coarse-graining, the entropy is a monotonic increasing function of the bin width for histogram or bandwidth for kernel estimation, which seems to be difficult to select an optimal bin width/bandwidth for computing the entropy. Fortunately, we notice that there exists a minimum of the first derivative of entropy for both histogram and kernel e…
In this paper, the exact distribution of the largest eigenvalue of a singular random matrix for multivariate analysis of variance (MANOVA) is discussed. The key to developing the distribution theory of eigenvalues of a singular random matrix is to use heterogeneous hypergeometric functions with two matrix arguments. In this study, we define the singular beta F-matrix and extend the distributions of a nonsingular beta F -matrix to the singular case. We also give the joint density of eigenvalues and the exact distribution of the largest eigenvalue in terms of heterogeneous hypergeometric functio…
Stochastic volatility processes are used in multivariate time-series analysis to track time-varying patterns in covariance matrices. Uhlig extended and beta-Bartlett processes are especially convenient for analyzing high-dimensional time-series because they are conjugate with Wishart likelihoods. In this article, we show that Uhlig extended and beta-Bartlett are closely related, but not equivalent: their hyperparameters can be matched so that they have the same forward-filtered posteriors and one-step ahead forecasts, but different joint (smoothed) posterior distributions. Under this circumsta…
We study theoretically, for the first time, the Dirichlet kernel estimator introduced by Aitchison and Lauder (1985) for the estimation of multivariate densities supported on the $d$-dimensional simplex. The simplex is an important case as it is the natural domain of compositional data and has been neglected in the literature on asymmetric kernels. The Dirichlet kernel estimator, which generalizes the (non-modified) unidimensional Beta kernel estimator from Chen (1999), is free of boundary bias and non-negative everywhere on the simplex. We show that it achieves the optimal convergence rate $O…
Conditional density estimation generalizes regression by modeling a full density f(yjx) rather than only the expected value E(yjx). This is important for many tasks, including handling multi-modality and generating prediction intervals. Though fundamental and widely applicable, nonparametric conditional density estimators have received relatively little attention from statisticians and little or none from the machine learning community. None of that work has been applied to greater than bivariate data, presumably due to the computational difficulty of data-driven bandwidth selection. We descri…
Estimating spot covariance is an important issue to study, especially with the increasing availability of high-frequency financial data. We study the estimation of spot covariance using a kernel method for high-frequency data. In particular, we consider first the kernel weighted version of realized covariance estimator for the price process governed by a continuous multivariate semimartingale. Next, we extend it to the threshold kernel estimator of the spot covariances when the underlying price process is a discontinuous multivariate semimartingale with finite activity jumps. We derive the asy…
In this work, we study a random orthogonal projection based least squares estimator for the stable solution of a multivariate nonparametric regression (MNPR) problem. More precisely, given an integer $d\geq 1$ corresponding to the dimension of the MNPR problem, a positive integer $N\geq 1$ and a real parameter $α\geq -\frac{1}{2},$ we show that a fairly large class of $d-$variate regression functions are well and stably approximated by its random projection over the orthonormal set of tensor product $d-$variate Jacobi polynomials with parameters $(α,α).$ The associated uni-variate Jacobi polyn…
This paper considers the asymptotic behavior in $β$-Hölder spaces, and under $L^p$ losses, of a Dirichlet kernel density estimator proposed by Aitchison and Lauder (1985) for the analysis of compositional data. In recent work, Ouimet and Tolosana-Delgado (2022) established the uniform strong consistency and asymptotic normality of this estimator. As a complement, it is shown here that the Aitchison-Lauder estimator can achieve the minimax rate asymptotically for a suitable choice of bandwidth whenever $(p,β) \in [1, 3) \times (0, 2]$ or $(p, β) \in \mathcal{A}_d$, where $\mathcal{A}_d$ is a sp…
In this paper, we consider the problem of estimating a conditional density in moderately large dimensions. Much more informative than regression functions, conditional densities are of main interest in recent methods, particularly in the Bayesian framework (studying the posterior distribution, finding its modes...). Considering a recently studied family of kernel estimators, we select a pointwise multivariate bandwidth by revisiting the greedy algorithm Rodeo (Regularisation Of Derivative Expectation Operator). The method addresses several issues: being greedy and computationally efficient by …
Laboratoire de Physique de l’Ecole Normale Supérieure（巴黎高等师范学院物理实验室）；Université PSL（巴黎文理研究大学）；CNRS（法国国家科学研究中心）；Sorbonne Université（索邦大学）；Université Paris-Diderot（巴黎第七大学）；Sorbonne Paris Cité（索邦巴黎西岱大学）；Bocconi University（博科尼大学）
In nonparametric regression problems involving multiple predictors, there is typically interest in estimating an anisotropic multivariate regression surface in the important predictors while discarding the unimportant ones. Our focus is on defining a Bayesian procedure that leads to the minimax optimal rate of posterior contraction (up to a log factor) adapting to the unknown dimension and anisotropic smoothness of the true surface. We propose such an approach based on a Gaussian process prior with dimension-specific scalings, which are assigned carefully-chosen hyperpriors. We additionally sh…
This study examines a nonparametric inference on a stationary Lévy-driven Ornstein-Uhlenbeck (OU) process $X = (X_{t})_{t \geq 0}$ with a compound Poisson subordinator. We propose a new spectral estimator for the Lévy measure of the Lévy-driven OU process $X$ under macroscopic observations. We also derive, for the estimator, multivariate central limit theorems over a finite number of design points, and high-dimensional central limit theorems in the case wherein the number of design points increases with an increase in the sample size. Built on these asymptotic results, we develop methods to co…
Estimation of the covariance matrix for high-dimensional multivariate datasets is a challenging and important problem in modern statistics. In this paper, we focus on high-dimensional Gaussian DAG models where sparsity is induced on the Cholesky factor L of the inverse covariance matrix. In recent work, ([Cao, Khare, and Ghosh, 2019]), we established high-dimensional sparsity selection consistency for a hierarchical Bayesian DAG model, where an Erdos-Renyi prior is placed on the sparsity pattern in the Cholesky factor L, and a DAG-Wishart prior is placed on the resulting non-zero Cholesky entr…
Associated kernels have been introduced to improve the classical continuous kernels for smoothing any functional on several kinds of supports such as bounded continuous and discrete sets. This work deals with the effects of combined associated kernels on nonparametric multiple regression functions. Using the Nadaraya-Watson estimator with optimal bandwidth matrices selected by cross-validation procedure, different behaviours of multiple regression estimations are pointed out according the type of multivariate associated kernels with correlation or not. Through simulation studies, there are no …
In this paper, we establish uniform-in-bandwidth limit laws of the logarithm for nonparametric Inverse Probability of Censoring Weighted (I.P.C.W.) estimators of the multivariate regression function under random censorship. A similar result is deduced for estimators of the conditional distribution function. The uniform-in-bandwidth consistency for estimators of the conditional density and the conditional hazard rate functions are also derived from our main result. Moreover, the logarithm laws we establish are shown to yield almost sure simultaneous asymptotic confidence bands for the functions…
We study location-scale mixture priors for nonparametric statistical problems, including multivariate regression, density estimation and classification. We show that a rate-adaptive procedure can be obtained if the prior is properly constructed. In particular, we show that adaptation is achieved if a kernel mixture prior on a regression function is constructed using a Gaussian kernel, an inverse gamma bandwidth, and Gaussian mixing weights.
In his 1986 book, Aitchison explains that compositional data is regularly mishandled in statistical analyses, a pattern that continues to this day. The Dirichlet Type I distribution is a multivariate distribution commonly used to model a set of proportions that sum to one. Aitchinson goes on to lament the difficulties of Dirichlet modelling and the scarcity of alternatives. While he addresses the second of these issues, we address the first. The Dirichlet Type II distribution is a transformation of the Dirichlet Type I distribution and is a multivariate distribution on the positive real number…
We study the problem of linear and convex aggregation of $M$ estimators of a density with respect to the mean squared risk. We provide procedures for linear and convex aggregation and we prove oracle inequalities for their risks. We also obtain lower bounds showing that these procedures are rate optimal in a minimax sense. As an example, we apply general results to aggregation of multivariate kernel density estimators with different bandwidths. We show that linear and convex aggregates mimic the kernel oracles in asymptotically exact sense for a large class of kernels including Gaussian, Silve…
