Mathematics

In this paper relations among some kinds of cumulative entropies and moments of order statistics are presented. By using some characterizations and the symmetry of a non negative and absolutely continuous random variable X, lower and upper bounds for entropies are obtained and examples are given.

We study a likelihood ratio test for detecting multiple weak changes in the mean of a class of CHARN models. The locally asymptotically normal (LAN) structure of the family of likelihoods under study is established. It results that the test is asymptotically optimal and an explicit form of its asymptotic local power is given as a function of candidates change locations. Weak changes locations estimates are obtained as the time indexes maximizing an estimate of the local power. A simulation study shows the good performance of our methods compared to some CUSUM approaches. Our results are also applied to three sets of real data.

In this article, a new natural discrete analog of the one parameter polynomial exponential(OPPE) distribution as a mixture of a number of negative binomial distributions has been proposed and is called as a natural discrete one parameter polynomial exponential (NDOPPE) distribution. This distribution is a generalized version of natural discrete Lindley (NDL) distribution, proposed and studied by Ahmed and Afify (2019). Two estimators viz., MLE and UMVUE of the PMF and the CDF of a NDOPPE distribution have been derived. The estimators have been compared with respect to their MSEs. Simulation study has been conducted to verify the consistency of the estimators. A real data illustration has been reported.

Recently Qiu et al. (2017) have introduced residual extropy as measure of uncertainty in residual lifetime distributions analogues to residual entropy (1996). Also, they obtained some properties and applications of that. In this paper, we study the extropy to measure the uncertainty in a past lifetime distribution. This measure of uncertainty is called past extropy. Also it is showed a characterization result about the past extropy of largest order statistics.

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?�p ) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. The measurements are correlated both time-wise and between rows. We show that the r largest eigenvalues of the wavelet random matrices, when appropriately rescaled, converge to scale invariant functions in the high-dimensional limit. By contrast, the remaining p?�r eigenvalues remain bounded. Under additional assumptions, we show that, up to a log transformation, the r largest eigenvalues of wavelet random matrices exhibit asymptotically Gaussian distributions. The results have direct consequences for statistical inference.

In this paper a new family of minimum divergence estimators based on the Bregman divergence is proposed. The popular density power divergence (DPD) class of estimators is a sub-class of Bregman divergences. We propose and study a new sub-class of Bregman divergences called the exponentially weighted divergence (EWD). Like the minimum DPD estimator, the minimum EWD estimator is recognised as an M-estimator. This characterisation is useful while discussing the asymptotic behaviour as well as the robustness properties of this class of estimators. Performances of the two classes are compared -- both through simulations as well as through real life examples. We develop an estimation process not only for independent and homogeneous data, but also for non-homogeneous data. General tests of parametric hypotheses based on the Bregman divergences are also considered. We establish the asymptotic null distribution of our proposed test statistic and explore its behaviour when applied to real data. The inference procedures generated by the new EWD divergence appear to be competitive or better that than the DPD based procedures.

This paper develops recurrence relations for integrals that relate the density of multivariate extended skew-normal (ESN) distribution, including the well-known skew-normal (SN) distribution introduced by Azzalini and Dalla-Valle (1996) and the popular multivariate normal distribution. These recursions offer a fast computation of arbitrary order product moments of the multivariate truncated extended skew-normal and multivariate folded extended skew-normal distributions with the product moments as a byproduct. In addition to the recurrence approach, we realized that any arbitrary moment of the truncated multivariate extended skew-normal distribution can be computed using a corresponding moment of a truncated multivariate normal distribution, pointing the way to a faster algorithm since a less number of integrals is required for its computation which result much simpler to evaluate. Since there are several methods available to calculate the first two moments of a multivariate truncated normal distribution, we propose an optimized method that offers a better performance in terms of time and accuracy, in addition to consider extreme cases in which other methods fail. The R MomTrunc package provides these new efficient methods for practitioners.

The problem of generating random samples of high-dimensional posterior distributions is considered. The main results consist of non-asymptotic computational guarantees for Langevin-type MCMC algorithms which scale polynomially in key quantities such as the dimension of the model, the desired precision level, and the number of available statistical measurements. As a direct consequence, it is shown that posterior mean vectors as well as optimisation based maximum a posteriori (MAP) estimates are computable in polynomial time, with high probability under the distribution of the data. These results are complemented by statistical guarantees for recovery of the ground truth parameter generating the data. Our results are derived in a general high-dimensional non-linear regression setting (with Gaussian process priors) where posterior measures are not necessarily log-concave, employing a set of local `geometric' assumptions on the parameter space, and assuming that a good initialiser of the algorithm is available. The theory is applied to a representative non-linear example from PDEs involving a steady-state Schrödinger equation.

Several regularization methods have been considered over the last decade for sparse high-dimensional linear regression models, but the most common ones use the least square (quadratic) or likelihood loss and hence are not robust against data contamination. Some authors have overcome the problem of non-robustness by considering suitable loss function based on divergence measures (e.g., density power divergence, gamma-divergence, etc.) instead of the quadratic loss. In this paper we shall consider a loss function based on the Rényi's pseudodistance jointly with non-concave penalties in order to simultaneously perform variable selection and get robust estimators of the parameters in a high-dimensional linear regression model of non-polynomial dimensionality. The desired oracle properties of our proposed method are derived theoretically and its usefulness is illustustrated numerically through simulations and real data examples.

We consider the problem of estimating the mean vector θ of a d -dimensional spherically symmetric distributed X based on balanced loss functions of the forms: {\bf (i)} $\omega \rho(\|\de-\de_{0}\|^{2}) +(1-\omega)\rho(\|\de - \theta\|^{2})$ and {\bf (ii)} $\ell\left(\omega \|\de - \de_{0}\|^{2} +(1-\omega)\|\de - \theta\|^{2}\right)$, where δ 0 is a target estimator, and where ? and ??are increasing and concave functions. For d?? and the target estimator δ 0 (X)=X , we provide Baranchik-type estimators that dominate δ 0 (X)=X and are minimax. The findings represent extensions of those of Marchand \& Strawderman (\cite{ms2020}) in two directions: {\bf (a)} from scale mixture of normals to the spherical class of distributions with Lebesgue densities and {\bf (b)} from completely monotone to concave ? ??and ????.