Huber's Minimax Approach in Distribution Classes with Bounded Variances and Subranges with Applications to Robust Detection of Signals

A brief survey of former and recent results on Huber's minimax approach in robust statistics is given. The least mformative distributions minimizing Fisher information for location over several distribution classes with upper-bounded variances and subranges are written down. These least informative distributions are qualitatively different from classical Huber's solution and have the following common structure: (i) with relatively small variances they are short-tailed, in particular normal; (ii) with relatively large variances they are heavytailed, in particular the Laplace; (iii) they are compromise with relatively moderate variances. These results allow to raise the efficiency of minimax robust procedures retaining high stability as compared to classical Huber's procedure for contaminated normal populations. In application to signal detection problems, the proposed minimax detection rule has proved to be robust and close to Huber's for heavy-tailed distributions and more efficient than Huber's for short-tailed ones both in asymptotics and on finite samples.     

Huber＇s Minimax Approach in Distribution Classes with Bounded Variances and Subranges with Applications to Robust Detection of Signals

A brief survey of former and recent results on Huber‘s minimax approach in robust statistics is given. The least informative distributions minimizing Fisher information for location over several distribution classes with upper-bounded variances and subranges are written down. These least informative distributions are qualitatively different from classical Huber‘s solution and have the following common structure: (i) with relatively small variances they are short-tailed, in particular normal；(ii) with relatively large variances they are heavytailed, in particular the Laplace; (iii) they are compromise with relatively moderate variances. These results allow to raise the efficiency of minimax robust procedures retaining high stability as compared to classical Huber‘s procedure for contaminated normal populations. In application to signal detection problems, the proposed minimax detection rule has proved to be robust and close to Huber‘s for heavy-tailed distributions and more efficient than Huber‘s for short-tailed ones both in asymptotics and on finite samples。     

Discrete analogs of Fisher information and least favorable distributions in robust estimation of location parameters

The analog of Fisher information for discrete lattice distributions is introduced. The variational problems of minimization of Fisher information are solved for the discrete analogs of the classes of finite distributions, nonsingular densities and ε-contaminated distributions. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 14–18, Perm, 1991.     

Bayes minimax estimators of the mean of a scale mixture of multivariate normal distributions

Bayes estimation of the mean of a variance mixture of multivariate normal distributions is considered under sum of squared errors loss. We find broad class of priors (also in the variance mixture of normal class) which result in proper and generalized Bayes minimax estimators. This paper extends the results of Strawderman [Minimax estimation of location parameters for certain spherically symmetric distribution, J. Multivariate Anal. 4 (1974) 255-264] in a manner similar to that of Maruyama [Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distribution, J. Multivariate Anal. 21 (2003) 69-78] but somewhat more in the spirit of Fourdrinier et al. [On the construction of bayes minimax estimators, Ann. Statist. 26 (1998) 660-671] for the normal case, in the sense that we construct classes of priors giving rise to minimaxity. A feature of this paper is that in certain cases we are able to construct proper Bayes minimax estimators satisfying the properties and bounds in Strawderman [Minimax estimation of location parameters for certain spherically symmetric distribution, J. Multivariate Anal. 4 (1974) 255-264]. We also give some insight into why Strawderman's results do or do not seem to apply in certain cases. In cases where it does not apply, we give minimax estimators based on Berger's [Minimax estimation of location vectors for a wide class of densities, Ann. Statist. 3 (1975) 1318-1328] results. A main condition for minimaxity is that the mixing distributions of the sampling distribution and the prior distribution satisfy a monotone likelihood ratio property with respect to a scale parameter.     

Minimax estimators for location vectors in elliptical distributions with unknown scale parameter and its application to variance reduction in simulation

In this paper, we give an ever wider and new class of minimax estimators for the location vector of an elliptical distribution (a scale mixture of normal densities) with an unknown scale parameter. The its application to variance reduction for Monte Carlo simulation when control variates are used is considered. The results obtained thus extend (i) Berger's result concerning minimax estimation of location vectors for scale mixtures of normal densities with known scale parameter and (ii) Strawderman's result on the estimation of the normal mean with common unknown variance.Research partially supported by National Science Foundation, Grant #DMS 8901922.     

Structure of Stieltjes classes of moment-equivalent probability laws

A Stieltjes class is a one-parameter family of moment-equivalent distribution functions constructed by modulation of a given indeterminate distribution function F, called the center of the class. Members of a Stieltjes class are mutually absolutely continuous, and conversely, any pair of moment-equivalent and mutually absolutely continuous distribution functions can be joined by a Stieltjes class. The center of a Stieltjes class is an equally weighted mixture of its extreme members, and this places restrictions on which distributions can belong to a Stieltjes class with a given center. The lognormal law provides interesting illustrations of the general ideas. In particular, it is possible for two moment equivalent infinitely divisible distributions to be joined by a Stieltjes class, and random scaling can be used to construct new Stieltjes classes from a given Stieltjes class.     

On the Parameters Estimators for a Discrete Analog of the Generalized Exponential Distribution

Nekoukhou et. al (Commun. Statist. Th. Meth., 2012) introduced a two-parameters discrete probability distribution so-called Discrete Analog of the Generalized Exponential Distribution (in short, DGED). We shall attempt to derive conditions under which a solution for the system of likelihood equations exists and coincides with the maximum likelihood (ML) estimators of the DGED. This kind of ML estimators are coincided with some moment estimators. An approximate computation based on Fisher’s accumulation method is presented in order for the ML estimations of the unknown parameters. Simulation study is also illustrated. Meanwhile, in the sequel two special cases of the DGED are considered. Some statistical properties for such special cases of the DGED are provided. We also propose a linear regression-type model for estimation of the parameter. Finally, we fit the DGED to a real data set and compare it with two other discrete distributions.     

Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation

Structure-preserving numerical schemes for a nonlinear parabolic fourth-order equation, modeling the electron transport in quantum semiconductors, with periodic boundary conditions are analyzed. First, a two-step backward differentiation formula (BDF) semi-discretization in time is investigated. The scheme preserves the nonnegativity of the solution, is entropy stable and dissipates a modified entropy functional. The existence of a weak semi-discrete solution and, in a particular case, its temporal second-order convergence to the continuous solution is proved. The proofs employ an algebraic relation which implies the G-stability of the two-step BDF. Second, an implicit Euler and $q$ -step BDF discrete variational derivative method are considered. This scheme, which exploits the variational structure of the equation, dissipates the discrete Fisher information (or energy). Numerical experiments show that the discrete (relative) entropies and Fisher information decay even exponentially fast to zero.     

Estimation of the ratio of the scale parameters of two exponential distributions with unknown location parameters

We consider the estimation of the ratio of the scale parameters of two independent two-parameter exponential distributions with unknown location parameters. It is shown that the best affine equivariant estimator (BAEE) is inadmissible under any loss function from a large class of bowl-shaped loss functions. Two new classes of improved estimators are obtained. Some values of the risk functions of the BAEE and two improved estimators are evaluated for two particular loss functions. Our results are parallel to those of Zidek (1973, Ann. Statist., 1, 264–278), who derived a class of estimators that dominate the BAEE of the scale parameter of a two-parameter exponential distribution.     

Estimation for discretely observed continuous state branching processes with immigration

We study the problem of parameter estimation for the continuous state branching processes with immigration, observed at discrete time points. The weighted conditional least square estimators (WCLSEs) are used for the drift parameters. Under the proper moment conditions, asymptotic distributions of the WCLSEs are obtained in the supercritical, sub- or critical cases.     

On the geometry of generalized Gaussian distributions

In this paper we consider the space of those probability distributions which maximize the q-Rényi entropy. These distributions have the same parameter space for every q, and in the q=1 case these are the normal distributions. Some methods to endow this parameter space with a Riemannian metric is presented: the second derivative of the q-Rényi entropy, the Tsallis entropy, and the relative entropy give rise to a Riemannian metric, the Fisher information matrix is a natural Riemannian metric, and there are some geometrically motivated metrics which were studied by Siegel, Calvo and Oller, Lovri?, Min-Oo and Ruh. These metrics are different; therefore, our differential geometrical calculations are based on a new metric with parameters, which covers all the above-mentioned metrics for special values of the parameters, among others. We also compute the geometrical properties of this metric, the equation of the geodesic line with some special solutions, the Riemann and Ricci curvature tensors, and the scalar curvature. Using the correspondence between the volume of the geodesic ball and the scalar curvature we show how the parameter q modulates the statistical distinguishability of close points. We show that some frequently used metrics in quantum information geometry can be easily recovered from classical metrics.     

一类由概率分布生成的新度量

主要得到了一类由概率分布生成的新度量.以信息理论中的重要概念-相对熵为基础,对前人文章中的重要结论进行推广,通过利用改进的初等方法在离散的可测空间中得到了这类新度量,并且证明了得到的新度量成立的充要条件.由此再将这类新度量推广到连续的可测空间中,得到了同样的结果.最后讨论了新度量的最值问题.     

Parametric Borwein-Preiss variational principle and applications

A parametric version of the Borwein-Preiss smooth variational principle is presented, which states that under suitable assumptions on a given convex function depending on a parameter, the minimum point of a smooth convex perturbation of it depends continuously on the parameter. Some applications are given: existence of a Nash equilibrium and a solution of a variational inequality for a system of partially convex functions, perturbed by arbitrarily small smooth convex perturbations when one of the functions has a non-compact domain; a parametric version of the Kuhn-Tucker theorem which contains a parametric smooth variational principle with constraints; existence of a continuous selection of a subdifferential mapping depending on a parameter.

The tool for proving this parametric smooth variational principle is a useful lemma about continuous -minimizers of quasi-convex functions depending on a parameter, which has independent interest since it allows direct proofs of Ky Fan's minimax inequality, minimax equalities for quasi-convex functions, Sion's minimax theorem, etc.

     

Estimation of a Location Parameter with Restrictions or “vague information” for Spherically Symmetric Distributions

In this article we consider estimating a location parameter of a spherically symmetric distribution under restrictions on the parameter. First we consider a general theory for estimation on polyhedral cones which includes examples such as ordered parameters and general linear inequality restrictions. Next, we extend the theory to cones with piecewise smooth boundaries. Finally we consider shrinkage toward a closed convex set K where one has vague prior information that θ is in K but where θ is not restricted to be in K. In this latter case we give estimators which improve on the usual unbiased estimator while in the restricted parameter case we give estimators which improve on the projection onto the cone of the unbiased estimator. The class of estimators is somewhat non-standard as the nature of the constraint set may preclude weakly differentiable shrinkage functions. The technique of proof is novel in the sense that we first deduce the improvement results for the normal location problem and then extend them to the general spherically symmetric case by combining arguments about uniform distributions on the spheres, conditioning and completeness.     

Optimal rates of convergence for estimating Toeplitz covariance matrices

Toeplitz covariance matrices are used in the analysis of stationary stochastic processes and a wide range of applications including radar imaging, target detection, speech recognition, and communications systems. In this paper, we consider optimal estimation of large Toeplitz covariance matrices and establish the minimax rate of convergence for two commonly used parameter spaces under the spectral norm. The properties of the tapering and banding estimators are studied in detail and are used to obtain the minimax upper bound. The results also reveal a fundamental difference between the tapering and banding estimators over certain parameter spaces. The minimax lower bound is derived through a novel construction of a more informative experiment for which the minimax lower bound is obtained through an equivalent Gaussian scale model and through a careful selection of a finite collection of least favorable parameters. In addition, optimal rate of convergence for estimating the inverse of a Toeplitz covariance matrix is also established.     

Loss Estimation for Spherically Symmetrical Distributions

In this paper we consider the problem of estimating the quadratic loss of point estimators of a location parameter for a family of spherically symmetric distributions. We compare the unbiased loss estimator of the minimax estimator with a new shrinkage type loss estimator. Conditions on the distributions for the domination of competing estimators are given. It is shown that, in addition to the class of scale mixtures of normal distributions, there exists a more general family for which the domination results hold.     

Minimax estimation of a variance

The nonparametric problem of estimating a variance based on a sample of sizen from a univariate distribution which has a known bounded range but is otherwise arbitrary is treated. For squared error loss, a certain linear function of the sample variance is seen to be minimax for eachn from 2 through 13, exceptn=4. For squared error loss weighted by the reciprocal of the variance, a constant multiple of the sample variance is minimax for eachn from 2 through 11. The least favorable distribution for these cases gives probability one to the Bernoulli distributions.     

Multivalued Parametric Variational Inequalities with α-Pseudomonotone Maps

In this paper, we consider variational inequalities with pseudomonotone maps which depend on a parameter and study the behavior of their solutions. The main result gives sufficient conditions for the stability of the initial variational inequality problem under small perturbations of the parameter. As an application, we obtain a stability result for a class of parametric optimization problems.     

Robust regression function estimation

A robust estimator of the regression function is proposed combining kernel methods as introduced for density estimation and robust location estimation techniques. Weak and strong consistency and asymptotic normality are shown under mild conditions on the kernel sequence. The asymptotic variance is a product from a factor depending only on the kernel and a factor similar to the asymptotic variance in robust estimation of location. The estimation is minimax robust in the sense of Huber (1964). Robust estimation of a location parameter. Ann. Math. Statist.33 73–101.     

Minimax estimates of a linear parameter function in a regression model under restrictions on the parameters and variance-covariance matrix

Minimax nonhomogeneous linear estimators of scalar linear parameter functions are studied in the paper under restrictions on the parameters and variance-covariance matrix. The variance-covariance matrix of the linear model under consideration is assumed to be unknown but from a specific set R of nonnegativedefinite matrices. It is shown under this assumption that, without any restriction on the parameters, minimax estimators correspond to the least-squares estimators of the parameter functions for the “worst” variance-covariance matrix. Then the minimax mean-square error of the estimator is derived using the Bayes approach, and finally the exact formulas are derived for the calculation of minimax estimators under elliptical restrictions on the parameter space and for two special classes of possible variance-covariance matrices R. For example, it is shown that a special choice of a constant q ₀ and a matrixW ₀ defining one of the above classes R leads to the well known Kuks—Olman admissible estimator (see [16]) with a known variance-covariance matrixW ₀. Bibliography:32 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 79–92.