On Bayes and unbiased estimators of loss

We consider estimation of loss for generalized Bayes or pseudo-Bayes estimators of a multivariate normal mean vector, θ. In 3 and higher dimensions, the MLEX is UMVUE and minimax but is inadmissible. It is dominated by the James-Stein estimator and by many others. Johnstone (1988, On inadmissibility of some unbiased estimates of loss,Statistical Decision Theory and Related Topics, IV (eds. S. S. Gupta and J. O. Berger), Vol. 1, 361–379, Springer, New York) considered the estimation of loss for the usual estimatorX and the James-Stein estimator. He found improvements over the Stein unbiased estimator of risk. In this paper, for a generalized Bayes point estimator of θ, we compare generalized Bayes estimators to unbiased estimators of loss. We find, somewhat surprisingly, that the unbiased estimator often dominates the corresponding generalized Bayes estimator of loss for priors which give minimax estimators in the original point estimation problem. In particular, we give a class of priors for which the generalized Bayes estimator of θ is admissible and minimax but for which the unbiased estimator of loss dominates the generalized Bayes estimator of loss. We also give a general inadmissibility result for a generalized Bayes estimator of loss. Research supported by NSF Grant DMS-97-04524.     

On a class of almost hypoelliptic polynomials

The paper gives criteria for almost hypoelliptic polynomials P of two and three variables, providing that P(ξ) → ∞ as |ξ| → +∞.     

Existence conditions for the uniformly minimum risk unbiased estimators in a class of linear models

This paper studies the existence of the uniformly minimum risk unbiased (UMRU) estimators of parameters in a class of linear models with an error vector having multivariate normal distribution or t-distribution, which include the growth curve model, the extended growth curve model, the seemingly unrelated regression equations model, the variance components model, and so on. The necessary and sufficient existence conditions are established for UMRU estimators of the estimable linear functions of regression coefficients under convex losses and matrix losses, respectively. Under the (extended) growth curve model and the seemingly unrelated regression equations model with normality assumption, the conclusions given in the literature can be derived by applying the general results in this paper. For the variance components model, the necessary and sufficient existence conditions are reduced as terse forms.     

On a class of almost regular Landsberg metrics

There is a long existing "unicorn" problem in Finsler geometry: whether or not any Landsberg metric is a Berwald metric? Some classes of metrics were studied in the past and no regular non-Berwaldian Landsberg metric was found. However, if the metric is almost regular(allowed to be singular in some directions),some non-Berwaldian Landsberg metrics were found in the past years. All of them are composed by Riemannian metrics and 1-forms. This motivates us to ?nd more almost regular non-Berwaldian Landsberg metrics in the class of general(α, β)-metrics. In this paper, we ?rst classify almost regular Landsberg general(α, β)-metrics into three cases and prove that those regular metrics must be Berwald metrics. By solving some nonlinear PDEs,some new almost regular Landsberg metrics are constructed which have not been described before.     

On a class of weakly almost periodic mappings

Communicated by Karl H. Hofmann     

On the optimal design of the randomized unbiased Monte Carlo estimators

We consider a class of unbiased Monte Carlo estimators and develop an efficient algorithm to produce the distribution of the optimal random truncation level. We establish the convergence and optimality results of the associated algorithm and also derive its exact complexity. We find this algorithm has a much lower complexity as compared to the existing one in the literature. The proposed algorithm is also applicable to optimization problems arising in supply chain management, such as economic reorder interval problem.     

On moment-type estimators for a class of log-symmetric distributions

In this paper, we propose three simple closed form estimators for a class of log-symmetric distributions on \({\mathbb {R}}^{+}\). The proposed methods make use of some key properties of this class of distributions. We derive the asymptotic distributions of these estimators. The performance of the proposed estimators are then compared with those of the maximum likelihood estimators through Monte Carlo simulations. Finally, some illustrative examples are presented to illustrate the methods of estimation developed here.     

On smooth solutions of a class of almost hypoelliptic equations

For a class of non-degenerate (regular) almost hypoelliptic equations, which are partially hypoelliptic with respect to certain variables, infinitely differentiable in a specific strip solutions are extracted.     

On the class of almost L-weakly and almost M-weakly compact operators

In this paper, we introduce and study new concepts of almost L-weakly and almost M-weakly compact operators.     

On the non existence of unbiased estimators of risk for spherically symmetric distributions

We study existence of unbiased estimators of risk for estimators of the location parameter of a spherically symmetric distribution, when a residual vector is available to estimate scale, under invariant quadratic loss. We show such existence often characterizes normality.     

Optimal unbiased estimators for finite families of distributions

Necessary and sufficient conditions for the existence of a nontrivial optimal algebra are established on the basis of a statistical structure with finitely many measures. A method is proposed for constructing an optimal unbiased estimator of a given function of a parameter, if it exists.Translated from Statisticheskie Metody, pp. 156–163, 1980.     

Inadmissibility and admissibility results for unbiased loss estimators based on gauss-markov estimators

LetY be distributed according to ann-variate normal distribution with a meanX and a nonsingular covariance matrix ² V, where bothX andV are known, R ^p is a parameter, > 0 is known or unknown. Denote and . Assume thatF is linearly estimable. When is known, it is proved that the unbiased loss estimator ²tr(F(XV ^–1 X)^– F) of is admissible for rank (F)=k4 and inadmissible fork 5 with the squared error loss . When is unknown and rank (X) <n, it is established that the loss estimatorcS ², wherec is any nonnegative constant, of is inadmissible and that the unbiased loss estimator tr(F(XV ^–1 X)^– F) of is admissible fork 4, and inadmissible fork 5 with squared error loss.This project is supported by the National Natural Science Foundation of China.     

On a class of almost hypoelliptic operators in generalized Sobolev spaces

The paper considers differential operators in the generalized Sobolev spaces. Differring necessary and sufficient conditions on the polyhedron ? are found, under which the function h _? ^s satisfies the Beurling condition for s great enough. They coincide (become necessary and sufficient) for polyhedrons in ? ₊ ² . The regularity of a class of differential equations is investigated.     

On the class of weak almost limited operators

Abstract

We introduce and study the class of weak almost limited operators. We establish a characterization of pairs of Banach lattices E, F for which every positive weak almost limited operator T : E→F is almost limited (resp. almost Dunford- Pettis). As consequences, we will give some interesting results.     

Optimisation of linear unbiased intensity estimators for point processes

A general non-stationary point process whose intensity function is given up to unknown numerical factor λ is considered. As an alternative to the conventional estimator of λ based on counting the points, we consider general linear unbiased estimators of λ given by sums of weights associated with individual points. A necessary and sufficient condition for a linear, unbiased estimator for the intensity λ to have the minimum variance is determined. It is shown that there are “nearly” no other processes than Poisson and Cox for which the unweighted estimator of λ, which counts the points only, is optimal. The properties of the optimal estimator are illustrated by simulations for the Matérn cluster and the Matérn hard-core processes. This research was partially supported by Grant Agency of Czech Republic, project No. 201/03/D062.     

On integrable with a weight solutions of a class of almost hypoelliptic equations

We prove that the solutions (from the weighted space L_2,δ) of almost hypoelliptic equations belong to the Gevrey classes.