This paper addresses the problem of estimating the mean vector of a singular multivariate normal distribution with an unknown singular covariance matrix. The maximum likelihood estimator is shown to be minimax relative to a quadratic loss weighted by the Moore-Penrose inverse of the covariance matrix. An unbiased risk estimator relative to the weighted quadratic loss is provided for a Baranchik...

We consider a sequence space model of statistical linear inverse problems where we need to estimate a function f from indirect noisy observations. Let a nite set of linear estimators be given. Our aim is to mimic the estimator in that has the smallest risk on the true f. Under general conditions, we show that this can be achieved by simple minimization of unbiased risk estimator, provided the s...

Correspondence: [email protected] Institute of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, P. R China Abstract In this article, we prove the existence of solutions for the general variational inequality j(x, y) ≥ f(x) f(y) and Minty type theorem by using the generalized KKM theorem in topological spaces without linear structure. Some properties of solutions set for t...

In the estimation of a multivariate normal mean, it is shown that the problem of deriving shrinkage estimators improving on the maximum likelihood estimator can be reduced to that of solving an integral inequality. The integral inequality not only provides a more general condition than a conventional differential inequality studied in the literature, but also handles non-differentiable or disco...

Recently, variational inequality theory has been extended and applied in various directions, see [6, 7, 8, 9, 10] and the references therein. In particular, Ding [1, 2, 3], and Ding and Tan [4] studied the existence of solutions for several nonlinear variational-like inequalities in reflexive Banach spaces. In this paper, a new class of general nonlinear variational-like inequalities in reflexi...

We compute the norm of the restriction of a Schur multiplier, arising from a multiplication operator, to a coordinate subspace. This result is used to generalize Wielandt’s minimax inequality. Furthermore, we compute various s-numbers of an elementary Schur multiplier and determine criteria for membership of such multipliers in certain operator ideals.

We investigate whether beliefs about the income distribution are associated with political positions for or against redistribution. Using a novel elicitation method, we assess individuals’ beliefs about the shape of the income distribution in the United States. We find that beliefs about inequality, measured in terms of income dispersion, play only a marginal role in political positions as well...

The information inequality has been shown to be an effective tool for providing lower bounds for the minimax risk. Bounds based on the chisquare distance can sometimes offer a considerable improvement especially when applied iteratively. This paper compares these two methods in four examples including the bounded normal mean problem as well as obervations from a Poisson distribution.

This paper provides necessary and sufficient conditions for fixed-point theorems, minimax inequalities and some related theorems defined on arbitrary topological spaces that may be discrete, continuum, non-compact or non-convex. We establish a single condition, γ-recursive transfer lower semicontinuity, which fully characterizes the existence of equilibrium of minimax inequality without imposin...