Estimator of Scale Parameter in a Subclass of the Exponential Family under Symmetric Entropy Loss

In this paper we investigate the estimator for the rth power of the scale parameter in a class of exponential family under symmetric entropy loss L（θ, δ） = v（θ/δ ＋ δ/θ - 2）. An exact form of the minimum risk equivariant estimator under symmetric entropy loss is given, and the minimaxity of the minimum risk equivariant estimator is proved. The results with regard to admissibility and inadmissibility of a class of linear estimators of the form cT（X） ＋ d are given, where T（X） Gamma（v, θ）.

All Admissible Linear Estimators under Quadratic Loss in Multivariate Model

For multivariate linear model Y=XΘ+ε, ~N(0, σ 2ΣV), this paper is concerned with the admissibility of linear estimators of estimable function SXΘ in the class of all estimators. All admissible linear estimators of SXΘ are given under each of four definitions of admissibility.

ADMISSIBILITY OF LINEAR ESTIMATORS OF REGRESSION COEFFICIENTS UNDER QUADRATIC LOSS

For the general fixed effects linear model: Y = X_T+ε, ε～N(0, V), V≥0, weobtain the necessary and sufficient conditions for LY +a to be admissible for a linear estimablefunction S_r in the class of all estimators under the loss function (d -- Sr)'D(d --Sr), whereD≥0 is known. For the general random effects linear model: Y = Xβ+ε,(βε)～N((Aα 0), (V_(11)V_(12)V_(21)V_(22))), ∧= XV_(11)X'+XV_(12)+ V_(21)X+V_(22)≥0, we also get the necessaryand sufficient conditions for LY+a to be admissible for a linear estimable function Sα+Qβin the class of all estimators under the loss function (d-Sα-Qβ)'D(d-Sα-Qβ).whereD≥0 is known.

A class of admissible estimators of multiple regression coefficient with an unknown variance

Suppose that we observe y|θ,τ∼N_(p)(Xθ,τ^(−1)I_(p)),where θ is an unknown vector with unknown precisionτ.Estimating the regression coefficient θ with known τ has been well studied.However,statistical properties such as admissibility in estimating θ with unknownτare not well studied.Han[(2009).Topics in shrinkage estimation and in causal inference(PhD thesis).Warton School,University of Pennsylvania]appears to be the first to consider the problem,developing sufficient conditions for the admissibility of estimating means of multivariate normal distributions with unknown variance.We generalise the sufficient conditions for admissibility and apply these results to the normal linear regression model.2-level and 3-level hierarchical models with unknown precisionτare investigated when a standard class of hierarchical priors leads to admissible estimators of θ under the normalised squared error loss.One reason to consider this problem is the importance of admissibility in the hierarchical prior selection,and we expect that our study could be helpful in providing some reference for choosing hierarchical priors.

ADMISSIBLE ESTIMATES IN THE IMPORTANT CLASS OF ESTIMATES OF THE COVARIANCE MATRIX

Let X1,X2(where N>m)be independent Nm(μ,∑) random vectors, and putwhere T is upper- triangular with positive diagonal elements. The author considers the problemof estimating ∑,and restricts his attention to the class of estimates is any diagonal matrix and b* is any nonnegative constant} because it has the following attractive features:(a) Its elements are all quadratic forms of the sufficient and complete statistics (X, T).(b) It contains all estimates of the form(and), which constructa complete subclass of the class of nonnegative quadratic estimates (where X = (X1,…, XN)') for any strict convex loss function.(c)It contains all invariant estimates under the transformation group of upper-triangularmatrices.The author obtains the Characteristics for an estimate of the formof ∑ to be admissible in when the loss function is chosed as tr(∑-1∑-Ⅰ)2, and shows,by an example, that(and)is admissible in can not imply itsadmissibility in .

The Cauchy Problem for Coupled Nonlinear Schrdinger Equations with Linear Damping： Local and Global Existence and Blowup of Solutions

The authors study, by applying and extending the methods developed by Cazenave(2003), Dias and Figueira(2014), Dias et al.(2014), Glassey(1994–1997), Kato(1987), Ohta and Todorova(2009) and Tsutsumi(1984), the Cauchy problem for a damped coupled system of nonlinear Schrdinger equations and they obtain new results on the local and global existence of H^1-strong solutions and on their possible blowup in the supercritical case and in a special situation, in the critical or supercritical cases.