Generalized Bayes minimax estimation of the normal mean matrix with unknown covariance matrix

This paper addresses the problem of estimating the normal mean matrix in the case of unknown covariance matrix. This problem is solved by considering generalized Bayesian hierarchical models. The resulting generalized Bayes estimators with respect to an invariant quadratic loss function are shown to be matricial shrinkage equivariant estimators and the conditions for their minimaxity are given.     

A class of multiple shrinkage estimators

Based on a sample of size n, we investigate a class of estimators of the mean of a p-variate normal distribution with independent components having unknown covariance. This class includes the James-Stein estimator and Lindley's estimator as special cases and was proposed by Stein. The mean squares error improves on that of the sample mean for p3. Simple approximations imations for this improvement are given for large n or p. Lindley's estimator improves on that of James and Stein if either n is large, and the coefficient of variation of is less than a certain increasing function of p, or if p is large. An adaptive estimator is given which for large samples always performs at least as well as these two estimators.     

A Berry-Esseen bound for least squares error variance estimators of regression parameters

We construct a precise Berry-Esseen bound for the least squares error variance estimators of regression parameters. Our bound depends explicitly on the sequence of design variables and is of the order O(N ^−1/2) if this sequence is “regular” enough. Supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 1–8, January–March, 1999.     

非平衡随机效应模型中方差分量的经验Bayes估计

对非平衡单向分类随机效应模型中方差分量找到了其最小充分统计量,在加权平方损失下导出了其Bayes估计,利用多元密度及其偏导数的核估计方法构造了方差分量的经验Bayes(EB)估计,并导出了其收敛速度.文末用例子说明了符合定理条件的先验分布是存在的.     

Stein-type improvements of confidence intervals for the generalized variance

Based on independent random matices X: p×m and S: p×p distributed, respectively, as N _pm (, I _m ) and W _p (n, ) with unknown and np, the problem of obtaining confidence interval for || is considered. Stein's idea of improving the best affine equivariant point estimator of || has been adapted to the interval estimation problem. It is shown that an interval estimator of the form |S|(b ^–1, a ^–1) can be improved by min{|S|, c|S +XX'|}(b ^–1, a ^–1) for a certain constant c depending on (a, b).     

Inequalities and monotonicity of ratios for generalized hypergeometric function

We find two-sided inequalities for the generalized hypergeometric function of the form _q+1F_q(−x) with positive parameters restricted by certain additional conditions. Both lower and upper bounds agree with the value of _q+1F_q(−x) at the endpoints of positive semi-axis and are asymptotically precise at one of the endpoints. The inequalities are derived from a theorem asserting the monotony of the quotient of two generalized hypergeometric functions with shifted parameters. The proofs hinge on a generalized Stieltjes representation of the generalized hypergeometric function. This representation also provides yet another method to deduce the second Thomae relation for ₃F₂(1) and leads to an integral representations of ₄F₃(x) in terms of the Appell function F₃. In the last section of the paper we list some open questions and conjectures.     

A class of minimax estimators of a normal quantile

The estimation problem of the quantiles of a normal distribution with both parameters unknown, is considered. We construct a class of minimax procedures each of which improves upon the traditional (best equivariant) estimator of a quantile different from the median. For this purpose a differential inequality and a family of its solutions is found.     

Bayes estimators whose range has convex hull of zero prior probability

It is shown that in a general statistical decision problem Bayes procedures under convex loss can possess the following property: The induced prior measure of the convex hull of the range of the corresponding Bayes rule is equal to zero. A stronger statement holds in the case of binomial distributions.     

Empirical Bayes Test for the Parameter of the Truncated-Type Distribution Families

The empirical Bayes test (EBT) is proposed for testing H₀ : ₀ H₁ : > ₀ in the truncated-type distribution families. It is found that the EBT proposed is obtained asymptotically optimal and its convergence rate is also obtained.AMS Subject Classification (2000) 62J10, 62G99     

Tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance

In this paper, the tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance are investigated. Based on the LaSalle’s invariant set theorem, a robust adaptive controller is contrived to acquire tracking control and generalized projective synchronization and parameter identification simultaneously. It is proved theoretically that the proposed scheme can allow us to drive the hyperchaotic system to any desired reference signals, including hyperchaotic signals, chaotic signals, periodic orbits or fixed value by the given scaling factor. The presented simulation results further demonstrate that the proposed method is effective and robust.     

基于屏蔽数据的部件可靠性指标的贝叶斯估计

基于屏蔽数据,在单元的屏蔽与失效不独立的情形下,研究系统中Burr XII部件可靠性指标的估计问题。在三种不同的损失函数下,推算出部件寿命参数及可靠性指标的贝叶斯估计。最后利用随机模拟对两种估计方法进行比较分析,并讨论屏蔽概率和屏蔽水平对估计结果的影响。     

广义Pareto分布参数的经验Bayes估计的收敛速度

在加权平方损失函数下,获得广义Pareto分布形状参数的经验Bayes(EB)估计,并得到了该估计的收敛速度.     

熵损失函数下两参数指数威布尔分布尺度参数的Bayes估计

本文给定一截尾样本,在熵损失函数下,研究了两参数指数威布尔分布尺度参数在先验伽玛分布下的Bayes估计,并给出了该参数的Bayes区间估计。     

An extended class of orthogonal polynomials defined by a Sturm-Liouville problem

We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as X₁-Jacobi and X₁-Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval [−1,1] or the half-line [0,∞), respectively, and they are a basis of the corresponding L² Hilbert spaces. Moreover, we prove a converse statement similar to Bochner's theorem for the classical orthogonal polynomial systems: if a self-adjoint second-order operator has a complete set of polynomial eigenfunctions , then it must be either the X₁-Jacobi or the X₁-Laguerre Sturm-Liouville problem. A Rodrigues-type formula can be derived for both of the X₁ polynomial sequences.     

New class of generating functions associated with generalized hypergeometric polynomials

In this paper authors prove a general theorem on generating relations for a certain sequence of functions. Many formulas involving the families of generating functions for generalized hypergeometric polynomials are shown here to be special cases of a general class of generating functions involving generalized hypergeometric polynomials and multiple hypergeometric series of several variables. It is then shown how the main result can be applied to derive a large number of generating functions involving hypergeometric functions of Kampé de Fériet, Srivastava, Srivastava-Daoust, Chaundy, Fasenmyer, Cohen, Pasternack, Khandekar, Rainville and other multiple Gaussian hypergeometric polynomials scattered in the literature of special functions.     

截断型分布族参数经验Bayes检验的收敛速度

§ 1.　Introduction　　SupposethatrandomvariableXhaspdf(forLebesguemeasure)f(x｜θ) =u(x)m(θ)I(θ ,b) (x) ,(1 )whereθ>a≥ 0 ,θisthetruncationparameterofourinterestandb≤+∞ ,u(x)ispositiveLebesgueintegrablefunctionon (θ,b) ,m(θ) =[∫bθ(u(x)dx] - 1 .ThehypothesistobetestedisH0 ;θ≤θ0 H1 :θ >θ0 , (2 )whereθ0 isaknownconstant.LetlossfunctionisL(θ ,a0 ) =b0 max(θ -θ0 ,0 )foracceptingH0 andL(θ,a1 ) =b0 max(θ0 -θ,0 )foracceptingH1 ,whereb0 isapositiverealnumber,D ={a0 ,a1 }isth…     

绝对值损失下单边截断型分布族参数的经验Bayes估计

本文在绝对值损失下,构造了单边截断型分布族参数的EB估计,并证明了在一组条件下,其Bayes风险的收敛速度为0((ln n/n)~(λγ/(2r+))·M_n),其中0<λ,γ≤1,M_n≤ln ln n(n充分大),M_n为一无穷大量。     

绝对损失下双边截断型分布族参数的渐近最优经验Bayes估计

本文在绝对损失下构造了双边截断型分布族参数的经验Bayes估计,并在合适的条件下证明了该估计的渐近最优性.最后,给出两个有关本文主要结果的例子.