Bayes empirical Bayes estimation for discrete exponential families

Bayes-empiric Bayes estimation of the parameter of certain one parameter discrete exponential families based on orthogonal polynomials on an interval (a, b) is introduced. The resulting estimator is shown to be asymptotically optimal. The application of this method to three special distributions, the binomial, Poisson and negative binomial, is discussed.The first author was supported by NSF grant DCR-8504620.     

Admissibility of goodness of fit tests for discrete exponential families

Consider testing the composite hypothesis that a population is a particular discrete exponential family. For example, a Poisson distribution with unknown parameter. We find a sufficient condition for the admissibility of goodness of fit tests. Included in the class of admissible tests is the usual chi — square test of goodness of fit.     

On some multivariate density estimates and empirical Bayes problems

This paper consider estimates of multidimensional density functions and their derivatives. The asymptotic unbiasedness and the convergence properties of these estimates are established.Some applications to empirical Bayes problems are considered.     

Monotone Empirical Bayes Test for the Exponential Distribution

We study the two-action problem in the exponential distribution via empirical Bayes (EB) approach. Based on typeⅡcensored samples, we construct an EB test rule and obtain an optimal rate of convergence which much improves the existing results in the literature.     

多参数离散指数族参数渐近最优的经验Bayes估计的收敛速度

本文构造了多参数离散指数族参数的渐近最优的经验Bayes（EB）估计,若记B_n（δ_n,G）为δn的全面Bayes风险,R_G最小Bayes风险,则在某些条件下c_1n~(-1）2成立,其中c_1,c_2为正的常数,     

一维双边截断型分布族中参数函数的经验Bayes估计及其收敛速度

对一维双边截断型分布族构造了参数函数的经验 Ｂａｙｅｓ 估计,在适当的条件下给出了相应的收敛速度,并说明此收敛速度可充分接近 １２ ．     

随机删失下刻度参数的单调经验贝叶斯检验

利用经验贝叶斯方法研究了刻度指数族的两行动问题, 提出了一个在历史样本被随机右删失的条件下收敛速度可以任意接近$O(n^{-1})$的单调经验贝叶斯检验.     

Bayes empirical Bayes estimation of a poisson mean

In the empirical Bayes (EB) decision problem consisting of squared error estimation of a Poisson mean, a prior distribution λ is placed on the gamma family of prior distributions to produce Bayes EB estimators which are admissible. A subclass of such estimators is shown to be asymptotically optimal (a.o.). The results of a Monte Carlo study are presented to demonstrate the favorable a.o. property of the Bayes EB estimators in comparison with other competitors.     

Frequency tests for discrete sequences

The paper examines some issues of practical application in hypothesis testing of statistics based on the frequencies of occurrence of certain combinations in a long realization of a sequence of random variables with a finite value set.Translated from Statisticheskie Metody, pp. 49–62, 1982.     

On empirical Bayes with sequential component

Laippala (1979, Scand. J. Statist., 6, 113–118, correction note, 7, 105; 1985, Ann. Inst. Statist. Math., 37, 315–327) has defined a concept within the empirical Bayes framework that he calls floating optimal sample size. We examine this concept and show that it is one of many possibilities resulting from restricting the class of component sampling procedures in the empirical Bayes decision problem with a sequential component. All ideas are illustrated with the finite state component.     

An empirical Bayes procedure for the credit granting decision

The use of a simple Stein-rule method is discussed for estimating the repayment probability of a new credit applicant. It is argued that this type of procedure could provide a superior algorithm when a multiperiod view of repayments is taken and when there is a desire to balance the categorised estimate from the common discriminant scoring procedure with a fairer reflection of the applicant's actual performance.     

Monotone iterative technique for initial-value problems of nonlinear singular discrete systems

This paper studies a class of initial-value problems of nonlinear singular discrete systems and obtains the existence theorem of extremal solutions by employing a monotone iterative technique combined with the method of upper and lower solutions.     

A note on Bayes empirical Bayes estimation by means of Dirichlet processes

Bayes estimators are derived by means of the Dirichlet process hyperprior approach for general empirical Bayes problems. For any sample size, these estimators are expressed concisely as ratios of two multidimensional integrals. A numerical example on Poisson sampling is given.     

Approximated Bayes and empirical Bayes confidence intervals—The known variance case

In this paper hierarchical Bayes and empirical Bayes results are used to obtain confidence intervals of the population means in the case of real problems. This is achieved by approximating the posterior distribution with a Pearson distribution. In the first example hierarchical Bayes confidence intervals for the Efron and Morris (1975, J. Amer. Statist. Assoc., 70, 311–319) baseball data are obtained. The same methods are used in the second example to obtain confidence intervals of treatment effects as well as the difference between treatment effects in an analysis of variance experiment. In the third example hierarchical Bayes intervals of treatment effects are obtained and compared with normal approximations in the unequal variance case.Financially supported by the CSIR and the University of the Orange Free State, Central Research Fund.     

Bayes and empirical Bayes iteration estimators in two seemingly unrelated regression equations

For a system of two seemingly unrelated regression equations given by (?)(y_1 is an m×1 vector and y_2 is an n×1 vector,m≠n),employ- ing the covariance adjusted technique,we propose the parametric Bayes and empirical Bayes iteration estimator sequences for regression coefficients.We prove that both the covariance matrices converge monotonically and the Bayes iteration estimator squence is consistent as well.Based on the mean square error (MSE) criterion,we elaborate the su- periority of empirical Bayes iteration estimator over the Bayes estimator of single equation when the covariance matrix of errors is unknown.The results obtained in this paper further show the power of the covariance adiusted approach.     

One-sided empirical Bayes test for location parameter in Gamma distribution

In this paper, we devote to constructing the one-sided empirical Bayes(EB) test for the location parameter in the Gamma distribution by nonparametric method. Under some mild conditions, we prove that the EB test is asymptotically optimal with the rate of the order O(n~(-δs/(2s+1))), where 1/2 ≤δ 1 and s 1 is a given natural number. An example is also given to illustrate that the conditions of the main theorems are easily satisfied.     

Minimax multivariate empirical Bayes estimators under multicollinearity

In this paper we consider the problem of estimating the matrix of regression coefficients in a multivariate linear regression model in which the design matrix is near singular. Under the assumption of normality, we propose empirical Bayes ridge regression estimators with three types of shrinkage functions, that is, scalar, componentwise and matricial shrinkage. These proposed estimators are proved to be uniformly better than the least squares estimator, that is, minimax in terms of risk under the Strawderman's loss function. Through simulation and empirical studies, they are also shown to be useful in the multicollinearity cases.     

Monotone difference schemes stabilized by discrete mollification for strongly degenerate parabolic equations

The discrete mollification method is a convolution‐based filtering procedure suitable for the regularization of ill‐posed problems and for the stabilization of explicit schemes for the solution of PDEs. This method is applied to the discretization of the diffusive terms of a known first‐order monotone finite difference scheme [Evje and Karlsen, SIAM J Numer Anal 37 (2000) 1838–1860] for initial value problems of strongly degenerate parabolic equations in one space dimension. It is proved that the mollified scheme is monotone and converges to the unique entropy solution of the initial value problem, under a CFL stability condition which permits to use time steps that are larger than with the unmollified (basic) scheme. Several numerical experiments illustrate the performance and gains in CPU time for the mollified scheme. Applications to initial‐boundary value problems are included. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 38–62, 2012     

Empirical Bayes sequential estimation for exponential families: The untruncated component

We consider the empirical Bayes decision problem where the component problem is the sequential estimation of the mean of one-parameter exponential family of distributions with squared error loss for the estimation error and a cost c>0 for each observation. The present paper studies the untruncated sequential component case. In particular, an untruncated asymptotically pointwise optimal sequential procedure is employed as the component. With sequential components, an empirical Bayes decision procedure selects both a stopping time and a terminal decision rule for use in the component with parameter . The goodness of the empirical Bayes sequential procedure is measured by comparing the asymptotic behavior of its Bayes risk with that of the component procedure as the number of past data increases to infinity. Asymptotic risk equivalence of the proposed empirical Bayes sequential procedure to the component procedure is demonstrated.This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant GP7987.