Empirical bayes shrinkage estimation of reliability in the exponential distribution

In this paper we propose two empirical Bayes shrinkage estimators for the reliability of the exponential distribution and study their properties. Under the uniform prior distribution and the inverted gamma prior distribution these estimators are developed and compared with a preliminary test estimator and with a shrinkage testimator in terms of mean squared error. The proposed empirical Bayes shrinkage estimator under the inverted gamma prior distribution is shown to be preferable to the preliminary test estimator and the shrinkage testimator when the prior value of mean life is clsoe to the true mean life.     

Empirical bayes estimation of functionals of unknown probability measures

The empirical Dayes approach to one and two sal-npie problcrns has beeir considered by Korwar and Hollander (1976), Holiander and Korwar (1976) and Phadia and Susarla (1979). In this article we essen- tially generalize their empirical Bayes results by replacing the inlicaro-functions of. the sets (?∞,x) and {X≦Y} by arbitrary mea5, irable functions h(x) and h(x,y). More speclfically, the ernpiricaion yes estimation of esrimabie paramerers of degree one ani KG,I;ti kliown probability measure Pon (R,R) is considered. The asymptotic optimality of the these estimators, obtaining the exact risk expressions, is established. Also the results of Dalal and Phad (1983) we extended to the estimation of an estimable parametric function of an unknow probability measure P on (R² , B²)     

Empirical bayes interval estimates involving uniform distributions

Bayesian and empirical Bayesian decision rules are exhibited for the interval estimation of the parameter 0 of a Uniform (0,θ) distribution. The estimate ?,δ>resulting in the interval [?,?+δ]suffers loss given by L(?,δ>,θ)=1-[?≦e≦?+δ]+c₁((?-θ)²+(?+δ?θ)²))+c₂δ. The solution is presented for prior distributions G which have bounded support, no point masses,∫θ^?mdG(θ)<∞ and for some integer m. An example is presented involving a particular parametric form for G and rates of risk convergence in the empirical Bayes problem for this example are calculated.     

Empirical bayes estimation of binomial parameter with symmetric priors

This paper deals with the problem of estimating the binomial parameter via the nonparametric empirical Bayes approach. This estimation problem has the feature that estimators which are asymptotically optimal in the usual empirical Bayes sense do not exist (Robbins (1958, 1964)), However, as pointed out by Liang (1934) and Gupta and Liang (1988), it is possible to construct asymptotically optimal empirical Bayes estimators if the unknown prior is symmetric about the point 1/2, In this paper, assuming symmetric priors a monotone empirical Bayes estimator is constructed by using the isotonic regression method. This estimator is asymptotically optimal in the usual empirical Bayes sense. The corresponding rate of convergence is investigated and shown to be of order n^-1, where n is the number of past observations at hand.     

Linear. empirical bayes estimation in the case of the wishart distribution

We consider independent pairs (X₁,∑₁), (X₂,∑₂),…,(X_n∑_n), where each Si is distributed according to some unknown density function g(∑) and, given ∑_i = ∑, X has a conditional density function g(x|∑) of the Wishart type. In each pair, the first component is observable but the second is not. After the (n + l)-th observation X_n+i is obtained, the objective is to estimate ∑ _n+i corresponding to X_n+i. This estimator is called an empirical Bayes (EB) estimator of ∑. We construct a linear EB estimator of ∑ and examine its precision.     

Extended linear empirical bayes estimation

In this paper, the linear empirical Bayes estimation method, which is based on approximation of the Bayes estimator by a linear function, is generalized to an extended linear empirical Bayes estimation technique which represents the Bayes estimator by a series of algebraic polynomials. The extended linear empirical Bayes estimators are elaborated in the case of a location or a scale parameter. The theory is illustrated by examples of its application to the normal distribution with a location parameter and the gamma distribution with a scale parameter. The linear and the extended linear empirical Bayes estimators are constructed in these two cases and, then, studied numerically via Monte Carlo simulations. The simulations show that the extended linear empirical Bayes estimators have better convergence rates than the traditional linear empirical Bayes estimators.     

Empirical bayes estimation of a distribution function with a gamma process prior

A sequence of empirical Bayes estimators is given for estimating a distribution function. It is shown that ‘i’ this sequence is asymptotically optimum relative to a Gamma process prior, ‘ii’ the overall expected loss approaches the minimum Bayes risk at a rate of n , and ‘iii’ the estimators form a sequence of proper distribution functions. Finally, the numerical example presented by Susarla and Van Ryzin ‘Ann. Statist., 6, 1978’ reworked by Phadia ‘Ann. Statist., 1, 1980, to appear’ has been analyzed and the results are compared to the numerical results by Phadia     

Empirical bayes approach to bayes sequential estimation: the poisson and bernoulli cases

The problem considered is the Bayes sequential estimation of the mean with quadratic loss and fixed cost per observation. Assume the prior distribution is not completely known. Some empirical Bayes procedures are proposed in the Poisson and Bernoulli cases, and they are shown to be asymptotically non-deficient in the sense of Woodroofe (1981).     

Convergence rates for empirical bayes estimators of parameters in multi-parameter exponential families

This paper obtains the convergence rates of the empirical Bayes estimators of parameters in the multi-parameter exponential families. The rates can approximate to 0(n=¹) arbitrarily. The paper presents the multivariate orthogonal polynomials which are continuous on the total space R^p.     

A note on the convergence rates for empirical bayes estimators of parameters in multiple-parameter exponential families

Under suitable conditions upon prior distribution, the convergence rates for empirical Bayes estimators of parameters in multi-parameter exponential families (M-PEF) are obtained. It is shown that the assumptions Tong (1996) imposed on the marginal density can be reduced. The above result can also be extended to more general forms of M-PEF. Finally, some examples which satisfy the conditions of the theorems are given.     

The umvue and bayes estimate of reliability of mixed failure time distribution

We study the reliability estimates of the non-standard mixture of degenerate (degenerated at zero) and exponential distributions. The Uniformly Minimum Variance Unbiased Estimator (UMVUE) and Bayes estimator of the reliability for some selective prior when the mixing proportion is known and unknown are derived. The Bayes risk is computed for each Bayes estimator of the reliability. A simulated study is carried out to assess the performance of the estimators alongwith the true and Maximum Likelihood Estimate (MLE) of the reliability. An example from Vannman (1991) is also discussed at the end of the paper.     

Nonparametric empirical bayes estimation of certain funtionals

For estimating functionals of the form ∫∫φ(x,y)dF(x) dF(y), nonparametric empirical Bayes estimators are developed which are competitors of the classical U-statistics. Asymptotic optimality of the proposed estimators is proved     

On asymptotic optimality of bayes empirical bayes estimators

In an empirical Bayes decision problem, a prior distribution ? is placed on a one-dimensfonal family G of priors G_w, wεΩ, to produce a Bayes empirical Bayes estimator, The asymptotic optimaiity of the Bayes estimator is established when the support of ? is Ω and the marginal distributions H_w have monotone likelihood ratio and continuous Kullback-Leibler information number.     

Bayes and empirical bayes estimation of the probability that z > x + y

Let X, Y and Z be independent random variables with common unknown distribution F. Using the Dirichlet process prior for F and squared erro loss function, the Bayes and empirical Bayes estimators of the parameters λ(F). the probability that Z > X + Y, are derived. The limiting Bayes estimator of λ(F) under some conditions on the parameter of the process is shown to be asymptotically normal. The aysmptotic optimality of the empirical Bayes estimator of λ(F) is established. When X, Y and Z have support on the positive real line, these results are derived for randomly right censored data. This problem relates to testing whether than used discussed by Hollander and Proshcan (1972) and Chen, Hollander and Langberg (1983).     

Empirical bayes estimation of the mean in a multivariate normal distribution

We consider the problem of estimating the mean vector of a multivariate normal distribution under a variety of assumed structures among the parameters of the sampling and prior distributions. We adopt a pragmatic approach. We adopt distributional familites, assess hyperparmeters, and adopt patterned mean and coveariance structures when it is relatively simple to do so; alternatively, we use the sample data to estimate hyperparameters of prior distributions when assessment is a formidable task; such as the task of assessing parameters of multidimensional problems. James-Stein-like estimators are found to result. In some cases, we've been abl to show that the estimators proposed uniformly dominate the MLE's when measured with respect to quadratic loss functions.     

Minimax linear empirical bayes estimation of the binomial parameter

The minimax linear Empirical Bayes estimators for a binomial parameter are obtained, assuming some information about the moments of the prior. The form of these estimates is used to propose a criterion which may be helpful in determining whether Empirical Bayes estimation is Indicated for a given problem.     

Empirical Bayes estimation in continuous one-parameter exponential families under associated samples

In this paper, we study the empirical Bayes (EB) estimation in continuous one-parameter exponential families under negatively associated (NA) samples and positively associated (PA) samples. Under certain regularity conditions, it is shown that the convergence rates of proposed EB estimators under NA or PA samples are the same as those of EB estimators under independent observations, which significantly improve the existing results in EB estimation under associated samples.     

On empirical bayes sequential estimation

We study the empirical Bayes approach to the sequential estimation problem. An empirical Bayes sequential decision procedure, which consists of a stopping rule and a terminal decision rule, is constructed for use in the component. Asymptotic behaviors of the empirical Bayes risk and the empirical Bayes stopping times are investigated as the number of components increase.     

A comparison of bayes and maximum likelihood estimation of the intraclass correlation coefficient

Two methods of estimating the intraclass correlation coefficient (p) for the one-way random effects model were compared in several simulation experiments using balanced and unbalanced designs. Estimates based on a Bayes approach and a maximum likelihood approach were compared on the basis of their biases (differences between estimates and true values of p) and mean square errors (mean square errors of estimates of p) in each of the simulation experiments. The Bayes approach used the median of a conditional posterior density as its estimator.     

Approximate bayes estimation of reliability of two parameter inverse gaussian distribution

This paper extends the result of Padgett (1981) and gives a Bayes estimate of the reliability function of two-parameter inverse Gaussian distribution using Jeffreys' non-informative joint prior and a squared error loss fun ction . A numerical example is given. Based on a Monte Carlo simulation, Bayes estimator of reliability is compared with its maximum likelihood counterpart.