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.     

A Generalized Bayes Rule for Prediction

In the case of prior knowledge about the unknown parameter, the Bayesian predictive density coincides with the Bayes estimator for the true density in the sense of the Kullback-Leibler divergence, but this is no longer true if we consider another loss function. In this paper we present a generalized Bayes rule to obtain Bayes density estimators with respect to any α-divergence, including the Kullback-Leibler divergence and the Hellinger distance. For curved exponential models, we study the asymptotic behaviour of these predictive densities. We show that, whatever prior we use, the generalized Bayes rule improves (in a non-Bayesian sense) the estimative density corresponding to a bias modification of the maximum likelihood estimator. It gives rise to a correspondence between choosing a prior density for the generalized Bayes rule and fixing a bias for the maximum likelihood estimator in the classical setting. A criterion for comparing and selecting prior densities is also given.     

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 Uncertain Bayes Methods in Area‐level Models

Random effects model can account for the lack of fitting a regression model and increase precision of estimating area‐level means. However, in case that the synthetic mean provides accurate estimates, the prior distribution may inflate an estimation error. Thus, it is desirable to consider the uncertain prior distribution, which is expressed as the mixture of a one‐point distribution and a proper prior distribution. In this paper, we develop an empirical Bayes approach for estimating area‐level means, using the uncertain prior distribution in the context of a natural exponential family, which we call the empirical uncertain Bayes (EUB) method. The regression model considered in this paper includes the Poisson‐gamma and the binomial‐beta, and the normal‐normal (Fay–Herriot) model, which are typically used in small area estimation. We obtain the estimators of hyperparameters based on the marginal likelihood by using a well‐known expectation‐maximization algorithm and propose the EUB estimators of area means. For risk evaluation of the EUB estimator, we derive a second‐order unbiased estimator of a conditional mean squared error by using some techniques of numerical calculation. Through simulation studies and real data applications, we evaluate a performance of the EUB estimator and compare it with the usual empirical Bayes estimator.     

Adaptive nonparametric empirical Bayes estimation via wavelet series: The minimax study

In the present paper, we derive lower bounds for the risk of the nonparametric empirical Bayes estimators. In order to attain the optimal convergence rate, we propose generalization of the linear empirical Bayes estimation method which takes advantage of the flexibility of the wavelet techniques. We present an empirical Bayes estimator as a wavelet series expansion and estimate coefficients by minimizing the prior risk of the estimator. As a result, estimation of wavelet coefficients requires solution of a well-posed low-dimensional sparse system of linear equations. The dimension of the system depends on the size of wavelet support and smoothness of the Bayes estimator. An adaptive choice of the resolution level is carried out using Lepski et al. (1997) method. The method is computationally efficient and provides asymptotically optimal adaptive EB estimators. The theory is supplemented by numerous examples.     

A note on james-stein and bayes empiricl bayes estimators

In an empirical Bayes decision problem, a simple class of estimators is constructed that dominate the James-Stein

estimator, A prior distribution A is placed on a restricted (normal) class G of priors to produce a Bayes empirical Bayes estimator, The Bayes empirical Bayes estimator is smooth, admissible, and asymptotically optimal. For certain A rate of convergence to minimum Bayes risk is 0(n^-1)uniformly on G. The results of a Monte Carlo study are presented to demonstrate the favorable risk bebhavior of the Bayes estimator In comparison with other competitors including the James-Stein estimator.     

Small sample study on estimators of life distributions and of mean survival times using randomly censored data

Whereas large-sample properties of the estimators of survival distributions using censored data have been studied by many authors, exact results for small samples have been difficult to obtain. In this paper we obtain the exact expression for the ath moment (a > 0) of the Bayes estimator of survival distribution using the censored data under proportional hazard model. Using the exact expression we compute the exact mean, variance and MSE of the Bayes estimator. Also two estimators ofthe mean survival time based on the Kaplan-Meier estimator and the Bayes estimator are compared for small samples under proportional hazards.     

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.     

ESTIMATION OF EXPONENTIATED WEIBULL SHAPE PARAMETERS UNDER LINEX LOSS FUNCTION

ABSTRACT

The paper deals with Bayes estimation of the exponentiated Weibull shape parameters under linex loss function when independent non-informative type of priors are available for the parameters. Generalized maximum likelihood estimators have also been obtained. Performances of the proposed Bayes estimator, generalized maximum likelihood estimators, posterior mean (i.e., Bayes estimator under squared error loss function) and maximum likelihood estimators have been studied on the basis of their risks under linex loss function. The comparison is based on a simulation study because the expressions for risk functions of these estimators cannot be obtained in nice closed forms.     

Mean likelihood estimators

The use of Mathematica in deriving mean likelihood estimators is discussed. Comparisons are made between the mean likelihood estimator, the maximum likelihood estimator, and the Bayes estimator based on a Jeffrey's noninformative prior. These estimators are compared using the mean-square error criterion and Pitman measure of closeness. In some cases it is possible, using Mathematica, to derive exact results for these criteria. Using Mathematica, simulation comparisons among the criteria can be made for any model for which we can readily obtain estimators.In the binomial and exponential distribution cases, these criteria are evaluated exactly. In the first-order moving-average model, analytical comparisons are possible only for n = 2. In general, we find that for the binomial distribution and the first-order moving-average time series model the mean likelihood estimator outperforms the maximum likelihood estimator and the Bayes estimator with a Jeffrey's noninformative prior. Mathematica was used for symbolic and numeric computations as well as for the graphical display of results. A Mathematica notebook which provides the Mathematica code used in this article is available: http://www.stats.uwo.ca/mcleod/epubs/mele. Our article concludes with our opinions and criticisms of the relative merits of some of the popular computing environments for statistics researchers.     

Relative squared error prediction in the generalized linear regression model

In linear regression models, predictors based on least squares or on generalized least squares estimators are usually applied which, however, fail in case of multicollinearity. As an alternative biased estimators like ridge estimators, Kuks-Olman estimators, Bayes or minimax estimators are sometimes suggested. In our analysis the relative instead of the generally used absolute squared error enters the objective function. An explicit minimax solution is derived which, in an important special case, can be viewed as a predictor based on a Kuks-Olman estimator.     

The Statistical Properties of the Equity Estimator

In this article we consider the Equity estimator proposed by Krishnamurthi and Rangaswamy. We show that this estimator is inconsistent and does not necessarily improve on the mean squared error (MSE) of the least squares (LS) estimator. We perform a Monte Carlo experiment based on the price-promotion model used in marketing research, with marketing data, comparing the MSE of the Equity estimator to that of two empirical Bayes estimators and the LS estimator. We find that the empirical Bayes estimators have substantially smaller MSE than the Equity estimator in almost every case.     

Nonparametric Bayes estimation of gap-time distribution with recurrent event data

Nonparametric Bayes (NPB) estimation of the gap-time survivor function governing the time to occurrence of a recurrent event in the presence of censoring is considered. In our Bayesian approach, the gap-time distribution, denoted by F, has a Dirichlet process prior with parameter α. We derive NPB and nonparametric empirical Bayes (NPEB) estimators of the survivor function F?=1?F and construct point-wise credible intervals. The resulting Bayes estimator of F? extends that based on single-event right-censored data, and the PL-type estimator is a limiting case of this Bayes estimator. Through simulation studies, we demonstrate that the PL-type estimator has smaller biases but higher root-mean-squared errors (RMSEs) than those of the NPB and the NPEB estimators. Even in the case of a mis-specified prior measure parameter α, the NPB and the NPEB estimators have smaller RMSEs than the PL-type estimator, indicating robustness of the NPB and NPEB estimators. In addition, the NPB and NPEB estimators are smoother (in some sense) than the PL-type estimator.     

Nonparametric hierarchical Bayes analysis of binomial data via Bernstein polynomial priors

For binomial data analysis, many methods based on empirical Bayes interpretations have been developed, in which a variance‐stabilizing transformation and a normality assumption are usually required. To achieve the greatest model flexibility, we conduct nonparametric Bayesian inference for binomial data and employ a special nonparametric Bayesian prior—the Bernstein–Dirichlet process (BDP)—in the hierarchical Bayes model for the data. The BDP is a special Dirichlet process (DP) mixture based on beta distributions, and the posterior distribution resulting from it has a smooth density defined on [0, 1]. We examine two Markov chain Monte Carlo procedures for simulating from the resulting posterior distribution, and compare their convergence rates and computational efficiency. In contrast to existing results for posterior consistency based on direct observations, the posterior consistency of the BDP, given indirect binomial data, is established. We study shrinkage effects and the robustness of the BDP‐based posterior estimators in comparison with several other empirical and hierarchical Bayes estimators, and we illustrate through examples that the BDP‐based nonparametric Bayesian estimate is more robust to the sample variation and tends to have a smaller estimation error than those based on the DP prior. In certain settings, the new estimator can also beat Stein's estimator, Efron and Morris's limited‐translation estimator, and many other existing empirical Bayes estimators. The Canadian Journal of Statistics 40: 328–344; 2012 © 2012 Statistical Society of Canada     

Admissibility of linear estimators with respect to restricted parameter sets

In this paper the admissibility of a linear estimator for a linear regression parameter is characterized for such cases, where the considered parameter varies in an ellipsoid. We obtain a certain subset of the set of all linear estimators which are admissible with respect to the unrestricted parameter set. Furthermore, various linear estimators which have been proposed for improving the least squares estimator in cases of a restricted parameter set are investigated for admissibility. It turns out that only some shrunken estimators and some estimators of ridge type are admissible, whereas the KUKS-OLMAN estimator and all estimators of MARQUARDT type can be improved.     

ASSESSING ERROR RATE ESTIMATORS: THE LEAVE-ONE-OUT METHOD RECONSIDERED

Many comparative studies of the estimators of error rates of supervised classification rules are based on inappropriate criteria. In particular, although they fix the Bayes error rate, their summary statistics aggregate a range of true error rates. This means that their conclusions about the performance of classification rules cannot be trusted. This paper discusses the general issues involved, and then focuses attention specifically on the leave-one-out estimator. The estimator is investigated in a simulation study, both in absolute terms and in comparison with a popular bootstrap estimator. An improvement to the leave-one-out estimator is suggested, but the bootstrap estimator appears to maintain superiority even when the criteria are adjusted.     

On the consistency between model-and design-based estimators in survey sampling

In finite population sampling, often a distinction is made between model-and design-based estimators of the parameters of interest (like the population total, population variance, etc.). The model-based estimators depend on the (known) parameters of the model, while the design-based estimators depend on the (known) selection probabilities of the different units in the population. It is shown in this paper that the two approaches are not necessarily incompatible, and indeed can often lead to the same estimator. Our ideas are illustrated with the Horvitz-Thompson, and the generalized Horvitz-Thompson estimator. These estimators are identified as hierarchical Bays estimators. Also, certain “stepwise-Bayes” estimators of Vardeman and Meeden (J. Stat. Inf. (1983), V7, pp 329-341) are unified from a hierarchical Bayes point of view.     

Estimation of p{y p> max(y1, y2, …, yp-1)}in theexponential case

This paper deals with the derivation of (i) the MLE (ii) the MVUE (iii) a Bayes estimator of the probability in the title, for the case p = 2. Simulation studies are carried out to compare these estimators. The results suggest that the MLE and the Bayes estimator are biased and the Bayes estimator have the smallest MSE. In the general case, explicit expression for the probability in the title is derived and the MLE and Bayes estimator are obtained. A general method of deriving the MVUE is pointed out. Because of the simulation studies for p = 2 it is recommended that the Bayes or predictive estimator should be used.     

The empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior under Stein's loss function

For the hierarchical Poisson and gamma model, we calculate the Bayes posterior estimator of the parameter of the Poisson distribution under Stein's loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein's Loss (PESL). We also obtain the Bayes posterior estimator of the parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior by two methods. In numerical simulations, we have illustrated: The two inequalities of the Bayes posterior estimators and the PESLs; the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters; the goodness-of-fit of the model to the simulated data. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, we exploit the attendance data on 314 high school juniors from two urban high schools to illustrate our theoretical studies.     

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.