Pseudo-bayes estimation of multinomial proportions

In the simultaneous estimation of multinomial proportions, two estimators are developed which incorporate prior means and a prior parameter which reflects the accuracy of the prior means. These estimators possess substantially smaller risk than the standard estimator in a region of the parameter space and are much more robust than the conjugate Bayes estimator with respect to parameter values far from the prior mean. When vague prior information is available, these estimators and confidence regions derived from them appear to be attractive alternatives to the procedures based on the standard estimator.     

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).     

Bayes minimax estimation of a bernoulli p in a restricted parameter space

In many estimation problems the parameter of interest is known,a priori, to belong to a proper subspace of the natural parameter space. Although useful in practice this type of additional information can lead to surprising theoretical difficulties. In this paper the problem of minimax estimation of a Bernoulli pwhen pis restricted to a symmetric subinterval of the natural parameter space is considered. For the sample sizes n = 1,2,3, and 4 least favorable priors with finite support are provided and the corresponding Bayes estimators are shown to be minimax. For n = 5 and 6 the usual constant risk minimax estimator is shown to be the Bayes minimax estimator corresponding to a least favorable prior with finite support, provided the restriction on the parameter space is not too tight.     

Minimax Estimation of the Bounded Parameter of Some Discrete Distributions Under LINEX Loss Function

For a class of discrete distributions, including Poisson(θ), Generalized Poisson(θ), Borel(m, θ), etc., we consider minimax estimation of the parameter θ under the assumption it lies in a bounded interval of the form [0, m] and a LINEX loss function. Explicit conditions for the minimax estimator to be Bayes with respect to a boundary supported prior are given. Also for Bernoulli(θ)-distribution, which is not in the mentioned class of discrete distributions, we give conditions for which the Bayes estimator of θ ∈ [0, m], m < 1 with respect to a boundary supported prior is minimax under LINEX loss function. Numerical values are given for the largest values of m for which the corresponding Bayes estimators of θ are minimax.     

Empirical Bayes estimation for the mean of the selected normal population when there are additional data

ABSTRACT

This paper is concerned with the problem of estimation for the mean of the selected population from two normal populations with unknown means and common known variance in a Bayesian framework. The empirical Bayes estimator, when there are available additional observations, is derived and its bias and risk function are computed. The expected bias and risk of the empirical Bayes estimator and the intuitive estimator are compared. It is shown that the empirical Bayes estimator is asymptotically optimal and especially dominates the intuitive estimator in terms of Bayes risk, with respect to any normal prior. Also, the Bayesian correlation between the mean of the selected population (random parameter) and some interested estimators are obtained and compared.     

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.     

Nonparametric estimators for shift in quantal bioassay

Let F(x) and F(x+θ) be log dose-response curves for a standard preparation and a test preparation, respectively, in a parallel quantal bioassay designed to test the relative potency of a drug, toxicant, or some other substance, and suppose the form of F is unknown. Several estimators of the shift parameter θ or relative potency, are compared, including some generalized and trimmed Spearman-Kärber estimators and a non parametric maximum likelihood estimator. Both point and interval estimation are discussed. Some recommendations concerning the choices of estimators are offered.     

Classical and Bayesian estimation of P(Y < X) for Kumaraswamy's distribution

The aim of this paper is to study the estimation of the reliability R=P(Y<X) when X and Y are independent random variables that follow Kumaraswamy's distribution with different parameters. If we assume that the first shape parameter is common and known, the maximum-likelihood estimator (MLE), the exact confidence interval and the uniformly minimum variance unbiased estimator of R are obtained. Moreover, when the first parameter is common but unknown, MLEs, Bayes estimators, asymptotic distributions and confidence intervals for R are derived. Furthermore, Bayes and empirical Bayes estimators for R are obtained when the first parameter is common and known. Finally, when all four parameters are different and unknown, the MLE of R is obtained. Monte Carlo simulations are performed to compare the different proposed methods and conclusions on the findings are given.     

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.     

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.     

MSE performance of the weighted average estimators consisting of shrinkage estimators when each individual regression coefficient is estimated

In this paper, we analytically derive the exact formula for the mean squared error (MSE) of two weighted average (WA) estimators for each individual regression coefficient. Further, we execute numerical evaluations to investigate small sample properties of the WA estimators, and compare the MSE performance of the WA estimators with the other shrinkage estimators and the usual OLS estimator. Our numerical results show that (1) the WA estimators have smaller MSE than the other shrinkage estimators and the OLS estimator over a wide region of parameter space; (2) the range where the relative MSE of the WA estimator is smaller than that of the OLS estimator gets narrower as the number of explanatory variables k increases.     

Improved confidence estimators for Fieller's confidence sets

Fieller's confidence set C_F for ratios of location parameters, although of great importance in practice, is often cited as an example to criticize frequentist theory. The reason is that the set can consist of the whole parameter space and yet the confidence is γ = 1 – α in any case. In this paper, we study the problem of constructing data-dependent estimators better than γ₊, A reasonable estimator appears to be γ₊, which is one if C_F is the whole parameter space and γ otherwise. By using an estimated confidence approach and a squared-error loss, it is shown that γ₊ dominates γ. The risk improvement of γ₊ over γ can be sizable. Also, by numerically comparing γ₊ with a generalized Bayes estimator γ_L, which is shown to be admissible when one or two ratios are concerned, it is shown that γ₊ is nearly admissible. We also conclude that the common practice of reporting 1 – α only when C_F is not the whole parameter space is nearly admissible.     

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     

The Bayes rule of the parameter in (0,1) under Zhang’s loss function with an application to the beta-binomial model

Abstract

For the restricted parameter space (0,1), we propose Zhang’s loss function which satisfies all the 7 properties for a good loss function on (0,1). We then calculate the Bayes rule (estimator), the posterior expectation, the integrated risk, and the Bayes risk of the parameter in (0,1) under Zhang’s loss function. We also calculate the usual Bayes estimator under the squared error loss function, and the Bayes estimator has been proved to underestimate the Bayes estimator under Zhang’s loss function. Finally, the numerical simulations and a real data example of some monthly magazine exposure data exemplify our theoretical studies of two size relationships about the Bayes estimators and the Posterior Expected Zhang’s Losses (PEZLs).     

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.     

On the Bayesian Estimation for the Uniform Scale Parameter via a Functional Equation

Bayes uniform model under the squared error loss function is shown to be completely identifiable by the form of the Bayes estimates of the scale parameter. This results in solving a specific functional equation. A complete characterization of differentiable Bayes estimators (BE) and generalized Bayes estimators (GBE) is given as well as relations between degrees of smoothness of the estimators and the priors. Characterizations of strong (generalized Bayes) Bayes sequence (SBS or SGBS) are also investigated. A SBS is a sequence of estimators (one for each sample size) where all its components are BE generated by the same prior measure. A complete solution is given for polynomial Bayesian estimation.     

Empirical Bayes sequential estimation of a finite population mean

In this article, we consider the Bayes and empirical Bayes problem of the current population mean of a finite population when the sample data is available from other similar (m-1) finite populations. We investigate a general class of linear estimators and obtain the optimal linear Bayes estimator of the finite population mean under a squared error loss function that considered the cost of sampling. The optimal linear Bayes estimator and the sample size are obtained as a function of the parameters of the prior distribution. The corresponding empirical Bayes estimates are obtained by replacing the unknown hyperparameters with their respective consistent estimates. A Monte Carlo study is conducted to evaluate the performance of the proposed empirical Bayes procedure.     

Combining poisson means

The problem of estimating the Poisson mean is considered based on the two samples in the presence of uncertain prior information (not in the form of distribution) that two independent random samples taken from two possibly identical Poisson populations. The parameter of interest is λ₁ from population I. Three estimators, i.e. the unrestricted estimator, restricted estimator and preliminary test estimator are proposed. Their asymptotic mean squared errors are derived and compared; parameter regions have been found for which restricted and preliminary test estimators are always asymptotically more efficient than the classical estimator. The relative dominance picture of the estimators is presented. Maximum and minimum asymptotic efficiencies of the estimators relative to the classical estimator are tabulated. A max-min rule for the size of the preliminary test is also discussed. A Monte Carlo study is presented to compare the performance of the estimator with that of Kale and Bancroft (1967).     

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.     

Optimal Nonparametric Quantile Estimators. Towards a General Theory. A Survey

“Nonparametric” in the title is used to say that observations X ₁,…,X _n come from an unknown distribution F ∈ ? with ? being the class of all continuous and strictly increasing distribution functions. The problem is to estimate the quantile of a given order q ∈ (0,1) of the distribution F. The class ? of distributions is very large; it is so large that even X _nq:n , where nq is an integer, may be very poor estimator of the qth quantile. To assess the performance of estimators no properties based on moments may be used: expected values of estimators should be replaced by their medians, their variances—by some characteristics of concentration of distributions around the median. If an estimator is median-biased for one of distributions, the bias of the estimator may be infinitely large for other distributions. In the note optimal estimators with respect to various criteria of optimality are presented. The pivotal function F(T) of the estimator T is introduced which enables us to apply the classical statistical approach.