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 Bayes rule of the variance parameter of the hierarchical normal and inverse gamma model under Stein's loss

For the variance parameter of the hierarchical normal and inverse gamma model, we analytically calculate the Bayes rule (estimator) with respect to a prior distribution IG (alpha, beta) under Stein's loss function. This estimator minimizes the posterior expected Stein's loss (PESL). We also analytically calculate the Bayes rule and the PESL under the squared error loss. Finally, the numerical simulations exemplify that the PESLs depend only on alpha and the number of observations. The Bayes rules and PESLs under Stein's loss are unanimously smaller than those under the squared error loss.     

Some General Comparative Points on Chao's and Zelterman's Estimators of the Population Size

Abstract. Two simple and frequently used capture–recapture estimates of the population size are compared: Chao's lower‐bound estimate and Zelterman's estimate allowing for contaminated distributions. In the Poisson case it is shown that if there are only counts of ones and twos, the estimator of Zelterman is always bounded above by Chao's estimator. If counts larger than two exist, the estimator of Zelterman is becoming larger than that of Chao's, if only the ratio of the frequencies of counts of twos and ones is small enough. A similar analysis is provided for the binomial case. For a two‐component mixture of Poisson distributions the asymptotic bias of both estimators is derived and it is shown that the Zelterman estimator can experience large overestimation bias. A modified Zelterman estimator is suggested and also the bias‐corrected version of Chao's estimator is considered. All four estimators are compared in a simulation study.     

An efficient algorithm for computing the noncentrality parameters of chi-squared tests

Use of Newton's method for computing the noncentrality parameter based on the specified power in sample size problems of chi-squared tests requires that we evaluate both the noncentral chi-squareddistribution function and its derivative with respect to the noncentrality parameter. A close relationship between computing formulas for them is revealed, by which their evaluations can be performed jointly. This property greatly reduces the amount of computation involved. The corresponding algorithm is provided in a step-by-step form.     

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).     

Corrections for Bias in Maximum Likelihood Parameter Estimates Due to Nuisance Parameters

Abstract

In his Fisher Lecture, Efron (Efron, B. R. A. (1998 Efron, B. R. A. 1998. Fisher in the 21st century (with discussion). Statistical Science, 13: 95–122. [Crossref], [Web of Science ®] , [Google Scholar]). Fisher in the 21st Century (with discussion). Statistical Science 13:95–122) pointed out that maximum likelihood estimates (MLE) can be badly biased in certain situations involving many nuisance parameters. He predicted that with modern computing equipment a computer-modified version of the MLE that was less biased could become the default estimator of choice in applied problems in the 21st century. This article discusses three modifications—Lindsay's conditional likelihood, integrated likelihood, and Bartlett's bias-corrected estimating function. Each is evaluated through a study of the bias and MSE of the estimates in a stratified Weibull model with a moderate number of nuisance parameters. In Lindsay's estimating equation, three different methods for estimation of the nuisance parameters are evaluated—the restricted maximum likelihood estimate (RMLE), a Bayes estimator, and a linear Bayes estimator. In our model, the conditional likelihood with RMLE of the nuisance parameters is equivalent to Bartlett's bias-corrected estimating function. In the simulation we show that Lindsay's conditional likelihood is in general preferred, irrespective of the estimator of the nuisance parameters. Although the integrated likelihood has smaller MSE when the precise nature of the prior distribution of the nuisance parameters is known, this approach may perform poorly in cases where the prior distribution of the nuisance parameters is not known, especially using a non-informative prior. In practice, Lindsay's method using the RMLE of the nuisance parameters is recommended.     

Robust estimation for the coefficient of a first order autoregressive process

Nonparametric methods, Theil's method and Hussain's method have been applied to simple linear regression problems for estimating the slope of the regression line.We extend these methods and propose a robust estimator to estimate the coefficient of a first order autoregressive process under various distribution shapes, A simulation study to compare Theil's estimator, Hus-sain's estimator, the least squares estimator, and the proposed estimator is also presented.     

Estimation and prediction of the Burr type XII distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.     

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     

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.     

A new approach to default priors and robust bayes methodology

Following the developments in DasGupta et al. (2000), the authors propose and explore a new method for constructing proper default priors and a method for selecting a Bayes estimate from a family. Their results are based on asymptotic expansions of certain marginal correlations. For ease of exposition, most results are presented for location families and squared error loss only. The default prior methodology amounts, ultimately, to the minimization of Fisher information, and hence, Bickel's prior works out as the default prior if the location parameter is bounded. As for the selected Bayes estimate, it corresponds to ‘Gaussian tilting’ of an initial reference prior.     

A shift parameter estimation based on smoothed Kolmogorov–Smirnov

A new procedure of shift parameter estimation in the two-sample location problem is investigated and compared with existing estimators. The proposed procedure smooths the empirical distribution functions of each random sample and replaces empirical distribution functions in the two-sample Kolmogorov–Smirnov method. The smoothed Kolmogorov–Smirnov is minimized with respect to an arbitrary shift variable in order to find an estimate of the shift parameter. The proposed procedure can be considered the smoothed version of a very little known method of shift parameter estimation from Rao-Schuster-Littell (RSL) [Rao et al., Estimation of shift and center of symmetry based on Kolmogorov–Smirnov statistics, Ann. Stat. 3(4) (1975), pp. 862–873]. Their estimator will be discussed and compared with the proposed estimator in this paper. An example and simulation studies have been performed to compare the proposed procedure with existing shift parameter estimators such as Hodges–Lehmann (H–L) and least squares in addition to RSL's estimator. The results show that the proposed estimator has lower mean-squared error as well as higher relative efficiency against RSL's estimator under normal or contaminated normal model assumptions. Moreover, the proposed estimator performs competitively against H–L and least-squares shift estimators. Smoother function and bandwidth selections are also discussed and several alternatives are proposed in the study.     

Spatially-adaptive Penalties for Spline Fitting

The paper studies spline fitting with a roughness penalty that adapts to spatial heterogeneity in the regression function. The estimates are p th degree piecewise polynomials with p − 1 continuous derivatives. A large and fixed number of knots is used and smoothing is achieved by putting a quadratic penalty on the jumps of the p th derivative at the knots. To be spatially adaptive, the logarithm of the penalty is itself a linear spline but with relatively few knots and with values at the knots chosen to minimize the generalized cross validation (GCV) criterion. This locally-adaptive spline estimator is compared with other spline estimators in the literature such as cubic smoothing splines and knot-selection techniques for least squares regression. Our estimator can be interpreted as an empirical Bayes estimate for a prior allowing spatial heterogeneity. In cases of spatially heterogeneous regression functions, empirical Bayes confidence intervals using this prior achieve better pointwise coverage probabilities than confidence intervals based on a global-penalty parameter. The method is developed first for univariate models and then extended to additive models.     

Bayes estimation and prediction of the two-parameter gamma distribution

In this article, the Bayes estimates of two-parameter gamma distribution are considered. It is well known that the Bayes estimators of the two-parameter gamma distribution do not have compact form. In this paper, it is assumed that the scale parameter has a gamma prior and the shape parameter has any log-concave prior, and they are independently distributed. Under the above priors, we use Gibbs sampling technique to generate samples from the posterior density function. Based on the generated samples, we can compute the Bayes estimates of the unknown parameters and can also construct HPD credible intervals. We also compute the approximate Bayes estimates using Lindley's approximation under the assumption of gamma priors of the shape parameter. Monte Carlo simulations are performed to compare the performances of the Bayes estimators with the classical estimators. One data analysis is performed for illustrative purposes. We further discuss the Bayesian prediction of future observation based on the observed sample and it is seen that the Gibbs sampling technique can be used quite effectively for estimating the posterior predictive density and also for constructing predictive intervals of the order statistics from the future sample.     

Estimating expected sojourn times for a markov renewal process

We consider the estimation of the expected sojourn time in a Markov renewal process under the data condition that only the counts of the exits from the states are available for fixed intervals of time. For analytical and illustrative purposes we concentrate on the two-state process case. We present least squares and method of moments estimators and compare their statistical properties both analytically and empirically. We also present modified estimators with improved properties based upon an overlapping interval sampling strategy. The major results indicate that the least squares estimator is biased in general with the bias depending on the size of the sampling interval and the first two moments of the sojourn time distribution function. The bias becomes negligible as the size of the sampling interval increases. Analytical and empirical results indicate that the method of moments estimator is less sensitive to the size of the sampling interval and has slightly better mean squared error properties than the least squares estimator.     

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.     

Bayesian shrinkage estimation based on Rayleigh type-II censored data

In this paper, we construct a Bayes shrinkage estimator for the Rayleigh scale parameter based on censored data under the squared log error loss function. Risk-unbiased estimator is derived and its risk is computed. A Bayes shrinkage estimator is obtained when a prior point guess value is available for the scale parameter. Risk-bias of the Bayes shrinkage estimator is considered. A comparison between the proposed Bayes shrinkage estimator and the risk-unbiased estimator is provided using calculation of the relative efficiency. A numerical example is presented for illustrative and comparative purposes.     

A Hybrid Approximation Bayesian Test of Variance Components for Longitudinal Data

The test of variance components of possibly correlated random effects in generalized linear mixed models (GLMMs) can be used to examine if there exists heterogeneous effects. The Bayesian test with Bayes factors offers a flexible method. In this article, we focus on the performance of Bayesian tests under three reference priors and a conjugate prior: an approximate uniform shrinkage prior, modified approximate Jeffreys' prior, half-normal unit information prior and Wishart prior. To compute Bayes factors, we propose a hybrid approximation approach combining a simulated version of Laplace's method and importance sampling techniques to test the variance components in GLMMs.     

Bayesian inference of three-parameter bathtub-shaped lifetime distribution

We investigate a Bayesian inference in the three-parameter bathtub-shaped lifetime distribution which is obtained by adding a power parameter to the two-parameter bathtub-shaped lifetime distribution suggested by Chen (2000). The Bayes estimators under the balanced squared error loss function are derived for three parameters. Then, we have used Lindley's and Tierney–Kadane approximations (see Lindley 1980; Tierney and Kadane 1986) for computing these Bayes estimators. In particular, we propose the explicit form of Lindley's approximation for the model with three parameters. We also give applications with a simulated data set and two real data sets to show the use of discussed computing methods. Finally, concluding remarks are mentioned.     

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.