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.     

Estimating the population median using only a central subset of observations

In estimating the population median, it is common to encounter estimators which are linear combinations of a small number of central observations. Sample medians, sample quasi medians, trimmed means, jackknifed (and delete‐d jackknifed) medians and jackknifed quasi medians are all familiar examples. The objective of this paper is to show that within this class the quasi medians turn out to have the best asymptotic mean squared error.     

Empirical Bayes Estimation of Small Area Means under a Nested Error Linear Regression Model with Measurement Errors in the Covariates

Abstract.  Previously, small area estimation under a nested error linear regression model was studied with area level covariates subject to measurement error. However, the information on observed covariates was not used in finding the Bayes predictor of a small area mean. In this paper, we first derive the fully efficient Bayes predictor by utilizing all the available data. We then estimate the regression and variance component parameters in the model to get an empirical Bayes (EB) predictor and show that the EB predictor is asymptotically optimal. In addition, we employ the jackknife method to obtain an estimator of mean squared prediction error (MSPE) of the EB predictor. Finally, we report the results of a simulation study on the performance of our EB predictor and associated jackknife MSPE estimators. Our results show that the proposed EB predictor can lead to significant gain in efficiency over the previously proposed EB predictor.     

Constrained Bayes and Empirical Bayes Estimation with Balanced Loss Functions

This article develops constrained Bayes and empirical Bayes estimators under balanced loss functions. In the normal-normal example, estimators of the mean squared errors of the EB and constrained EB estimators are provided which are correct asymptotically up to O(m ^?1), m denoting the number of strata.     

A relationship between generalized and integrated mean square errors

Generalised Mean squared error is a flexible measure of the adequancy of ? repression estimator. It allows specific characteristics of the regression model and its intended use to be In-corportated in the measure itself. Similarly, integrated mean squared error enables a researcher to stipulate particular regions of interest and wi ighting functions in the assessment of a prediction equation. The appeal of both measures is their ability to allow design or model characteristics to directly influence the evaluation of fitted regression models. In this note an e-quivalence of the two measures is established for correctly specified models.     

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.     

Some finite-sample-size properties of rosenblatt density estimates

The results of a Monte Carlo study of the sensitivity of Rosenblatt density estimates to the scale factor are presented.     

Further remarks on estimating the parameter of a noncentral chi-square distribution

Perlman and Rasmussen (1975) have found estimators of the non-centrality parameter of a noncentral chi-square distribution which have lower mean square error than the maximum likelihood estimator. This paper studies some extensions of their estimators and some related problems.     

A generalized reflection method of boundary correction in kernel density estimation

The kernel method of estimation of curves is now popular and widely used in statistical applications. Kernel estimators suffer from boundary effects, however, when the support of the function to be estimated has finite endpoints. Several solutions to this problem have already been proposed. Here the authors develop a new method of boundary correction for kernel density estimation. Their technique is a kind of generalized reflection involving transformed data. It generates a class of boundary corrected estimators having desirable properties such as local smoothness and nonnegativity. Simulations show that the proposed method performs quite well when compared with the existing methods for almost all shapes of densities. The authors present the theory behind this new methodology, and they determine the bias and variance of their estimators.     

FOURIER SERIES ESTIMATION FOR LENGTH BIASED DATA

This paper proposes and investigates Fourier series estimators for length biased data. Specifically, two Fourier series estimators are constructed and studied based on ideas of Jones (1991) and Bhattacharyya et al. (1988) in the case of kernel density estimation. Approximate expressions for mean squared errors and integrated mean squared errors are obtained and compared, and some simulated examples are investigated. The Fourier series estimator based on the proposal of Jones seems to have the more desirable properties of the two. The paper concludes with some comments that put this work in a wider context.     

Estimating the variance of a normal population by utilizing the information in a sample from a second related normal population

A class of estimators of the variance σ₁ ² of a normal population is introduced, by utilization the information in a sample from a second normal population with different mean and variance σ₂ ², under the restriction that σ₁ ²?≤?σ₂ ². Simulation results indicate that some members of this class are more efficient than the usual minimum variance unbiased estimator (MVUE) of σ₁ ², Stein estimator and Mehta and Gurland estimator. The case of known and unknown means are considered.     

Adaptively transformed mixed‐model prediction of general finite‐population parameters

For estimating area‐specific parameters (quantities) in a finite population, a mixed‐model prediction approach is attractive. However, this approach strongly depends on the normality assumption of the response values, although we often encounter a non‐normal case in practice. In such a case, transforming observations to make them suitable for normality assumption is a useful tool, but the problem of selecting a suitable transformation still remains open. To overcome the difficulty, we here propose a new empirical best predicting method by using a parametric family of transformations to estimate a suitable transformation based on the data. We suggest a simple estimating method for transformation parameters based on the profile likelihood function, which achieves consistency under some conditions on transformation functions. For measuring the variability of point prediction, we construct an empirical Bayes confidence interval of the population parameter of interest. Through simulation studies, we investigate the numerical performance of the proposed methods. Finally, we apply the proposed method to synthetic income data in Spanish provinces in which the resulting estimates indicate that the commonly used log transformation would not be appropriate.     

Correction of Density Estimators that are not Densities

Abstract. Several old and new density estimators may have good theoretical performance, but are hampered by not being bona fide densities; they may be negative in certain regions or may not integrate to 1. One can therefore not simulate from them, for example. This paper develops general modification methods that turn any density estimator into one which is a bona fide density, and which is always better in performance under one set of conditions and arbitrarily close in performance under a complementary set of conditions. This improvement-for-free procedure can, in particular, be applied for higher-order kernel estimators, classes of modern h ⁴ bias kernel type estimators, superkernel estimators, the sinc kernel estimator, the k -NN estimator, orthogonal expansion estimators, and for various recently developed semi-parametric density estimators.     

Minimax Rates for Estimating the Variance and its Derivatives in Non–Parametric Regression

In this paper the van Trees inequality is applied to obtain lower bounds for the quadratic risk of estimators for the variance function and its derivatives in non–parametric regression models. This approach yields a much simpler proof compared to previously applied methods for minimax rates. Furthermore, the informative properties of the van Trees inequality reveal why the optimal rates for estimating the variance are not affected by the smoothness of the signal g . A Fourier series estimator is constructed which achieves the optimal rates. Finally, a second–order correction is derived which suggests that the initial estimator of g must be undersmoothed for the estimation of the variance.     

Non-parametric estimation of reciprocal coordinate subtangent for right censored dependent scheme

The concept of reciprocal coordinate subtangent (RCST) has been used as a useful tool to study the monotone behavior of a continuous density function and for characterizing probability distributions. In this paper, we propose a non-parametric estimator for RCST based on the censored dependent data. Asymptotic properties of the estimator are established under suitable regularity conditions. A simulation study is carried out to examine the performance of the estimator. The usefulness of the estimator is also examined through a real data.     

Issues for stratified randomization based on a factor derived from a continuous baseline variable

Stratified randomization based on the baseline value of the primary analysis variable is common in clinical trial design. We illustrate from a theoretical viewpoint the advantage of such a stratified randomization to achieve balance of the baseline covariate. We also conclude that the estimator for the treatment effect is consistent when including both the continuous baseline covariate and the stratification factor derived from the baseline covariate. In addition, the analysis of covariance model including both the continuous covariate and the stratification factor is asymptotically no less efficient than including either only the continuous baseline value or only the stratification factor. We recommend that the continuous baseline covariate should generally be included in the analysis model. The corresponding stratification factor may also be included in the analysis model if one is not confident that the relationship between the baseline covariate and the response variable is linear. In spite of the above recommendation, one should always carefully examine relevant historical data to pre-specify the most appropriate analysis model for a perspective study.     

On shrinkage estimation of normal population variance

Some shrunken estimators of the normal population variance 2 are proposed and compared with the usual estimator, s², in terms of mean squared error.     

Shrinkage estimation of scale in exponential distribution from censored samples

In this paper some shrunken and pretest shrunken estimators are suggested for the scale parameter of an exponential distribution, when observations become available from life test experiments. These estimators are shown to be more efficient than the usual estimator when a guessed value is nearer to the true value.