Optimal minimax squared error risk estimation of the mean of a multivariate normal distribution

Minimax squared error risk estimators of the mean of a multivariate normal distribution are characterized which have smallest Bayes risk with respect to a spherically symmetric prior distribution for (i) squared error loss, and (ii) zero-one loss depending on whether or not estimates are consistent with the hypothesis that the mean is null. In (i), the optimal estimators are the usual Bayes estimators for prior distributions with special structure. In (ii), preliminary test estimators are optimal. The results are obtained by applying the theory of minimax-Bayes-compromise decision problems.     

Mem squared error estimation in finite populations under nonlinear models

Four estimators of the prediction mean squared error (MSB) of an estimated finite population total for a zero-one characteristic are examined. The characteristic associated with each population unit is modeled as the realization of a Bernoulli random variable whose expected value is a nonlinear function of a parameter vector and a set of known auxiliary variables. To compare the estimators, a simulation study is conducted using a population of hospitals. The MSB estimator Implied by the form of the assumed model underestimates the mean squared error in each of the cases studied and produces confidence lntervals with less than the nominal coverage probabilities. Of the three alternative MSE estimators presented, a linear approximation to the jackknife produces the best results and improves upon the model-specific estimator.     

Estimation of poisson means under weighted squared error loss

In estimating the means of several independent Poisson distributions, we show that the maximum likelihood estimator is inadmissible when general weighted squared error loss is the criterion. Using this result, we extend the known results on estimation of several Poisson means (Peng 1975, Hudson 1978) to the case where possibly more than one observation is taken from each Poisson distribution and the samples are not necessarily of the same size.     

E-Bayesian estimation and its E-MSE under the scaled squared error loss function,for exponential distribution as example

This paper is concerned with using the E-Bayesian method for computing estimates of exponential distribution. In order to measure the estimated error, based on the E-Bayesian estimation, we proposed the definition of E-MSE(expected mean square error). Moreover, the formulas of E-Bayesian estimation and formulas of E-MSE are given respectively, these estimations are derived based on a conjugate prior distribution for the unknown parameter under the scaled squared error loss function. The properties of E-MSE under different scaled parameters are also provided. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and a real data set have been analysed for illustrative purposes. Results are compared on the basis of E-MSE.     

Minimax estimation of a probability of success under LINEX loss

Minimax estimation of a binomial probability under LINEX loss function is considered. It is shown that no equalizer estimator is available in the statistical decision problem under consideration. It is pointed out that the problem can be solved by determining the Bayes estimator with respect to a least favorable distribution having finite support. In this situation, the optimal estimator and the least favorable distribution can be determined only by using numerical methods. Some properties of the minimax estimators and the corresponding least favorable prior distributions are provided depending on the parameters of the loss function. The properties presented are exploited in computing the minimax estimators and the least favorable distributions. The results obtained can be applied to determine minimax estimators of a cumulative distribution function and minimax estimators of a survival function.     

The exact mean squared error risks of preliminary test type estimators for the multivariate normal mean

The exact mean squared error risks of the preliminary test estimtor and the Sclove modified Stein rule estimator (Sclove, Morris and Radhakrishnan, 1972) for the multivariate normal mean are computed and their risks are compared with the risks of Stein estimators.     

Shrinkage Estimators for Prediction Out-of-Sample: Conditional Performance

We find that, in a linear model, the James–Stein estimator, which dominates the maximum-likelihood estimator in terms of its in-sample prediction error, can perform poorly compared to the maximum-likelihood estimator in out-of-sample prediction. We give a detailed analysis of this phenomenon and discuss its implications. When evaluating the predictive performance of estimators, we treat the regressor matrix in the training data as fixed, i.e., we condition on the design variables. Our findings contrast those obtained by Baranchik (1973 Baranchik , A. J. ( 1973 ). Inadmissibility of maximum likelihood estimators in some multiple regression problems with three or more independent variables . Ann. Statist. 1 ( 2 ): 312 – 321 .[Crossref], [Web of Science ®] , [Google Scholar]) and, more recently, by Dicker (2012 Dicker , L. ( 2012 ). Dense signals, linear estimators, and out-of-sample prediction for high-dimensional linear models. arXiv:1102.2952 [math.ST].  [Google Scholar]) in an unconditional performance evaluation.     

New types of shrinkage estimators of Poisson means under the normalized squared error loss

In estimating p( ? 2) independent Poisson means, Clevenson and Zidek (1975) have proposed a class of estimators that shrink the unbiased estimator to the origin and dominate the unbiased one under the normalized squared error loss. This class of estimators was subsequently enlarged in several directions. This article deals with the problem and proposes new classes of dominating estimators using prior information pertinently. Dominance is shown by partitioning the sample space into disjoint subsets and averaging the loss difference over each subset. Estimation of several Poisson mean vectors is also discussed. Further, simultaneous estimation of Poisson means under order restriction is treated and estimators which dominate the isotonic regression estimator are proposed for some types of order restrictions.     

Multiparameter estimation in truncated power series distributions under the stein's loss

LetX ₁,…,X _p be p(≥2)independent random variables, where each X.has a distribution belonging to a one parameter truncated power series

distribution. The problem is to estimate simultaneously the unknown parameters under asymmetric loss developed by James and Stein (Proc. Fourth Berkeley Symp. Math. Statist. Prob. 1, 361-380). Several new classes of dominating estimators are obtained by solving a certain difference inequality.     

On robust Bayesian estimation under some asymmetric and bounded loss function

The problem of Bayesian and robust Bayesian estimation with some bounded and asymmetric loss function ABL is considered for various models. The prior distribution is not exactly specified and covers the conjugate family of prior distributions. The posterior regret, most robust and conditional Γ-minimax estimators are constructed and a preliminary comparison with square-error loss and LINEX loss is presented.     

Minimax estimation of a lower-bounded scale parameter of a gamma distribution for scale-invariant squared-error loss

Let X have a gamma distribution with known shape parameter θr;aL and unknown scale parameter θ. Suppose it is known that θ ≥ a for some known a > 0. An admissible minimax estimator for scale-invariant squared-error loss is presented. This estimator is the pointwise limit of a sequence of Bayes estimators. Further, the class of truncated linear estimators C = {θ_ρ|θ_ρ(x) = max(a, ρ), ρ > 0} is studied. It is shown that each θ_ρ is inadmissible and that exactly one of them is minimax. Finally, it is shown that Katz's [Ann. Math. Statist., 32, 136–142 (1961)] estimator of θ is not minimax for our loss function. Some further properties of and comparisons among these estimators are also presented.     

Comparison of the Stein and the usual estimators for the regression error variance under the Pitman nearness criterion when variables are omitted

This paper compares the Stein and the usual estimators of the error variance under the Pitman nearness (PN) criterion in a regression model which is mis-specified due to missing relevant explanatory variables. The exact expression of the PN-probability is derived and numerically evaluated. Contrary to the well-known result under mean squared errors (MSE), with the PN criterion the Stein variance estimator is uniformly dominated by the usual estimator when no relevant variables are excluded from the model. With an increased degree of model mis-specification, neither estimator strictly dominates the other. The authors are grateful to two anonymous referees for their valuable comments. Also, the first author is grateful to the Japan Society for the Promotion of Science for partial financial support.     

Bayesian estimation for randomly censored generalized exponential distribution under asymmetric loss functions

This paper deals with the Bayesian estimation of generalized exponential distribution in the proportional hazards model of random censorship under asymmetric loss functions. It is well known for the two-parameter lifetime distributions that the continuous conjugate priors for parameters do not exist; we assume independent gamma priors for the scale and the shape parameters. It is observed that the closed-form expressions for the Bayes estimators cannot be obtained; we propose Tierney–Kadane's approximation and Gibbs sampling to approximate the Bayes estimates. Monte Carlo simulation is carried out to observe the behavior of the proposed methods and one real data analysis is performed for illustration. Bayesian methods are compared with maximum likelihood and it is observed that the Bayes estimators perform better than the maximum-likelihood estimators in some cases.     

Bayes sequential estimation for a Poisson process under a LINEX loss function

In this paper, within the framework of a Bayesian model, we consider the problem of sequentially estimating the intensity parameter of a homogeneous Poisson process with a linear exponential (LINEX) loss function and a fixed cost per unit time. An asymptotically pointwise optimal (APO) rule is proposed. It is shown to be asymptotically optimal for the arbitrary priors and asymptotically non-deficient for the conjugate priors in a similar sense of Bickel and Yahav [Asymptotically pointwise optimal procedures in sequential analysis, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, University of California Press, Berkeley, CA, 1967, pp. 401–413; Asymptotically optimal Bayes and minimax procedures in sequential estimation, Ann. Math. Statist. 39 (1968), pp. 442–456] and Woodroofe [A.P.O. rules are asymptotically non-deficient for estimation with squared error loss, Z. Wahrsch. verw. Gebiete 58 (1981), pp. 331–341], respectively. The proposed APO rule is illustrated using a real data set.     

Comparison of procedures for estimation of error rates in discriminant analysis under nonnormal population

The parametric and nonparametric methods for estimating the error rates in linear discriminant analysis are examined both in normal and in nonnormal situations. A Monte Carlo experiment was carried out under the assumption that two population distributions were characterized by a mixture of two multivariate normal distributions. The bootstrap bias-corrected apparent error rate compares favourably to other available estimators for nonnormal populations with small Mahalanobis distance. The methods for error estimation are also applied to a practical problem in medical diagnosis     

Focused estimation for noisy and small data sets: a Bayesian minimum expected loss estimator approach

Central to many inferential situations is the estimation of rational functions of parameters. The mainstream in statistics and econometrics estimates these quantities based on the plug‐in approach without consideration of the main objective of the inferential situation. We propose the Bayesian Minimum Expected Loss (MELO) approach focusing explicitly on the function of interest, and calculating its frequentist variability. Asymptotic properties of the MELO estimator are similar to the plug‐in approach. Nevertheless, simulation exercises show that our proposal is better in situations characterised by small sample sizes and/or noisy data sets. In addition, we observe in the applications that our approach gives lower standard errors than frequently used alternatives when data sets are not very informative.     

A balanced approach to region estimation with tables for the normal model

Practitioners of statistics are too often guilty of routinely selecting a 95% confidence level in interval estimation and ignoring the sample size and the expected size of the interval. One way to balance coverage and size is to use a loss function in a decision problem. Then either the Bayes risk or usual risk (if a pivotal quantity exists) may be minimized. It is found that some non-Bayes solutions are equivalent to Bayes results based on non-informative priors. The decision theory approach is applied to the mean and standard deviation of the univariate normal model and the mean of the multivariate normal. Tables are presented for critical values, expected size, confidence and sample size.