Pitman nearness comparisons of Stein-type estimators for regression coefficients in replicated experiments

This paper presents a comparative study of the performance properties of one unbiased and two Stein-type estimators for combining the estimates of coefficients in a linear regression model when data sets are available from replicated experiments conducted at possibly different stations.     

Comparison of regression estimators using Pitman measures of nearness

Received: August 5, 1999; revised version: June 14, 2000     

A comparison of biased regression estimators using a pitman nearness criterion

Biased regression estimators have traditionally benn studied using the Mean Square Error (MSE) criterion. Usually these comparisons have been based on the sum of the MSE's of each of the individual parameters, i.e., a scaler valued measure that is the trace of the MSE matrix. However, since this summed MSE does not consider the covariance structure of the estimators, we propose the use of a Pitman Measure of Closeness (PMC) criterion (Keating and Gupta, 1984; Keating and Mason, 1985). In this paper we consider two versions of PMC. One of these compares the estimates and the other compares the resultant predicted values for 12 different regression estimators. These estimators represent three classes of estimators, namely, ridge, shrunken, and principal component estimators. The comparisons of these estimators using the PMC criteria are contrasted with the usual MSE criteria as well as the prediction mean square error. Included in the estimators is a relatively new estimator termed the generalized principal component estimator proposed by Jolliffe. This estimator has previously received little attention in the literature.     

Estimating powers of the generalized variance under the Pitman closeness criterion

For estimating powers of the generalized variance under a multivariate normal distribution with an unknown mean, the inadmissibility of the closest affine equivariant estimator is shown for the Pitman closeness criterion.     

On pitman nearness and variance of estimators

In this article, some conditions on variances are presented under which the (Generalized) Pitman Nearness Criterion Would Prefer one estimator over another. Results for univariate as well as multivariate cases are derived. An exact expression for a result of Rao, Keating and Mason (1985) is provided.     

Comparison of estimators of a dispersion matrix under generalized pitman nearness criterion

Some equivariant estimators of the dispersion matrix of a multivariate normal population are compared using the generalized Pitman nearness criterion based on the entropy loss function. It is shown that, under the group of lower triangular transformations, a best equivariant estimator does not exist. Existence of best estimators in certain subclasses are discussed and the performances of two commonly used estimators are compared. Some properties of central chi-square distributions are obtained and used to derive the main results.     

Optimal non-diagonal-type estimators in linear regression under the prediction error sum of squares criterion

This article considers the notion of the non-diagonal-type estimator (NDTE) under the prediction error sum of squares (PRESS) criterion. First, the optimal NDTE in the PRESS sense is derived theoretically and applied to the cosmetics sales data. Second, we make a further study to extend the NDTE to the general case of the covariance matrix of the model and then give a Bayesian explanation for this extension. Third, two remarks concerned with some potential shortcomings of the NDTE are presented and an alternative solution is provided and illustrated by means of simulations.     

Pitman closeness,monotonicity and consistency of best linear unbiased and invariant estimators for exponential distribution under Type II censoring

Comparisons of best linear unbiased estimators with some other prominent estimators have been carried out over the last 50 years since the ground breaking work of Lloyd [E.H. Lloyd, Least squares estimation of location and scale parameters using order statistics, Biometrika 39 (1952), pp. 88–95]. These comparisons have been made under many different criteria across different parametric families of distributions. A noteworthy one is by Nagaraja [H.N. Nagaraja, Comparison of estimators and predictors from two-parameter exponential distribution, Sankhyā Ser. B 48 (1986), pp. 10–18], who made a comparison of best linear unbiased (BLUE) and best linear invariant (BLIE) estimators in the case of exponential distribution. In this paper, continuing along the same lines by assuming a Type II right censored sample from a scaled-exponential distribution, we first compare BLUE and BLIE of the exponential mean parameter in terms of Pitman closeness (nearness) criterion. We show that the BLUE is always Pitman closer than the BLIE. Next, we introduce the notions of Pitman monotonicity and Pitman consistency, and then establish that both BLUE and BLIE possess these two properties.     

General linear estimators under the prediction error sum of squares criterion in a linear regression model

In this paper, the notion of the general linear estimator and its modified version are introduced using the singular value decomposition theorem in the linear regression model y=X β+e to improve some classical linear estimators. The optimal selections of the biasing parameters involved are theoretically given under the prediction error sum of squares criterion. A numerical example and a simulation study are finally conducted to illustrate the superiority of the proposed estimators.     

Comparison of the estimators of the intra-cluster correlation for the nested error regression model

The intra-cluster correlation is insisted on nested error regression model that, in practice, is rarely known. This article demonstrates the size in generalized least squares (GLS) F-test using Fuller–Battese transformation and modification F-test. For the balanced case, the former using strictly positive, analysis of covariance (ANCOVA) and analysis of variance (ANOVA) estimators of intra-cluster correlation can control the size for moderate intra-cluster correlations. For small intra-cluster correlation, they perform well when the numbers of cluster are large. The latter using the ANOVA estimator performs well except for small numbers of cluster. When intra-cluster correlation is large, it cannot control the size. For the unbalanced case, the GLS F-test using the Fuller–Battese transformation and the modification F-test using the strictly positive, the ANCOVA and the ANOVA estimators maintain the significance level for small total sample size and small intra-cluster correlations when there is a large variation in cluster sizes, but they perform well in controlling the size for large total sample size and small different variation in cluster sizes. Besides, Henderson’s method 3 estimator maintains the significance level for a few situations.     

Stein type confidence interval of the disturbance variance in a linear regression model with multivariate student-t distributed errors

This paper considers the interval estimation of the disturbance variance in a linear regression model with multivariate Student-t errors. The distribution function of the Stein type estimator under multivariate Student-t errors is derived, and the coverage probability of the Stein type confidence interval which is constructed under the normality assumption is numerically calculated under the multivariate Student-t distribution. It is shown that the coverage probability of the Stein type confidence interval is sometimes much smaller than the nominal level, and that it is larger than that of the usual confidence interval in almost all cases. For the case when it is known that errors have a multivariate Student-t distribution, sufficient conditions under which the Stein type confidence interval improves on the usual confidence interval are given, and the coverage probability of the stein type confidence interval is numerically evaluated.     

A Note on the Comparison of the Stein Estimator and the James-Stein Estimator

The seminal work of Stein (1956 Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proc. Third Berkeley Symp. Mathemat. Statist. Probab., University of California Press, 1:197–206. [Google Scholar]) showed that the maximum likelihood estimator (MLE) of the mean vector of a p-dimensional multivariate normal distribution is inadmissible under the squared error loss function when p ? 3 and proposed the Stein estimator that dominates the MLE. Later, James and Stein (1961 James, W., Stein, C. (1961). Estimation with quadratic loss. Proc. Fourth Berkeley Symp. Mathemat. Statist. Probab., University of California Press, 1:361–379. [Google Scholar]) proposed the James-Stein estimator for the same problem and received much more attention than the original Stein estimator. We re-examined the Stein estimator and conducted an analytic comparison with the James-Stein estimator. We found that the Stein estimator outperforms the James-Stein estimator under certain scenarios and derived the sufficient conditions.     

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.     

Comparison of risks of estimators under the LINEX loss for a family of truncated distributions

In most cases, we use a symmetric loss such as the quadratic loss in a usual estimation problem. But, in the non-regular case when the regularity conditions do not necessarily hold, it seems to be more reasonable to choose an asymmetric loss than the symmetric one. In this paper, we consider the Bayes estimation under the linear exponential (LINEX) loss which is regarded as a typical example of asymmetric loss. We also compare the Bayes risks of estimators under the LINEX loss for a family of truncated distributions and a location parameter family of truncated distributions.     

Inflation of Type I error rate in multiple regression when independent variables are measured with error

When independent variables are measured with error, ordinary least squares regression can yield parameter estimates that are biased and inconsistent. This article documents an inflation of Type I error rate that can also occur. In addition to analytic results, a large‐scale Monte Carlo study shows unacceptably high Type I error rates under circumstances that could easily be encountered in practice. A set of smaller‐scale simulations indicate that the problem applies to various types of regression and various types of measurement error. The Canadian Journal of Statistics 37: 33‐46; 2009 © 2009 Statistical Society of Canada     

Improved estimators for the location of double exponential distribution

In this paper, attention is focused on estimation of the location parameter in the double exponential case using a weighted linear combination of the sample median and pairs of order statistics, with symmetric distance to both sides from the sample median. Minimizing with respect to weights and distances we get smaller asymptotic variance in the second order. If the number of pairs is taken as infinite and the distances as null we attain the least asymptotic variance in this class of estimators. The Pitman estimator is also noted. Similarly improved estimators are scanned over their probability of concentration to investigate its bound. Numerical comparison of the estimators is shown.     

Comparison of the Iterative Stein-Rule and the Usual Estimators of the Disturbance Variance Under the Pitman Nearness Criterion in a Linear Regression Model with Proxy Variables

In a linear regression model with proxy variables, the iterative Stein-rule estimator and the usual estimator of the disturbance variance is compared under the Pitman Nearness Criterion. The exact expression of Pitman Nearness probability is derived and numerically evaluated.     

Stein’s method and the distribution of the product of zero mean correlated normal random variables

Abstract

Over the last 80?years there has been much interest in the problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables. Motivated by this historical interest, we use a recent technique from the Stein’s method literature to obtain a simple new proof, which also serves as an exposition of a general method that may be useful in related problems.     

Deviation of order p for estimators of the variance in first-order stochastic differential equation (SDE)

In this work, we consider a non-parametric estimator of the variance in one-dimensional diffusion models or, more generally, in Itô processes with a deterministic diffusion term and a general non-anticipative drift. The estimation is based on the quadratic variation of discrete time observations over a finite interval. In particular, a central limit theorem (CLT) is proved for the deviation in L ^p norm (p≥; 1) between the variance and this estimator. The method of the proof consists in writing the L ^p norm of the deviation, when the drift term is equal to zero, as a sum of 4-dependent random variables. The moments are then computed by means of a Gaussian approximation and a CLT for m-dependent random variables is applied. The convergence is stable in law, this allows the result for processes with general drifts to be obtained, by using Girsanov's formula.     

Risk comparison of the Stein-rule estimator in a linear regression model with omitted relevant regressors and multivariate<Emphasis Type="Italic">t</Emphasis> errors under the Pitman nearness criterion

In this paper we consider a linear regression model with omitted relevant regressors and multivariatet error terms. The explicit formula for the Pitman nearness criterion of the Stein-rule (SR) estimator relative to the ordinary least squares (OLS) estimator is derived. It is shown numerically that the dominance of the SR estimator over the OLS estimator under the Pitman nearness criterion can be extended to the case of the multivariatet error distribution when the specification error is not severe. It is also shown that the dominance of the SR estimator over the OLS estimator cannot be extended to the case of the multivariatet error distribution when the specification error is severe. This research is partially supported by the Grants-in-Aid for 21st Century COE program.