Shrunken estimators via a pitman nearness criterion

Shrunken estimators have traditionally been developed and studied using mean square error (MSE). Recent research on Pitman nearness (PN), however, indicates that it is an interesting, “intrinsic”, alternative to the mean square error (MSE) criterion for investigating estimators. Thus, we develop a shrunken estimator for the mean of a multivariate normal distribution based on minimizing PN, instead of MSE, Further, since the shrinkage factor of this estimator depends on unknown parameters, we examine two approaches for determining this factor: (1) “plug-in” estimates, (2) a range of values for the factor based on an approximate cońfidence interval for the Pitman Nearness probability. A numerical example is given.     

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.     

PITMAN NEARNESS COMPARISONS OF ESTIMATES OF TWO ORDERED NORMAL MEANS

Maximum likelihood estimates of ordered means of two normal distributions having common variance have been shown to be better than the usual maximum likelihood estimates (i.e. corresponding sample means) with respect to Pitman Nearness criterion. The maximum likelihood estimate of common variance taking into consideration the order restriction of the means is shown to have smaller mean square error than the unrestricted maximum likelihood estimate of the common variance. These two estimators have also been compared with respect to Pitman Nearness criterion.     

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.     

Simulated deficiencies of robust estimators of a proportionality constant

In linear regression, robust methods are at the beginning of their use in practice. In the small sample case, such robust methods provide a necessary measure of protection against deviations from the assumed error distribution. This paper studies through simulation the deficiencies of bioptimal estimators and compares them with more common methods like Huber's estimator or Tukey's estimator. Polyoptimal estimators are convex combinations of Pitman estimators and are optimally robust for a confrontation containing several shapes. The word confrontation is due to J.W. Tukey. It expresses the situation when compromising two or several error distributions. The paper uses the confrontation containing the Gaussian distribution along with a symmetric heavy-tailed distribution having a tail of order 0(t^-2) as t→ ±∞.     

Estimation of signal to noise ratio using mahalanobis distance

In this paper, we propose some alternative estimatiors to that given by C. G. Khatri and C. R. Rao (1985), for estimating Signal to Noise ratio. Using Pitman Nearness, Condition for prefering one estimator over the other is estabilished. It is shown numerically that estimators corresponding to Entropy loss function are better more oftern than those corresponding to Squared Error loss.     

On intra-class correlation coefficient estimation

Consider the problem of estimating the intra-class correlation coefficient of a symmetric normal distribution. In a recent article (Pal and Lim (1999)) it has been shown that the three popular estimators, namely—the maximum likelihood estimator (MLE), the method of moments estimator (MME) and the unique minimum variance unbiased estimator (UMVUE), are second order admissible under the squared error loss function. In this paper we study the performance of the above mentioned estimators in terms of Pitman Nearness Criterion (PNC) as well as Stochastic Domination Criterion (SDC). We then apply the aforementioned estimators to two real life data sets with moderate to large sample sizes, and bootstrap bias as well as mean squared errors are computed to compare the estimators. In terms of overall performance the MME seems most appealing among the three estimators considered here and this is the main contribution of our paper. Formerly University of Southewestern Louisisna     

Comparison of some common ratio estimators using pitman measure of nearness

In this article we compare some common ratio estimators for estimating the population total of a given characteristic. The sampling schemes considered are simple random sampling (S.R.S.) and S.R.S.under stratification. The comparisons are made using the Pitman Nearness criterion under the model-based approach. The error term is assumed normal with mean zero and variance σg(x). The function g(x) is a known function of the auxiliary variable x. Special interest is on the cases of g(x) =l and x. The result is found the same as that using MSE criterion, although the PN is very different from the MSE intrinsically.     

On characterizations of pitman closeness of some shrinkage estimators

For the multivariate normal mean (vector) estimation problem, some characterizations of the Pitman closest property of a general class of shrinkage (or Stein-rule) estimators (including the so called positive-rule versions) are studied. Further, for the same model when the parameter is restricted to a positively homogeneous cone, Pitman closeness of restricted shrinkage maximum likelihood estimators is established.     

Some Pitman closeness properties pertinent to symmetric populations

In this paper, we focus on Pitman closeness probabilities when the estimators are symmetrically distributed about the unknown parameter θ. We first consider two symmetric estimators θ?₁ and θ?₂ and obtain necessary and sufficient conditions for θ?₁ to be Pitman closer to the common median θ than θ?₂. We then establish some properties in the context of estimation under the Pitman closeness criterion. We define Pitman closeness probability which measures the frequency with which an individual order statistic is Pitman closer to θ than some symmetric estimator. We show that, for symmetric populations, the sample median is Pitman closer to the population median than any other independent and symmetrically distributed estimator of θ. Finally, we discuss the use of Pitman closeness probabilities in the determination of an optimal ranked set sampling scheme (denoted by RSS) for the estimation of the population median when the underlying distribution is symmetric. We show that the best RSS scheme from symmetric populations in the sense of Pitman closeness is the median and randomized median RSS for the cases of odd and even sample sizes, respectively.     

Bayes Estimation of Weibull Distribution Parameters Using Ranked Set Sampling

Estimation of the parameters of Weibull distribution is considered using different methods of estimation based on different sampling schemes namely, Simple Random Sample (SRS), Ranked Set Sample (RSS), and Modified Ranked Set Sample (MRSS). Methods of estimation used are Maximum Likelihood (ML), Method of moments (Mom), and Bayes. Comparison between estimators is made through simulation via their Biases, Relative Efficiency (RE), and Pitman Nearness Probability (PN). Estimators based on RSS and MRSS have many advantages over those that are based on SRS.     

some properties of posterior pitman closeness

This article presents some results on a Bayesian notion of Pitman Closeness, defined in Ghosh and Sen (1991) and called Posterior Pitman Closeness (PPC). Their criterion avoids some of the drawbacks of the well-known (frequentist) Pitman closeness criterion, as introduced by Pitman (1937). It is shown that, if two estimators have the same posterior distribution of the distance from θ, the posterior distribution of θ has to be symmetric. This implies, in particular, that the estimators are Posterior Pitman equivalent. It is also shown that the PPC criterion does not suffer from another paradoxical property illustrated by Blyth and Pathak (1985) - that of an estimator δ₁ being stochastically closer to a parameter θ than another estimator δ₂ and yet being Pitman closer to θ than δ₁. It turns out that, if δ₁ is stochastically closer to θ than δ₂, conditional on x, then it is also Posterior Pitman closer.

We show that the original multivariate concept of PPC is no longer transitive. We provide necessary and sufficient conditions for a Posterior Pitman closest estimator to exist, thus generalizing Theorems 2 and 3 of Ghosh and Sen (1991). We show that a Posterior Pitman closest estimator does not always exist in several dimensions.     

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.     

The successive raising estimator and its relation with the ridge estimator

The raised estimators are used to reduce collinearity in linear regression models by raising a column in the experimental data matrix which may be nearly linear with the other columns. The raising procedure has two components, namely stretching and rotating, which we can analyze separately. We give the relationship between the raised estimators and the classical ridge estimators. Using a case study, we show how to determine the perturbation parameter for the raised estimators by controlling the amount of precision to be retained in the original data.     

James-Stein Estimation from an Alternative Perspective

In the comparison of estimators, the typical viewpoint of mean squared error has been challenged by C. R. Rao. In this article we propose a method of selecting estimators in normal populations based on their regions of preference. These regions of preference are a natural consequence of Rao's emphasis on Pitman nearness. We apply the method in the case of estimation of the mean of a bivariate normal through the James-Stein class.     

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.     

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.     

Maximum-likelihood estimates and likelihood-ratio criteria for multivariate elliptically contoured distributions

For a class of multivariate elliptically contoured distributions the maximum-likelihood estimators of the mean vector and covariance matrix are found under certain conditions. Likelihood-ratio criteria are obtained for a class of null hypotheses. These have the same form as in the normal case.     

Estimation of parameters of exponential distribution in the truncated space using asymmetric loss function

This paper is concerned with estimation of location and scale parameters of an exponential distribution when the location parameter is bounded above by a known constant. We propose estimators which are better than the standard estimators in the unrestricted case with respect to the suitable choice of LINEX loss. The admissibility of the modified Pitman estimators with respect to the LINEX loss is proved. Finally the theory developed is applied to the problem of estimating the location and scale parameters of two exponential distributions when the location parameters are ordered.     

Pitman closeness comparison of linear estimators:a canonical form

A general form is presented for the comparison of two linear estimators of a common parameter by means of the Pitman measure of closeness. Several asymptotic results are given. The case in which the estimators are linear combinations of the order statistics is discussed. The asymptotic comparison of the sample mean versus the sample median is derived for the Laplace distribution, and two other examples are given.