Pitman Nearness Research Articles

A Comparison of the Maximum Likelihood Estimator and the Posterior Mean in the Single-Parameter Case

Abstract Consider estimation of a scalar parameter θ of a distribution on the basis of a random sample of size n. Possible estimators include the maximum likelihood estimator, n , and the posterior mean with respect to some prior distribution, n . In this article n and n are compared by considering a large-sample version of the Pitman nearness criterion; conditions are given under which n is preferable to n in this sense. Several examples are provided that show that comparisons of n , and n based on Pitman nearness are often in conflict with comparisons based on mean squared error.

A Comparison of the Maximum Likelihood Estimator and the Posterior Mean in the Single-Parameter Case

Abstract Consider estimation of a scalar parameter θ of a distribution on the basis of a random sample of size n. Possible estimators include the maximum likelihood estimator, n , and the posterior mean with respect to some prior distribution, n . In this article n and n are compared by considering a large-sample version of the Pitman nearness criterion; conditions are given under which n is preferable to n in this sense. Several examples are provided that show that comparisons of n , and n based on Pitman nearness are often in conflict with comparisons based on mean squared error.

Pitman nearness in statistical estimation: A panel discussion on recent developments

A panel discussion including an introduction by Sen, a paper by Keating and Mason: «Closeness comparison of classical and inverse regression estimator» a second one by Sen on recent developments in Pitman closeness and its applications and some final remarks by Rao

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.

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 confidence interval for the Pitman Nearness probability. A numerical example is given.

Ridge regression versus ols by pitman's closeness under quadratic and fisher's loss

The existence of values of the ridge parameter such that ridge regression is preferable to OLS by the Pitman nearness criterion under both the quadratic and the Fisher's loss is shown. Preference regions of the two estimators under the above loss functions are found. An upper bound for the value of the Pitman's measure of closeness, independent of a deterministic or stochastic choice of the ridge parameter, is given.

Estimation of location and scale parameters using generalized Pitman nearness criterion

Abstract A generalization of the Pitman nearness criterion for comparing estimators when the criterion for accuracy is based on a loss function is developed. The concept is applicable to multi-parameter problems. The generalized criterion is then used with invariance to find best estimators of a location or scale parameter. These best estimators do not depend on the form of the loss function provided that it satisfies some reasonable conditions. Thus, the generalized criterion is useful for finding a best estimator when the exact loss function is not known. Some examples are given which establish an additional optimality property of some commonly used estimators.

Comparisons of changepoint estimators

Statistical comparisons of several nonparametric changepoint estimators which are currently offered in the literature are undertaken. The criteria for comparison are Pitman nearness, probability concentration and mean squared error. Recommendations for small, moderate and large sample cases are given.

On the characterization of Pitman measure of nearness

Two sufficient conditions for characterizing Pitman nearness (PN) are derived. It is found that one of the conditions is a standardized decision criterion. Two examples are given, including comparison of the James—Stein estimator with the usual unbiased estimator. It is also shown that stochastic domination implies PN for comparing any two multivariate normal estimators. Moreover, they are equivalent if two estimators are unbiased.

A Comparison of james–sten regression with least squares in the pitman nearness sense

The least squares ectimatnr and the James-Stein estimator of rhe vector-valued parameter in a multiple linear regression model are compared in thc Pitman searness sense. A table of the comparison is presented, and the James—Siein esiiriiaior is fad to be uniform!y preferred te the least squares in this sense. In many practical situations the James-Stein estimate is extremely close to the least squares estimate, and so is barely preferred in the Pitman nearness sense. However, there are practical situations in which the James—Stein estimate is sufficiently different from the least squares estimate to be substantially better in the Pitman nearness sense.

Estimation of matrix valued realized signal to noise ratio

In this paper we derive several estimators of matrix valued realized signal to noise ratio as defined by Khatri and Rao (1987, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, No. 5 671–679) for real and complex cases. To do so we define the matrix valued confluent hypergeometric distribution and establish some of its properties. Also we derive unique admissible estimates under generalized Pitman nearness. Finally a discussion of confidence interval estimation for signal to noise ratio is given.

On the inadmssibelity of ridge estimator in a linear model

The necessary and sufficient conditions for the inadmissibility of the ridge regression is discussed under two different criteria, namely, average loss and Pitman nearness. Although the two criteria are very different, same conclusions are obtained. The loss functions considered in this article are th likelihood loss function and the Mahalanobis loss function. The two loss functions are motivated from the point of view of classification of two normal populations. Under the Mahalanobis loss it is demonstrated that the ridge regression is always inadmissible as long as the errors are assumed to be symmetrically distributed about the origin.

James-Stein Estimation from an Alternative Perspective

Abstract 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.

Predictive power of weekly money stock announcements

The objective of this study is to test if the weekly money stock can help to predict the monthly money stock using the data in the period of 1974 to 1985. Pitman's nearness measure and a statistic which is similar to the linear feedback statistic are used, respectively, for the qualitative and quantitative investigations. Other variables, such as interest rates and stock prices are also considered in the model. The results indicate that the weekly money supply provides little information concerning the monthly money stock.

A short note on pitman nearness for elliptically symmetric estimators

The Pitman nearness criterion for estimators with elliptically symmetric distributions is considered. Certain results in Peddada and Khattree (1986) are generalized for this family of distributions.

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.

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 Comparison Of Estimates Of Dispersion Using Generalized Pitman Nearness Criterion

In this paper, estimates QP dispersion matrix and its functions are compared based on generalized Pitman nearness criterion, Various Iosa functions are considered for the purpose. Locally superior estimates are defined and obtained. Comparison of these estimates are made with other standard ones. It is snown that within certain classes, defined in the paper, these are the best estimatcrs ia the generalized Fitman nearness sense

The pitman nearness criterion and its determination

A general method for determining Pitman Nearness is given In the case of univariate estimators. This method is then applied to some estimation problems. The concept of Pitman Nearness is also generalized to the multivariate case. The James-Stein estimators are used to illustrate the multivariate comparison.

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.