Pitman's Measure Of Closeness Research Articles

A new least squares method for estimation and prediction based on the cumulative Hazard function

<abstract><p>In this paper, the cumulative hazard function is used to solve estimation and prediction problems for generalized ordered statistics (defined in a general setup) based on any continuous distribution. The suggested method makes use of Rényi representation. The method can be used with type Ⅱ right-censored data as well as complete data. Extensive simulation experiments are implemented to assess the efficiency of the proposed procedures. Some comparisons with the maximum likelihood (ML) and ordinary weighted least squares (WLS) methods are performed. The comparisons are based on both the root mean squared error (RMSE) and Pitman's measure of closeness (PMC). Finally, two real data sets are considered to investigate the applicability of the presented methods.</p></abstract>

The generalized Pitman measure of similarity and hierarchical clustering

Pitman measure of closeness (PMC) is a criterion to show how much an estimator is close to its parameter with respect to another estimator. We develop a new similarity measure which uses PMC on the basis of the similarity measure for hierarchical clustering algorithm that has better performance for heavy-tailed data.

A Pitman measure of similarity in k-means for clustering heavy-tailed data

ABSTRACTOne of the most popular methods and algorithms to partition data to k clusters is k-means clustering algorithm. Since this method relies on some basic conditions such as, the existence of mean and finite variance, it is unsuitable for data that their variances are infinite such as data with heavy tailed distribution. Pitman Measure of Closeness (PMC) is a criterion to show how much an estimator is close to its parameter with respect to another estimator. In this article using PMC, based on k-means clustering, a new distance and clustering algorithm is developed for heavy tailed data.

On the estimation of missing values in AR(1) model with exponential innovations

ABSTRACTWe consider estimation of a missing value for a stationary autoregressive process of order one with exponential innovations and compare two methods of estimation of the missing value, with respect to Pitman's measure of closeness (PMC).

Comparison of preliminary test estimators based on generalized order statistics from proportional hazard family using Pitman measure of closeness

ABSTRACTIn this article, based on generalized order statistics from a family of proportional hazard rate model, we use a statistical test to generate a class of preliminary test estimators and shrinkage preliminary test estimators for the proportionality parameter. These estimators are compared under Pitman measure of closeness (PMC) as well as MSE criteria. Although the PMC suffers from non transitivity, in the first class of estimators, it has the transitivity property and we obtain the Pitman-closest estimator. Analytical and graphical methods are used to show the range of parameter in which preliminary test and shrinkage preliminary test estimators perform better than their competitor estimators. Results reveal that when the prior information is not too far from its real value, the proposed estimators are superior based on both mentioned criteria.

Robust Bayesian Pitman closeness

In this paper, the robust Bayesian methodology has been developed in the sense of Pitman measure of closeness. To do this, the definition of Pitman posterior closeness, introduced by Ghosh and Sen (Commun Stat Theory Methods 20:3659–3678, 1991) and simultaneous closeness are integrated. First, the \(\varGamma \)-minimax problem is developed in the sense of Pitman’s criterion and the basic results and definitions are provided. Then, several results regarding Pitman \(\varGamma \)-minimax have been proved. Some examples have been presented to illustrate the application of the findings. Finally, other aspect of robust Bayesian methodology such as: Pitman stable rules and Pitman regret type estimators have been proposed.

Comparison of order statistics in two-sample problem in the sense of Pitman closeness

In this paper, we study the Pitman measure of closeness of order statistics of two independent samples from the same distribution to population quantiles. We then derive various exact expressions of the probability closeness of order statistics from the X and Y samples. Some distribution-free results for the median of the sampling distribution are obtained. Exact and explicit expressions are presented for Uniform(−1, 1) and exponential distributions. Numerical results for illustrative purposes are also provided.

Comparisons among some predictors of exponential distributions using Pitman closeness

The bias, mean squared error and likelihood optimality criteria are used frequently to compare estimators and predictors. Recently, the probability of nearness around a statistic (estimator or predictor) has received a considerable attention in the literature. In this article, we adopt the Pitman's measure of closeness (PMC) as an optimality criterion to compare the maximum likelihood, best linear unbiased, best linear invariant, median unbiased and conditional median predictors of a future ordered statistic based on a type II censored sample from an exponential distribution with unknown scale parameter. Numerical computations of the PMC for all comparisons among these predictors are performed and presented.

A note on Pitman's measure of closeness with balanced loss function

In this note, we consider the problem of estimating an unknown parameter θ in the sense of the Pitman's measure of closeness (PMC) using the balanced loss function (BLF). We show that the PMC comparison of estimators under the BLF can be reduced to the PMC comparison under the usual absolute error loss. The Pitman-closest estimators of the location and scale parameters under BLF are also characterized. Illustrative examples are given to show the broad range applications of the obtained results.

Estimating the lifetime performance index with Weibull distribution based on progressive first-failure censoring scheme

An important topic in the manufacturing industries is the assessing of the lifetime performance. In this paper, it is supposed the lifetimes of products are independent and have a common Weibull distribution with a known shape parameter. The lifetime performance index (CL) proposed by Montgomery (1985) [1], is used for evaluating the performance of a process with respect to a lower specification limit (L). The maximum likelihood estimate of CL is obtained on the basis of the progressive first-failure censored data. This estimate is then used for developing a confidence interval for CL. The behavior of the confidence interval for the parameter CL given a significance level is investigated and also two illustrative examples and a sensitivity analysis are given. For the exponential distribution as a special case of Weibull distribution, a comparison study for various estimates of CL based on mean squared error (MSE) and Pitman measure of closeness (PMC) criteria is done.

Estimation with left-truncated and right censored data: A comparison study

Estimation based on the left-truncated and right randomly censored data arising from a general family of distributions is considered. In the special case, when the random variables satisfy a proportional hazard model, the maximum likelihood estimators (MLEs) as well as the uniformly minimum variance unbiased estimators (UMVUEs) of the unknown parameters are obtained. Explicit expressions for the MLEs are obtained when the random variables follow an exponential distribution. In the latter case, three different estimators for the parameter of interest are proposed. These estimators are compared using the criteria of mean squared error (MSE) and Pitman measure of closeness (PMC). It is shown that shrinking does not always yield a better estimator.

Estimating Common Mean of a Bivariate Normal Population with Order Restricted Variances

Let ( X1, Y1), ( X2, Y2),…,( X n, Y n) be a random sample of size n from a bivariate normal population with a common mean μ variances [Formula: see text] and [Formula: see text] and correlation coefficient ρ The estimation of the common mean μ has been considered when it is known a priori that the variances are ordered, that is, [Formula: see text] A new estimator has been proposed which dominates the maximum likelihood estimator (without restrictions) stochastically and also in terms of Pitman measure of closeness. Further an inadmissibility condition has been obtained for affine equivariant estimators when the variances are ordered. A numerical comparison of various estimators has been carried out.

ON THE ESTIMATION OF σ AND THE PROCESS CAPABILITY INDICES Cp AND Cpm

We consider the problem of estimation of the standard deviation σ and the process capability indices Cp and Cpm for a normally distributed process. The problem is addressed with the objective of obtaining optimal estimators under the classic criterion of minimum variance unbiased estimation, as well as under the Pitman measure of closeness.

On estimation of the common mean of k (⩾ 2) normal populations with order restricted variances

For estimating the common mean of k (⩾ 2) normal populations with order restricted unknown variances, using the theory of isotomic regression, we propose a simple estimator which is better than the usual Graybill-Deal estimator in terms of stochastic dominance and the Pitman measure of closeness.

Estimation of an exponential quantile under pitman's measure of closeness

AbstractWe consider the problem of estimating a quantile of an exponential distribution with unknown location and scale parameters under Pitman's measure of closeness (PMC). The loss function is required to satisfy some mild conditions but is otherwise arbitrary. An optimal estimator is obtained in the class of location‐scale‐equivariant estimators, and its admissibility in the sense of PMC is investigated.

Further investigations of berkson's example

Berkson's bioassay problem is investigated in detail. In addition to maximum likelihood estimators and Berkson's proposed minimum logit χ2 estimators, several modified estimators are considered. Two Rao-Blackwellized estimators - a conditional mean and conditional median, given the sufficient statistics - as well as Amemiya's bias-corrected and partially bias-correct estimators are included. The estimators are compared with respect to bias, variance, mean squared error, and Pitman's measure of closeness. The experiment is examined at four different levels of the parameters, and is re-examined subject to a monotonicity condition imposed on the sample space. The results provide a new insight into the controversy over the comparison of minimum χ2 and maximum likelihood estimators.

Karlin's corollary: a topological approach to pitman's measure

In this article, many of the known univariate results about Pitman's Measure of Closeness (PMC) are synthesized through a topological approach. The proofs of many known results are simplified and clarified. The approach extends some previous results established under other restrictions. Connections between PMC and Bayesian estimation are discussed but the inherent interpretations differ. A discourse on this connection can be found in the article of Ghosh and Sen (1991). A transitiveness property for ordered estimators is established and a counter example is given for unordered ones. These results help distinguish between the Bayesian and classical interpretations of Pitman's measure.

Preface

Abstract The appearance of this issue will mark almost two years since its inception by the late Editor, Professor Don B. Owen. Professor Malay Ghosh suggested a special issue on Pitman's Measure of Closeness (PMC) to Professor Owen in autumn of 1989. After a thorough review process the issue was finalized in June 1991. It is with remorse that we publish this special issue in memory of Professor Owen. The completion of the issue coincided with a special conference, “Pitman's Measure of Cfoseness: Celebrating a Decade of Renaissance,” held on June 15, 1991 at the University of Texas at San Antonio (UTSA). The papers in this special issue, except those of Professor Kubokawa and Drs. Bertuzzi and Gandolfi, were presented and discussed. The conference participants included C.R. Rao (Pennsylvania State University), Colin Blyth (Queen's University), and H.T. David (Iowa State University). Further, P.K. Sen (University of North Carolina) and Malay Ghosh (University of Florida) gave keynote addresses that respectively set the themes for the morning and afternoon sessions. The conference banquet held in the Regent's Room at the University of Texas at San Antonio, featured stimulating addresses by C.R. Rao and Colin Blyth on some of the major controversies of PMC such as intransitiveness and Berkson's conjecture. We are grateful to the University of Texas at San Antonio for making this conference a reality. In particular, we thank Professor Shair Ahmad, Director of the Division of Mathematics, Computer Science, and Statistics at UTSA for funding travel expenses not only for the invited speakers but also for many of the younger researchers. We also acknowledge University President Samuel Kirkpatrick who made the KIVA room available for the technical sessions and the Regent's Room for the Banquet. We also acknowledge the financial support of Bell Helicopter Textron, Inc. as a cosponsor of the conference.

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.

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...