Jerome P. Keating

Pitman's measure of closeness : a comparison of statistical estimators

Preface Part I. Introduction 1. Evolution of Estimation Theory Least Squares Method of Moments Maximum Likelihood Uniformly Minimum Variance Unbiased Estimation Biased Estimation Bayes and Empirical Bayes Influence Functions and Resampling Techniques Future Directions 2. PMC Comes of Age PMC: A Product of Controversy PMC as an Intuitive Criterion 3. The Scope of the Book History, Motivation, and Controversy of PMC A Unified Development of PMC Part II. Development of Pitmans Measure of Closeness: 1. The Intrinsic Appeal of PMC Use of MSE Historical Development of PMC Convenience Store Example 2. The Concept of Risk Renyis Decomposition of Risk How Do We Understand Risk? 3. Weakness in the Use of Risk When MSE Does Not Exist Sensitivity to the Choice of the Loss Function The Golden Standard 4. Joint Versus Marginal Information Comparing Estimators with an Absolute Ideal Comparing Estimators with One Another 5. Concordance of PMC with MSE and MAD Part III. Anomalies with PMC: 1. Living in an Intransitive World Round-Robin Competition Voting Preferences Transitiveness 2. Paradoxes Among Choice The Pairwise-Worst Simultaneous-Best Paradox The Pairwise-Best Simultaneous-Worst Paradox Politics: The Choice of Extremes 3. Raos Phenomenom 4. The Question of Ties Equal Probability of Ties Correcting the Pitman Criterion A Randomized Estimator 5. The Rao-Berkson Controversy Minimum Chi-Square and Maximum Likelihood Model Inconsistency Remarks Part 4. Pairwise Comparisons 1. Geary-Rao Theorem 2. Applications of the Geary-Rao Theorem 3. Karlins Corollary 4. A Special Case of the Geary-Rao Theorem Surjective Estimators The MLR Property 5. Applications of the Special Case 6. Transitiveness Transitiveness Theorem Another Extension of Karlins Corollary Part V. Pitman-Closest Estimators: 1. Estimation of Location Parameters 2. Estimators of Scale 3. Generalization via Topological Groups 4. Posterior Pitman Closeness 5. Linear Combinations 6. Estimation by Order Statistics Part 6. Asymptotics and PMC 1. Pitman Closeness of BAN Estimators Modes of Convergence Fisher Information BAN Estimates are Pitman Closet 2. PMC by Asymptotic Representations A General Proposition 3. Robust Estimation of a Location Parameter L-Estimators M-Estimators R-Estimators 4. APC Characterizations of Other Estimators Pitman Estimators Examples of Pitman Estimators PMC Equivalence Bayes Estimators 5. Second-Order Efficiency and PMC Asymptotic Efficiencies Asymptotic Median Unbiasedness Higher-Order PMC Index Bibliography.

Practical Relevance of an Alternative Criterion in Estimation

Abstract Some pedagogical and practical examples, both univariate and bivariate, are given in which Pitmans measure of closeness is more relevant than mean squared error. The examples also illustrate interesting characteristics of Pitmans measure of closeness and provide some practical insight into the Rao—Berkson controversy in estimation theory. These examples are given in the hope that some moderation might be practiced in the teaching of mean squared error in modern statistical education.

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 Raos 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.

On the Determination of Critical Values for Bartlett's Test

Abstract The exact critical values for Bartletts test for homogeneity of variances based on equal sample sizes from several normal populations are tabulated. It is also shown how these values may be used to obtain highly accurate approximations to the critical values for unequal sample sizes. An application is given that deals with the variability of log bids on a group of federal offshore oil and gas leases.

Pitman Closeness of Order Statistics to Population Quantiles

In this article, Pitman closeness of sample order statistics to population quantiles of a location-scale family of distributions is discussed. Explicit expressions are derived for some specific families such as uniform, exponential, and power function. Numerical results are then presented for these families for sample sizes n = 10,15, and for the choices of p = 0.10, 0.25, 0.75, 0.90. The Pitman-closest order statistic is also determined in these cases and presented.

Comparison of Linear Estimators Using Pitman's Measure of Closeness

Abstract A method is given for the comparison of two linear forms of a common random vector under the criterion of Pitmans measure of closeness. Assuming multivariate normality of the random vector, one can determine exact expressions for the closeness probabilities. The applicability of the theory is illustrated on the comparison of ridge regression estimators.

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.

Pitman Closeness Comparison of Best Linear Unbiased and Invariant Predictors for Exponential Distribution in One- and Two-Sample Situations

Best linear unbiased, best linear invariant, and maximum likelihood predictors are commonly used in reliability studies for predicting either censored failure times or lifetimes from a future life-test. In this article, by assuming a Type-II right-censored sample from an exponential distribution, we compare best linear unbiased (BLUP) and best linear invariant (BLIP) predictors of the censored order statistics in the one-sample case and order statistics from a future sample in the two-sample case, in terms of Pitman closeness criterion. Some specific conclusions are drawn and supporting numerical results are presented.

The Pitman comparison of unbiased linear estimators

The canonical form for the comparison of certain linear estimators using Pitmans Measure of Closeness is generalized to the class of all linear estimators. Under the assumption of normality, the equivalence of Pitman-closest linear unbiased estimators and best linear unbiased estimators is shown. A sufficient condition is given for which the BLUE will be Pitman-closer than the best linear equivalent estimator (BLEE).

VOLCANISM AND MESOAMERICAN ARCHAEOLOGY

Drought and drought-induced famine are recurring phenomena in Mesoamerica that have devastated populations in the region repeatedly during the past two millennia. Although it is counterintuitive to conceive of the idea that volcanic eruptions anywhere in the world might affect the lives of people in Mesoamerica, we examine the reports of drought and famine during the period A.D. 1440 to 1840 and compare them with known, large volcanic eruptions. We then apply non-parametric statistical techniques to determine whether the coincidences seen between worldwide volcanic eruptions and Mesoamerican drought within the following two years were due to random chance or whether there was a direct, mathematically verifiable correlation. We find a direct correlation to a probability of 56 in 100 million. We conclude that due to its unique geographical position, Mesoamerica was repeatedly devastated by drought and subsequent famine between 1440 and 1840 due to the indirect climatic effects of large volcanic eruptions that could be located anywhere in the world.