Pranab Kumar Sen

Estimates of the Regression Coefficient Based on Kendall's Tau

Abstract The least squares estimator of a regression coefficient β is vulnerable to gross errors and the associated confidence interval is, in addition, sensitive to non-normality of the parent distribution. In this paper, a simple and robust (point as well as interval) estimator of β based on Kendalls [6] rank correlation tau is studied. The point estimator is the median of the set of slopes (Yj - Yi )/(tj-ti ) joining pairs of points with ti ≠ ti , and is unbiased. The confidence interval is also determined by two order statistics of this set of slopes. Various properties of these estimators are studied and compared with those of the least squares and some other nonparametric estimators.

Handbook of sequential analysis

Sequential analysis refers to the body of statistical theory and methods where the sample size may depend in a random manner on the accumulating data. A formal theory in which optimal tests are derived for simple statistical hypotheses in such a framework was developed by Abraham Wald in the early 1

Nonparametric methods in general linear models

Distribution theory of rank statistics: Distribution Theory of Linear Rank-Order Statistics Distribution Theory of Signed Rank Order Statistics Distribution Theory of Multivariate Linear Rank-Order Statistics Nonparametric inference in linear models: Distribution-Free Rank-Order Tests for Some Linear Hypotheses Rank-Order Estimation Theory in Some Linear Models Asymptotically Distribution-Free Aligned Rank- Order Tests for Some General Linear Hypotheses Rank-Order Tests for Miscellaneous Problems in Linear Models Appendix.

Constrained statistical inference : inequality, order, and shape restrictions.

Dedication. Preface. 1. Introduction. 1.1 Preamble. 1.2 Examples. 1.3 Coverage and Organization of the Book. 2. Comparison of Population Means and Isotonic Regression. 2.1 Ordered Hypothesis Involving Population Means. 2.2 Test of Inequality Constraints. 2.3 Isotonic Regression. 2.4 Isotonic Regression: Results Related to Computational Formulas. 3. Two Inequality Constrained Tests on Normal Means. 3.1 Introduction. 3.2 Statement of Two General Testing Problems. 3.3 Theory: The Basics in 2 Dimensions. 3.4 Chi-bar-square Distribution. 3.5 Computing the Tail Probabilities of chi-bar-square Distributions. 3.6 Detailed Results relating to chi-bar-square Distributions. 3.7 LRT for Type A Problems: V is known. 3.8 LRT for Type B Problems: V is known. 3.9 Inequality Constrained Tests in the Linear Model. 3.10 Tests When V is known. 3.11 Optimality Properties. 3.12 Appendix 1: Convex Cones. 3.13 Appendix B. Proofs. 4. Tests in General Parametric Models. 4.1 Introduction. 2.2 Preliminaries. 4.3 Tests of Rtheta = 0 against Rtheta 0. 4.4 Tests of h(theta) = 0. 4.5 An Overview of Score Tests with no Inequality Constraints. 4.6 Local Score-type Tests of Ho : psi = 0 vs H1 : psi &epsis PSI. 4.7 Approximating Cones and Tangent Cones. 4.8 General Testing Problems. 4.9 Properties of the mle When the True Value is on the Boundary. 5. Likelihood and Alternatives. 5.1 Introduction. 5.2 The Union-Intersection principle. 5.3 Intersection Union Tests (IUT). 5.4 Nanparametrics. 5.5 Restricted Alternatives and Simes-type Procedures. 5.6 Concluding Remarks. 6. Analysis of Categorical Data. 6.1 Motivating Examples. 6.2 Independent Binomial Samples. 6.3 Odds Ratios and Monotone Dependence. 6.4 Analysis of 2 x c Contingency Tables. 6.5 Test to Establish that Treatment is Better than Control. 6.6 Analysis of r x c Tables. 6.7 Square Tables and Marginal Homogeneity. 6.8 Exact Conditional Tests. 6.9 Discussion. 7. Beyond Parametrics. 7.1 Introduction. 7.2 Inference on Monotone Density Function. 7.3 Inference on Unimodal Density Function. 7.4 Inference on Shape Constrained Hazard Functionals. 7.5 Inference on DMRL Functions. 7.6 Isotonic Nonparametric Regression: Estimation. 7.7 Shape Constraints: Hypothesis Testing. 8. Bayesian Perspectives. 8.1 Introduction. 8.2 Statistical Decision Theory Motivations. 8.3 Steins Paradox and Shrinkage Estimation. 8.4 Constrained Shrinkage Estimation. 8.5 PC and Shrinkage Estimation in CSI. 8.6 Bayes Tests in CSI. 8.7 Some Decision Theoretic Aspects: Hypothesis Testing. 9. Miscellaneous Topics. 9.1 Two-sample Problem with Multivariate Responses. 9.2 Testing that an Identified Treatment is the Best: The mini-test. 9.3 Cross-over Interaction. 9.4 Directed Tests. Bibliography. Index.

Robust statistical procedures : asymptotics and interrelations

ASYMPTOTICS AND INTERRELATIONS Preliminaries Robust Estimation of Location and Regression Asymptotic Representations for L-Estimators Asymptotic Representations for M-Estimators Asymptotic Representations for R-Estimators Asymptotic Interrelations of Estimators ROBUST STATISTICAL INFERENCE Robust Sequential and Recursive Point Estimation Robust Confidence Sets and Intervals Robust Statistical Tests Appendix References Indexes.

Time-dependent coefficients in a Cox-type regression model

Estimation of a time-varying coefficient in a Cox-type parameterization of the stochastic intensity of a point process is considered. A sieve estimation procedure (Grenander, 1981) is used to estimate the coefficient. A rate of convergence in probability for the sieve estimator is given and a functional CLT for the integrated sieve estimator is proved.

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.

On the Bahadur representation of sample quantiles for sequences of φ-mixing random variables☆

The object of the present investigation is to show that the elegant asymptotic almost-sure representation of a sample quantile for independent and identically distributed random variables, established by Bahadur [1] holds for a stationary sequence of [phi]-mixing random variables. Two different orders of the remainder term, under different [phi]-mixing conditions, are obtained and used for proving two functional central limit theorems for sample quantiles. It is also shown that the law of iterated logarithm holds for quantiles in stationary [phi]-mixing processes.

Nonparametric Testing Under Progressive Censoring

Summary Progressive censoring schemes (allowing a continuous monitoring of experimentation until a terminal decision is reached) are often adopted in clinical trials and life testing problems. In this paper, a general class of rank order tests for progressive censoring is proposed. Along with a basic martinga:le property and a Brownian motion approximation for a related rank order process, asymptotic distribution theory of the proposed statistics is developed. Asymptotic performance characteristics of the proposed tests (in the light of Bahadur efficiency and the stochastic smallness of the stopping variables) are studied.

Non-Parametric Tests for the Bivariate Two-Sample Location Problem

Summary In this paper, the Wilcoxon-Mann-Whitney rank-sum test and Moods median test for the univariate two-sample location problem have been extended to the case of two variables, following a conditional approach. Various properties of the proposed tests have been studied and their asymptotic powers compared.