Selected Topics in Statistical Inference: Theory and Applications

This book focuses exclusively on the domain of parametric inference and that, too, from a reader’s perspective, i.e., covering only point estimation of parameter(s). It covers those topics in parametric inference which need clarity of exposure to students, researchers, and teachers alike; mere statements of theorems and proofs may not always reveal the inner beauty and significance of some aspects of inference. To ensure clarity, the book discusses the following topics at an advanced level―(1) sequential (unbiased) point estimation of ‘p’ and its functions; generalization to trinomial and tetranomial populations; (2) some aspects of the use of additional resources in finite population inference; (3) the concept of sufficiency vis-à-vis the notion of sufficient experiments and comparison of experiments; (4) estimation of the size of a finite population with special features; and (5) unbiased estimation of reliability in exponential samples and other settings. This book provides a platform for thought-provoking, creative, and challenging discussions on a variety of topics in statistical estimation theory, it is also ideal for research methodology course for statistics research scholars, and for clarification of basic ideas in topics discussed at basic/advanced levels. Preface Acknowledgements Contents About the Authors 1 Glimpses of the Book 1.1 Introduction 1.2 Chapter-Wise Description of Contents 2 Sequential Binomial Estimation 2.1 Introduction 2.2 Notations and Nomenclature 2.3 Bernoulli Sampling Plans 2.3.1 Closure of a Sampling Plan 2.3.2 Concept of Path Counting 2.3.3 Concept of Pull-Down and Push-Up Plans 2.4 Unbiased Estimation of Functions of p 2.4.1 Unbiased Estimation of p 2.4.2 Unbiased Estimation of p - 1 2.4.3 Unbiased Estimation of p, q, and pq under Two Non-traditional Sampling Plans 2.4.4 Unbiased Estimation of pa qb 2.5 Unbiased Estimation of Parameters in a Trinomial Set-Up 2.5.1 Parameter Estimation in a Trinomial Distribution 2.5.2 Parameter Estimation in an Inverse Trinomial Distribution 2.5.3 Unbiased Estimation of pa qb rc 2.5.4 Unbiased Estimation of 1pq and 1pqr under an Experiment Performed until (a, a, a) is achieved, a being a Specified Positive Integer 2.6 Unbiased Estimation of Parameters in a Tetranomial Set-Up 2.6.1 Sequential Sampling Plan 2.6.2 Unbiased Estimation of Parameters References 3 Use of Additional Resources in Finite Population Inference 3.1 Introduction 3.2 Basics in Finite Population Inference 3.3 A Basic Result 3.4 Case of SRSWOR Designs 3.5 Case of Arbitrarily Specified upper F upper S left parenthesis n right parenthesisFS(n) Connected Design and an Extended Mixture Design 3.6 Further Issues Involving the Use of Additional Resources 3.7 Understanding Lanke's Formula 3.8 Understanding Lanke's Formula: All That Glitters! 3.9 Conclusion References 4 Notion of Sufficiency in Statistical Inference—Theory and Applications 4.1 Introduction: Understanding Sufficiency 4.2 Examples of Sufficiency in Statistical Experiments 4.3 The Notion of Sufficient Experiments in Linear Regression Set-Up 4.4 The Notion of Sufficient Experiments in Bivariate and Trivariate Normal Populations 4.5 The Notion of Sufficient Experiments in One Way ANOVA Models with Random Effects 4.6 Conclusion References 5 Estimation of the Size of a Finite Population with Special Features 5.1 Introduction and Literature Review 5.2 Special Features of Finite Populations 5.2.1 Concepts of Reference Units and Ultimate Units 5.2.2 Concepts of a Bipartite Graph and a Network of Reference Units and Ultimate Units 5.2.3 Sampling from Reference Units and Creation of a Sample Network 5.3 Unbiased Estimation of the Size of a Specially Featured Population 5.3.1 Concepts of Probing and No-Probing 5.3.2 Estimation of N Under Probing: Theory and Examples 5.3.3 Estimation of upper NN Without Probing: Theory and Examples 5.4 Estimation of upper NN: Heuristic Principle References 6 Unbiased Estimation of Reliability in Exponential Samples 6.1 Introduction 6.2 Unbiased Estimation of μ 6.2.1 Unbiased Estimation of μ Under Type I Censoring 6.2.2 Unbiased Estimation of μ Under Type II Censoring 6.3 Unbiased Estimation of Reliability 6.3.1 Unbiased Estimator of R(t) Based only on r 6.3.2 Unbiased Estimator of R(t) based on both r and the Observed Lifetimes 6.4 Unbiased Estimation of Reliability Based on a Few Selected Order Statistics 6.4.1 Unbiased Estimation of R(t) Based on a Single Order Statistic 6.4.2 Unbiased Estimation of R(t) Based on a Few Order Statistics 6.5 Interval Estimation of Reliability Function 6.6 Unbiased Estimation of Stress-Strength Reliability Under One-Parameter Exponential Distribution References Exercises Author Index Subject Index