The Career of a Research Statistician: From Consulting to Theoretical Development (Statistics for Industry, Technology, and Engineering)

"This monograph highlights the connection between the theoretical work done by research statisticians and the impact that work has on various industries. Drawing on decades of experience as an industry consultant, the author details how his contributions have had a lasting impact on the field of statistics as a whole. Aspiring statisticians and data scientists will be motivated to find practical applications for their knowledge, as they see how such work can yield breakthroughs in their field. Each chapter highlights a consulting position the author held that resulted in a significant contribution to statistical theory. Topics covered include tracking processes with change points, estimating common parameters, crossing fields with absorption points, military operations research, sampling surveys, stochastic visibility in random fields, reliability analysis, applied probability, and more. Notable advancements within each of these topics are presented by analyzing the problems facing various industries, and how solving those problems contributed to the development of the field. The Career of a Research Statistician is ideal for researchers, graduate students, or industry professionals working in statistics. It will be particularly useful for up-and-coming statisticians interested in the promising connection between academia and industry"--Provided by publisher Preface 7 Acknowledgments 9 Contents 10 Abbreviations and Acronyms 15 1 Introduction 17 1.1 Background and Motivation 17 1.2 Tracking Normal Processes Subjected to Change Points 18 1.3 Estimating the Common Parameters of Several Distributions 19 1.4 Survival Probabilities in Crossing Fields Having Absorption Points 19 1.5 Logistics and Operations Analysis for the Military 20 1.6 Foundation of Sampling Surveys 21 1.7 Visibility in Fields with Obscuring Elements 21 1.8 Sequential Testing and Estimation for Reliability Analysis 22 1.9 Random Choice of Fractional Replication 22 1.10 Sequential Search for the Maximum Tolerated Dose in Clinical Trials 23 1.11 Contributions to Applied Probability 24 1.12 Current Challenges in Statistics 24 2 Tracking Processes with Change Points 27 2.1 Tracking Procedures 27 2.2 The Bayesian AMOC Tracking Procedure 29 2.2.1 The Bayesian Framework for the AMOC Procedure 29 2.2.2 Bayesian AMOC for Poisson Random Variables 31 2.2.3 Bayesian AMOC for Binomial Random Variables 32 2.2.4 Bayesian AMOC for Normal Random Variables 35 2.3 Sequential Detection of Change Points 37 2.3.1 Bayesian Sequential Detection, When the Distributions Before and After Are Known 38 2.3.2 Bayesian Detection When the Distributions Before and After Are Unknown 40 2.4 Statistical Process Control 44 2.4.1 The Cumulative Sum Procedure 45 3 Estimating the Common Parameter of Several Distributions 48 3.1 Estimating the Common Mean of Two Normal Distributions with Samples of Equal Size 49 3.1.1 Bayesian Estimator 50 3.1.2 Pretest Estimator 52 3.2 Bayes Equivariant Estimators 53 3.3 Minimax Estimators 55 3.4 Multivariate Extension 56 3.5 Estimating the Common Variance of Correlated Normal Distributions 57 3.5.1 Scale Equivariant Estimators of σ2 59 3.5.2 Bayesian Equivariant Estimators 60 4 Survival Probabilities in Crossing Fields with Absorption Points 62 4.1 Probabilistic Models 62 4.2 Survival Probabilities 63 4.3 Bayesian Strategies for Choosing Alternative Paths 66 4.4 Telegraph Process with Elastic Boundary at the Origin 68 5 Logistics and Operations Analysis for the Military 71 5.1 One-Echelon Inventory System 71 5.1.1 The Inventory Model 71 5.1.2 Prior, Posterior, and Marginal Distributions 72 5.1.3 Bayesian Determination of Stock Levels 73 5.2 Two-Echelon Inventory System 75 5.2.1 The Inventory Model 75 5.2.2 The Cost Structure 76 5.3 Readiness Evaluation 77 5.3.1 Marine Corps Combat Readiness Evaluation 77 5.3.2 Data Structure 78 5.3.3 The Statistical Model 78 5.3.4 Pairwise Maximum Likelihood Estimators (PMLE) 79 5.4 Empirical Bayes Applied to MCCRES Data 83 5.4.1 Stein-Type Estimators 83 5.4.2 Empirical Bayes Estimators 84 5.4.3 Numerical Comparison of the Estimators 87 5.5 Time of Entrance to the Wear-Out Phase 88 5.5.1 The Exponential-Weibull Wear-Out Life Distribution 89 5.5.2 Bayesian Estimates of τ 90 6 Foundation of Sampling Surveys 93 6.1 Foundations of the Design-Approach 94 6.2 Foundation of the Modeling-Approach 95 6.3 Optimal Predication of Population Quantities 96 6.3.1 Best Linear Unbiased Predictors 96 6.4 Bayes Prediction in Finite Populations 98 6.4.1 General Theory 98 6.4.2 Bayesian Prediction of the Population Variance 99 6.4.3 Bayes Equivariant Predictors of the Variance 99 6.5 Bayesian Allocation of Stratified Sampling for Estimating the Population Proportion 100 6.6 Bayesian Sequential Estimation of the Size of a FinitePopulation 103 6.6.1 The Statistical Model 103 6.6.2 The Bayesian Framework 104 6.6.3 Bayesian Estimator of N 104 6.6.4 Computing the Bayes Estimator and the Bayes Risk for Poisson Prior 105 6.6.5 Stopping Rules 107 7 Visibility in Random Fields 110 7.1 Poisson Random Fields and Visibility Probabilitiesin the Plane 110 7.2 Visibility Probabilities on Star-Shaped Curves 114 7.3 Some Special Cases 117 7.3.1 The K-Functions 117 7.3.2 The Annular Region 117 7.3.3 The Moments 118 7.4 Random Coverage of a Circle and Distribution of Length of Visible Arcs 120 7.4.1 Coverage of a Circle by Random Black Arcs 120 7.4.2 The Measure of Vacancy and Its Moments 122 7.5 Applications to Shadowing Problem 123 8 Sequential Testing and Estimation for Reliability Analysis 125 8.1 Introductory Comments 125 8.2 Sequential Testing Procedures 126 8.2.1 Wald SPRT for Poisson Processes 126 8.2.2 Truncated SPRT of Type I 128 8.2.2.1 The Distribution of Ts(1) 129 8.2.3 Truncated SPRT of Type II 131 8.2.3.1 Operating Characteristics 132 8.3 Sequential Estimation 136 8.3.1 Distributions of Stopping Times 136 8.3.2 Distribution of First Crossing Concave Boundaries 137 8.3.3 Estimating β Under N(1) 139 8.4 Reliability Analysis for Damage Processes 139 8.4.1 Compound Cumulative Damage Processes 140 8.4.2 Estimation of Parameters 141 8.4.2.1 The Method of Moments Equations 142 8.4.2.2 Maximum Likelihood Estimator of ς When (λ,ν) Are Known 143 9 Random Choice of Fractional Replications 144 9.1 Models for Factorial Experiments 144 9.1.1 The Linear Case, p=2 145 9.2 Fractional Replications 146 9.2.1 Classical Notation for Small Size 2s Experiments 146 9.2.1.1 The Algebra of 2s Fractional Replications 149 9.3 Randomly Chosen Fractional Replications 150 9.3.1 Randomization Procedure R.P.I. 150 9.3.1.1 Randomization Procedure R.P.II 151 9.4 Testing Hypotheses in R.P.I 152 9.5 The ANOVA Scheme Under R.P.I 155 9.6 Randomized Fractional Weighing Designs 156 9.6.1 Randomized Fractional Weighing Designs and Unbiased Estimation 157 9.6.2 The Variance of an Unbiased Estimator 158 10 Sequential Search of an Optimal Dosage 160 10.1 The Linear Models 160 10.1.1 A Search Procedure for Model 1 161 10.1.2 A Search Procedure for Model 2 162 10.1.3 Bayes Procedures 163 10.1.4 Numerical Comparison 164 10.2 Categorical Model for Dosage Toxicity 165 10.2.1 Bayesian Adaptive Search for the MTD 166 11 Contributions to Applied Probability 170 11.1 The Compound Poisson Process 170 11.2 First Crossing Linear Boundaries 172 11.2.1 Nonincreasing Linear Boundaries 172 11.2.2 Increasing Linear Boundaries 174 11.2.2.1 Case I: Negative Intercept 174 11.2.2.2 Case II: B(t)=β+t,β>0 175 11.3 Fraction of Time in a Mode 177 11.4 Compound Poisson Process Subordinated by a PoissonProcess 179 11.5 Telegraph Processes 181 11.6 Applications 182 11.6.1 Inventory Control 182 11.6.2 Queuing Systems 183 11.6.3 Application in Insurance 184 12 Current Challenges in Statistics 187 12.1 Teaching Statistics to Data Scientists 188 12.2 Additional Comments About Data Analysis 189 Appendix: R Programs 190 A.1 Chapter 2 191 A.2 Chapter 3 194 A.3 Chapter 4 197 A.4 Chapter 5 198 A.5 Chapter 8 199 A.6 Chapter 10 204 A.7 Chapter 11 205 References 207 Author Index 213 Subject Index 215