Computational probability applications / [edited by] Andrew G. Glen, Lawrence M. Leemis

Includes bibliographical references and index.

Preface; Contents; 1 Accurate Estimation with One Order Statistic; 1.1 Introduction; 1.2 The Case of the Exponential Distribution; 1.3 An Example for the Exponential Distribution; 1.4 The Rayleigh and Weibull Distribution Extensions; 1.5 Simulations and Computational Issues; 1.6 Implications for Design of Life Tests; 1.7 Conclusions; 2 On the Inverse Gamma as a Survival Distribution ; 2.1 Introduction; 2.2 Probabilistic Properties; 2.3 Statistical Inference; 2.3.1 Complete Data Sets; 2.3.2 Censored Data Sets; 2.4 Conclusions; 3 Order Statistics in Goodness-of-Fit Testing ; 3.1 Introduction

3.2 P-Vector3.3 Computation of the P-Vector ; 3.4 Goodness-of-Fit Testing; 3.5 Power Estimates for Test Statistics; 3.6 Further Research; 4 The ``Straightforward'' Nature of Arrival Rate Estimation?; 4.1 Introduction; 4.1.1 Sampling Plan 1: Time Sampling; 4.1.2 Sampling Plan 2: Count Sampling; 4.1.3 Sampling Plan 3: Limit Both Time and Arrivals; 4.2 Conclusions; 5 Survival Distributions Based on the Incomplete Gamma Function Ratio ; 5.1 Introduction; 5.2 Properties and Results; 5.3 Examples; 5.4 Conclusions

6 An Inference Methodology for Life Tests with Full Samples or Type II Right Censoring 6.1 Introduction and Literature Review; 6.2 The Methodology for Censored Data; 6.3 The Uniformity Test Statistic; 6.4 Implementation Using APPL; 6.5 Power Simulation Results; 6.6 Some Applications and Implications; 6.7 Conclusions and Further Research; 7 Maximum Likelihood Estimation Using Probability Density Functions of Order Statistics ; 7.1 Introduction; 7.2 MLEOS with Complete Samples; 7.3 Applying MLEOS to Censored Samples; 7.4 Conclusions and Further Research; 8 Notes on Rank Statistics

8.1 Introduction8.2 Explanation of the Tests; 8.3 Distribution of the Test Statistic Under H0; 8.4 Wilcoxon Power Curves for n = 2; 8.5 Generalization to Larger Sample Sizes; 8.6 Comparisons and Analysis; 8.7 The Wilcoxon-Mann-Whitney Test; 8.8 Explanation of the Test; 8.9 Three Cases of the Distribution of W Under H0; 8.9.1 Case I: No Ties; 8.9.2 Case II: Ties Only Within Each Sample; 8.9.3 Case III: Ties Between Both Samples; 8.10 Conclusions; 9 Control Chart Constants for Non-normal Sampling ; 9.1 Introduction; 9.2 Constants d2, d3; 9.3 Constants c4, c5; 9.3.1 Normal Sampling

9.3.2 Non-normal Sampling9.4 Conclusions; 10 Linear Approximations of Probability DensityFunctions; 10.1 Approximating a PDF; 10.2 Methods for Endpoint Placement; 10.2.1 Equal Spacing; 10.2.2 Placement by Percentiles; 10.2.3 Curvature-Based Approach; 10.2.4 Optimization-Based Approach; 10.3 Comparison of the Methods; 10.4 Application; 10.4.1 Convolution Theorem; 10.4.2 Monte Carlo Approximation; 10.4.3 Convolution of Approximate PDFs; 10.5 Conclusions; 11 Univariate Probability Distributions ; 11.1 Introduction; 11.2 Discussion of Properties; 11.3 Discussion of Relationships

This focuses on the developing field of building probability models with the power of symbolic algebra systems. The book combines the uses of symbolic algebra with probabilistic/stochastic application and highlights the applications in a variety of contexts. The research explored in each chapter is unified by the use of A Probability Programming Language (APPL) to achieve the modeling objectives. APPL, as a research tool, enables a probabilist or statistician the ability to explore new ideas, methods, and models. Furthermore, as an open-source language, it sets the foundation for future algorithms to augment the original code. Computational Probability Applications is comprised of fifteen chapters, each presenting a specific application of computational probability using the APPL modeling and computer language. The chapter topics include using inverse gamma as a survival distribution, linear approximations of probability density functions, and also moment-ratio diagrams for univariate distributions. These works highlight interesting examples, often done by undergraduate students and graduate students that can serve as templates for future work. In addition, this book should appeal to researchers and practitioners in a range of fields including probability, statistics, engineering, finance, neuroscience, and economics