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E-book: Computational Probability Applications

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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 univ

ariate 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.

Accurate Estimation with One Order Statistic.- On the Inverse Gamma as a Survival Distribution.- Order Statistics in Goodness-of-Fit Testing.- The "Straightforward" Nature of Arrival Rate Estimation .- Survival Distributions Based on the Incomplete Gamma Function Ratio.- An Inference Methodology for Life Tests with Full Samples or Type II Right Censoring.- Maximum Likelihood Estimation Using Probability Density Functions of Order Statistics.- Notes on Rank Statistics.- Control Chart Constants for Non-Normal Sampling.- Linear Approximations of Probability Density Functions.- Univariate Probability Distributions.- Moment-Ratio Diagrams for Univariate Distributions.- The Distribution of the Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling Test Statistics for Exponential Populations with Estimated Parameters.- Parametric Model Discrimiation for Heavily Censored Survival Data.- Lower Confidence Bounds for System Reliability from Binary Failure Data Using Bootstrapping.

Reviews

This volume successfully merges the use of symbolic algebra with stochastic applications and displays its applications in a host of situations. The book organized sequentially, well structured, and chapters are self-contained. The book is a good source as a reference book in a multitude fields. this is a good contribution, providing up-to-date coverage on selected topics in a logical and systematic manner. Variability and diversity in research is the spice of the life! (S. Ejaz Ahmed, Technometrics, Vol. 59 (3), July, 2017)

1 Accurate Estimation with One Order Statistic
1(14)
1.1 Introduction
2(1)
1.2 The Case of the Exponential Distribution
3(4)
1.3 An Example for the Exponential Distribution
7(2)
1.4 The Rayleigh and Weibull Distribution Extensions
9(2)
1.5 Simulations and Computational Issues
11(1)
1.6 Implications for Design of Life Tests
12(1)
1.7 Conclusions
13(2)
2 On the Inverse Gamma as a Survival Distribution
15(16)
2.1 Introduction
16(1)
2.2 Probabilistic Properties
17(5)
2.3 Statistical Inference
22(6)
2.3.1 Complete Data Sets
22(3)
2.3.2 Censored Data Sets
25(3)
2.4 Conclusions
28(3)
3 Order Statistics in Goodness-of-Fit Testing
31(10)
3.1 Introduction
32(1)
3.2 V-Vector
33(2)
3.3 Computation of the V-Vector
35(1)
3.4 Goodness-of-Fit Testing
35(2)
3.5 Power Estimates for Test Statistics
37(1)
3.6 Further Research
38(3)
4 The "Straightforward" Nature of Arrival Rate Estimation?
41(10)
4.1 Introduction
42(7)
4.1.1 Sampling Plan 1: Time Sampling
43(1)
4.1.2 Sampling Plan 2: Count Sampling
44(3)
4.1.3 Sampling Plan 3: Limit Both Time and Arrivals
47(2)
4.2 Conclusions
49(2)
5 Survival Distributions Based on the Incomplete Gamma Function Ratio
51(8)
5.1 Introduction
51(2)
5.2 Properties and Results
53(3)
5.3 Examples
56(2)
5.4 Conclusions
58(1)
6 An Inference Methodology for Life Tests with Full Samples or Type II Right Censoring
59(16)
6.1 Introduction and Literature Review
60(2)
6.2 The Methodology for Censored Data
62(1)
6.3 The Uniformity Test Statistic
63(1)
6.4 Implementation Using APPL
64(2)
6.5 Power Simulation Results
66(1)
6.6 Some Applications and Implications
67(1)
6.7 Conclusions and Further Research
68(7)
7 Maximum Likelihood Estimation Using Probability Density Functions of Order Statistics
75(12)
7.1 Introduction
75(2)
7.2 MLEOS with Complete Samples
77(2)
7.3 Applying MLEOS to Censored Samples
79(6)
7.4 Conclusions and Further Research
85(2)
8 Notes on Rank Statistics
87(20)
8.1 Introduction
88(1)
8.2 Explanation of the Tests
89(1)
8.3 Distribution of the Test Statistic Under H0
90(1)
8.4 Wilcoxon Power Curves for n = 2
91(3)
8.5 Generalization to Larger Sample Sizes
94(2)
8.6 Comparisons and Analysis
96(2)
8.7 The Wilcoxon-Mann-Whitney Test
98(1)
8.8 Explanation of the Test
99(1)
8.9 Three Cases of the Distribution of W Under H0
100(6)
8.9.1 Case I: No Ties
100(2)
8.9.2 Case II: Ties Only Within Each Sample
102(2)
8.9.3 Case III: Ties Between Both Samples
104(2)
8.10 Conclusions
106(1)
9 Control Chart Constants for Non-normal Sampling
107(12)
9.1 Introduction
107(1)
9.2 Constants d2, d3
108(4)
9.3 Constants C4, C5
112(4)
9.3.1 Normal Sampling
112(2)
9.3.2 Non-normal Sampling
114(2)
9.4 Conclusions
116(3)
10 Linear Approximations of Probability Density Functions
119(14)
10.1 Approximating a PDF
119(2)
10.2 Methods for Endpoint Placement
121(4)
10.2.1 Equal Spacing
121(1)
10.2.2 Placement by Percentiles
121(1)
10.2.3 Curvature-Based Approach
122(1)
10.2.4 Optimization-Based Approach
123(2)
10.3 Comparison of the Methods
125(1)
10.4 Application
125(4)
10.4.1 Convolution Theorem
126(1)
10.4.2 Monte Carlo Approximation
126(2)
10.4.3 Convolution of Approximate PDFs
128(1)
10.5 Conclusions
129(4)
11 Univariate Probability Distributions
133(16)
11.1 Introduction
134(4)
11.2 Discussion of Properties
138(2)
11.3 Discussion of Relationships
140(2)
11.3.1 Special Cases
140(1)
11.3.2 Transformations
140(1)
11.3.3 Limiting Distributions
141(1)
11.3.4 Bayesian Models
141(1)
11.4 The Binomial Distribution
142(2)
11.5 The Exponential Distribution
144(2)
11.6 Conclusions
146(3)
12 Moment-Ratio Diagrams for Univariate Distributions
149(16)
12.1 Introduction
150(3)
12.1.1 Contribution
152(1)
12.1.2 Organization
152(1)
12.2 Reading the Moment-Ratio Diagrams
153(2)
12.3 The Skewness-Kurtosis Diagram
155(1)
12.4 The CV-Skewness Diagram
156(1)
12.5 Application
157(3)
12.6 Conclusions and Further Research
160(5)
13 The Distribution of the Kolmogorov--Smirnov, Cramer--von Mises, and Anderson--Darling Test Statistics for Exponential Populations with Estimated Parameters
165(26)
13.1 The Kolmogorov--Smirnov Test Statistic
166(10)
13.1.1 Distribution of D1 for Exponential Sampling
167(1)
13.1.2 Distribution of D2 for Exponential Sampling
168(8)
13.2 Other Measures of Fit
176(5)
13.2.1 Distribution of W2/1 and A2/1 for Exponential Sampling
177(1)
13.2.2 Distribution of W2/2 and A2/2 for Exponential Sampling
178(3)
13.3 Applications
181(10)
14 Parametric Model Discrimination for Heavily Censored Survival Data
191(26)
14.1 Introduction
192(1)
14.2 Literature Review
193(3)
14.3 A Parametric Example
196(1)
14.4 Methodology
197(11)
14.4.1 Uniform Kernel Function
198(5)
14.4.2 Triangular Kernel Function
203(5)
14.5 Monte Carlo Simulation Analysis
208(2)
14.6 Conclusions and Further Work
210(7)
15 Lower Confidence Bounds for System Reliability from Binary Failure Data Using Bootstrapping
217(22)
15.1 Introduction
217(1)
15.2 Single-Component Systems
218(1)
15.3 Multiple-Component Systems
219(5)
15.4 Perfect Component Test Results
224(6)
15.5 Simulation
230(5)
15.6 Conclusions
235(4)
References 239(10)
Index 249
Dr. Andrew Glen is a Professor Emeritus of Operations Research from the United States Military Academy, in West Point, NY. He is currently a visiting professor at The Colorado College in Colorado Springs, Colorado. He is a retired colonel from the US Army, and spend 16 years on faculty at West Point. He has published three books and dozens of scholarly articles, mostly on the subject of computational probability. His research and teaching interests are in computational probability and statistical modeling.  Lawrence Leemis is a professor in the Department of Mathematics at The College of William & Mary in Williamsburg, Virginia, U.S.A. He received his BS and MS degrees in mathematics and his PhD in operations research from Purdue University. He has also taught courses at Purdue University, The University of Oklahoma, and Baylor University. He has served as Associate Editor for the IEEE Transactions on Reliability, Book Review Editor for the Journal of Quality Technology, and an Associate Editor for Naval Research Logistics. He has published six books and over 100 research articles, proceedings papers, and book chapters. His research and teaching interests are in reliability, simulation, and computational probability.
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