Computational Probability Applications

258 Pages·2017·5.25 MB·English

by  Andrew G. Glen, Lawrence M. Leemis (eds.)

International Series in
Operations Research & Management Science
Andrew G. Glen
Lawrence M. Leemis Editors
Computational
Probability
Applications
International Series in Operations Research
& Management Science
Volume247
SeriesEditor
CamilleC.Price
StephenF.AustinStateUniversity,TX,USA
AssociateSeriesEditor
JoeZhu
WorcesterPolytechnicInstitute,MA,USA
FoundingSeriesEditor
FrederickS.Hillier
StanfordUniversity,CA,USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/6161
Andrew G. Glen • Lawrence M. Leemis
Editors
Computational Probability
Applications
123
Editors
AndrewG.Glen LawrenceM.Leemis
DepartmentofMathematics DepartmentofMathematics
andComputerScience TheCollegeofWilliamandMary
TheColoradoCollege Williamsburg,VA,USA
ColoradoSprings,CO,USA
ISSN0884-8289 ISSN2214-7934 (electronic)
InternationalSeriesinOperationsResearch&ManagementScience
ISBN978-3-319-43315-8 ISBN978-3-319-43317-2 (eBook)
DOI10.1007/978-3-319-43317-2
LibraryofCongressControlNumber:2016960577
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Preface
In the spring of 1994 at the College of William & Mary, we started work
on a project that would end up being a long-lasting source of research. We
explored the idea of combining a computer algebra system (Maple V at the
time) and probability results to see if the computer could be useful in per-
forming operations on random variables and finding new distributions. Over
the next 4 years, a series of procedures written in Maple started to form its
own programming language, soon to be called A Probability Programming
Language (APPL). Furthermore, the language and the results that the lan-
guage helped produce were starting to contribute to a field of research we
called computational probability. The programAPPL, unlike statistical soft-
ware that works ondata values, is designed to workon randomvariables and
the various functions that describe their distribution. APPL helps derive dis-
tributions of functions of random variables, probabilistic models, and other
transformations. Soon after, Diane Evans joined the team and wrote proce-
duresfordiscretedistributions.Thetwosetsofprocedureswereputtogether,
and in a 2001 article in The American Statistician [60], the launch of this
open-source software began.
In 2008, John Drew, along with Evans, Glen, and Leemis, put together a
monograph explaining the creation of APPL and some of the important re-
sultsfromtheresearch.Thisbook,ComputationalProbability:Algorithmsand
Applications in the Mathematical Sciences[46],establishedthestateofAPPL
at the time, primarilyhow it evolvedandits majoralgorithms. Camille Price
contacted us recently and requested that we update the original monograph
and write a secondmonographthat summarizes some more recent work. The
purpose of this, the second monograph,is twofold. First, we wantto combine
in this one document some of the recent results that have come about with
the language. Second, we want to inspire future users, professors, students,
and researchers to bring APPL into their work, their classroom, and their
mindset. Just as Word, Excel, LATEX, and PowerPoint are vital yet ubiqui-
tous elements to many researchers, we hope that APPL will become such a
V
VI Preface
research tool that enables a probabilist or statistician the ability to explore
new ideas, methods, and models.
Much of what is contained in the chapters that follow was published in
journals over the last 20 years. Some of the works in the monograph are
original efforts, yet to be published. These works highlight interesting exam-
ples, often done by undergraduate students and graduate students, that can
serveastemplates for future work.Eachchapteris astand-alonepublication,
with the authors recognized, and a short description of the importance that
APPL had in the research. Furthermore, as an open-source language, it sets
the foundation for future algorithms to augment the original code. Some pa-
pers heavily rely on APPL procedures; others enjoy the ease of use of data
structures. Still others have added procedures to the base language.
The editors would like to thank the many people who have contributed,
supported, and encouraged this effort. Each chapter author clearly has been
instrumental in furthering this cause, and they are recognized at the start of
eachchapter.Manyfriendsandcolleagueshavealsobeenimmenselysupport-
iveovertheyears.Wewouldespeciallyliketorecognizethelifelongsupportof
ourwives,JillLeemisandLisaGlen,whohaveputupwithourwildideas,even
though if often meant more work for them in other areas. Our children Lind-
sey, Mark, Logan, Andrea, Rebecca, Mary, Grace, Gabriel, Anna, Michael,
and Claire have all been supportive and patient “listeners” to their fathers.
Our many colleagues over the years deserve our heartfelt thanks: Richard
Bell, Roger Berger, Barry Bodt, Fr. Gabriel Costa, Kevin Cumminskey, Sam
Ellis, James Fritz, Ben Garlick, Grant Hartman, Steven Horton, Ted Hro-
madka, Michael Huber, Steven Janke, Rex Kincaid, Chris Marks, Joe Myers,
BillPulleyblank,TessPowers,MatthewRobinson,MickSmith,AlexStodala,
Rod Sturdivant, Fred Tinsley, Dave Webb, Chris Weld, Joanne Whitner, and
Wei Yin-Loh.
TheeditorsgratefullyacknowledgesupportfromtheArmyResearchOffice
for providing funding in their grant number 67906-MA.
Colorado Springs, CO, USA Andrew G. Glen
Williamsburg, VA, USA Lawrence M. Leemis
Contents
1 Accurate Estimation with One Order Statistic............. 1
1.1 Introduction ............................................ 2
1.2 The Case of the Exponential Distribution................... 3
1.3 An Example for the Exponential Distribution ............... 7
1.4 The Rayleigh and Weibull Distribution Extensions........... 9
1.5 Simulations and Computational Issues ..................... 11
1.6 Implications for Design of Life Tests ....................... 12
1.7 Conclusions............................................. 13
2 On the Inverse Gamma as a Survival Distribution ........ 15
2.1 Introduction ............................................ 16
2.2 Probabilistic Properties .................................. 17
2.3 Statistical Inference...................................... 22
2.3.1 Complete Data Sets ............................... 22
2.3.2 Censored Data Sets................................ 25
2.4 Conclusions............................................. 28
3 Order Statistics in Goodness-of-Fit Testing ............... 31
3.1 Introduction ............................................ 32
3.2 P-Vector ............................................... 33
3.3 Computation of the P-Vector ............................ 35
3.4 Goodness-of-Fit Testing .................................. 35
3.5 Power Estimates for Test Statistics ........................ 37
3.6 Further Research ........................................ 38
4 The “Straightforward” Nature of Arrival Rate
Estimation? ............................................... 41
4.1 Introduction ............................................ 42
4.1.1 Sampling Plan 1: Time Sampling.................... 43
VII
VIII Contents
4.1.2 Sampling Plan 2: Count Sampling ................... 44
4.1.3 Sampling Plan 3: Limit Both Time and Arrivals....... 47
4.2 Conclusions............................................. 49
5 Survival Distributions Based on the Incomplete Gamma
Function Ratio ............................................ 51
5.1 Introduction ............................................ 51
5.2 Properties and Results ................................... 53
5.3 Examples............................................... 56
5.4 Conclusions............................................. 58
6 An Inference Methodology for Life Tests with Full
Samples or Type II Right Censoring ...................... 59
6.1 Introduction and Literature Review........................ 60
6.2 The Methodology for Censored Data....................... 62
6.3 The Uniformity Test Statistic ............................. 63
6.4 Implementation Using APPL ............................. 64
6.5 Power Simulation Results................................. 66
6.6 Some Applications and Implications ....................... 67
6.7 Conclusions and Further Research ......................... 68
7 Maximum Likelihood Estimation Using Probability
Density Functions of Order Statistics ..................... 75
7.1 Introduction ............................................ 75
7.2 MLEOS with Complete Samples........................... 77
7.3 Applying MLEOS to Censored Samples .................... 79
7.4 Conclusions and Further Research ......................... 85
8 Notes on Rank Statistics .................................. 87
8.1 Introduction ............................................ 88
8.2 Explanation of the Tests ................................. 89
8.3 Distribution of the Test Statistic Under H0 ................. 90
8.4 Wilcoxon Power Curves for n=2.......................... 91
8.5 Generalization to Larger Sample Sizes...................... 94
8.6 Comparisons and Analysis................................ 96
8.7 The Wilcoxon–Mann–Whitney Test........................ 98
8.8 Explanation of the Test .................................. 99
8.9 Three Cases of the Distribution of W Under H0 .............100
8.9.1 Case I: No Ties ...................................100
8.9.2 Case II: Ties Only Within Each Sample ..............102
8.9.3 Case III: Ties Between Both Samples ................104
8.10 Conclusions.............................................106
Contents IX
9 Control Chart Constants for Non-normal Sampling .......107
9.1 Introduction ............................................107
9.2 Constants d2, d3 ........................................108
9.3 Constants c4, c5 .........................................112
9.3.1 Normal Sampling..................................112
9.3.2 Non-normal Sampling..............................114
9.4 Conclusions.............................................116
10 Linear Approximations of Probability Density
Functions..................................................119
10.1 Approximating a PDF ...................................119
10.2 Methods for Endpoint Placement..........................121
10.2.1 Equal Spacing ....................................121
10.2.2 Placement by Percentiles ...........................121
10.2.3 Curvature-Based Approach .........................122
10.2.4 Optimization-Based Approach ......................123
10.3 Comparison of the Methods...............................125
10.4 Application.............................................125
10.4.1 Convolution Theorem..............................126
10.4.2 Monte Carlo Approximation ........................126
10.4.3 Convolution of Approximate PDFs ..................128
10.5 Conclusions.............................................129
11 Univariate Probability Distributions ......................133
11.1 Introduction ............................................134
11.2 Discussion of Properties ..................................138
11.3 Discussion of Relationships ...............................140
11.3.1 Special Cases .....................................140
11.3.2 Transformations...................................140
11.3.3 Limiting Distributions .............................141
11.3.4 Bayesian Models ..................................141
11.4 The Binomial Distribution................................142
11.5 The Exponential Distribution .............................144
11.6 Conclusions.............................................146
12 Moment-Ratio Diagrams for Univariate Distributions......149
12.1 Introduction ............................................150
12.1.1 Contribution......................................152
12.1.2 Organization......................................152
12.2 Reading the Moment-Ratio Diagrams ......................153
12.3 The Skewness-Kurtosis Diagram...........................155
12.4 The CV-Skewness Diagram ...............................156
12.5 Application.............................................157
12.6 Conclusions and Further Research .........................160
X Contents
13 The Distribution of the Kolmogorov–Smirnov,
Cramer–von Mises, and Anderson–Darling
Test Statistics for Exponential Populations with
Estimated Parameters .....................................165
13.1 The Kolmogorov–SmirnovTest Statistic....................166
13.1.1 Distribution of D1 for Exponential Sampling..........167
13.1.2 Distribution of D2 for Exponential Sampling..........168
13.2 Other Measures of Fit....................................176
13.2.1 Distribution of W2 and A2 for Exponential
1 1
Sampling.........................................177
13.2.2 Distribution of W2 and A2 for Exponential
2 2
Sampling.........................................178
13.3 Applications ............................................181
14 Parametric Model Discrimination for Heavily Censored
Survival Data .............................................191
14.1 Introduction ............................................192
14.2 Literature Review .......................................193
14.3 A Parametric Example...................................196
14.4 Methodology............................................197
14.4.1 Uniform Kernel Function...........................198
14.4.2 Triangular Kernel Function .........................203
14.5 Monte Carlo Simulation Analysis ..........................208
14.6 Conclusions and Further Work ............................210
15 Lower Confidence Bounds for System Reliability
from Binary Failure Data Using Bootstrapping ...........217
15.1 Introduction ............................................217
15.2 Single-Component Systems ...............................218
15.3 Multiple-Component Systems .............................219
15.4 Perfect Component Test Results...........................224
15.5 Simulation..............................................230
15.6 Conclusions.............................................235
References.....................................................239
Index..........................................................249