Handbook of statistical distributions with applications

404 Pages·2016·2.409 MB·English

by  Krishnamoorthy, Kalimuthu

Handbook of
Statistical Distributions
with Applications
Second Edition
©2016byTaylor&FrancisGroup,LLC
© 2016 by Taylor & Francis Group, LLC
Handbook of
Statistical Distributions
with Applications
Second Edition
K. Krishnamoorthy
University of Louisiana
Lafayette, Louisiana, USA
©2016byTaylor&FrancisGroup,LLC
CRC Press
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© 2016 by Taylor & Francis Group, LLC
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In Memory of My Parents

© 2016 by Taylor & Francis Group, LLC
© 2016 by Taylor & Francis Group, LLC
Contents
List of Figures xviii
List of Tables xix
Symbols xx
Preface to the First Edition xxiii
Preface to the Second Edition xxv
1 StatCalc 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Contents of StatCalc . . . . . . . . . . . . . . . . . . . . . . 5
2 Preliminaries 9
2.1 Random Variables and Expectations . . . . . . . . . . . . . . 9
2.2 Moments and Other Functions . . . . . . . . . . . . . . . . . 11
2.2.1 Measures of Central Tendency . . . . . . . . . . . . . 12
2.2.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Measures of Variability . . . . . . . . . . . . . . . . . 13
2.2.4 Measures of Relative Standing . . . . . . . . . . . . . 13
2.2.5 Other Measures . . . . . . . . . . . . . . . . . . . . . . 14
2.2.6 Moment Generating Function . . . . . . . . . . . . . . 14
2.3 Some Functions Relevant to Reliability . . . . . . . . . . . . 15
2.4 Model Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 Q–Q Plot . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.2 The Chi-Square Goodness-of-Fit Test . . . . . . . . . 17
2.5 Methods of Estimation . . . . . . . . . . . . . . . . . . . . . 17
2.6 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6.1 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . 19
2.6.2 Interval Estimation. . . . . . . . . . . . . . . . . . . . 22
2.7 Pivotal-BasedMethods for Location-Scale Families . . . . . 23
2.7.1 Pivotal Quantities . . . . . . . . . . . . . . . . . . . . 24
2.7.2 Generalized Pivotal Quantities (GPQ) . . . . . . . . . 25
2.8 Method of Variance Estimate Recovery . . . . . . . . . . . . 27
2.9 Modified Normal-Based Approximation . . . . . . . . . . . . 28
2.9.1 Linear Combination of Independent Random Variables 28
vii
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viii Contents
2.9.2 Ratio of Two Independent Random Variables . . . . . 29
2.10 Random Number Generation . . . . . . . . . . . . . . . . . . 29
2.11 Some Special Functions . . . . . . . . . . . . . . . . . . . . . 30
Discrete Distributions 33
3 Discrete Uniform Distribution 35
3.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Binomial Distribution 37
4.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Probabilities, Percentiles, and Moments . . . . . . . . . . . . 38
4.4 Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4.1 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4.2 Power and Sample Size Calculation. . . . . . . . . . . 43
4.4.3 Confidence Intervals . . . . . . . . . . . . . . . . . . . 45
4.4.4 Sample Size Calculation for a Given Precision . . . . . 47
4.5 Prediction Intervals . . . . . . . . . . . . . . . . . . . . . . . 49
4.6 Tolerance Intervals . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6.1 Equal-Tailed and Two-Sided Tolerance Intervals . . . 52
4.6.2 Tolerance Intervals Based on Approximate Quantiles . 53
4.7 Tests for the Difference between Two Proportions . . . . . . 54
4.7.1 Approximate Tests . . . . . . . . . . . . . . . . . . . . 55
4.7.2 Fisher’s Exact Test . . . . . . . . . . . . . . . . . . . . 56
4.7.3 Powers and Sample Size Calculation . . . . . . . . . . 56
4.8 Two-Sample Confidence Intervals for Proportions . . . . . . 58
4.8.1 Difference . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.8.2 Relative Risk and Odds Ratio . . . . . . . . . . . . . . 60
4.9 Confidence Intervals for a Linear Combination of Proportions 64
4.10 Properties and Results . . . . . . . . . . . . . . . . . . . . . 66
4.10.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.10.2 Relation to Other Distributions . . . . . . . . . . . . . 67

4.10.3 Approximations. . . . . . . . . . . . . . . . . . . . . . 68
4.11 Random Number Generation . . . . . . . . . . . . . . . . . . 68
4.12 Computation of Probabilities . . . . . . . . . . . . . . . . . . 69
5 Hypergeometric Distribution 73
5.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3 Probabilities, Percentiles, and Moments . . . . . . . . . . . . 74
5.4 Point Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5 Test for the Proportion and Power Calculation . . . . . . . . 79
5.6 Confidence Interval and Sample Size Calculation . . . . . . . 80
5.7 Test for Comparing Two Proportions . . . . . . . . . . . . . 82
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Contents ix
5.8 Properties and Results . . . . . . . . . . . . . . . . . . . . . 85
5.9 Random Number Generation . . . . . . . . . . . . . . . . . . 85
5.10 Computation of Probabilities . . . . . . . . . . . . . . . . . . 86
6 Poisson Distribution 89
6.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3 Probabilities, Percentiles, and Moments . . . . . . . . . . . . 92
6.4 Model Fitting with Examples . . . . . . . . . . . . . . . . . . 92
6.5 One-Sample Inference . . . . . . . . . . . . . . . . . . . . . . 94
6.6 Test for the Mean . . . . . . . . . . . . . . . . . . . . . . . . 94
6.7 Confidence Intervals for the Mean . . . . . . . . . . . . . . . 96
6.8 Prediction Intervals . . . . . . . . . . . . . . . . . . . . . . . 98
6.9 Tolerance Intervals . . . . . . . . . . . . . . . . . . . . . . . . 99
6.10 Tests for Comparing Two Means and Power Calculation . . . 101
6.11 Confidence Intervals for the Ratio of Two Means . . . . . . . 104
6.12 Confidence Intervals for the Difference between Two Means . 106
6.13 Inference for a Weighted Sum of Poisson Means . . . . . . . 108
6.14 Properties and Results . . . . . . . . . . . . . . . . . . . . . 109
6.14.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.14.2 Relation to Other Distributions . . . . . . . . . . . . . 110
6.14.3 Approximations. . . . . . . . . . . . . . . . . . . . . . 110
6.15 Random Number Generation . . . . . . . . . . . . . . . . . . 110
6.16 Computation of Probabilities . . . . . . . . . . . . . . . . . . 111
7 Geometric Distribution 115
7.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.3 Probabilities, Percentiles, and Moments . . . . . . . . . . . . 116
7.4 Properties and Results . . . . . . . . . . . . . . . . . . . . . 117
7.5 Random Number Generation . . . . . . . . . . . . . . . . . . 117
8 Negative Binomial Distribution 119
8.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.3 Probabilities, Percentiles, and Moments . . . . . . . . . . . . 120
8.4 Point Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.5 Test for the Proportion . . . . . . . . . . . . . . . . . . . . . 123
8.6 Confidence Intervals for the Proportion . . . . . . . . . . . . 124
8.7 Properties and Results . . . . . . . . . . . . . . . . . . . . . 124
8.7.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.7.2 Relation to Other Distributions . . . . . . . . . . . . . 125
8.8 Random Number Generation . . . . . . . . . . . . . . . . . . 125
8.9 Computation of Probabilities . . . . . . . . . . . . . . . . . . 126
© 2016 by Taylor & Francis Group, LLC
x Contents
9 Logarithmic Series Distribution 129
9.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.3 Probabilities, Percentiles, and Moments . . . . . . . . . . . . 130
9.4 Inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.4.1 Point Estimation . . . . . . . . . . . . . . . . . . . . . 132
9.4.2 Interval Estimation. . . . . . . . . . . . . . . . . . . . 133
9.5 Properties and Results . . . . . . . . . . . . . . . . . . . . . 134
9.6 Random Number Generation . . . . . . . . . . . . . . . . . . 134
9.7 Computation of Probabilities . . . . . . . . . . . . . . . . . . 135
Continuous Distributions 137
10 Continuous Uniform Distribution 139
10.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10.3 Inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
10.4 Properties and Results . . . . . . . . . . . . . . . . . . . . . 141
10.5 Random Number Generation . . . . . . . . . . . . . . . . . . 141
11 Normal Distribution 143
11.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
11.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
11.3 Probabilities, Percentiles, and Moments . . . . . . . . . . . . 145
11.4 One-Sample Inference . . . . . . . . . . . . . . . . . . . . . . 149
11.4.1 Test for the Mean and Power Computation . . . . . . 149
11.4.2 Confidence Interval for the Mean . . . . . . . . . . . . 151
11.4.3 Confidence Interval for the Coefficient of Variation and
Survival Probability . . . . . . . . . . . . . . . . . . . 152
11.4.4 Prediction Intervals . . . . . . . . . . . . . . . . . . . 155
11.4.5 Test for Variance . . . . . . . . . . . . . . . . . . . . . 157
11.5 Two-Sample Inference . . . . . . . . . . . . . . . . . . . . . . 159
11.5.1 Inference for the Ratio of Variances . . . . . . . . . . 159
11.5.2 Inference for the Difference between Two Means . . . 160
11.5.3 Inference for the Difference between Two Means when
Variances Are Unknown and Arbitrary . . . . . . . . . 163
11.5.4 Comparison of Two Coefficients of Variation . . . . . 164
11.6 Tolerance Intervals . . . . . . . . . . . . . . . . . . . . . . . . 166
11.6.1 Two-Sided Tolerance Intervals . . . . . . . . . . . . . 166
11.6.2 One-Sided Tolerance Limits . . . . . . . . . . . . . . . 167
11.6.3 Equal-Tailed Tolerance Intervals . . . . . . . . . . . . 169
11.6.4 Simultaneous Hypothesis Testing for Quantiles . . . . 169
11.7 Inference Based on Censored Samples . . . . . . . . . . . . . 170
11.7.1 Confidence Intervals for the Mean and Variance . . . . 172
11.7.2 Tolerance Intervals . . . . . . . . . . . . . . . . . . . . 173
© 2016 by Taylor & Francis Group, LLC