Statistical Tolerance Regions Theory,

The topic of tolerance intervals and tolerance regions has undergone significant growth during recent years, with applications arising in various areas such as quality control, industry, and environmental monitoring. Statistical Tolerance Regions presents the theoretical development of tolerance intervals and tolerance regions through computational algorithms and the illustration of numerous practical uses and examples. This is the first book of its kind to successfully balance theory and practice, providing a state-of-the-art treatment on tolerance intervals and tolerance regions.

K. Krishnamoorthy, PhD, is Professor in the Department of Mathematics at the University of Louisiana at Lafayette. He is Associate Editor of Communications in Statistics and has published numerous journal articles in his areas of research interest, which include tolerance regions, multivariate analysis, and statistical computing.

Thomas Mathew, PhD, is Professor in the Department of Mathematics and Statistics at the University of Maryland, Baltimore County. He currently focuses his research on tolerance regions, inference in linear mixed and random models, and bioequivalence testing. A Fellow of the Institute of Mathematical Statistics and the American Statistical Association, Dr. Mathew is the coauthor of Statistical Tests for Mixed Linear Models, also published by Wiley.

List of Tables	p. xiii
Preface	p. xvii
Preliminaries	p. 1
Introduction	p. 1
One-Sided Tolerance Intervals	p. 2
Tolerance Intervals	p. 4
Survival Probability and Stress-Strength Reliability	p. 5
Some Technical Results	p. 7
The Modified Large Sample (MLS) Procedure	p. 11
The Generalized P-value and Generalized Confidence Interval	p. 13
Description	p. 14
GPQs for a Location-Scale Family	p. 16
Some Examples	p. 17
Exercises	p. 20
Univariate Normal Distribution	p. 25
Introduction	p. 25
One-Sided Tolerance Limits for a Normal Population	p. 26
Two-Sided Tolerance Intervals	p. 30
Tolerance Intervals	p. 30
Equal-Tailed Tolerance Intervals for a Normal Distribution	p. 33
Simultaneous Hypothesis Testing about Normal Quantiles	p. 34
Tolerance Limits for X1 - X2	p. 38
Exact One-Sided Tolerance Limits for the Distribution of X1 - X2 When the Variance Ratio Is Known	p. 39
One-Sided Tolerance Limits for the Distribution of X1 - X2 When the Variance Ratio Is Unknown	p. 40
Hypothesis Testing About the Quantiles of X1 - X2	p. 43
Comparison of the Approximate Methods for Making Inference about Quantiles of X1 - X2	p. 44
Applications of Tolerance Limits for X1 - X2 with Examples	p. 45
Simultaneous Tolerance Limits for Normal Populations	p. 50
Simultaneous One-Sided Tolerance Limits	p. 50
Simultaneous Tolerance Intervals	p. 51
Exercises	p. 54
Univariate Linear Regression Model	p. 59
Notations and Preliminaries	p. 59
One-Sided Tolerance Intervals and Simultaneous Tolerance Intervals	p. 62
One-Sided Tolerance Intervals	p. 62
One-Sided Simultaneous Tolerance Intervals	p. 66
Two-Sided Tolerance Intervals and Simultaneous Tolerance Intervals	p. 69
Two-Sided Tolerance Intervals	p. 69
Two-Sided Simultaneous Tolerance Intervals	p. 74
The Calibration Problem	p. 78
Exercises	p. 81
The One-Way Random Model with Balanced Data	p. 85
Notations and Preliminaries	p. 85
Two Examples	p. 87
One-Sided Tolerance Limits for N $$	p. 88
The Mee-Owen Approach	p. 89
Vangel's Approach	p. 91
The Krishnamoorthy-Mathew Approach	p. 93
Comparison of Tolerance Limits	p. 97
Examples	p. 97
One-Sided Confidence Limits for Exceedance Probabilities	p. 100
One-Sided Tolerance Limits When the Variance Ratio Is Known	p. 103
One-Sided Tolerance Limits for $$	p. 104
Two-Sided Tolerance Intervals for $$	p. 105
Mee's Approach	p. 106
The Liao-Lin-Iyer Approach	p. 107
Two-Sided Tolerance Intervals for $$	p. 111
Exercises	p. 113
The One-Way Random Model with Unbalanced Data	p. 117
Notations and Preliminaries	p. 117
Two Examples	p. 118
One-Sided Tolerance Limits for $$	p. 120
The Krishnamoorthy and Mathew Approach	p. 120
The Liao, Lin and Iyer Approach	p. 123
One-Sided Confidence Limits for Exceedance Probabilities	p. 128
One-Sided Tolerance Limits for N $$	p. 130
The Krishnamoorthy and Mathew Approach	p. 131
The Liao, Lin and Iyer Approach	p. 131
Two-Sided Tolerance Intervals	p. 133
A Two-Sided Tolerance Interval for $$	p. 133
A Two-Sided Tolerance Interval for $$	p. 134
Exercises	p. 135
Some General Mixed Models	p. 137
Notations and Preliminaries	p. 137
Some Examples	p. 141
Tolerance Intervals in a General Setting	p. 144
One-Sided Tolerance Intervals	p. 145
Two-Sided Tolerance Intervals	p. 147
A General Model with Two Variance Components	p. 151
One-Sided Tolerance Limits	p. 154
Two-Sided Tolerance Intervals	p. 156
A One-Way Random Model with Covariates and Unequal Variances	p. 158
Testing Individual Bioequivalence	p. 163
Exercises	p. 169
Some Non-Normal Distributions	p. 173
Introduction	p. 173
Lognormal Distribution	p. 174
Gamma Distribution	p. 175
Normal Approximation to a Gamma Distribution	p. 176
Tolerance Intervals and Survival Probability	p. 177
Applications with an Example	p. 178
Stress-Strength Reliability	p. 181
Two-Parameter Exponential Distribution	p. 182
Some Preliminary Results	p. 183
One-Sided Tolerance Limits	p. 184
Estimation of Survival Probability	p. 189
Stress-Strength Reliability	p. 192
Weibull Distribution	p. 195
Some Preliminaries	p. 195
The Maximum Likelihood Estimators and Their Distributions	p. 196
Generalized Pivotal Quantities for Weibull Parameters	p. 198
One-Sided Tolerance Limits	p. 199
A GPQ for a Survival Probability	p. 200
Stress-Strength Reliability	p. 201
Exercises	p. 204
Nonparametric Tolerance Intervals	p. 207
Notations and Preliminaries	p. 207
Order Statistics and Their Distributions	p. 208
One-Sided Tolerance Limits and Exceedance Probabilities	p. 211
Tolerance Intervals	p. 212
Confidence Intervals for Population Quantiles	p. 214
Sample Size Calculation	p. 215
Sample Size for Tolerance Intervals of the Form $$	p. 215
Sample Size for Tolerance Intervals of the Form $$	p. 217
Nonparametric Multivariate Tolerance Regions	p. 220
Exercises	p. 222
The Multivariate Normal Distribution	p. 225
Introduction	p. 225
Notations and Preliminaries	p. 226
Some Approximate Tolerance Factors	p. 228
Methods Based on Monte Carlo Simulation	p. 232
Simultaneous Tolerance Intervals	p. 238
Tolerance Regions for Some Special Cases	p. 242
Exercises	p. 246
The Multivariate Linear Regression Model	p. 249
Preliminaries	p. 249
The Model	p. 249
Some Examples	p. 251
Approximations for the Tolerance Factor	p. 252
Accuracy of the Approximate Tolerance Factors	p. 257
Methods Based on Monte Carlo Simulation	p. 258
Application to the Example	p. 260
Multivariate Calibration	p. 261
Problem Formulation and the Pivot Statistic	p. 261
The Confidence Region	p. 263
Computation of the Confidence Region	p. 264
A Generalization	p. 267
An Example and Some Numerical Results	p. 268
Exercises	p. 273
Bayesian Tolerance Intervals	p. 275
Notations and Preliminaries	p. 275
The Univariate Normal Distribution	p. 277
Tolerance Intervals Under the Non-Informative Prior	p. 278
Tolerance Intervals Under the Conjugate Prior	p. 279
The One-Way Random Model with Balanced Data	p. 281
Two Examples	p. 284
Exercises	p. 291
Miscellaneous Topics	p. 293
Introduction	p. 293
ß-Expectation Tolerance Regions	p. 293
ß-Expectation Tolerance Intervals for the Normal Distribution	p. 294
ß-Expectation Tolerance Intervals for the One-Way Random Model with Balanced Data	p. 295
ß-Expectation Tolerance Intervals for the One-Way Random Model with Unbalanced Data	p. 300
ß-Expectation Tolerance Intervals for a General Mixed Effects Model with Balanced Data	p. 301
Multivariate ß-Expectation Tolerance Regions	p. 303
Bayesian ß-Expectation Tolerance Intervals	p. 304
Tolerance Limits for a Ratio of Normal Random Variables	p. 305
An Upper Tolerance Limit Based on an Approximation to the cdf	p. 308
Tolerance Limits Based on the Exact cdf	p. 310
An Application	p. 311
Sample Size Determination	p. 312
Sample Size Determination for a $$ Two-Sided Tolerance Interval for a Normal Population	p. 312
Sample Size Determination for a ß-Expectation Two-Sided Tolerance Interval for a Normal Population	p. 314
Reference Limits and Coverage Intervals	p. 315
Tolerance Intervals for Binomial and Poisson Distributions	p. 316
Binomial Distribution	p. 318
Poisson Distribution	p. 322
Two-Sided Tolerance Intervals for Binomial and Poisson Distributions	p. 324
Tolerance Intervals Based on Censored Samples	p. 326
Normal and Related Distributions	p. 327
Two-Parameter Exponential Distribution	p. 336
Weibull and Extreme Value Distributions	p. 340
Exercises	p. 343
Data Sets	p. 349
Tables	p. 355
References	p. 441
Index	p. 457
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Statistical Tolerance Regions Theory, Applications, and Computation

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