Statistical Tolerance Regions: Theory, Applications, and ComputationISBN: 9780470380260
461 pages
April 2009

Description
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 stateoftheart treatment on tolerance intervals and tolerance regions.
The book begins with the key definitions, concepts, and technical results that are essential for deriving tolerance intervals and tolerance regions. Subsequent chapters provide indepth coverage of key topics including:
 Univariate normal distribution
 Nonnormal distributions
 Univariate linear regression models
 Nonparametric tolerance intervals
 The oneway random model with balanced data
 The multivariate normal distribution
 The oneway random model with unbalanced data
 The multivariate linear regression model
 General mixed models
 Bayesian tolerance intervals
A final chapter contains coverage of miscellaneous topics including tolerance limits for a ratio of normal random variables, sample size determination, reference limits and coverage intervals, tolerance intervals for binomial and Poisson distributions, and tolerance intervals based on censored samples. Theoretical explanations are accompanied by computational algorithms that can be easily replicated by readers, and each chapter contains exercise sets for reinforcement of the presented material. Detailed appendices provide additional data sets and extensive tables of univariate and multivariate tolerance factors.
Statistical Tolerance Regions is an ideal book for courses on tolerance intervals at the graduate level. It is also a valuable reference and resource for applied statisticians, researchers, and practitioners in industry and pharmaceutical companies.
Table of Contents
Preface.
1 Preliminaries.
1.1 Introduction.
1.2 Some Technical Results.
1.3 The Modified Large Sample (MLS) Procedure.
1.4 The Generalized Pvalue and Generalized Confidence Interval.
1.5 Exercises.
2 Univariate Normal Distribution.
2.1 Introduction.
2.2 OneSided Tolerance Limits for a Normal Population.
2.3 TwoSided Tolerance Intervals.
2.4 Tolerance Limits for X_{1}  X_{2}.
2.5 Simultaneous Tolerance Limits for Normal Populations.
2.6 Exercises.
3 Univariate Linear Regression Model.
3.1 Notations and Preliminaries.
3.2 OneSided Tolerance Intervals and Simultaneous Tolerance Intervals.
3.3 Twosided Tolerance Intervals and Simultaneous Tolerance Intervals.
3.4 The Calibration Problem.
3.5 Exercises.
4 The OneWay Random Model with Balanced Data.
4.1 Notations and Preliminaries.
4.2 Two Examples.
4.3 OneSided Tolerance Limits for N(µ, σ²τ + σ²τ_{e)}.
4.4 OneSided Tolerance Limits for N(µ, σ²τ¨).
4.5 TwoSided Tolerance Intervals for N(µ, σ²τ + σ²τ_{e}).
4.6 TwoSided Tolerance Intervals for N(µ, σ²τ¨).
4.7 Exercises.
5 The OneWay Random Model with Unbalanced Data.
5.1 Notations and Preliminaries.
5.2 Two Examples.
5.3 OneSided Tolerance Limits for N(µ, σ²τ + σ²_{e}).
5.4 OneSided Tolerance Limits for N(µ, σ²τ).
5.5 TwoSided Tolerance Intervals.
5.6 Exercises.
6 Some General Mixed Models.
6.1 Notations and Preliminaries.
6.2 Some Examples.
6.3 Tolerance Intervals in a General Setting.
6.4 A General Model with Two Variance Components.
6.5 A OneWay Random Model with Covariates and Unequal Variances.
6.5 Testing Individual Bioequivalence.
6.6 Exercises.
7 Some NonNormal Distributions.
7.1 Introduction.
7.2 Lognormal Distribution.
7.3 Gamma Distribution.
7.4 TwoParameter Exponential Distribution.
7.5 Weibull Distribution.
7.6 Exercises.
8 Nonparametric Tolerance Intervals.
8.1 Notations and Preliminaries.
8.2 Order Statistics and Their Distributions.
8.3 OneSided Tolerance Limits and Exceedance Probabilities.
8.4 Tolerance Intervals.
8.5 Confidence Intervals for Population Quantiles.
8.6 Sample Size Calculation.
8.7 Nonparametric Multivariate Tolerance Regions.
8.8 Exercises.
9 The Multivariate Normal Distribution.
9.1 Introduction.
9.2 Notations and Preliminaries.
9.3 Some Approximate Tolerance Factors.
9.4 Methods Based on Monte Carlo Simulation.
9.5 Simultaneous Tolerance Intervals.
9.6 Tolerance Regions for Some Special Cases.
9.7 Exercises.
10 The Multivariate Linear Regression Model.
10.1 Preliminaries.
10.2 Approximations for the Tolerance Factor.
10.3 Accuracy of the Approximate Tolerance Factors.
10.4 Methods Based on Monte Carlo Simulation.
10.5 Application to the Example.
10.6 Multivariate Calibration.
10.7 Exercises.
11 Bayesian Tolerance Intervals.
11.1 Notations and Preliminaries.
11.2 The Univariate Normal Distribution.
11.3 The OneWay Random Model With Balanced Data.
11.4 Two Examples.
11.5 Exercises.
12 Miscellaneous Topics.
12.1 Introduction.
12.2 βExpectation Tolerance Regions.
12.3 Tolerance Limits for a Ratio of Normal Random Variables.
12.4 Sample Size Determination.
12.5 Reference Limits and Coverage Intervals.
12.6 Tolerance Intervals for Binomial and Poisson Distributions.
12.7 Tolerance Intervals Based on Censored Samples.
12.8 Exercises.
Appendix A: Data Sets.
Appendix B: Tables.
References.
Index.
Author Information
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.
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