# Statistical Tolerance Regions: Theory, Applications, and Computation

# Statistical Tolerance Regions: Theory, Applications, and Computation

ISBN: 978-0-470-47390-0

Nov 2008

512 pages

## Description

**A modern and comprehensive treatment of tolerance intervals and regions**

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.

The book begins with the key definitions, concepts, and technical results that are essential for deriving tolerance intervals and tolerance regions. Subsequent chapters provide in-depth coverage of key topics including:

- Univariate normal distribution
- Non-normal distributions
- Univariate linear regression models
- Nonparametric tolerance intervals
- The one-way random model with balanced data
- The multivariate normal distribution
- The one-way 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.

Preface.

**1 Preliminaries.**

1.1 Introduction.

1.2 Some Technical Results.

1.3 The Modified Large Sample (MLS) Procedure.

1.4 The Generalized P-value and Generalized Confidence Interval.

1.5 Exercises.

**2 Univariate Normal Distribution.**

2.1 Introduction.

2.2 One-Sided Tolerance Limits for a Normal Population.

2.3 Two-Sided 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 One-Sided Tolerance Intervals and Simultaneous Tolerance Intervals.

3.3 Two-sided Tolerance Intervals and Simultaneous Tolerance Intervals.

3.4 The Calibration Problem.

3.5 Exercises.

**4 The One-Way Random Model with Balanced Data.**

4.1 Notations and Preliminaries.

4.2 Two Examples.

4.3 One-Sided Tolerance Limits for N(µ, σ²τ + σ²τ_{e)}.

4.4 One-Sided Tolerance Limits for N(µ, σ²τ¨).

4.5 Two-Sided Tolerance Intervals for N(µ, σ²τ + σ²τ_{e}).

4.6 Two-Sided Tolerance Intervals for N(µ, σ²τ¨).

4.7 Exercises.

**5 The One-Way Random Model with Unbalanced Data.**

5.1 Notations and Preliminaries.

5.2 Two Examples.

5.3 One-Sided Tolerance Limits for N(µ, σ²τ + σ²_{e}).

5.4 One-Sided Tolerance Limits for N(µ, σ²τ).

5.5 Two-Sided 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 One-Way Random Model with Covariates and Unequal Variances.

6.5 Testing Individual Bioequivalence.

6.6 Exercises.

**7 Some Non-Normal Distributions.**

7.1 Introduction.

7.2 Lognormal Distribution.

7.3 Gamma Distribution.

7.4 Two-Parameter 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 One-Sided 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 One-Way Random Model With Balanced Data.

11.4 Two Examples.

11.5 Exercises.

**12 Miscellaneous Topics.**

12.1 Introduction.

12.2 β-Expectation 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.