Statistical Tolerance Regions (eBook)

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.

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 X1 - X2.

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, sigma²tau + sigma²taue).

4.4 One-Sided Tolerance Limits for N(µ, sigma²tau¨).

4.5 Two-Sided Tolerance Intervals for N(µ, sigma²tau + sigma²taue).

4.6 Two-Sided Tolerance Intervals for N(µ, sigma²tau¨).

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, sigma²tau + sigma²e).

5.4 One-Sided Tolerance Limits for N(µ, sigma²tau).

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 beta-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.