Statistical Tolerance Regions: Theory, Applications and Computation

Transcription

1 Statistical Tolerance Regions: Theory, Applications and Computation K. KRISHNAMOORTHY University of Louisiana at Lafayette THOMAS MATHEW University of Maryland Baltimore County

2 Contents List of Tables Preface xiii xix 1 Preliminaries Introduction One-Sided Tolerance Intervals Tolerance Intervals Survival Probability and Stress-Strength Reliability Some Technical Results The Modified Large Sample (MLS) Procedure The Generalized P-value and Generalized Confidence Interval Description GPQs for a Location-Scale Family Some Examples Exercises Univariate Normal Distribution Introduction One-Sided Tolerance Limits for a Normal Population Two-Sided Tolerance Intervals Tolerance Intervals Equal-Tailed Tolerance Intervals for a Normal Distribution Simultaneous Hypothesis Testing about Normal Quantiles Tolerance Limits for X 1 X Exact One-Sided Tolerance Limits for the Distribution of X 1 X 2 When the Variance Ratio Is Known vii

3 viii Contents One-Sided Tolerance Limits for the Distribution of X 1 X 2 When the Variance Ratio Is Unknown Hypothesis Testing About the Quantiles of X 1 X Comparison of the Approximate Methods for Making Inference about Quantiles of X 1 X Applications of Tolerance Limits for X 1 X 2 with Examples Simultaneous Tolerance Limits for Normal Populations Simultaneous One Sided Tolerance Limits Simultaneous Tolerance Intervals Exercises Univariate Linear Regression Model Notations and Preliminaries One-Sided Tolerance Intervals and Simultaneous Tolerance Intervals One-Sided Tolerance Intervals One-Sided Simultaneous Tolerance Intervals Two-Sided Tolerance Intervals and Simultaneous Tolerance Intervals Two-Sided Tolerance Intervals Two-Sided Simultaneous Tolerance Intervals The Calibration Problem Exercises The One-Way Random Model with Balanced Data Notations and Preliminaries Two Examples One-Sided Tolerance Limits for N(μ, στ 2 + σ2 e ) The Mee-Owen Approach Vangel s Approach The Krishnamoorthy-Mathew Approach Comparison of Tolerance Limits Examples One-Sided Confidence Limits for Exceedance Probabilities One-Sided Tolerance Limits When the Variance Ratio Is Known One-Sided Tolerance Limits for N(μ, στ 2 ) Two-Sided Tolerance Intervals for N(μ, στ 2 + σe) Mee s Approach

4 Contents ix The Liao-Lin-Iyer Approach Two-Sided Tolerance Intervals for N(μ, σ 2 τ ) Exercises The One-Way Random Model with Unbalanced Data Notations and Preliminaries Two Examples One-Sided Tolerance Limits for N(μ, σ 2 τ + σ 2 e) The Krishnamoorthy and Mathew Approach The Liao, Lin and Iyer Approach One-Sided Confidence Limits for Exceedance Probabilities One-Sided Tolerance Limits for N(μ, σ 2 τ ) The Krishnamoorthy and Mathew Approach The Liao, Lin and Iyer Approach Two-Sided Tolerance Intervals A Two-Sided Tolerance Interval for N(μ, σ 2 τ + σ 2 e) A Two-Sided Tolerance Interval for N(μ, σ 2 τ ) Exercises Some General Mixed Models Notations and Preliminaries Some Examples Tolerance Intervals in a General Setting One-Sided Tolerance Intervals Two-Sided Tolerance Intervals A General Model with Two Variance Components One-Sided Tolerance Limits Two-Sided Tolerance Intervals A One-Way Random Model with Covariates and Unequal Variances Testing Individual Bioequivalence Exercises Some Non-Normal Distributions Introduction Lognormal Distribution Gamma Distribution Normal Approximation to a Gamma Distribution Tolerance Intervals and Survival Probability

5 x Contents Applications with an Example Stress-Strength Reliability Two-Parameter Exponential Distribution Some Preliminary Results One-Sided Tolerance Limits Estimation of Survival Probability Stress-Strength Reliability Weibull Distribution Some Preliminaries The Maximum Likelihood Estimators and Their Distributions Generalized Pivotal Quantities for Weibull Parameters One-Sided Tolerance Limits A GPQ for a Survival Probability Stress-Strength Reliability Exercises Nonparametric Tolerance Intervals Notations and Preliminaries Order Statistics and Their Distributions One-Sided Tolerance Limits and Exceedance Probabilities Tolerance Intervals Confidence Intervals for Population Quantiles Sample Size Calculation Sample Size for Tolerance Intervals of the Form (X (1),X (n) ) Sample Size for Tolerance Intervals of the Form (X (r),x (s) ) Nonparametric Multivariate Tolerance Regions Exercises The Multivariate Normal Distribution Introduction Notations and Preliminaries Some Approximate Tolerance Factors Methods Based on Monte Carlo Simulation Simultaneous Tolerance Intervals Tolerance Regions for Some Special Cases

6 Contents xi 9.7 Exercises The Multivariate Linear Regression Model Preliminaries The Model Some Examples Approximations for the Tolerance Factor Accuracy of the Approximate Tolerance Factors Methods Based on Monte Carlo Simulation Application to the Example Multivariate Calibration Problem Formulation and the Pivot Statistic The Confidence Region Computation of the Confidence Region A Generalization An Example and Some Numerical Results Exercises Bayesian Tolerance Intervals Notations and Preliminaries The Univariate Normal Distribution Tolerance Intervals Under the Non-Informative Prior Tolerance Intervals Under the Conjugate Prior The One-Way Random Model with Balanced Data Two Examples Exercises Miscellaneous Topics Introduction β Expectation Tolerance Regions β Expectation Tolerance Intervals for the Normal Distribution β Expectation Tolerance Intervals for the One-Way Random Model with Balanced Data β Expectation Tolerance Intervals for the One-Way Random Model with Unbalanced Data β Expectation Tolerance Intervals for a General Mixed Effects Model with Balanced Data

7 xii Contents Multivariate β Expectation Tolerance Regions Bayesian β ExpectationToleranceIntervals Tolerance Limits for a Ratio of Normal Random Variables An Upper Tolerance Limit Based on an Approximation to the cdf Tolerance Limits Based on the Exact cdf An Application Sample Size Determination Sample Size Determination for a (p, 1 α) Two-Sided Tolerance Interval for a Normal Population Sample Size Determination for a β Expectation Two-Sided Tolerance Interval for a Normal Population Reference Limits and Coverage Intervals Tolerance Intervals for Binomial and Poisson Distributions Binomial Distribution Poisson Distribution Two-Sided Tolerance Intervals for Binomial and Poisson Distributions Tolerance Intervals Based on Censored Samples Normal and Related Distributions Two-Parameter Exponential Distribution Weibull and Extreme Value Distributions Exercises Appendix A: Data Sets 349 Appendix B: Tables 355 References 441 Index 457