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### Summary

### Summary

Full coverage of statistical techniques for developing and implementing precedence-type tests

Precedence-Type Tests and Applications provides a comprehensive overview of theoretical and applied approaches to a variety of problems in which precedence-type test procedures can be used. The authors clearly demonstrate the effectiveness of these tests in life-testing situations designed for making quick and reliable decisions in the early stages of an experiment. Most of the text's examples use life-time data; however, theoretical properties are also discussed in the context of precedence testing. Monte Carlo studies are used to illustrate important results.

Following the authors' careful step-by-step instructions and guidance, readers master the wide range of statistical techniques involved in the development and implementation of precedence-type tests. The book covers the foundations of precedence testing research from the early 1960s up to the most recent theory and applications, including the authors' current contributions to the field. The book features the following parts:

* Part A deals with the original precedence test and some properties of precedence and related test procedures

* Part B explores alternatives to precedence testing, including maximal precedence, weighted forms of precedence and maximal precedence, and Wilcoxon-type rank-sum precedence tests and their properties

* Part C compares the extension of precedence, maximal precedence, and Wilcoxon-type rank-sum precedence tests to situations in which the sample arising from the life-testing experiment is progressively Type-II censored

* Part D examines precedence-type tests in multi-sample situations and selection problems

Tables are presented throughout the book to facilitate the application of the tests to practical problems. Helpful examples illustrate all of the precedence-type procedures, and an extensive bibliography enables readers to explore specialized topics in greater depth.

This book is a recommended reference for researchers and practitioners in reliability and life-time data analysis, applied probabilists, and engineers. It also serves as a supplemental text for courses in nonparametric statistics and reliability.

### Author Notes

N. Balakrishnan, PhD, is Professor of Mathematics and Statistics at McMaster University

H. K. Tony Ng, PhD, is Assistant Professor in the Department of Statistical Science at Southern Methodist University

### Table of Contents

List of Tables | p. xiii |

List of Figures | p. xxi |

Preface | p. xxiii |

1 Introduction | p. 1 |

1.1 Problems of Interest | p. 1 |

1.2 Special Considerations | p. 1 |

1.3 Special Form of Testing | p. 2 |

1.4 Precedence Tests | p. 2 |

1.5 Developments | p. 3 |

2 Preliminaries | p. 7 |

2.1 Life-Test Data | p. 7 |

2.2 Order Statistics | p. 8 |

2.2.1 Joint Densities and Markovian Property | p. 8 |

2.2.2 Marginal Densities | p. 9 |

2.2.3 Moments | p. 11 |

2.2.4 Results for the Uniform Distribution | p. 11 |

2.2.5 Results for the Exponential Distribution | p. 13 |

2.3 Censored Data | p. 15 |

2.3.1 Type-I Censoring | p. 15 |

2.3.2 Type-II Censoring | p. 16 |

2.4 Progressively Censored Data | p. 17 |

2.4.1 General Properties | p. 17 |

2.4.2 Results for the Uniform Distribution | p. 19 |

2.4.3 Results for the Exponential Distribution | p. 20 |

2.5 Some Useful Lifetime Distributions | p. 23 |

2.5.1 Exponential Distribution | p. 23 |

2.5.2 Gamma Distribution | p. 24 |

2.5.3 Weibull Distribution | p. 25 |

2.5.4 Extreme Value Distribution | p. 26 |

2.5.5 Lognormal Distribution | p. 27 |

2.6 Wilcoxon's Rank-Sum Statistic | p. 28 |

2.7 Randomized Test | p. 29 |

3 Precedence Test | p. 31 |

3.1 Introduction | p. 31 |

3.2 Concept of Precedence Test | p. 32 |

3.3 Exact Null Distribution | p. 34 |

3.4 Exact Power Function Under Lehmann Alternative | p. 35 |

3.5 Monte Carlo Simulation Under Location-Shift Alternative | p. 39 |

3.6 Evaluation and Comparative Remarks | p. 40 |

3.7 Properties of Precedence and Related Tests | p. 52 |

3.7.1 Powerful Precedence Tests | p. 52 |

3.7.2 Median Tests | p. 53 |

3.7.3 Precedence-type Tests for Complete and Censored Data | p. 54 |

3.7.4 Exceedance Statistics and Placement Statistics | p. 55 |

3.8 Illustrative Examples | p. 56 |

4 Maximal Precedence Test | p. 61 |

4.1 Introduction | p. 61 |

4.2 Exact Null Distribution | p. 62 |

4.3 Exact Power Function Under Lehmann Alternative | p. 66 |

4.4 Monte Carlo Simulation Under Location-Shift Alternative | p. 72 |

4.5 Evaluation and Comparative Remarks | p. 74 |

4.6 Illustrative Examples | p. 85 |

5 Weighted Precedence and Weighted Maximal Precedence Tests | p. 87 |

5.1 Introduction | p. 87 |

5.2 Test Statistics and Exact Null Distributions | p. 89 |

5.3 Exact Power Function Under Lehmann Alternative | p. 90 |

5.4 Monte Carlo Simulation Under Location-Shift Alternative | p. 93 |

5.5 Illustrative Examples | p. 94 |

6 Wilcoxon-type Rank-sum Precedence Tests | p. 101 |

6.1 Introduction | p. 101 |

6.2 Test Statistics and Exact Null Distributions | p. 102 |

6.2.1 Wilcoxon-type Rank-sum Precedence Test Statistics | p. 102 |

6.2.2 Null Distributions | p. 104 |

6.3 Large-Sample Approximation for the Null Distributions | p. 111 |

6.4 Exact Power Function Under Lehmann Alternative | p. 113 |

6.5 Monte Carlo Simulation Under Location-Shift Alternative | p. 126 |

6.6 Evaluation and Comparative Remarks | p. 149 |

6.7 Illustrative Examples | p. 150 |

7 Extension to Progressive Censoring | p. 153 |

7.1 Introduction | p. 153 |

7.2 Test Statistics and Exact Null Distributions | p. 154 |

7.3 Exact Power Function Under Lehmann Alternative | p. 156 |

7.4 Monte Carlo Simulation Under Location-Shift Alternative | p. 163 |

7.5 Illustrative Example | p. 172 |

8 Generalization to k-Sample Situation | p. 175 |

8.1 Introduction | p. 175 |

8.2 Some Pertinent Problems | p. 176 |

8.3 Comparing Treatments with Control | p. 176 |

8.4 Comparison of Treatments | p. 179 |

9 Selecting the Best Population Using a Test for Equality Based on Precedence Statistic | p. 181 |

9.1 Introduction | p. 181 |

9.2 Two-Sample Selection Problem | p. 182 |

9.2.1 Equal Sample Size Situation | p. 182 |

9.2.2 Unequal Sample Size Situation | p. 184 |

9.2.3 Performance Under Lehmann Alternative | p. 190 |

9.3 k-Sample Selection Problem | p. 194 |

9.3.1 Selection Procedure and Null Distribution | p. 194 |

9.3.2 Handling Ties | p. 197 |

9.4 Monte Carlo Simulation Under Location-Shift | p. 203 |

9.5 Evaluation and Comparative Remarks | p. 226 |

9.6 Illustrative Examples | p. 226 |

10 Selecting the Best Population Using a Test for Equality Based on Minimal Wilcoxon Rank-sum Precedence Statistic | p. 231 |

10.1 Introduction | p. 231 |

10.2 Two-Sample Selection Problem | p. 232 |

10.2.1 Equal Sample Size Situation | p. 232 |

10.2.2 Unequal Sample Size Situation | p. 234 |

10.2.3 Performance Under Lehmann Alternative | p. 237 |

10.3 k-Sample Selection Problem | p. 240 |

10.3.1 Selection Procedure and Null Distribution | p. 240 |

10.3.2 Handling Ties | p. 244 |

10.4 Monte Carlo Simulation Under Location-Shift | p. 247 |

10.5 Evaluation and Comparative Remarks | p. 248 |

10.6 Illustrative Examples | p. 265 |

Appendix | p. 269 |

Bibliography | p. 275 |

Author Index | p. 283 |

Subject Index | p. 285 |