Progressive Censoring

N. BALAKRISHNAN, PhD, is Professor of Mathematics and Statistics at McMaster University in Hamilton, Ontario, Canada.V. B. NEVZOROV, PhD, DS, is Professor of Probability and Statistics at St. Petersburg State University in St. Petersburg, Russia.

This new book offers a thorough guide to the theory and methods of progressive censoring for practitioners and professionals in applied statistics, quality control, life testing and reliability testing. In many industrial experiments involving lifetimes of machines or units, experiments have to be terminated early due to a variety of circumstances. Samples that arise from such experiments are called censored samples, and a new, efficient alternative method is referred to as 'progressive censoring' (where the removal of live units at time of failure is employed). Progressive Censoring first introduces progressive sampling foundations, then discusses various properties of progressive samples. It also describes how to make exact or approximate inferences for the different statistical models with samples based on progressive censoring schemes. With many concrete examples, the book points out the greater efficiency gained by using this scheme instead of classical right-censoring methods.

With its accessible style and concrete examples, the book is anessential resource and guide to the use of progressive censoringmethodology in applied statistics and statistical reliability testing.Advanced students, practitioners and professionals will find it auseful guide to understanding and using progressive censoring conceptsand methods in their work.

1 Introduction.- 1.1 The Big Picture.- 1.2 Genesis.- 1.3 The Need for Progressive Censoring.- 1.4 A Relatively Unexplored Idea.- 1.5 Mathematical Notations.- 1.6 A Friendly Note.- 2 Mathematical Properties of Progressively Type-II Right Censored Order Statistics.- 2.1 General Continuous Distributions.- 2.1.1 Introduction.- 2.1.2 Results.- 2.2 The Exponential Distribution: Spacings.- 2.2.1 Introduction.- 2.2.2 Progressively Type-II Right Censored Spacings.- 2.2.3 Deriving Moments Using Independent Spacings.- 2.3 The Uniform Distribution: Ratios.- 2.3.1 Introduction.- 2.3.2 Independent Ratios.- 2.3.3 Deriving Moments Using Independent Ratios.- 2.4 The Pareto Distribution: Ratios.- 2.4.1 Introduction.- 2.4.2 Independent Ratios.- 2.4.3 Deriving Moments Using Independent Ratios.- 2.5 Bounds for Means and Variances.- 3 Simulational Algorithms.- 3.1 Introduction.- 3.2 Simulation Using the Uniform Distribution.- 3.3 Simulation Using the Exponential Distribution.- 3.4 General Progressively Type-II Censored Samples.- 3.4.1 Arbitrary Continuous Distributions.- 3.4.2 The Exponential Distribution.- 3.4.3 The Uniform Distribution.- 4 Recursive Computation and Algorithms.- 4.1 Introduction.- 4.2 The Exponential Distribution.- 4.2.1 Recurrence Relations for Single Moments.- 4.2.2 Recurrence Relations for Product Moments.- 4.2.3 Recursive Algorithm.- 4.3 The Doubly Truncated Exponential Distribution.- 4.3.1 Recurrence Relations for Single Moments.- 4.3.2 Recurrence Relations for Product Moments.- 4.3.3 Recursive Algorithm.- 4.4 The Pareto Distribution and Truncated Forms.- 4.4.1 Recurrence Relations for Single Moments.- 4.4.2 Recurrence Relations for Product Moments.- 4.4.3 Recursive Algorithm.- 4.5 The Power Function Distribution and Truncated Forms.- 5 Alternative Computational Methods.- 5.1 Introduction.- 5.2 Formulas in Terms of Moments of Usual Order Statistics.- 5.3 Formulas in the Case of Symmetric Distributions.- 5.3.1 Progressive Withdrawal.- 5.3.2 Properties of Progressively Type-II Left Withdrawn Order Statistics.- 5.3.3 Moments of Progressively Type-II Right Censored Order Statistics from Symmetric Distributions.- 5.4 Other Relations for Moments.- 5.5 First-Order Approximations to the Moments.- 6 Linear Inference.- 6.1 One-Parameter (Scale) Models.- 6.1.1 Introduction.- 6.1.2 The Exponential Distribution.- 6.1.3 The Uniform Distribution.- 6.1.4 The Pareto Distribution.- 6.1.5 First-Order Approximation to the BLUE.- 6.2 Two-Parameter (Location-Scale) Models.- 6.2.1 Introduction.- 6.2.2 The Exponential Distribution.- 6.2.3 The Uniform Distribution.- 6.2.4 The Pareto Distribution.- 6.2.5 The Laplace Distribution.- 6.2.6 The Extreme Value Distribution.- 6.2.7 First-Order Approximations to the BLUEs.- 6.3 Best Linear Invariant Estimation.- 7 Likelihood Inference: Type-I and Type-II Censoring.- 71. Introduction.- 7.2 General Continuous Distributions.- 7.3 Specific Continuous Distributions.- 7.3.1 The Normal Distribution.- 7.3.2 The Exponential Distribution.- 7.3.3 The Weibull Distribution.- 7.3.4 The Uniform Distribution.- 7.3.4 The Pareto Distribution.- 7.3.6 The Laplace Distribution.- 7.3.7 Other Distributions (Log-Normal, Gamma, Burr).- 8 Linear Prediction.- 8.1 Introduction.- 8.2 The Exponential Case.- 8.3 Case of General Distributions.- 8.3.1 Scale-Parameter Distributions.- 8.3.2 Location-Scale Distributions.- 8.4 A Simple Approach Based on BLUEs.- 8.5 First-Order Approximations to BLUPs.- 8.6 Prediction Intervals.- 8.7 Illustrative Examples.- 9 Conditional Inference.- 9.1 Introduction.- 9.2 Inference for Location and Scale Parameters.- 9.3 Inference for Quantiles and Reliability and Prediction Intervals.- 9.3.1 Inference for Quantiles.- 9.3.2 Inference for Reliability.- 9.3.3 Prediction Intervals for Future Failures.- 9.4 Results for Extreme Value Distribution.- 9.5 Results for Exponential Distribution.- 9.6 Illustrative Examples.- 9.7 Results for Pareto Distribution.- 10 Optimal Censoring Schemes.- 10.1 Introduction.- 10.2 The Ex