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Birkhäuser - Birkhäuser Applied Probability and Statistics | Progressive Censoring - Theory, Methods, and Applications

Progressive Censoring

Theory, Methods, and Applications

Balakrishnan, N., Aggarwala, Rita

2000, XV, 248 p.

A product of Birkhäuser Basel
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  • About this book

Censored sampling arises in a life-testing experiment whenever the experimenter does not observe (either intentionally or unintentionally) the failure times of all units placed on a life-test. Inference based on censored sampling has been studied during the past 50 years by numerous authors for a wide range of lifetime distributions such as normal, exponential, gamma, Rayleigh, Weibull, extreme value, log-normal, inverse Gaussian, logistic, Laplace, and Pareto. Naturally, there are many different forms of censoring that have been discussed in the literature. In this book, we consider a versatile scheme of censoring called progressive Type-II censoring. Under this scheme of censoring, from a total of n units placed on a life-test, only m are completely observed until failure. At the time of the first failure, Rl of the n - 1 surviving units are randomly withdrawn (or censored) from the life-testing experiment. At the time of the next failure, R2 of the n - 2 -Rl surviving units are censored, and so on. Finally, at the time of the m-th failure, all the remaining Rm = n - m -Rl - . . . - Rm-l surviving units are censored. Note that censoring takes place here progressively in m stages. Clearly, this scheme includes as special cases the complete sample situation (when m = nand Rl = . . . = Rm = 0) and the conventional Type-II right censoring situation (when Rl = . . . = Rm-l = 0 and Rm = n - m).

Content Level » Research

Keywords » Censoring - Likelihood - Norm - Normal distribution - Simulation - Variance - censored samples - life testing - progressive censoring - quality - quality control - reliability testing - statistics

Related subjects » Birkhäuser Applied Probability and Statistics

Table of contents 

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 Exponential Distribution.- 10.3 The Normal Distribution.- 10.3.1 Discussion of Results.- 10.4 The Extreme Value Distribution.- 10.4.1 Discussion of Results.- 10.5 The Extreme Value (II) Distribution.- 10.5.1 Discussion of Results.- 10.6 The Log-Normal Distribution.- 10.6.1 Discussion of Results.- 10.7 Tables.- 11 Acceptance Sampling Plans.- 11.1 Introduction.- 11.2 The Exponential Distribution.- 11.2.1 One-Sided Sampling Plans.- 11.2.2 Two-Sided Sampling Plans.- 11.3 The Log-Normal Distribution.- Author Index.

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