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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 At the time of the first failure, Rl of the n - 1 surviving units are randomly withdrawn (or censored) from the life-testing experiment.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).

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
The importance of product reliability is greater than ever at the present time. As more and more products are introduced to the market, consumers now have the luxury of demanding high quality and long life in the products they purchase. In such a highly demanding and competitive market, one way by which manufacturers (of computers, automobiles, and electronic items, for example) attract consumers to their products is by providing warranties on product lifetimes. In order to design a cost-effective warranty, a manufacturer must have sound knowledge about product failure-time distributions. To gain this knowledge, life-testing and reliability experiments are carried out before (and while) products are put on the market. Of course, the information gained through life-testing experiments is also used for other purposes in addition to determining effective warranties; for example, in pharmaceutical applications, the lifetimes of drugs may be studied in order to determine appropriate dosage administration and expiry dates. Furthermore, continuous improvement of products becomes essential and even critical in a competitive market. Life-testing experimentation is one way by which product improvement and product quality can be gauged.
N. Balakrishnan, Rita Aggarwala

2. Mathematical Properties of Progressively Type-II Right Censored Order Statistics

Abstract
We mentioned earlier in Chapter 1 that marginal distributions of progressively Type-II right censored order statistics (except for the first one, X1:m:n ), unlike usual order statistics, are quite complicated and cumbersome and are not easily obtained (Try it — you won’t like it!). However, certain joint marginal distributions are readily obtained and they do aid in studying mathematical properties of progressively Type-II censored order statistics and also in developing inferential procedures based on progressively censored samples. Furthermore, they aid in providing some efficient algorithms for simulating progressively Type-II censored samples.
N. Balakrishnan, Rita Aggarwala

3. Simulational Algorithms

Abstract
Many statistical problems are far too complex to be solved explicitly using pure paper-and-pencil mathematics. For example, many industrial systems consist of hundreds of inter-related variables with possibly different probability distributions. In modelling such a complex system mathematically, without the use of heavy computation, many assumptions will be made and many variables assumed constant or negligible in their contribution in order to arrive at a tractable model. However, with high speed computers (getting higher in speed every few months, it seems) readily available to virtually all practitioners and researchers, more and more complex systems may be modelled via statistical simulation in which many variables can be made to vary simultaneously, and the process may be studied with surprisingly high accuracy; we can finally “fit a model to the data” rather than fitting the data to a “friendly” model. As one becomes more familiar with problems in statistics and mathematics, and we see that even innocent-looking problems are sometimes impossible to solve explicitly, the appreciation for simulation and computing power grows rapidly. Many of us have already mused, “It is a wonder that any problems at all were solved without the use of computers.”
N. Balakrishnan, Rita Aggarwala

4. Recursive Computation and Algorithms

Abstract
In the next two chapters, considerable emphasis will be placed on obtaining moments of progressively Type-II censored order statistics. One reason for this will become clear in Chapters 6 and 10 when the focus is on developing efficient inferential procedures based on progressive Type-II censoring. We have seen in the last chapter that moments of progressively censored order statistics from some distributions can be obtained explicitly. However, very often, it is not possible to obtain explicit expressions for these moments. The idea of obtaining moments of usual order statistics in a recursive manner has been explored for a number of distributions; for example, see Balakrishnan, Malik and Ahmed (1988), and Balakrishnan and Sultan (1998). In this chapter, we establish several recurrence relations satisfied by the single and product moments of progressively Type-II right censored order statistics from the exponential, Pareto and power function distributions as well as their truncated forms. It is important when establishing such recurrence relations that the relations be complete,in the sense that they may be used in a simple recursive manner in order to compute all the single and product moments of all progressively T-II right censored order statistics from the distributions of interest for all sample sizes n and all censoring schemes (R 1,R 2, ⋯, R), m ≤ n. This, in fact, is the case for all the distributions mentioned above.
N. Balakrishnan, Rita Aggarwala

5. Alternative Computational Methods

Abstract
Clever transformations and efficient recursive algorithms are often useful in obtaining moments and for establishing mathematical properties of progressively Type-II right censored order statistics from a number of distributions, as we have already seen. However, such elegant methods are not possible for all distributions that may be of interest to a practitioner. For this reason, alternative methods for computing moments of progressively censored order statistics must be sought. In this chapter, we present two such methods for the computation of moments of progressively Type-II right censored order statistics. The first applies to an arbitrary continuous distributions for which the moments of usual order statistics are known, and the second applies specifically to symmetric distributions for which the moments of progressively Type-II right censored order statistics from the corresponding folded distribution are known. These two methods will further enhance our repertoire of distributions that we can consider as models for lifetime data, and hence will make the use of progressive censoring in real-life situations much more attractive. Finally, we present some first-order approximations to the means, variances and covariances of progressively Type-II right censored order statistics based on Taylor series expansions. These expressions will be used later in Chapter 6 in order to develop and illustrate first-order approximations to the best linear unbiased estimators of location and scale parameters of any distribution of interest.
N. Balakrishnan, Rita Aggarwala

6. Linear Inference

Abstract
We have to this point developed a number of mathematical results for progressively Type-II censored order statistics, some of which are quite elegant and intuitively pleasing, and others which may be classified by many as “the reason why mathematics is so unpopular.” However, all of the results obtained thus far enjoy real and practical. applications. One such application has already been considered, namely, computer simulation of progressively censored order statistics. We will consider in this chapter the problem of linear inference for the parameters of lifetime distributions when observed samples are progressively censored. Many results presented will be for general progressively Type-II censored samples which were introduced in Section 1.5. To obtain results for progressively Type-II right censored order statistics, we would simply set r =0.
N. Balakrishnan, Rita Aggarwala

7. Likelihood Inference: Type-I and Type-II Censoring

Abstract
In Chapter 6, we considered linear inference based on progressively Type-II censored order statistics. Linear inference is popular because, in addition to having many desirable properties which we associate with good estimators, linear estimators have a very simple form, viz, the estimators are linear combinations of observed data values. As a result, these estimators are often quite simple to interpret from a practitioner’s point of view. Furthermore, we are able to very easily calculate the variances and covariance of linear estimators. This is not always true for other types of estimators, such as maximum likelihood and moment estimators. However, the linear inference that we have discussed applies only to scale- or location-scale families of distributions. If a new parameter, such as a shape parameter or a threshold parameter that can not be written as a simple shift from the standard distribution, is introduced, other methods of estimating these parameters need to be considered.
N. Balakrishnan, Rita Aggarwala

8. Linear Prediction

Abstract
Until now, we have discussed many properties of progressively Type-II right censored order statistics and also the estimation of location and scale parameters of different distributions based on progressively censored samples.
N. Balakrishnan, Rita Aggarwala

9. Conditional Inference

Abstract
So far, all the inferential methods we have discussed are unconditional in nature. In this chapter, we will demonstrate how exact confidence intervals or prediction intervals may be obtained using the conditional method. Conditional inference, first proposed by Fisher (1934), has been successfully applied by Lawless (1973, 1978, 1982) to develop inference based on complete as well as conventionally Type-II right censored samples. As a matter of fact, Lawless (1982, p. 199) indicated the use of conditional inference based on progressively Type-II right censored data; but a full length account of this topic has been provided by Viveros and Balakrishnan (1994) which naturally forms a basis for much of the discussion in this chapter.
N. Balakrishnan, Rita Aggarwala

10. Optimal Censoring Schemes

Abstract
Up to this point, we have considered mathematical properties and problems of inference for progressively censored samples when a particular censoring scheme is to be employed. In reading this far, perhaps you yourself have asked the question, “How does a practitioner decide what the censoring scheme should be?” Is the decision made strictly on the basis of convenience, or can we choose a scheme which makes the most sense in some more statistical or mathematical setting? The question of choosing optimal values of R1, R2, ⋯, Rm when considering a progressive Type-II right censoring scheme is certainly an important one to consider from a practical point of view, and as it turns out, it also gives rise to a number of interesting mathematical problems, in the areas of optimization, numerical analysis, simulation and programming, among others. We consider progressively Type-II right censored samples for the most part, since Type-II censored samples are far more tractable and interesting to consider from the point of linear inference and other mathematical properties, as we have already seen, and right censored samples will arise most frequently in life-testing applications, where it is possible and generally sensible to observe and monitor failures from the onset of experimentation at time t = 0.
N. Balakrishnan, Rita Aggarwala

11. Acceptance Sampling Plans

Abstract
In this Chapter, we discuss the construction of acceptance sampling plans based on progressively Type-II right censored samples. For this purpose, we may either use exact (ML/BLU) estimators of the parameters and their exact distributional properties or use first-order approximate estimators of the parameters and their distributional properties. We consider the exponential and log-normal distributions for illustration with the exponential being an example of the first kind and the log-normal being an example of the second kind.
N. Balakrishnan, Rita Aggarwala

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