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.