L-Moments of Residual Life and Partial Moments

Abstract

The residual life distribution and various descriptive measures derived from it form the basis of modelling, characterization and ageing concepts in reliability theory. Of these, the moment-based descriptive measures such as mean, variance and coefficient of variation of residual life and their quantile forms were all discussed earlier in Chaps. 2 and 4. The role of L-moments as alternatives to conventional moments in all forms of statistical analysis was also highlighted in Chap. 1. L-moments generally outperform the usual moments in providing smaller sampling variance, robustness against outliers and easier characterization of distributional characteristics, especially in the case of models with explicit quantile functions but no tractable distribution functions. For this reason, we discuss in this chapter the properties of the first two L-moments of residual life. After introducing the definitions, several identities that connect L-moments of residual life with the hazard quantile function, and mean and variance of residual quantile function, are derived. A comparison between the second L-moment and variance of residual life points out the situations in which the former is better. Expressions for the L-moments of residual life of quantile function models of Chap. 3 are derived and their behaviour is discussed in relation to the mean residual quantile function. Characterization of lifetime models based on the functional form of the second L-moment as well as in terms of its relationship with the hazard and mean residual quantile functions are also presented. The upper and lower partial moments have been found to be of use in reliability analysis, economics, insurance and risk theory. Quantile-based definitions of these moments and their relationships with various reliability functions are presented in this chapter. Many of the results on L-moments of residual life have potential applications in economics. For example, income distributions can be characterized by means of some simple properties of concepts like income gap ratio, truncated Gini index and poverty measures. Quantile forms of all these measures are defined and their usefulness in establishing characterizations are explored.