Moments of order statistics and k-record values arising from the complementary beta distribution with application

Abstract

The order statistics (OSs) arise in both practical and theoretical aspects including goodness-of-fit tests, characterizations of probability distributions and some estimation approaches such as the L-moments method. Most of these instances are connected with moments of OSs. The records and k-record statistics, also called kth record values in the literature, are other important topics related to the OSs. This kind of data appears in some practical circumstances, e.g. shock models, industrial stress testing and meteorological data analysis. This paper deals with deriving recurrence relations for the single and the product moments of OSs as well as k-record values from the complementary beta (CB) distribution defined by Jones (2002). The CB model, as a distribution on the unit interval, is more flexible than the well-known beta distribution in problems involving moments of OSs. This fact suggests that the former distribution can be employed rather than the latter when OSs are of the main interest. The results obtained are applied on four real data sets for the issues of parameter estimation as well as providing a visual inspection for goodness-of-fit of the CB model.