Muhammad Qaiser Shahbaz

Dual Generalized Order Statistics

In previous chapters we have discussed some popular models of ordered random variables when the sample is arranged in increasing order. The simplest of these models is Order Statistics and is discussed in detail in Chap. 2. The comprehensive model for ordered random variables arranged in ascending order is Generalized Order Statistics and is discussed in detail in Chap. 4. The generalized order statistics produce several models for ordered random variables arranged in ascending order as special cases for various choices of the parameters involved. Often it happen that the sample is arranged in descending order for example the life length of an electric bulb arranged from highest to lowest. In such situations the distributional properties of variables can not be studied by using the models of ordered random variables discussed earlier. The study of distributional properties of such random variables is studied by using the inverse image of Generalized Order Statistics and is popularly known as Dual Generalized Order Statistics (DGOS).

Some Uses of Ordered Random Variables

In previous chapters we have discussed popular models of ordered random variables alongside their common distributional properties. Ordered random variables are useful in several areas of statistics like reliability analysis and censoring. In this chapter we will discuss some common uses of ordered random variables.

The Generalized Order Statistics

In Chap. 1 we have given a brief overview of some of the possible models for ordered random variables.

Characterizations of Distribution

Characterization of probability distributions play important role in probability and statistics. Before a probability distribution is applied to a real set of data it is necessary to know the distribution that fits the data by characterization. A probability distribution can be characterized by various methods, see for example Ahsanullah et al. (2014). In this chapter we will characterize probability distributions by various properties of ordered data. We will consider order statistics, record values and generalized order statistics to characterize the probability distributions.