Characterization of distributions and related results

Abstract

In applied statistics, one observes a random quantity X , a number of times and based on these observations, one would like to conclude facts about the distribution function ) (x F of X . The usual approach is to start with a family and#61510;and#61472;of distributions and to select from this family a distribution ) (x F which is the most acceptable one in a given sense. Unfortunately in many cases and#61510;and#61472;simply consist of a single function which is dependent on one or several parameters and the observations are used merely to approximate its parameters. The function thus obtained is chosen as ) (x F [Galambos and Kotz, 1978]. Characterization is a condition involving certain properties of a random variables ) , , , ( 2 1 n X X X X K and#61501;and#61472;, which identifies the associated distribution function ) (x F . The property that uniquely determines ) (x F may be based on a function of random variables whose joint distribution is related to that of ) , , , ( 2 1 n X X X X K and#61501;and#61472;. newlineThe only method of finding distribution function ) (x F exactly, which avoids the subjective choice, is a characterization theorem. A theorem is on a characterization of a distribution function if it concludes that a set of conditions is satisfied by ) (x F and only by ) (x F . Another important consequence of characterization theorem is that these results help us in better understanding the structures and implications of the choice of distribution for a special problem. With this in view, some distributions here are characterized through records, order statistics, generalized order statistics and dual generalized order statistics.Main contributors for characterizations through order statistics are: newlineFerguson (1967), Shanbhag (1970), Hamdan (1972), Beg and Kirmani (1978), Khan and Khan (1986), Khan and Beg (1987), Khan and Ali (1987), Nagaraja (1988), Khan and Abu-Salih (1989), Ouyang and Wu (1996), Franco and Ruiz (1995, 1997), Blaquez and Rebollo (1997), Wesolowski and Ahsanullah (1997), Dembiand#324;ska and Wesolowski (1998) and Khan and Abouammoh (2000), Lee et al. (2002)