CHARACTERIZATION OF DISTRIBUTIONS BY EQUALITIES OF TWO GENERALIZED OR DUAL GENERALIZED ORDER STATISTICS

Generalized order statistics have been introduced by Kamps [1] as an integrated approach of varied models of ascending ordered random variables (rv’s) including, ordinary order statistics, sequential order statistics, progressive type II censored order statistics, record values, and Pfeifer’s records. The gos Y (i, n, m̃n, k), i = 1, 2, ..., n with n > 2, based on a df F are defined via their joint probability density function (jpdf), f1,2,...,n(y1, y2, ..., yn) = k n−1 ∏

A characterization in statistics is a specific distributional property of a statistic that uniquely identify related parametric family of distributions.Classical results in characterizations can be found in [6], [7] and [8].For a comprehensive survey of characterizations on the basis of functions of order statistics, see Gather et al [9].Since gos have been introduced in [1], characterization of probability distributions based on ordered rv's receives an increasing attention by several authors including, [10], [11] and [12], among others.
In this paper, the results of [13] for ordinary order statistics, are extended to gos and dgos models.

Characterization of Distribution Based on gos
In this section, all possible situations when two different gos have the same distribution function are discussed.In other words, if what are the conclusions that can be obtained concerning F ? Before answering this question, we define the following set of integers, and let L C , denote the complement set of L. The main results in this section are formulated in theorems 2.1 and 2.2.
.., n 2 are gos based on the same df F, where (n 1 , n 2 , r, s) ∈ L C .Suppose also that (2.1) holds and the following condition, is satisfied.Then F is degenerate df.
The proof of the theorem is split into three lemmas.The first lemma includes some ordered relations which will be needed in the sequel, while the second lemma is formulated in the general case and include three cases in which the df F is degenerate.
Lemma 2.1.Let n, r, t be positive integers and let m j −1 be real numbers for all j = 1, 2, ..., n , where U is a standard uniform random variable.From the assumptions of the lemma, I get Hence, we get
Lemma 2.2.Let (2.1) be satisfied with (n 1 , n 2 , r, s) ∈ L C .Then F is degenerate df, if one of the following three conditions holds.
Proof.For the first case, if C 1 is satisfied, (2.1) reads P (Y (r, n, mn , k) y) = P (Y (s, n, mn , k) y) with r < s, and y ∈ R, which by (1.1) can be written as Since r < s and U ∼ U (0, 1), V r (n) > V s (n) with probability one.Thus by (2.6)I get which holds only when F (y) = 0 or 1.To prove the second case, first note that under condition C 2 , (1.1) and (2.1) yield (2.7) In view of (2.3), V r (n 1 ) < V r (n 2 ) almost sure.Consequently, which corresponds to degenerate distribution.If condition C 3 holds, (1.1) implies Again, an application of (2.3) yields, V r (n 2 ) > V s (n 1 ) with probability one.Thus Hence F is degenerate df.The lemma is thus established.
Lemma 2.3.Under the same conditions of Theorem 2.1, if relation (2.1) is satisfied for Then F is degenerate df.
Proof.According to Lemma 2.1, the following ordered relation holds true ) Hence by the first case of Lemma 2.3, F must be degenerate df.
In the previous four cases, it has been shown that F is degenerate df.The last case is given in the following theorem.
In order to prove the theorem, a simple representation for the marginal df of the rth gos, Y (r, n, m, k), with less restrictive conditions than m−gos, is given in the following lemma. where For we have m r,n = m and m * r,n = 0, which leads to where N = n + k m+1 .Proof.According to lemma 1.1 of [16], I get where If α is an integer, it can be proved that The lemma immediately follows by putting α = r and β = N r,n − R r,n in (2.12).

Characterization of Distribution Based on dgos
Precisely, descending ordered rv's such as lower record values cannot be included in the gos model.Burkschat, et al [17] have introduced dgos as a unified model of descending ordered random variables like reversed order statistics, lower k−records and lower Pfeirfer' records, through a combined approach.By analogy with (1.1), the dgos, Y d (r, n, mn , k), r = 1, 2, ..., n, have been defined in [17], as What are conclusions that can be gained about F ?The answer of this question is presented in Theorems 3.1 and 3.2.The ordered relations presented in Lemma 3.1 is necessary for proving Theorem 3.1, while Lemma 3.2 will be used in the proof of Theorem 3.2.