Generalized Order Statistics from Generalized Exponential Distributions in Explicit Forms

The generalized order statistics which introduced by [1] are studied 
in the present paper. The Gompertz distribution is widely used to describe the 
distribution of adult deaths, and some related models used in the economic 
applications [2]. Previous works concentrated on formulating approximate 
relationships to characterize it [3-5]. The main aim of this paper is to obtain the 
distribution of single, two, and all generalized order statistics from Gompertz 
distribution with some special cases. In addition the conditional distribution 
of two generalized order statistics from the same distribution is obtained. The Gompertz 
distribution has a continuous probability density function with location 
parameter a and shape parameter b, , where x restricted by the interval . The nth moment generated 
function of the Gompertz distributed random variable X is given on the form: where, is the generalized 
integro-exponential function [6]. In this paper we shall obtain joint 
distribution, distribution of product of two generalized order statistics from 
the Gompertz distribution, and then derive some useful formulas of these 
distributions as special cases.


Introduction
In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson. In other words, it is a process in which events occur continuously and independently at a constant average rate. The ordered random variables such as order statistics play an important roles in many branches of statistics and applied probability. The concept of Generalized Order Statistics (GOS) is introduced in Kamps (1995) and showed that order statistics, record values, and some other ordered random variables can be considered as special cases of generalized order statistics. Several recurrence relations satisfied by the single and the product moments for order statistics from the Generalized Exponential Distribution (GED) are discussed in Ragab (2004). The relationships can be written in terms of polygamma and hypergeometric functions and used in a simple recursive manner in on the cone F −1 (0) < X 1 ≤ X 2 ≤ . . . ≤ X n < F −1 (1) of R n , with parameters n ∈ N, n ≥ 2, k > 0, m = (m 1 , m 2 , . . . m n−1 ) ∈ R n−1 , M r = n−1 j=r m j , such that γ r = k + n − r + M r > 0 for all r ∈ {1, 2, . . . , n − 1}, let c r−1 = r j=1 γ j , r = 1, 2, . . . , n − 1 and γ n = k.

Definition 1.3
A random variable X is said to have a LED with two non-negative parameters θ and λ, if it's pdf and cdf are given, respectively, by Mahmoud and Al-Nagar (2009) This paper is organized as follows: Sections 2 and 3 present the joint distribution for all and two GOS from GED and LED; Sections 4 displays the distribution of single GOS; section 5 demonstrates conditional distribution of GOS from GED and LED; and Section 6 summarizes the important results and offers suggestions for future research.

Joint Distribution of all Generalized Order Statistics
In this section, we derive the joint distribution of all GOS for generalized exponential distributions and linear exponential distributions.

Joint Distribution of all Generalized Order Statistics for Generalized Exponential Distribution
Now, we shall obtain the joint pdf of X (1, n, m, k) , . . . , X (n, n, m, k) for GED and discuss some special cases.
Theorem 2.1 The joint pdf of X (1, n, m, k) , . . . , X (n, n, m, k) for GED is given by Proof: Using the pdf and cdf given in (2) and (3) in (1) and collecting terms we get (6) and that completes the proof. We discuss some special cases in Corollaries 2.2 and 2.3.

Joint Distribution of all Generalized Order Statistics for Linear Exponential Distribution
Now, we shall obtain the joint pdf of X (1, n, m, k) , . . . , X (n, n, m, k) for LED and discuss some special cases.
Theorem 2.4 The joint pdf of X (1, n, m, k) , . . . , X (n, n, m, k) for LED is given by Proof: Using the pdf and cdf given in (4) and (5) in (1) we get Collecting terms we get (9) and that completes the proof.

Samir Khaled Safi, AlSheikh Ahmed Rehab H. AlSheikh Ahmed
We discuss some special cases in Corollaries 2.5 and 2.6.

Corollary 2.5 (The joint pdf of all GOS for LED)
In equation (9), let θ = 1, and collecting terms then the joint pdf of all GOS X (1, n, m, k) , . . . , X (n, n, m, k) for LED is given by Corollary 2.6 (The joint pdf of all ordinary order statistics for LED) In equation (10), let k = 1 and m = 0 and collecting terms then the joint pdf of all ordinary order statistics X (1, n, 0, 1) , . . . , X (n, n, 0, 1) for LED is

Joint Distribution of Two Generalized Order Statistics
In this section, we derive the joint distribution of two GOS for generalized Exponential distribution and linear Exponential distribution.
Definition 3.1 The joint pdf of i th and j th GOS X (i, n, m, k) and X (j, n, m, k) and , is given by Garg (2009).
for all x ∈ (0, 1) and for all m with g −1 (x) = lim Electronic Journal of Applied Statistical Analysis 227

Joint Distribution of Two Generalized Order Statistics for Generalized Exponential Distribution
Now, we shall obtain the joint pdf of two GOS for GED and discuss some special cases. In Theorem 3.2, we derive the joint pdf of X (i, n, m, k) and X (j, n, m, k) for GED.
Theorem 3.2 Using the pdf and cdf given in (2) and (3) in (12) we get joint pdf of X (i, n, m, k) and X (j, n, m, k) for GED is given by m + 1 and collecting terms, the joint pdf of X (i, n, m, k) and X (j, n, m, k) for GED is We discuss some special cases in Corollaries 3.3 and 3.4. Corollary 3.3 (The joint pdf of two GOS for Exponential Distribution) In equation (13) , let θ = 1 and collecting terms, then the joint pdf of two GOS X (i, n, m, k) and X (j, n, m, k) for Exponential distribution is Corollary 3.4 (The joint pdf of two ordinary order statistics for Exponential Distribution) In equation (14), let k = 1 and m = 0 and collecting terms then the joint pdf of two ordinary order statistics X (i, n, 0, 1) and X (j, n, 0, 1) for Exponential distribution is where, γ j − 1 = n − j when k = 1 and m = 0.

Joint Distribution of Two Generalized Order Statistics for Linear Exponential Distribution
Now, we shall obtain the joint pdf of two GOS for LED and discuss some special cases. In Theorem 3.5, we derive the joint pdf of X (i, n, m, k) and X (j, n, m, k) for LED.
Theorem 3.5 Using the pdf and cdf given in (4) and (5) in (12), we get the joint pdf of X (i, n, m, k) and X (j, n, m, k) for LED is given by Collecting terms, the joint pdf of X (i, n, m, k) and X (j, n, m, k) for LED is given by where, We discuss some special cases in Corollaries 3.6 and 3.7.
Corollary 3.6 (The joint pdf of two GOS for LED) In equation (16) , let θ = 1, and collecting terms then the joint pdf of two GOS X (i, n, m, k) and X (j, n, m, k) for LED is given by where, g * m (F (x i )) = g m (F (x i )) when θ = 1.
Corollary 3.7 (The joint pdf of two ordinary order statistics for LED) In equation (17) , let k = 1 and m = 0, and collecting terms then the joint pdf of two ordinary order statistics X (i, n, 0, 1) and X (j, n, 0, 1) for LED is given by where γ j − 1 = n − j when k = 1 and m = 0.

Distribution of Single Generalized Order Statistics
In this section, we derive the pdf of the minimum and maximum GOS and consider GED and LED.
Lemma 4.3 The pdf of the maximum generalized order statistic is Proof: Using Definition (4.1), let r = n, then g m (F ( get (21) and that completes the proof.

Distribution of Single Generalized Order Statistics for Generalized Exponential Distribution
Now, we shall obtain the pdf of minimum and maximum GOS for GED and discuss some special cases.
Lemma 4.4 (The pdf of the minimum GOS for GED) Using the pdf and cdf given in (2) and (3) in (20), and collecting terms, we get the pdf of the minimum GOS for GED is given by We discuss some special cases in Corollaries 4.5 and 4.6.

Corollary 4.5 (The pdf of the minimum GOS for Exponential Distribution)
In equation (22) , let θ = 1 and collecting terms, then the pdf of the minimum generalized order statistic for Exponential Distribution is Corollary 4.6 (The pdf of the minimum ordinary statistics for Exponential Distribution) In equation (23), let k = 1 and m = 0 then, f 1,n,0,1 (x) = nλ [exp (−λx 1 )] n , which is the well known pdf of the minimum ordinary order statistic for Exponential Distribution.
Lemma 4.7 (The pdf of the maximum GOS for GED) Using the pdf and cdf given in (2) and (3) in (21), and collecting terms we get pdf of the maximum GOS for GED, We discuss some special cases in Corollaries 4.8 and 4.9.
Corollary 4.8 (The pdf of the maximum GOS for Exponential Distribution) In equation (24) , let θ = 1, then the pdf of the maximum GOS for Exponential Distribution is given by Corollary 4.9 (The pdf of the maximum ordinary statistics for Exponential
Equation (26) is the well known pdf of the maximum ordinary order statistic for Exponential Distribution.

Distribution of Single Generalized Order Statistics for Linear Exponential Distribution
Now, we shall obtain the pdf of minimum and maximum GOS for LED and discuss some special cases. Lemma 4.10 (The pdf of the minimum GOS for LED) Using the pdf and cdf given in (4) and (5) in (20), and collecting terms, the pdf of single GOS for LED is given by We discuss some special cases in Corollaries 4.11 and 4.12.
Corollary 4.11 (The pdf of the minimum GOS for LED for θ = 1) In equation (27), let θ = 1 and collecting terms then the pdf of the minimum generalized order statistic for LED is given by Corollary 4.12 (The pdf of the minimum ordinary statistics for LED) In equation (28), let k = 1 and m = 0 then The pdf of the minimum ordinary statistics for LED is given by Equation (29) is the well known pdf of the minimum ordinary order statistic for LED.
Lemma 4.13 (The pdf of the maximum GOS for LED) Using the pdf and cdf given in (4) and (5) in (21), and collecting terms we get the pdf of the maximum GOS for LED is given by We discuss some special cases in Corollaries 4.14 and 4.15.
Corollary 4.14 (The pdf of the maximum GOS for LED for ) In equation (30) , let θ = 1, then the pdf of the maximum generalized order statistic for LED is given by Corollary 4.15 (The pdf of the maximum ordinary statistics for LED) In equation (31), let k = 1 and m = 0, and collecting terms then the pdf of the maximum ordinary statistics for LED is given by where, [k + (n − 1) (m + 1)] [k + (n − 2) (m + 1)] . . .
Equation (32) is the well known pdf of the maximum ordinary order statistic for LED.

Conditional Distribution of Generalized Order Statistics
In this section, we introduce the conditional distribution of GOS and ordinary order statistic. We consider the conditional distribution of GOS for GED and LED.
Theorem 5.1 Let X 1 , X 2 , . . . , X n be a random sample from a continuous population with cdf , F (x) and pdf, f (x) . Let X (1, n, m, k) , . . . , X (n, n, m, k) denote the GOS obtained from this sample.
Then the conditional pdf of X (s, n, m, k) |X (r, n, m, k) = x for r < s is given by Samuel (2008), Some special cases are derived in Corollaries 5.7 and 5.8.

Conclusion and Future Research
In this paper, we have derived the joint pdf s of GOS for Generalized and Linear Exponential distributions in explicit forms. In addition, the pdf of the conditional distribution of GOS from those distributions is given. Furthermore, some special cases have been discussed.

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Many opportunities of future research are available. The plan for the future research on GOS from Generalized and Linear Exponential distributions can be split into two main areas. Estimation and hypothesis testing of Generalized Exponential parameters based on generalized order statistics. Aknowledgment I am grateful for the referees for their valuable comments and suggestions on earlier draft of this paper.