Abstract
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.
Highlights
Order statistics appears in many statistical applications and is widely used in statistical modeling and inference
On the other hand, generalized order statistics (GOS) have been of interest in the past ten years because they are more flexible in reliability theory, statistical modeling and inference, the generalized order statistics have been introduced as a unified distribution theoretical set-up which contains a variety of models of ordered random variables with different interpretations
Generalized order statistics (GOS) have been of interest in the past ten years because they are more flexible in reliability theory, statistical modeling and inference [12], Uniform generalized order statistics is defined via some joint density function on a cone of Rn. (GOS) based on an arbitrary distribution function F is defined by means of the inverse function of F, as in the following: Definition (1) Let F x denotes an absolutely continuous distribution function with density function f(x), the sequence of random variables X1:n,m,k, X 2:n,m,k,Λ, X n:n,m,k is called “n” Generalized Order Statistics (GOS), where (k ≥ 1, m is a real number) [3]
Summary
Order statistics appears in many statistical applications and is widely used in statistical modeling and inference. On the other hand, generalized order statistics (GOS) have been of interest in the past ten years because they are more flexible in reliability theory, statistical modeling and inference, the generalized order statistics have been introduced as a unified distribution theoretical set-up which contains a variety of models of ordered random variables with different interpretations. The Generalized Exponential Distribution (GED) with two non-negative parameters and is considered to be one of those distributions which have real attention from researchers It has been studied and introduced by [12]. Generalized order statistics (GOS) have been of interest in the past ten years because they are more flexible in reliability theory, statistical modeling and inference [12], Uniform generalized order statistics is defined via some joint density function on a cone of Rn. If m 0 and k 1 it gives the joint pdf of “n” ordinary order statistics X1,n , X 2,n ,Λ, X n,n , if m 1, k 1 , it gives the joint pdf of the first “n” upper records of the independent identically distributed random variables
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