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Characterization of Distributions by Using the Conditional Expectations of Generalized Order Statistics

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Author(s): Tuğba Yıldız | Ismihan Bairamov

Journal: Selçuk Journal of Applied Mathematics
ISSN 1302-7980

Volume: 9;
Issue: 2;
Start page: 19;
Date: 2008;
Original page

Keywords: Order statistics | progressively Type II censored order statistics | generalized order statistics | characterization of distributions.

ABSTRACT
In this study, some continuous distributions through the properties of conditional expectations of generalized order statistics are characterized. Let X_{1:n:m:k},...,X_{n:n:m:k} be the generalized order statistics, where n∈ℕ,k>0,m₁,...,m_{n-1}∈ℝ, M_{r}=∑_{j=r}ⁿ⁻¹m_{j},1≤r≤n-1,γ_{r}=k+n-r+M_{r}>0 for all r∈{1,...,n-1} and let m={m₁,...,m_{n-1}}, if n≥2, m∈ℝ, arbitrary, if n=1. Characterization theorems for a general class of distributions are presented in terms of the function E{g(X_{j:n:m:m+1})∣X_{j-p:n:m:m+1}=x,X_{j+q:n:m:m+1}=y}=A(x,y), where k=m+1, p and q are positive integers such that p+1≤j≤n-q and g(.), A(.,.) is a real valued function satisfying certain regularity conditions.
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