Abstract
In this paper, we show a unified approach to the problem of characterizing general distribution functions based on the conditional expectation between adjacent order statistics, ξ ( x ) = E ( h ( X r , n ) ∣ X r + 1 , n = x ) \xi (x)=E(h(X_{r,n})\mid X_{r+1,n}=x) or ξ ¯ ( x ) = E ( h ( X r + 1 , n ) ∣ X r , n = x ) \overline {\xi }(x)=E(h(X_{r+1,n})\mid X_{r,n}=x) , where h h is a real, continuous and strictly monotonic function. We have the explicit expression of the distribution function F F from the above order mean function, ξ \xi and ξ ¯ \overline {\xi } , and we give necessary and sufficient conditions so that any real function can be an order mean function. Our results generalize the results given for the discrete, absolutely continuous and continuous cases. Further, we show stability theorems for these characterizations.
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