Distributions determined by conditioning on a pair of order statistics

Abstract

LetX ₁,X ₂, ...,X _n (n≥3) be a random sample on a random variableX with distribution functionF having a unique continuous inverseF ⁻¹ over (a,b), −∞≤a<b≤∞ the support ofF. LetX _1:n <X _2:n <...<X _n:n be the corresponding order statistics. Letg be a nonconstant continuous function over (a,b). Then for some functionG over (a, b) and for some positive integersr ands, 1<r+1<s≤n

$$E\left\{ {\frac{1}{{s - r + 1}}\sum\limits_{i = r}^s {g(X_{i:n} )|X_{r:n} = x,X_{s:n} = y} } \right\} = \frac{{G(x) + G(y)}}{2},\forall x,y \in (a,b)$$

iffg andG are bounded, increasing and continuous,G=g andF(x)=g(x)−g(a+) / g(b−)−g(a+). This leads to characterization of several distributions.