Characterizations of distributions via two expected values of order statistics
Let \(X_{k,n}\) be the k-th smallest order statistic of a random sample of size n from a distribution F(x) with \(E| X| <\infty\). Then in general two expected values of order statistics, say E \(X_{k,n}\) and E \(X_{\ell,m}\), are not enough to characterize F in the class of all distributions on the real line \(R\equiv (-\infty,\infty)\). However, in the following class of distributions associated with any fixed non- degenerate distribution \(F_ 0\) through the location and scale parameters, \[ {\mathcal F}_{F_ 0}=\{F_{\mu,\sigma}:\quad F_{\mu,\sigma}(x)=F_ 0((x-\mu)/\sigma)\quad on\quad R,\quad \mu \in R,\quad \sigma >0\}, \] F is characterized by any pair E \(X_{k,n}\) and E \(X_{\ell,m}\) with indices satisfying the condition: \[ (\ell -k, m- n)\not\in \{(a,b):\quad a=b=0\quad or\quad 0<a<b\quad or\quad b<a<0\}. \] Three special cases concerning the uniform, exponential and logistic distributions are investigated completely.
scientific article; zbMATH DE number 4068062