Exponential families are not preserved by the formation of order statistics

Let FF be a natural exponential family (NEF) generated by a measure μμ and X=(X1,…,Xn) a random sample with a common distribution belonging to FF. Consider the set of order statistics X(1)≤X(2)≤⋯≤X(n)X(1)≤X(2)≤⋯≤X(n) and let Gr,nGr,n denote the family of distributions induced by the rr-th order statistic X(r)X(r), r=1,…,nr=1,…,n. The main problem of the paper, namely, the closedness of NEF’s under the formation of order statistics, can be posed as follows: for which NEF’s FF, the set of distributions Gr,nGr,n constitutes, for all n∈Nn∈N and for some r∈{1,…,n}r∈{1,…,n}, an NEF on RR? If Gr,nGr,n is an NEF, we shall say that FF is closed under the rr-th order statistic. A comprehensive answer to this problem seems to be rather difficult when μμ is an arbitrary measure. However, if μμ is a continuous measure we show that if 1