Exponential families are not preserved by the formation of order statistics

Let F be a natural exponential family (NEF) generated by a measure μ and X = (X₁, ..., X_n) a random sample with a common distribution belonging to F. Consider the set of order statistics X₍₁₎ ≤ X₍₂₎ ≤ ⋯ ≤ X_(n) and let G_{r, n} denote the family of distributions induced by the r-th order statistic X_(r), r = 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 F, the set of distributions G_{r, n} constitutes, for all n ∈ N and for some r ∈ {1, ..., n}, an NEF on R? If G_{r, n} is an NEF, we shall say that F is closed under the r-th order statistic. A comprehensive answer to this problem seems to be rather difficult when μ is an arbitrary measure. However, if μ is a continuous measure we show that if 1 < r < n, then G_{r, n} is not an NEF. The remaining cases r = 1 or r = n are equivalent under an appropriate affine transformation. For r = n we prove that G_{n, n} is an NEF if and only if F is the family of exponential distributions supported on R^-; or, equivalently, for r = 1, G_{1, n} is an NEF if and only if F is the family of exponential distributions supported on R⁺.