Ordering distributions by scaled order statistics

Marco Scarsini (1, 2) , Moshe Shaked (3)

1 GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC
2 Dipartimento di Scienze Economiche e Aziendali
3 University of Arizona

Motivated by applications in reliability theory, we define a preordering (X sub 1, ..., X sub n) about or less than (k) (Y sub 1, ..., Y sub n) of nonnegative random vectors by requiring the k-th order statistic of a sub 1 X sub 1, ..., a sub n X sub n to be stochastically smaller than the k-th order statistic of a sub 1 Y sub 1, ..., a sub n Y sub n for all choices of a sub i > O, i = 1, 2, ..., n. We identify a class of functions M sub k, n such that X about or less than (k) Y if and only if E phi (X) < E phi (Y) for all phi epsilon M sub k,n. Some preservation results related to the ordering about or less than (k) are obtained. Some examples and applications of the results are given.

Keywords: Reliability; Order statistics; Lifetime; Stochastic orderings

Domains: Economy and decision science

Marco Scarsini, Moshe Shaked. Ordering distributions by scaled order statistics. Mathematical Methods of Operations Research, 1987, Vol. 31, N°1, pp. A1-A13. ⟨10.1007/BF01258603⟩. ⟨hal-00542242⟩