On the Markov property of order statistics

Characterization of the exponential distribution function by Properties of the differenceX k+s∶n−X k∶n of order statistics

“…To begin with, we explain that we need the assumption of continuity of F. It was shown in Arnold et al (1984) that the order statistics corresponding to a sample of size n > 3 possess the Markov property if and only if F i's continuous for 0 < F(x) < 1. But because of the assumption F(0) = 0 it is the Markov property that leads to (2.4a), see Arnold et al (1984).…”

“…But because of the assumption F(0) = 0 it is the Markov property that leads to (2.4a), see Arnold et al (1984).…”

“…we obtain from (2.3a) and well-known properties of order statistics (see Arnold et al (1984) Comparing (2.6) with (2.3b) we get the following formula (2.7). On the other hand developing v ~-~ --[(v -1) + 1] '-~ in (2.1) into a binomial sum and concluding analogously, we get (2.8).…”

Generalized densities of order statistics

“…As applications we given in Section 6 an expeditious proof of a characterization in ARNOLD et al (1984) of when the order statistics M r n form a Markov sequence in r, and, in Section 7, bounds for the distributions of trimmed sums which have been used in various applications. Other authors often approach order statistics for general distributions through quantile constructions based on uniformly distributed samples that live on an enriched probability space.…”

Characterization based on conditional expectations of adjacent order statistics: A unified approach

“…And let X 1,n , X 2,n , ..., X n,n be the associated order statistics with X r,n < X r+1,n , which prevents the case of ties when F is discrete, so the event {X r,n = X r+1,n } 862 M. FRANCO AND J. M. RUIZ has zero probability when F is discrete, and the X r,n 's possess Markovian structure (see Arnold et al [3] and Nagaraja [10]). …”