Order statistics from a logistic distribution and applications to survival and reliability analysis

Joint moments involving arbitrary powers of order statistics are the main concern. Consider order statistics u/sub 1/ < u/sub 2/ < ... < u/sub k/ coming from a simple random sample of size n from a real continuous population where u/sub 1/ = x/sub r(1):n/ is order-statistic #r/sub 1/, u/sub 2/ = x/sub r(1)+r(2):n/ is order statistic #(r/sub 1/ + r/sub 2/), et al., and u/sub k/ = x/sub r(1)+...+r(k):n/ is order statistic #(r/sub 1/ +...+ r/sub k/). Product moments are examined of the type E[u/sub 1//sup (alpha)(1)/ . u/sub 2//sup (alpha)(2)//sub . ..../u/sub k//sup (alpha)(k)/] where (alpha)/sub 1/, ..., (alpha)/sub k/ are arbitrary quantities that might be complex numbers, and E [.] denotes the s-expected value. Some explicit evaluations are considered for a logistic population. Detailed evaluations of all integer moments of u/sub 1/ and recurrence relations, recurring only on the order of the moments, are given. Connections to survival functions in survival analysis, hazard functions in reliability situations, real type-1, type-2 (beta) and Dirichlet distributions are also examined. Arbitrary product moments for the survival functions are evaluated. Very general results are obtained which can be used in many problems in various areas.