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 u₁ ≤ u₂ ≤ ··· ≤ u_k coming from a simple random sample of size n from a real continuous population where u₁ = x_r(1):n is order-statistic #r₁, u₂ = x_r(1)+r(2):n is order statistic #(r₁ + r₂), et al., and u_k = x_{r(1)+···+r(k):n} is order statistic #(r₁ +···+ r_k). Product moments are examined of the type E[u₁^α(1) · u₂^α(2)_{· ····}u_k^α(k)] where α₁, ..., α_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 u₁ 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 β 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.