Stochastic properties of INID progressive Type-II censored order statistics

Let X1:n<=X2:n<=...<=Xn:n denote the order statistics of random variables X1,X2,...,Xn which are independent but not necessarily identically distributed (INID), and let K1,K2 be two integer-valued random variables, independent of {X1,...,Xn}, such that 1<=K1<=K2<=n. It is shown that if K1 has a log-concave probability function and SI(K2K1) then RTI(XK2:nXK1:n), and if K2 has a log-concave probability function and SI(K1K2) then LTD(XK1:nXK2:n), where SI, RTI and LTD are three notions of bivariate positive dependence. Based on these, we obtain that RTI and LTD whenever 1<=i<j<=m, where are progressive Type-II censored order statistics from INID random variables {X1,...,Xn}. Furthermore, one result concerning the likelihood ratio ordering of the progressive Type-II censored order statistics is also given.