Stochastic properties of INID progressive Type-II censored order statistics

Let X1:n≤X2:n≤⋯≤Xn:nX1:n≤X2:n≤⋯≤Xn:n denote the order statistics of random variables X1,X2,…,XnX1,X2,…,Xn which are independent but not necessarily identically distributed (INID), and let K1,K2K1,K2 be two integer-valued random variables, independent of {X1,…,Xn}{X1,…,Xn}, such that 1≤K1≤K2≤n1≤K1≤K2≤n. It is shown that if K1K1 has a log-concave probability function and SI(K2|K1)(K2|K1) then RTI(XK2:n|XK1:n)(XK2:n|XK1:n), and if K2K2 has a log-concave probability function and SI(K1|K2)(K1|K2) then LTD(XK1:n|XK2:n)(XK1:n|XK2:n), where SI, RTI and LTD are three notions of bivariate positive dependence. Based on these, we obtain that RTI(Xj:m,nR|Xi:m,nR) and LTD(Xi:m,nR|Xj:m,nR) whenever 1≤i