Stochastic properties of INID progressive Type-II censored order statistics

Let X 1 : n ≤ X 2 : n ≤ ⋯ ≤ X n : n denote the order statistics of random variables X 1 , X 2 , … , X n which are independent but not necessarily identically distributed (INID), and let K 1 , K 2 be two integer-valued random variables, independent of { X 1 , … , X n } , such that 1 ≤ K 1 ≤ K 2 ≤ n . It is shown that if K 1 has a log-concave probability function and SI ( K 2 | K 1 ) then RTI ( X K 2 : n | X K 1 : n ) , and if K 2 has a log-concave probability function and SI ( K 1 | K 2 ) then LTD ( X K 1 : n | X K 2 : n ) , where SI, RTI and LTD are three notions of bivariate positive dependence. Based on these, we obtain that RTI ( X j : m , n R | X i : m , n R ) and LTD ( X i : m , n R | X j : m , n R ) whenever 1 ≤ i < j ≤ m , where { X 1 : m , n R , … , X m : m , n R } are progressive Type-II censored order statistics from INID random variables { X 1 , … , X n } . Furthermore, one result concerning the likelihood ratio ordering of the progressive Type-II censored order statistics is also given.