Progressive Type II censored order statistics for multivariate observations

For a sequence of independent and identically distributed random vectors X i = ( X i 1 , X i 2 , … , X i p ) , i = 1 , 2 , … , n , we consider the conditional ordering of these random vectors with respect to the magnitudes of N ( X i ) , i = 1 , 2 , … , n , where N is a p-variate continuous function defined on the support set of X 1 and satisfying certain regularity conditions. We also consider the Progressive Type II right censoring for multivariate observations using conditional ordering. The need for the conditional ordering of random vectors exists for example, in reliability analysis when a system has n independent components each consisting of p arbitrarily dependent and parallel connected elements. Let the vector of life lengths for the ith component of the system be X i = ( X i 1 , X i 2 , … , X i p ) , i = 1 , 2 , … , n , where X i j denotes the life length of the jth element of the ith component. Then the first failure in the system occurs at time min max ( X 1 1 , X 1 2 , … , X 1 p ) , max ( X 2 1 , X 2 2 , … , X 2 p ) , … , max ( X n 1 , X n 2 , … , X n p ) , and for this case N ( X i ) = max ( X i 1 , X i 2 , … , X i p ) . In this paper we introduce the conditionally ordered and Progressive Type II right-censored conditionally ordered statistics for multivariate observations and to study their distributional properties.