Stochastic comparisons of order statistics and their concomitants

Let X1:n≤X2:n⋯≤Xn:nX1:n≤X2:n⋯≤Xn:n be the order statistics from some sample, and let Y[1:n],Y[2:n],…,Y[n:n]Y[1:n],Y[2:n],…,Y[n:n] be the corresponding concomitants. One purpose of this paper is to obtain results that stochastically compare, in various senses, the random vector (Xr:n,Y[r:n])(Xr:n,Y[r:n]) to the random vector (Xr+1:n,Y[r+1:n])(Xr+1:n,Y[r+1:n]), r=1,2,…,n−1r=1,2,…,n−1. Such comparisons are called one-sample comparisons  . Next, let S1:n≤S2:n⋯≤Sn:nS1:n≤S2:n⋯≤Sn:n be the order statistics constructed from another sample, and let T[1:n],T[2:n],…,T[n:n]T[1:n],T[2:n],…,T[n:n] be the corresponding concomitants. Another purpose of this paper is to obtain results that stochastically compare, in various senses, the random vector (Xr:n,Y[r:n])(Xr:n,Y[r:n]) with the random vector (Sr:n,T[r:n])(Sr:n,T[r:n]), r=1,2,…,nr=1,2,…,n. Such comparisons are called two-sample comparisons. It is shown that some of the results in this paper strengthen previous results in the literature. Some applications in reliability theory are described.