Let \((X_i,\boldsymbol{Y}_i)=(X_i,Y_i^{(1)},Y_i^{(2)},\dots,Y_i^{(k)})\), \(i=1,2,\dots\), be a sequence of independent and identically distributed \((k+1)\)-dimensional random vectors. Furthermore, let \(X_{1:n}\le X_{2:n}\dots\le X_{n:n}\) be the order statistics based on the first coordinates of the first \(n\) elements in the sequence, and let \(\boldsymbol{Y}_{[1:n]},\boldsymbol{Y}_{[2:n]},\dots,\boldsymbol{Y}_{[n:n]}\) be the corresponding \(k\)-dimensional concomitants. Several results that compare \((X_{j:m},\boldsymbol{Y}_{[j:m]})\) and \((X_{i:n},\boldsymbol{Y}_{[i:n]})\), with respect to various multivariate stochastic orders, are obtained. Next, let \((S_i,\boldsymbol{T}_i)=(S_i,T_i^{(1)},T_i^{(2)},\dots,T_i^{(k)})\), \(i=1,2,\dots\), be another sequence of independent and identically distributed \((k+1)\)-dimensional random vectors. Furthermore, let \(S_{1:m}\le S_{2:m}\dots\le S_{m:m}\) be the order statistics based on the first coordinates of the first \(m\) elements in the sequence, and let \(\boldsymbol{T}_{[1:m]},\boldsymbol{T}_{[2:m]},\dots,\boldsymbol{T}_{[m:m]}\) be the corresponding \(k\)-dimensional concomitants. Several results that compare \((X_{j:m},\boldsymbol{Y}_{[j:m]})\) and \((S_{i:n},\boldsymbol{T}_{[i:n]})\), with respect to various multivariate stochastic orders, are established. It is shown that some of the results in this paper strengthen previous results in the literature. An application in reliability theory is described. A derivation of computable bounds, on various probabilistic quantities of interest involving multivariate concomitants, illustrates the theory.
