Comparison of the multivariate skew-normal random vectors based on the integral stochastic ordering

Integral stochastic ordering of the multivariate normal mean-variance and the skew-normal scale-shape mixture models

“…By Lemma 7 and (11), and Lemma 8 and (2), respectively, for the cases (25) and (27), we have X 1 + X 2 ≼ icx Y 1 + Y 2 if and only if σ X,11 + σ X,22 + 2σ X,12 ≤ σ Y,11 + σ Y,22 + 2σ X,12 . Since σ X,11 = σ Y,11 and σ X,22 = σ Y,22 , it becomes equivalent to ρ X ≤ ρ Y , as required.…”

“…To deal with cases when the covariance matrix does not exist, a generalized correlation coefficient for bivariate vectors in (25) with…”

“…Consider the bivariate SSMSN random vectors X and Y in (25). Then, X ≼ conc Y if and only if ρ X ≤ ρ Y .…”

“…, where X and Y are random vectors for both cases (25) and (27). Then, X 1 + X 2 ≼ icx Y 1 + Y 2 if and only if ρ X ≤ ρ Y .…”

“…Stochastic ordering for the univariate SN distribution and its extensions have been discussed recently, which include some reliability orderings for the generalized SN distribution [23], ordering of univariate SN distribution and general skew-symmetric distributions [9] and characterizations of likelihood ratio order and usual stochastic order for the univariate skew-symmetric distributions [24]. Jamali et al [25] extended some orderings in [34] for the multivariate SN distributions.…”