Tail asymptotics for the bivariate skew normal in the general case
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The present paper is a sequel to and generalization of Fung and Seneta (2016) whose main result gives the asymptotic behaviour as u → 0^+ of λ_L(u) = P(X_1 ≤ F_1^-1(u) | X_2 ≤ F_2^-1(u)), when X∼ SN_2(α, R) with α_1 = α_2 = α, that is: for the bivariate skew normal distribution in the equi-skew case, where R is the correlation matrix, with off-diagonal entries ρ, and F_i(x), i=1,2 are the marginal cdf's of X. A paper of Beranger et al. (2017) enunciates an upper-tail version which does not contain the constraint α_1=α_2= α but requires the constraint 0 <ρ <1 in particular. The proof, in their Appendix A.3, is very condensed. When translated to the lower tail setting of Fung and Seneta (2016), we find that when α_1=α_2= α the exponents of u in the regularly varying function asymptotic expressions do agree, but the slowly varying components, always of asymptotic form const (-log u)^τ, are not asymptotically equivalent. Our general approach encompasses the case -1 <ρ < 0, and covers all possibilities.
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