Tail Asymptotics for the Bivariate Skew Normal in the General Case

Abstract The present paper is a sequel to and generalization of [10] whose main result gives the asymptotic behaviour as $$ u \rightarrow 0^{+}$$ u → 0 + of $$\lambda _L(u) = P(X_1 \le F_1^{-1}(u) | X_2 \le F_2^{-1}(u)),$$ λ L ( u ) = P ( X 1 ≤ F 1 - 1 ( u ) | X 2 ≤ F 2 - 1 ( u ) ) , when $${\textbf {X}} \sim SN_2(\varvec{\alpha }, R)$$ X ∼ S N 2 ( α , R ) with $$\alpha _1 = \alpha _2 = \alpha $$ 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 $$\rho $$ ρ , and $$F_i(x)$$ F i ( x ) , $$i=1,2$$ i = 1 , 2 are the marginal cdf’s of $${\textbf {X}}$$ X . In the present paper we show in full generality that $$ \lambda _L(u) \sim Ku^{\theta }(-\log u)^{\tau }$$ λ L ( u ) ∼ K u θ ( - log