2022/10/04 by Fung, Thomas, Seneta, Eugene
#FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2210.01284
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 λL(u) = P(X1 ≤ F1-1(u) | X2 ≤ F2-1(u)), when \bfX ∼ SN2(\boldsymbolα, R) with α1 = α2 = 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 Fi(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 α1=α2= 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 α1=α2= 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.