Higher-order expansions of sample extremal quotient from skew-normal distribution

Consider the sequence {Xn,n≥1} of independent random variables, sharing an identical skew-normal distribution, and define Mn and mn as the partial maximum and minimum of the sequence. This paper establishes the limiting distribution of the normalized sample extremal quotient mn/Mn. With appropriate normalizing constants, we derive higher-order asymptotic expansions for the distribution of normalized sample extremal quotient, from which the associated convergence rate to its limiting distribution is obtained. Numerical analysis is provided to compare the accuracy of different asymptotics.