A Higher Order Expansion for the Joint Density of the Sum and the Maximum under the Gumbel Domain of Attraction

An asymptotic expansion is developed for the joint density of the sum and maximum of an i.i.d. sequence when the parent distribution is in the domain of extreme value attraction of the Gumbel distribution. Previous results by Chow and Teugels, extended by Anderson and Turkman, show that in this situation, the normalized sum and normalized maximum of the sample converge to independent normal and Gumbel distributions, but they have not characterized the rate of convergence. The present development proceeds via three technical propositions. The first extends previous results by Cohen and by Smith to derive the rate of convergence of the density of the sample maximum to a limiting Gumbel density. The second technical proposition is a conditional Edgeworth expansion for the sum given the maximum. The third concerns the expansion of conditional means and variances. By combining these propositions, a leading-term expansion is developed for the dependence between the sum and the maximum, and uniform convergence is proved over an expanding sequence of subsets of the plane. Simulations allow us to assess the practical applicability of the results. They show that for moderate sample sizes, the sum and the maximum are far from being independent, but that the leading-term asymptotic expansion is a substantial improvement over independence of the two random variables.