Uniform convergence rates of skew-normal extremes

TL;DR

This paper derives the rate at which the maximum of i.i.d. skew-normal variables converges to its extreme value distribution, showing a rate proportional to 1/log n with optimal normalization.

Contribution

It provides the first explicit derivation of the uniform convergence rate for skew-normal extremes with optimal normalization constants.

Findings

Convergence rate is proportional to 1/log n.

Optimal normalizing constants are identified.

Results extend extreme value theory to skew-normal distributions.

Abstract

Let $M_n=\max \left(X_1, X_2, \ldots, X_n \right)$ denote the partial maximum of an independent and identically distributed skew-normal random sequence. In this paper, the rate of uniform convergence of skew-normal extremes is derived. It is shown that with optimal normalizing constants the convergence rate of $\left(M_{n}-b_n\right)/a_n$ to its ultimate extreme value distribution is proportional to $1/\log n$.