Rates of convergence of extremes from skew normal samples

TL;DR

This paper investigates how quickly the distribution of the maximum of skew normal samples converges to the Gumbel distribution, providing explicit rates and asymptotic expansions.

Contribution

It derives the convergence rates and asymptotic expansion for the maximum of skew normal samples towards the Gumbel distribution, identifying the optimal rate as proportional to 1/log n.

Findings

Convergence rate is proportional to 1/log n.

Asymptotic expansion of the normalized maximum distribution is provided.

Optimal norming constants are identified for convergence.

Abstract

For a skew normal random sequence, convergence rates of the distribution of its partial maximum to the Gumbel extreme value distribution are derived. The asymptotic expansion of the distribution of the normalized maximum is given under an optimal choice of norming constants. We find that the optimal convergence rate of the normalized maximum to the Gumbel extreme value distribution is proportional to $1/\log n$.