Another look at Second order condition in Extreme Value Theory

TL;DR

This paper compares two methods for establishing normality theorems in univariate Extreme Value Theory, analyzing their assumptions, auxiliary functions, and applications to understand their relative advantages.

Contribution

It provides a detailed comparison of the second order condition and the representational approach, showing how to transition between them and their implications for statistical applications.

Findings

Both approaches are valid under certain conditions.

Auxiliary functions are explicitly computed and compared.

The paper demonstrates applications using both methods simultaneously.

Abstract

This note compares two approaches both alternatively used when establishing normality theorems in univariate Extreme Value Theory. When the underlying distribution function ($df$) is the extremal domain of attraction, it is possible to use representations for the quantile function and regularity conditions (RC), based on these representations, under which strong and weak convergence are valid. It is also possible to use the now fashion second order condition (SOC), whenever it holds, to do the same. Some authors usually favor the first approach (the SOC one) while others are fond of the second approach that we denote as the representational one. This note aims at comparing the two approaches and show how to get from one to the other. The auxiliary functions used in each approach are computed and compared. Statistical applications using simultaneously both approaches are provided. A final comparison is provided.