# Intermediate efficiency of some weighted goodness-of-fit statistics

## TL;DR

This paper compares weighted goodness-of-fit tests, like Anderson-Darling and Eicker-Jaeschke, to the classical Kolmogorov-Smirnov test, focusing on tail detection and providing a quantitative, analytic evaluation of their efficiency.

## Contribution

It introduces a tractable method for comparing weighted tests to the classical test and proposes a modified statistic within the Eicker-Jaeschke class as a competitive alternative.

## Key findings

- Weighted tests show improved tail detection over classical tests.
- Analytic comparison confirms the efficiency of proposed modifications.
- Finite sample results support theoretical conclusions.

## Abstract

This paper compares the Anderson-Darling and some Eicker-Jaeschke statistics to the classical unweighted Kolmogorov-Smirnov statistic. The goal is to provide a quantitative comparison of such tests and to study real possibilities of using them to detect departures from the hypothesized distribution that occur in the tails. This contribution covers the case when under the alternative a moderately large portion of probability mass is allocated towards the tails. It is demonstrated that the approach allows for tractable, analytic comparison between the given test and the benchmark, and for reliable quantitative evaluation of weighted statistics. Finite sample results illustrate the proposed approach and confirm the theoretical findings. In the course of the investigation we also prove that a slight and natural modification of the solution proposed by Borovkov and Sycheva (1968) leads to a statistic which is a member of Eicker-Jaeschke class and can be considered an attractive competitor of the very popular supremum-type Anderson-Darling statistic.