High moments Jarque-Bera tests for arbitrary distribution functions

TL;DR

This paper generalizes the Jarque-Bera test to arbitrary distributions with many moments, revealing its dependence on the first eight moments and introducing new chi-square tests, supported by simulations.

Contribution

It extends the Jarque-Bera test to higher moments and arbitrary distributions using modern empirical process techniques, providing new theoretical insights and practical tests.

Findings

The classical JB test depends on the first eight moments.

Generalized JB tests can handle any distribution with at least 4k moments.

Simulation results compare new tests with Kolmogorov-Smirnov and classical JB tests.

Abstract

The Jarque-Bera's fitting test for normality is a celebrated and powerful one. In this paper, we consider general Jarque-Bera tests for any distribution function df having at least 4k finite moments for k greater than 2. The tests use as many moments as possible whereas the JB classical test is supposed to test only skewness and kurtosis for normal variates. But our results unveil the relations between the coeffients in the JB classical test and the moments, showing that it really depends on the first eight moments. This is a new explanation for the powerfulness of such tests. General Chi-square tests for an arbitraty model, not only normal, are also derived. We make use of the modern functional empirical processes approach that makes it easier to handle statistics based on the high moments and allows the generalization of the JB test both in the number of involved moments and in the underlying distribution. Simulation studies are provided and comparison cases with the Kolmogorov-Smirnov's tests and the classical JB test are given.