# A goodness-of-fit test for elliptical distributions with diagnostic capabilities

## TL;DR

This paper introduces a flexible, invariant goodness-of-fit test for elliptical distributions that includes diagnostic tools to identify specific deviations, demonstrated on various distributions including the multivariate normal.

## Contribution

It presents a new adaptive omnibus test with diagnostic capabilities for elliptical distributions, generalizing existing methods and applicable to multiple distribution types.

## Key findings

- Test is invariant to affine transformations
- Diagnostic components help identify specific distribution deviations
- Simulation confirms effectiveness of diagnostic tools

## Abstract

This paper develops a smooth test of goodness-of-fit for elliptical distributions. The test is adaptively omnibus, invariant to affine-linear transformations and has a convenient expression that can be broken into components. These components have diagnostic capabilities and can be used to identify specific departures. This helps in correcting the null model when the test rejects. As an example, the results are applied to the multivariate normal distribution for which the R package ECGofTestDx is available. It is shown that the proposed test strategy encompasses and generalizes a number of existing approaches. Some other cases are studied, such as the bivariate Laplace, logistic and Pearson type II distribution. A simulation experiment shows the usefulness of the diagnostic tools.