Omnibus goodness-of-fit tests for univariate continuous distributions based on trigonometric moments

TL;DR

This paper introduces a new omnibus goodness-of-fit test for univariate continuous distributions based on trigonometric moments, improving calibration and power over existing methods.

Contribution

It fully exploits the covariance structure of trigonometric statistics, providing a well-calibrated test with expanded applicability to 11 distribution families.

Findings

Test statistic converges to a chi-squared distribution under null hypothesis.

Simulation studies show accurate size and strong power.

Application to weather forecast errors demonstrates practical utility.

Abstract

We propose a new omnibus goodness-of-fit test based on trigonometric moments of probability-integral-transformed data. The test builds on the framework of the LK test introduced by Langholz and Kronmal [J. Amer. Statist. Assoc. 86 (1991), 1077-1084], but fully exploits the covariance structure of the associated trigonometric statistics. As a result, our test statistic converges under the null hypothesis to a $\chi_2^2$ distribution, even in the presence of nuisance parameters, yielding a well-calibrated rejection region. We derive the exact asymptotic covariance matrix required for normalization and propose a unified approach to computing the LK normalizing scalar. The applicability of both the proposed test and the LK test is substantially expanded by providing implementation details for 11 families of continuous distributions, covering most commonly used parametric models. Simulation studies demonstrate accurate empirical size, close to the nominal level, and strong power properties, yielding fully plug-and-play procedures. Further insight is provided by an analysis under local alternatives. The methodology is illustrated using surface temperature forecast errors from a numerical weather prediction model.