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Omnibus goodness-of-fit tests based on trigonometric moments

Alain Desgagné, Frédéric Ouimet

TL;DR

This paper develops an omnibus goodness-of-fit framework based on trigonometric moments of probability-integral-transformed data, anchored by the LK test but enhanced to exploit the full covariance structure of the trigonometric statistics. It introduces the $T_n$ statistic, which converges to a $\chi^2_2$ limit under the null even with nuisance parameters, and provides an exact form for the asymptotic covariance $\Sigma(\boldsymbol{\theta})$ and a unified way to compute the LK normalizing scalar. The authors extend applicability to 11 parametric families (yielding 53 testing configurations), supply detailed implementation guidance, and demonstrate accurate size control and competitive power through extensive simulations, local-alternative analysis, and a meteorological data example. The results deliver fully plug-and-play GOF procedures with strong practical relevance for model validation across disciplines, and offer clear directions for future multivariate and discrete-data extensions.

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 $χ_2^2$ 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.

Omnibus goodness-of-fit tests based on trigonometric moments

TL;DR

This paper develops an omnibus goodness-of-fit framework based on trigonometric moments of probability-integral-transformed data, anchored by the LK test but enhanced to exploit the full covariance structure of the trigonometric statistics. It introduces the statistic, which converges to a limit under the null even with nuisance parameters, and provides an exact form for the asymptotic covariance and a unified way to compute the LK normalizing scalar. The authors extend applicability to 11 parametric families (yielding 53 testing configurations), supply detailed implementation guidance, and demonstrate accurate size control and competitive power through extensive simulations, local-alternative analysis, and a meteorological data example. The results deliver fully plug-and-play GOF procedures with strong practical relevance for model validation across disciplines, and offer clear directions for future multivariate and discrete-data extensions.

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 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.

Paper Structure

This paper contains 27 sections, 6 theorems, 304 equations, 7 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Goodness-of-fit test based on trigonometric moments
  3. Trigonometric tests for specific distributions
  4. Empirical size and power
  5. Empirical size under H0
  6. Empirical power for the Laplace distribution
  7. Asymptotic power under sequences of local alternatives
  8. Real-data application
  9. Summary and outlook
  10. Reproducibility
  11. Constants required for the Tn and LK test statistics
  12. Supplement for "Omnibus goodness-of-fit tests based on trigonometric moments"
  13. Proof for the EPD tests in the ML- and MM-based cases
  14. Proof for the half-EPD test
  15. Proof for the SN test
  16. ...and 12 more sections

Key Result

proposition 1

If $\hat{\boldsymbol{\theta}}_n$ denotes the maximum likelihood estimator and the usual regularity conditions hold, then under $\mathcal{H}_0$ and as $n \to \infty$, where $\bigl[C_n(\boldsymbol{\theta}), S_n(\boldsymbol{\theta})\bigr]^{\top}$ is defined in eq:Cn.Sn, $I_2$ denotes the $2\times2$ identity matrix, and

Figures (7)

  • Figure 1: The functions $u\mapsto \cos(2\pi u)$ (solid) and $u\mapsto \sin(2\pi u)$ (dashed) plotted on the interval $[0,1]$. The line color indicates the function's sign: black for positive values and gray for negative values.
  • Figure 2: Asymptotic power of our test $T_n(\hat{\boldsymbol{\theta}}_n)$ and of the GLRT/score-test benchmark for testing the gamma distribution (with $\lambda_0 = 1$) and the EPD distribution (with $\lambda_0 = 1.5$) under local alternatives. The nominal significance level is $0.05$.
  • Figure 3: Histogram of $n=96$ temperature forecast errors with seven fitted models.
  • Figure 4: The observed values of ${[\sqrt{n} C_n(\hat{\boldsymbol{\theta}}_n), \sqrt{n} S_n(\hat{\boldsymbol{\theta}}_n)]^{\top}}$ with $95\%$-confidence ellipses for the normal, EPD, and Gumbel models. For each panel, the outer vertical dotted lines mark $\pm Z_{0.025} \sqrt{[b]{[\Sigma(\hat{\boldsymbol{\theta}}_n)]_{_{1,1}}}}$, while the horizontal lines mark $\pm Z_{0.025} \sqrt{[b]{[\Sigma(\hat{\boldsymbol{\theta}}_n)]_{_{2,2}}}}$, visually reproducing the $95\%$ univariate $Z$-score thresholds.
  • Figure 5: Power curves of the $T_n$, LK, AD, CvM, Ku, and Wa tests for assessing normality (with $\mu_0$ and $\sigma_0$ unknown) under $\mathrm{EPD}(0.4\leq \lambda \leq 2, \mu, \sigma)$, $\mathrm{EPD}(2 \leq \lambda \leq 25, \mu, \sigma)$, and $\mathrm{APD}_{\lambda=2}(0.5 \leq \alpha < 1, \rho = 2, \mu, \sigma)$ alternatives. The nominal significance level is $0.05$, and the sample size is $n = 50$.
  • ...and 2 more figures

Theorems & Definitions (16)

  • proposition 1
  • Remark 1
  • proposition 2
  • proposition 3
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • proposition 4
  • ...and 6 more