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Score-type tests for normal mixtures

Author

Listed:
  • Dante Amengual
  • Xinyue Bei
  • Marine Carrasco
  • Enrique Sentana
Abstract
Testing normality against discrete normal mixtures is complex because some parameters turn increasingly underidentified along alternative ways of approaching the null, others are inequality constrained, and several higher-order derivatives become identically 0. These problems make the maximum of the alternative model log-likelihood function numerically unreliable. We propose score-type tests asymptotically equivalent to the likelihood ratio as the largest of two simple intuitive statistics that only require estimation under the null. One novelty of our approach is that we treat symmetrically both ways of writing the null hypothesis without excluding any region of the parameter space. We derive the asymptotic distribution of our tests under the null and sequences of local alternatives. We also show that their asymptotic distribution is the same whether applied to observations or standardized residuals from heteroskedastic regression models. Finally, we study their power in simulations and apply them to the residuals of Mincer earnings functions. Tester la normalité contre un mélange discret de normales est complexe car certains paramètres sont sous-identifiés, d’autres sont contraints par une inégalité et certaines dérivées sont identiquement nulles. Ces problèmes rendent la maximisation de la vraisemblance peu fiable numériquement. Nous proposons des tests du type score qui sont asymptotiquement équivalents au rapport de vraisemblance et ne nécessitent que l’estimation sous l’hypothèse nulle. Une nouveauté de notre approche est que l’on traite symétriquement les deux manières d’écrire l’hypothèse nulle sans exclure de régions de l’espace paramétrique. Nous établissons la distribution asymptotique de nos tests sous la nulle et les alternatives locales. Nous montrons que leur distribution asymptotique est la même que l’on utilise les données ou les résidus obtenus à partir d’une régression hétéroscédastique. Enfin, nous étudions leur puissance en simulations et les appliquons au résidu de la fonction de revenu de Mincer.

Suggested Citation

  • Dante Amengual & Xinyue Bei & Marine Carrasco & Enrique Sentana, 2023. "Score-type tests for normal mixtures," CIRANO Working Papers 2023s-02, CIRANO.
    • Handle: RePEc:cir:cirwor:2023s-02
    as

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    File URL: https://cirano.qc.ca/files/publications/2023s-02.pdf
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    References listed on IDEAS

    as
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    More about this item

    Keywords

    Generalized extremum tests; Higher-order identifiability; Like-lihood ratio test; Mincer equations; tests extremum généralisés; identification à ordre supérieur; test de rapport de vraisemblance; équations de Mincer;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions

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