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Profile Likelihood and Incomplete Data

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  • Zhiwei Zhang

Abstract

According to the law of likelihood, statistical evidence is represented by likelihood functions and its strength measured by likelihood ratios. This point of view has led to a likelihood paradigm for interpreting statistical evidence, which carefully distinguishes evidence about a parameter from error probabilities and personal belief. Like other paradigms of statistics, the likelihood paradigm faces challenges when data are observed incompletely, due to non‐response or censoring, for instance. Standard methods to generate likelihood functions in such circumstances generally require assumptions about the mechanism that governs the incomplete observation of data, assumptions that usually rely on external information and cannot be validated with the observed data. Without reliable external information, the use of untestable assumptions driven by convenience could potentially compromise the interpretability of the resulting likelihood as an objective representation of the observed evidence. This paper proposes a profile likelihood approach for representing and interpreting statistical evidence with incomplete data without imposing untestable assumptions. The proposed approach is based on partial identification and is illustrated with several statistical problems involving missing data or censored data. Numerical examples based on real data are presented to demonstrate the feasibility of the approach. Selon la loi des vraisemblances, les preuves statistiques sont représentées par des fonctions de vraisemblance et leur solidité est mesurée par des rapports de vraisemblance. Ce point de vue a conduit à un paradigme de la vraisemblance destinéà interpréter les preuves statistiques et qui fait soigneusement la distinction entre les preuves d'un paramètre et les probabilités d'erreur, ainsi que les croyances personnelles. Comme c'est le cas pour d'autres paradigmes de statistiques, le paradigme de la vraisemblance est confrontéà des défis en cas d'observation incomplète de données en raison de l'absence de réponses ou de censure, par exemple. Les méthodes classiques destinées à générer des fonctions de vraisemblance dans de telles circonstances, exigent généralement des suppositions sur le mécanisme régissant l'observation incomplète de données, suppositions qui reposent habituellement sur des informations extérieures ne pouvant pas être validées par les données observées. Sans informations externes fiables, l'usage de suppositions non vérifiables entraînées par la commodité pourrait potentiellement compromettre l'interprétation de la vraisemblance qui en résulte, en tant que représentation objective des preuves observées. Cet article propose une approche de vraisemblance de profil pour représenter et interpréter des preuves statistiques avec des données incomplètes, sans imposer de suppositions non vérifiables. L'approche proposée repose sur une identification partielle et elle est illustrée au moyen de plusieurs problèmes statistiques mettant en jeu des données manquantes ou censurées. Des exemples numériques reposant sur des données réelles sont présentés, afin de démontrer la faisabilité de l'approche.

Suggested Citation

  • Zhiwei Zhang, 2010. "Profile Likelihood and Incomplete Data," International Statistical Review, International Statistical Institute, vol. 78(1), pages 102-116, April.
  • Handle: RePEc:bla:istatr:v:78:y:2010:i:1:p:102-116
    DOI: 10.1111/j.1751-5823.2010.00107.x
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    References listed on IDEAS

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    1. Horowitz, Joel L. & Manski, Charles F., 1998. "Censoring of outcomes and regressors due to survey nonresponse: Identification and estimation using weights and imputations," Journal of Econometrics, Elsevier, vol. 84(1), pages 37-58, May.
    2. Arie Beresteanu & Francesca Molinari, 2008. "Asymptotic Properties for a Class of Partially Identified Models," Econometrica, Econometric Society, vol. 76(4), pages 763-814, July.
    3. Richard Royall & Tsung‐Shan Tsou, 2003. "Interpreting statistical evidence by using imperfect models: robust adjusted likelihood functions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 391-404, May.
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