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The Coverage Properties of Confidence Regions After Model Selection

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  • Paul Kabaila

Abstract

It is very common in applied frequentist (“classical”) statistics to carry out a preliminary statistical (i.e. data‐based) model selection by, for example, using preliminary hypothesis tests or minimizing AIC. This is usually followed by the inference of interest, using the same data, based on the assumption that the selected model had been given to us a priori. This assumption is false and it can lead to an inaccurate and misleading inference. We consider the important case that the inference of interest is a confidence region. We review the literature that shows that the resulting confidence regions typically have very poor coverage properties. We also briefly review the closely related literature that describes the coverage properties of prediction intervals after preliminary statistical model selection. A possible motivation for preliminary statistical model selection is a wish to utilize uncertain prior information in the inference of interest. We review the literature in which the aim is to utilize uncertain prior information directly in the construction of confidence regions, without requiring the intermediate step of a preliminary statistical model selection. We also point out this aim as a future direction for research. En statistiques appliquées de l'approche fréquentiste (“classique”), il est courant de procéder à une sélection préliminaire du modèle statistique (c'est‐à‐dire basée sur des données) en utilisant, par exemple, des tests préliminaires fondés sur des hypothèses ou en minimisant AIC. Ceci est généralement suivi par l'inférence d'intérêt, où les mêmes données sont utilisées, et qui suppose que le modèle choisi nous avait été donnéà priori. Cette supposition est erronée et peut entraîner une inférence inexacte et trompeuse. Nous examinons un cas primordial où l'inférence d'intérêt constitue une région de confiance. Nous étudions la documentation qui indique que les régions de confiance qui en résultent ont en principe des propriétés d'application réduites. Nous examinons également de manière succincte les écrits en étroite relation qui décrivent les propriétés d'application des intervalles de prédiction après la sélection préliminaire du modèle statistique. Il est possible que la motivation sous‐tendant la sélection préliminaire du modèle statistique représente un désir d'utilizer des renseignements préalables incertains dans l'inférence d'intérêt. Nous étudions la documentation où l'objectif est d'utilizer des renseignements préalables incertains directement dans l'élaboration de régions de confiance, sans exiger de recourir à l'étape intermédiaire de sélection préliminaire du modèle statistique. Nous précisons également que cet objectif constitue un axe de recherche future.

Suggested Citation

  • Paul Kabaila, 2009. "The Coverage Properties of Confidence Regions After Model Selection," International Statistical Review, International Statistical Institute, vol. 77(3), pages 405-414, December.
  • Handle: RePEc:bla:istatr:v:77:y:2009:i:3:p:405-414
    DOI: 10.1111/j.1751-5823.2009.00089.x
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    File URL: https://doi.org/10.1111/j.1751-5823.2009.00089.x
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    References listed on IDEAS

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    1. Giles, David E. A. & Srivastava, Virendra K., 1993. "The exact distribution of a least squares regression coefficient estimator after a preliminary t-test," Statistics & Probability Letters, Elsevier, vol. 16(1), pages 59-64, January.
    2. Farchione, David & Kabaila, Paul, 2008. "Confidence intervals for the normal mean utilizing prior information," Statistics & Probability Letters, Elsevier, vol. 78(9), pages 1094-1100, July.
    3. Kabaila, Paul & Giri, Khageswor, 2009. "Large-sample confidence intervals for the treatment difference in a two-period crossover trial, utilizing prior information," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 652-658, March.
    4. Kabaila, Paul & Leeb, Hannes, 2006. "On the Large-Sample Minimal Coverage Probability of Confidence Intervals After Model Selection," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 619-629, June.
    5. Kabaila, Paul, 1995. "The Effect of Model Selection on Confidence Regions and Prediction Regions," Econometric Theory, Cambridge University Press, vol. 11(3), pages 537-549, June.
    6. Chiou, Paul, 1997. "Interval estimation of scale parameters following a pre-test for two exponential distributions," Computational Statistics & Data Analysis, Elsevier, vol. 23(4), pages 477-489, February.
    7. Leeb, Hannes & Pötscher, Benedikt M., 2005. "Model Selection And Inference: Facts And Fiction," Econometric Theory, Cambridge University Press, vol. 21(1), pages 21-59, February.
    8. Casella, George & Hwang, Jiunn Tzon, 1987. "Employing vague prior information in the construction of confidence sets," Journal of Multivariate Analysis, Elsevier, vol. 21(1), pages 79-104, February.
    9. Kabaila, Paul, 1998. "Valid Confidence Intervals In Regression After Variable Selection," Econometric Theory, Cambridge University Press, vol. 14(4), pages 463-482, August.
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    Cited by:

    1. Gueuning, Thomas & Claeskens, Gerda, 2016. "Confidence intervals for high-dimensional partially linear single-index models," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 13-29.
    2. Barber Benjamin & Weschle Simon & Pierskalla Jan, 2014. "Lobbying and the collective action problem: comparative evidence from enterprise surveys," Business and Politics, De Gruyter, vol. 16(2), pages 1-26, August.
    3. Leeb, Hannes & Pötscher, Benedikt M. & Ewald, Karl, 2014. "On various confidence intervals post-model-selection," MPRA Paper 58326, University Library of Munich, Germany, revised 2014.
    4. Kabaila, Paul, 2016. "The finite sample performance of the two-stage analysis of a two-period crossover trial," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 118-127.
    5. Paul Kabaila & A. H. Welsh & Waruni Abeysekera, 2016. "Model-Averaged Confidence Intervals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(1), pages 35-48, March.

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