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Depth estimators and tests based on the likelihood principle with application to regression

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  • Müller, Christine H.

Abstract

We investigate depth notions for general models which are derived via the likelihood principle. We show that the so-called likelihood depth for regression in generalized linear models coincides with the regression depth of Rousseeuw and Hubert (J. Amer. Statist. Assoc. 94 (1999) 388) if the dependent observations are appropriately transformed. For deriving tests, the likelihood depth is extended to simplicial likelihood depth. The simplicial likelihood depth is always a U-statistic which is in some cases not degenerated. Since the U-statistic is degenerated in the most cases, we demonstrate that nevertheless the asymptotic distribution of the simplicial likelihood depth and thus asymptotic [alpha]-level tests for general types of hypotheses can be derived. The tests are distribution-free. We work out the method for linear and quadratic regression.

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  • Müller, Christine H., 2005. "Depth estimators and tests based on the likelihood principle with application to regression," Journal of Multivariate Analysis, Elsevier, vol. 95(1), pages 153-181, July.
    • Handle: RePEc:eee:jmvana:v:95:y:2005:i:1:p:153-181
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    1. Van Aelst, Stefan & Rousseeuw, Peter J., 2000. "Robustness of Deepest Regression," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 82-106, April.
    2. R. W. Farebrother, 1984. "The Distribution of a Linear Combination of Central X2 Random Variables a Remark on as 153: Pan's Procedure for the Tail Probabilities of the Durbin–Watson Statistic," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 33(3), pages 363-366, November.
    3. Dümbgen, Lutz, 1992. "Limit theorems for the simplicial depth," Statistics & Probability Letters, Elsevier, vol. 14(2), pages 119-128, May.
    4. Van Aelst, Stefan & Rousseeuw, Peter J. & Hubert, Mia & Struyf, Anja, 2002. "The Deepest Regression Method," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 138-166, April.
    5. Hubert, Mia & Rousseeuw, Peter J., 1998. "The Catline for Deep Regression," Journal of Multivariate Analysis, Elsevier, vol. 66(2), pages 270-296, August.
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    1. Denecke, Liesa & Müller, Christine H., 2011. "Robust estimators and tests for bivariate copulas based on likelihood depth," Computational Statistics & Data Analysis, Elsevier, vol. 55(9), pages 2724-2738, September.
    2. Wellmann, Robin & Müller, Christine H., 2010. "Tests for multiple regression based on simplicial depth," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 824-838, April.
    3. Christine Müller, 2011. "Data depth for simple orthogonal regression with application to crack orientation," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(2), pages 135-165, September.
    4. Wellmann, Robin & Müller, Christine H., 2010. "Depth notions for orthogonal regression," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2358-2371, November.
    5. Davy Paindaveine & Germain Van Bever, 2017. "Tyler Shape Depth," Working Papers ECARES ECARES 2017-29, ULB -- Universite Libre de Bruxelles.

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