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Initial robust estimation in generalized linear models. (English) Zbl 07027269
Summary: Generalized Linear Models are routinely used in data analysis. Classical estimators are based on the maximum likelihood principle and it is well known that the presence of outliers can have a large impact on them. Several robust procedures have been presented in the literature, being redescending M-estimators the most widely accepted. Based on non-convex loss functions, these estimators need a robust initial estimate, which is often obtained by subsampling techniques. However, as the number of unknown parameters increases, the number of subsamples needed in order for this method to be robust, soon makes it infeasible. Furthermore the subsampling procedure provides a non deterministic starting point. A new method for computing a robust initial estimator is proposed. This method is deterministic and demands a relatively short computational time, even for large numbers of covariates. The proposed method is applied to M-estimators based on transformations. In addition, an iteratively reweighted least squares algorithm is proposed for the computation of the final estimates. The new methods are studied by means of Monte Carlo experiments.
MSC:
62-XX Statistics
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[1] Agostinelli, C., Valdora, M., Yohai, V., 2018. Robust m-estimators based on transformations for poisson model, https://cran.r-project.org/web/packages/poissonMT/index.html, R package version 0.3-5.
[2] Alqallaf, F.; Agostinelli, C., Robust inference in generalized linear models, Comm. Statist. Simulation Comput., 45, 9, 3053-3073, (2016) · Zbl 1348.62094
[3] Bergesio, A.; Yohai, V., Projection estimators for generalized linear models, J. Amer. Statist. Assoc., 106, 661-671, (2011) · Zbl 1232.62085
[4] Bianco, A.; Boente, G.; Rodrigues, I., Resistant estimators in poisson and Gamma models with missing responses and an application to outlier detection, J. Multivariate Anal., 114, 209-226, (2013) · Zbl 1255.62206
[5] Cantoni, E.; Ronchetti, E., Robust inference for generalized linear models, J. Amer. Statist. Assoc., 96, 1022-1030, (2001) · Zbl 1072.62610
[6] Connors, A.; Speroff, T.; Dawson, N.; Thomas, C.; Harrell, F.; Wagner, D.; Desbiens, N.; Goldman, L.; Wu, A.; Califf, R.; Fulkerson, W.; Vidaillet, H.; Broste, S.; Bellamy, P.; Lynn, J.; Knaus, W., The effectiveness of right heart catheterization in the initial care of critically iii patients, J. Am. Med. Assoc., 276, 11, 889-897, (1996)
[7] Cook, R., Detection of influential observation in linear regression, Technometrics, 19, 1, 15-18, (1977) · Zbl 0371.62096
[8] Dahl, D.B., 2016. xtable: Export Tables to LaTeX or HTML, https://CRAN.R-project.org/package=xtable, R package version 1.8-2.
[9] Künsch, H.; Stefanski, L.; Carroll, R., Conditionally unbiased bounded-influence estimation in general regression models, with applications to generalized linear models, J. Amer. Statist. Assoc., 84, 460-466, (1989) · Zbl 0679.62024
[10] Lang, M., checkmate: Fast Argument Checks for Defensive R Programming, R J., 9, 1, 437-445, (2017)
[11] Maechler, M., Rousseeuw, P., Croux, C., Todorov, V., Ruckstuhl, A., Salibian-Barrera, M., Verbeke, T., Koller, M., Conceicao, E., Anna di Palma, M., 2018. robustbase: Basic Robust Statistics, http://robustbase.r-forge.r-project.org/, R package version 0.93-1.
[12] Marazzi, A., 2018a. bcbi: Conditionally Unbiased Bounded Influence Estimates, https://CRAN.R-project.org/package=robcbi, R package version 1.1-2.
[13] Marazzi, A., 2018b. robeth: R Functions for Robust Statistics, https://CRAN.R-project.org/package=robeth, R package version 2.7-2.
[14] Maronna, R.; Martin, R.; Yohai, V., Robust Statistics. Theorey an Methods, (2006), Wiley
[15] McCullagh, P.; Nelder, J., Generalized Linear Models, (1989), Chapman and Hall/CRC · Zbl 0744.62098
[16] Peña, D.; Yohai, V., A fast procedure for outlier diagnostics in large regression problems, J. Amer. Statist. Assoc., 94, 434-445, (1999) · Zbl 1072.62618
[17] R Core Team, R: a language and environment for statistical computing, (2018), R Foundation for Statistical Computing: R Foundation for Statistical Computing Vienna, Austria
[18] Rousseeuw, P.; Leroy, A., Robust Regression and Outlier Detection, (1987), Wiley and Sons · Zbl 0711.62030
[19] Tierney, L., Rossini, A.J., Li, N., Sevcikova, H., 2016. snow: Simple Network of Workstations, https://CRAN.R-project.org/package=snow, R package version 0.4-2.
[20] Valdora, M.; Yohai, V., Robust estimators for generalized linear models, J. Statist. Plann. Inference, 146, 31-48, (2014) · Zbl 1279.62148
[21] Venables, W. N.; Ripley, B. D., Modern Applied Statistics with S, (2002), Springer: Springer New York, ISBN 0-387-95457-0 · Zbl 1006.62003
[22] Wang, J., Zamar, R., Marazzi, A., Yohai, V., Salibian-Barrera, M., Maronna, R., Zivot, E., Rocke, D., Martin, D., Maechler, M., Konis., K., 2017. robust: Port of the S+ ”Robust Library”, https://CRAN.R-project.org/package=robust, R package version 0.4-18.
[23] Wei, T., Simko, V., 2017. R package ”corrplot”: visualization of a correlation matrix, https://github.com/taiyun/corrplot, (Version 0.84).
[24] Yu, H., Rmpi: parallel statistical computing in r, R News, 2, 2, 10-14, (2002)
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