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Robust joint modeling of mean and dispersion through trimming. (English) Zbl 1239.62018
Summary: The Maximum Likelihood Estimator (MLE) and Extended Quasi-Likelihood (EQL) estimator have commonly been used to estimate the unknown parameters within the joint modeling of mean and dispersion framework. However, these estimators can be very sensitive to outliers in the data. In order to overcome this disadvantage, the usage of the maximum Trimmed Likelihood Estimator (TLE) and the maximum Extended Trimmed Quasi-Likelihood (ETQL) estimator is recommended to estimate the unknown parameters in a robust way. The superiority of these approaches in comparison with the MLE and EQL estimator is illustrated by an example and a simulation study. As a prominent measure of robustness, the finite sample Breakdown Point (BDP) of these estimators is characterized in this setting.

MSC:
62F10 Point estimation
62J12 Generalized linear models (logistic models)
62F35 Robustness and adaptive procedures (parametric inference)
65C60 Computational problems in statistics (MSC2010)
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[1] Cantoni, E.; Ronchetti, E., Robust inference for generalized linear models, J. amer. statist. assoc., 96, 1022-1030, (2001) · Zbl 1072.62610
[2] Cheng, T.-C., Robust diagnostics for the heteroscedastic regression model, Comput. statist. data anal., 55, 1845-1866, (2011) · Zbl 1328.65024
[3] Cuesta-Albertos, J.A.; Matrán, C.; Mayo-Iscar, A., Robust estimation in the normal mixture model based on robust clustering, J. R. stat. soc. ser. B, 70, 779-802, (2008) · Zbl 05563369
[4] Čížek, P., General trimmed estimation: robust approach to nonlinear and limited dependent variable models, Econometric theory, 24, 1500-1529, (2008) · Zbl 1231.62026
[5] Cox, D.R.; Reid, N., Parameter orthogonality and approximate conditional inference, J. R. stat. soc. ser. B, 49, 1-39, (1987) · Zbl 0616.62006
[6] Dimova, R.; Neykov, N.M., Generalized d-fullness technique for breakdown point study of the trimmed likelihood estimator with applications, (), 83-91 · Zbl 1088.62045
[7] Dunn, P., 2009. Tweedie exponential family models. http://cran.R-project.org/doc/packages/tweedie.pdf.
[8] Efron, B., Double exponential families and their use in generalized linear regression, J. amer. statist. assoc., 81, 709-721, (1986) · Zbl 0611.62072
[9] Gallegos, M.T.; Ritter, G., A robust method for cluster analysis, Ann. statist., 33, 347-380, (2005) · Zbl 1064.62074
[10] Gallegos, M.T.; Ritter, G., Using combinatorial optimization in model-based trimmed clustering with cardinality constraints, Comput. statist. data anal., 54, 637-654, (2010) · Zbl 05689619
[11] Garcia-Escudero, L.A.; Gordaliza, A.; Matrán, C.; Mayo-Iscar, A., A general trimming approach to robust cluster analysis, Ann. statist., 36, 1324-1345, (2008) · Zbl 1360.62328
[12] Green, P.J., Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistant alternatives, J. R. stat. soc. ser. B, 46, 149-192, (1984) · Zbl 0555.62028
[13] Hampel, F.R.; Ronchetti, E.M.; Rousseeuw, P.J.; Stahel, W.A., ()
[14] Hawkins, D.M.; Khan, D.M., A procedure for robust Fitting in nonlinear regression, Comput. statist. data anal., 53, 4500-4507, (2009) · Zbl 05689196
[15] Hawkins, D.M.; Olive, D.J., Inconsistency of resampling algorithms for high-breakdown regression estimators and a new algorithm (with discussions), J. amer. statist. assoc., 97, 136-159, (2002) · Zbl 1073.62546
[16] Herwindiati, D.E.; Djauhari, M.A.; Mashuri, M., Robust multivariate outlier labeling, Comm. statist. simulation comput., 36, 1287-1294, (2007) · Zbl 1126.62042
[17] Jørgensen, B., The theory of dispersion models, (1997), Chapman and Hall London · Zbl 0928.62052
[18] Lee, Y.; Nelder, J.A., The relationship between double-exponential families and extended quasi-likelihood families, with application to modelling geissler’s human sex ration data, Appl. stat., 49, 413-419, (2000)
[19] Lee, Y.; Nelder, J.A., Generalized linear models for the analysis of quality-improvement experiments, Canad. J. statist., 26, 95-105, (1998) · Zbl 0899.62088
[20] Lee, Y.; Nelder, J.A.; Pawitan, Y., Generalized linear models with random effects: unified analysis via \(H\)-likelihood, (2006), Chapman and Hall, CRC London · Zbl 1110.62092
[21] Markatou, M.; Basu, A.; Lindsay, B., Weighted likelihood estimating equations: the discrete case with applications to logistic regression, J. statist. plann. inference, 57, 215-232, (1997) · Zbl 1003.62506
[22] Maronna, R.A.; Martin, R.D.; Yohai, V.J., Robust statistics: theory and methods, (2006), John Wiley and Sons New York · Zbl 1094.62040
[23] McCullagh, P.; Nelder, J.A., Generalized linear models, (1989), Chapman and Hall London · Zbl 0744.62098
[24] Müller, C.H.; Neykov, N.M., Breakdown points of the trimmed likelihood and related estimators in generalized linear models, J. statist. plann. inference, 116, 503-519, (2003) · Zbl 1178.62074
[25] Nelder, J.A.; Pregibon, D., An extended quasi-likelihood function, Biometrika, 74, 221-232, (1987) · Zbl 0621.62078
[26] Neykov, N.M.; Müller, C.H., Breakdown point and computation of trimmed likelihood estimators in generalized linear models, (), 277-286 · Zbl 05280058
[27] Neykov, N.M., Neytchev, P., 1990. A robust alternative of the maximum likelihood estimators. COMPSTAT’90—Short Communications, Dubrovnik, Yugoslavia, pp. 99-100.
[28] Neykov, N.M.; Filzmoser, P.; Dimova, R.; Neytchev, P.N., Robust Fitting of mixtures using the trimmed likelihood estimator, Comput. statist. data anal., 52, 299-308, (2007) · Zbl 1328.62033
[29] R Development Core Team, 2006. R: a language and environment for statistical computing. R Foundation for Statistical Computing. Vienna, Austria. ISBN 3-900051-07-0.
[30] Ribatet, M., Iooss, B., 2009. Joint modeling of mean and dispersion package. http://cran.R-project.org/doc/packages/JointModeling.pdf.
[31] Rousseeuw, P.J., Least Median of squares regression, J. amer. statist. assoc., 79, 851-857, (1984)
[32] Rousseeuw, P.J.; Van Driessen, K., A fast algorithm for the minimum covariance determinant estimator, Technometrics, 41, 212-223, (1999)
[33] Rousseeuw, P.J.; Van Driessen, K., Computing least trimmed of squares regression for large data sets, Estadistica, 54, 163-190, (1999)
[34] Smyth, G.K., Generalized linear models with varying dispersion, J. R. stat. soc. ser. B, 51, 47-60, (1989)
[35] Smyth, G.K., 2009a. Double generalized linear models. http://cran.R-project.org/doc/packages/dglm.pdf.
[36] Smyth, G.K., 2009b. Statistical modeling. http://cran.R-project.org/doc/packages/statmod.pdf.
[37] Smyth, G.K.; Verbyla, A.P., Adjusted likelihood methods for modelling dispersion in generalized linear models, Environmetrics, 10, 696-709, (1999)
[38] Vandev, D.L.; Neykov, N.M., About regression estimators with high breakdown point, Statistics, 32, 111-129, (1998) · Zbl 1077.62513
[39] Vandev, D.L.; Neykov, N.M., Robust maximum likelihood in the Gaussian case, (), 259-264 · Zbl 0819.62049
[40] Visek, J.A., On high breakdown point estimation, Comput. statist., 11, 137-146, (1996) · Zbl 0933.62015
[41] Zuliani, U.; Mandras, A.; Beltrami, G.F.; Bonetti, A.; Montani, G.; Novarini, A., Metabolic modifications caused by sport activity: effect in leisure-time cross-country skiers, J. sports med. phys. fitness, 23, 385-392, (1983)
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