## Adaptively truncated maximum likelihood regression with asymmetric errors.(English)Zbl 1040.62057

Summary: We consider robust estimators for the linear regression model with asymmetric (or symmetric) error distribution. We assume that the error model belongs to a location-scale family of distributions. Since in the asymmetric case the mean response is very often the parameter of interest and scale is a main component of mean, we do not assume that scale is a nuisance parameter. First, we show how to convert an ordinary robust estimate for the usual model with symmetric errors to an estimate for the more general model with asymmetric errors. Then, in order to improve efficiency, we introduce the truncated maximum likelihood or TML-estimator. A TML-estimate is computed in three steps: first, an initial high breakdown point estimate is computed; then, observations that are unlikely under the estimated model are rejected; finally, the maximum likelihood estimate is computed with the retained observations.
The rejection rule used in the second step is based on a cut-off parameter that can be tuned to attain the desired efficiency while maintaining the breakdown point of the initial estimator (e.g., 50%). Optionally, one can use a new adaptive cut-off that, asymptotically, does not reject any observation when the data are generated according to the model. Under the model, the influence function of this adaptive TML-estimator (or ATML-estimator) coincides with the influence function of the maximum likelihood estimator. The ATML-estimator is, therefore, fully efficient at the model; nevertheless, its breakdown point is not smaller than the breakdown point of the initial estimator. We evaluate the TML- and ATML-estimators for finite sample sizes with the help of simulations and discuss an example with real data.

### MSC:

 62J05 Linear regression; mixed models 62F35 Robustness and adaptive procedures (parametric inference) 62F10 Point estimation
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### References:

 [1] Bednarski, T.; Clarke, B.R., Trimmed likelihood estimation of location and scale of the normal distribution, Austral. J. statist., 35, 2, 141-153, (1993) · Zbl 0798.62043 [2] Carroll, R.J.; Ruppert, D., Transformation and weighting in regression, (1988), Chapman & Hall New York · Zbl 0666.62062 [3] Cantoni, E.; Ronchetti, E., Robust inference for generalized linear models, J. amer. statist. assoc., 96, 1022-1030, (2001) · Zbl 1072.62610 [4] Clarke, B.R., An adaptive method of estimation and outlier detection in regression applicable for small to moderate sample sizes, Probab. statist., 20, 25-50, (2000) · Zbl 0971.62034 [5] Field, C.; Smith, B., Robust estimation—a weighted maximum likelihood approach, Internat. statist. rev., 62, 3, 405-424, (1994) · Zbl 0825.62428 [6] Gervini, D.; Yohai, V.J., A class of robust and fully efficient regression estimates, Ann. statist., 30, 2, 583-616, (2002) · Zbl 1012.62073 [7] Hampel, F.R.; Ronchetti, E.M.; Rousseeuw, P.J.; Stahel, W.A., Robust statistics: an approach based on the influence function, (1986), Wiley New York · Zbl 0593.62027 [8] Huber, P., Robust statistics, (1981), Wiley New York · Zbl 0536.62025 [9] Hettmansperger, T.P.; McKean, J.W., Robust nonparametric statistical methods, (1998), Arnold London · Zbl 0887.62056 [10] Marazzi, A.; Ruffieux, C., Implementing M-estimators of the gamma distribution, () · Zbl 0846.62031 [11] Marazzi, A.; Ruffieux, C., The truncated Mean of an asymmetric distribution, Comput. statist. data anal., 32, 1, 79-100, (1999) [12] Markatou, M.; Basu, A.; Lindsay, B.G., Weighted likelihood equations with bootstrap search, J. amer. statist. assoc., 93, 740-750, (1998) · Zbl 0918.62046 [13] McCullagh, P.; Nelder, J.A., Generalized linear models, (1989), Chapman & Hall New York [14] Rousseeuw, P.J.; Leroy, A.M., Robust regression and outlier detection, (1987), Wiley New York [15] Victoria-Feser, M.P.; Ronchetti, E., Robust methods for personal-income distribution models, Canad. J. statist., 22, 247-258, (1994) · Zbl 0801.62099 [16] Victoria-Feser, M.P.; Ronchetti, E., Robust estimation for grouped data, J. amer. statist. assoc., 92, 437, 333-340, (1997) · Zbl 1090.62521 [17] Williams, M.S., A regression technique accounting for heteroscedastic and asymmetric errors, J. agri., biol. environ. statist., 2, 1, 108-129, (1997)
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