×

zbMATH — the first resource for mathematics

Robust estimators in a generalized partly linear regression model under monotony constraints. (English) Zbl 1439.62122
Summary: In this paper, we consider the situation in which the observations follow an isotonic generalized partly linear model. Under this model, the mean of the responses is modelled, through a link function, linearly on some covariates and nonparametrically on an univariate regressor in such a way that the nonparametric component is assumed to be a monotone function. A class of robust estimates for the monotone nonparametric component and for the regression parameter, related to the linear one, is defined. The robust estimators are based on a spline approach combined with a score function which bounds large values of the deviance. As an application, we consider the isotonic partly linear log-Gamma regression model. Under regularity conditions, we derive consistency results for the nonparametric function estimators as well as consistency and asymptotic distribution results for the regression parameter estimators. Besides, the empirical influence function allows us to study the sensitivity of the estimators to anomalous observations. Through a Monte Carlo study, we investigate the performance of the proposed estimators under a partly linear log-Gamma regression model with increasing nonparametric component. The proposal is illustrated on a real data set.

MSC:
62G35 Nonparametric robustness
62G30 Order statistics; empirical distribution functions
62J05 Linear regression; mixed models
65D07 Numerical computation using splines
62P10 Applications of statistics to biology and medical sciences; meta analysis
62P20 Applications of statistics to economics
Software:
robustbase
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aït Sahalia Y (1995) The delta method for nonparametric kernel functionals. Ph.D. dissertation, University of Chicago
[2] Álvarez, E.; Yohai, J., \(M\)-estimators for isotonic regression, J Stat Plan Inference, 142, 2241-2284 (2012) · Zbl 1244.62023
[3] Bianco, A.; Boente, G., Robust estimators in semiparametric partly linear regression models, J Stat Plan Inference, 122, 229-252 (2004) · Zbl 1043.62030
[4] Bianco, Ana M.; Yohai, Víctor J., Robust Estimation in the Logistic Regression Model, Robust Statistics, Data Analysis, and Computer Intensive Methods, 17-34 (1996), New York, NY: Springer New York, New York, NY · Zbl 0839.62030
[5] Bianco, A.; García Ben, M.; Yohai, V., Robust estimation for linear regression with asymmetric errors, Can J Stat, 33, 511-528 (2005) · Zbl 1098.62084
[6] Bianco, A.; Boente, G.; Rodrigues, I., Resistant estimators in Poisson and Gamma models with missing responses and an application to outlier detection, J Multivar Anal, 114, 209-226 (2013) · Zbl 1255.62206
[7] Bianco, A.; Boente, G.; Rodrigues, I., Robust tests in generalized linear models with missing responses, Comput Stat Data Anal, 65, 80-97 (2013) · Zbl 06958966
[8] Birke, M.; Dette, H., Testing strict monotonicity in nonparametric regression, Math Methods Stat, 16, 110-123 (2007) · Zbl 1283.62092
[9] Boente, G.; Rodríguez, D., Robust inference in generalized partially linear models, Comput Stat Data Anal, 54, 2942-2966 (2010) · Zbl 1284.62195
[10] Boente, G.; He, X.; Zhou, J., Robust estimates in generalized partially linear models, Ann Stat, 34, 2856-2878 (2006) · Zbl 1114.62032
[11] Boente G, Rodríguez D, Vena P (2018) Robust estimators in a generalized partly linear regression model under monotony constraints. https://arxiv.org/abs/1802.07998
[12] Cantoni, E.; Ronchetti, E., Robust inference for generalized linear models, J Am Stat Assoc, 96, 1022-1030 (2001) · Zbl 1072.62610
[13] Cantoni, E.; Ronchetti, E., A robust approach for skewed and heavy tailed outcomes in the analysis of health care expenditures, J Health Econ, 25, 198-213 (2006)
[14] Croux, C.; Haesbroeck, G., Implementing the Bianco and Yohai estimator for logistic regression, Comput Stat Data Anal, 44, 273-295 (2002) · Zbl 1429.62317
[15] Du, J.; Sun, Z.; Xie, T., \(M\)-estimation for the partially linear regression model under monotonic constraints, Stat Probab Lett, 83, 1353-1363 (2013) · Zbl 1277.62092
[16] Gijbels, I.; Hall, P.; Jones, M.; Koch, I., Tests for monotonicity of a regression mean with guaranteed level, Biometrika, 87, 663-673 (2000) · Zbl 0956.62039
[17] Härdle, W.; Liang, H.; Gao, J., Partially linear models (2000), Wurzburg: Physica-Verlag, Wurzburg
[18] He, X.; Shi, P., Bivariate tensor-product \(B\)-spline in a partly linear model, J Multivar Anal, 58, 162-181 (1996) · Zbl 0865.62027
[19] He, X.; Shi, P., Monotone B-spline smoothing, J Am Stat Assoc, 93, 643-650 (1998) · Zbl 1127.62322
[20] He, X.; Zhu, Z.; Fung, W., Estimation in a semiparametric model for longitudinal data with unspecified dependence structure, Biometrika, 89, 579-590 (2002) · Zbl 1036.62035
[21] Heritier, S.; Cantoni, E.; Copt, S.; Victoria-Feser, Mp, Robust methods in biostatistics. Wiley series in probability and statistics (2009), New York: Wiley, New York
[22] Huang, J., A note on estimating a partly linear model under monotonicity constraints, J Stat Plan Inference, 107, 343-351 (2002) · Zbl 1095.62505
[23] Künsch, H.; Stefanski, L.; Carroll, R., Conditionally unbiased bounded influence estimation in general regression models with applications to generalized linear models, J Am Stat Assoc, 84, 460-466 (1989) · Zbl 0679.62024
[24] Lu, M., Spline-based sieve maximum likelihood estimation in the partly linear model under monotonicity constraints, J Multivar Anal, 101, 2528-2542 (2010) · Zbl 1198.62032
[25] Lu, M., Spline estimation of generalised monotonic regression, J Nonparametr Stat, 27, 19-39 (2015) · Zbl 1328.62248
[26] Lu, M.; Zhang, Y.; Huang, J., Estimation of the mean function with panel count data using monotone polynomial splines, Biometrika, 94, 705-718 (2007) · Zbl 1135.62069
[27] Mallows, C., On some topics in robustness (1974), Murray Hill: Memorandum Bell Laboratories, Murray Hill
[28] Manchester, L., Empirical influence for robust smoothing, Aust J Stat, 38, 275-296 (1996) · Zbl 0878.62002
[29] Marazzi, A.; Yohai, V., Adaptively truncated maximum likelihood regression with asymmetric errors, J Stat Plan Inference, 122, 271-291 (2004) · Zbl 1040.62057
[30] Maronna, R.; Martin, D.; Yohai, V., Robust statistics: theory and methods (2006), New York: Wiley, New York · Zbl 1094.62040
[31] Mccullagh, P.; Nelder, J., Generalized linear models (1989), London: Champman and Hall, London
[32] Ramsay, J., Monotone regression splines in action, Stat Sci, 3, 425-441 (1988)
[33] Schumaker, L., Spline functions: basic theory (1981), New York: Wiley, New York · Zbl 0449.41004
[34] Schwarz, G., Estimating the dimension of a model, Ann Stat, 6, 461-464 (1978) · Zbl 0379.62005
[35] Shen, X.; Wong, Wh, Convergence rate of sieve estimates, Ann Stat, 22, 580-615 (1994) · Zbl 0805.62008
[36] Stefanski, L.; Carroll, R.; Ruppert, D., Bounded score functions for generalized linear models, Biometrika, 73, 413-424 (1986) · Zbl 0616.62043
[37] Stone, Cj, The dimensionality reduction principle for generalized additive models, Ann Stat, 14, 590-606 (1986) · Zbl 0603.62050
[38] Sun, Z.; Zhang, Z.; Du, J., Semiparametric analysis of isotonic errors-in-variables regression models with missing response, Commun Stat Theory Methods, 41, 2034-2060 (2012) · Zbl 1270.62073
[39] Tamine J (2002) Smoothed influence function: another view at robust nonparametric regression. Discussion paper 62 Sonderforschungsbereich 373, Humboldt-Universität zu Berlin
[40] Van Der Geer, S., Empirical processes in \(M\)-estimation (2000), Cambridge: Cambridge University Press, Cambridge
[41] Van Der Vaart, A., Asymptotic statistics. Cambridge series in statistical and probabilistic mathematics (1998), Cambridge: Cambridge University Press, Cambridge
[42] Van Der Vaart, A.; Wellner, J., Weak convergence and empirical processes. With applications to statistics (1996), New York: Springer, New York · Zbl 0862.60002
[43] Wang, Y.; Huang, J., Limiting distribution for monotone median regression, J Stat Plann Infer, 108, 281-287 (2002) · Zbl 1016.62009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.