Nonparametric regression methods provide an alternative approach to parametric estimation that requires only weak identification assumptions and thus minimizes the risk of model misspecification. In this article, we survey some nonparametric regression techniques, with an emphasis on kernel-based estimation, that are additionally robust to atypical and outlying observations. While the main focus lies on robust regression estimation, robust bandwidth selection and conditional scale estimation are discussed as well. Robust estimation in popular nonparametric models such as additive and varying-coefficient models is summarized too. The performance of the main methods is demonstrated on a real dataset. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Robust Methods Statistical and Graphical Methods of Data Analysis > Nonparametric Methods K E Y W O R D S nonparametric regression, outliers, robust estimation 1 | INTRODUCTIONNonparametric regression models have received substantial attention in the theoretical and applied statistics literature over the last few decades. The underlying reason for their growing popularity and importance is that they do not rely on any particular form of the regression function characterizing the relationship between the dependent and explanatory variables and that they are thus robust to model misspecification. On the other hand, many classical nonparametric estimators are not robust in the sense that they are sensitive to atypical and outlying observations in the data. To address this, many authors have explored robust nonparametric regression estimation. As many nonparametric estimators of the regression function are local versions of the estimators of the location-scale model or the linear regression model, many initially proposed robust nonparametric regression estimators were inspired by the developments and construction of robust estimators in those two simple models, which we briefly recall in Section 1.1. Later, we introduce the main concepts of nonparametric regression in Section 1.2 and discuss the robust nonparametric methods from Section 2 on. | Robust estimation of the location and regression modelsConsider the simple univariate location-scale model Y = μ + σε, where Y is a continuously distributed univariate random variable, μ and σ are the location and scale parameters, respectively, and ϵ is an error term. For a random sample

In this paper, an extension of the indirect inference methodology to semiparametric estimation is explored in the context of censored regression. Motivated by weak small-sample performance of the censored regression quantile estimator proposed by Powell (J Econom 32:143-155, 1986a), two-and three-step estimation methods were introduced for estimation of the censored regression model under conditional quantile restriction. While those stepwise estimators have been proven to be consistent and asymptotically normal, their finite sample performance greatly depends on the specification of an initial estimator that selects the subsample to be used in subsequent steps. In this paper, an alternative semiparametric estimator is introduced that does not involve a selection procedure in the first step. The proposed estimator is based on the indirect inference principle and is shown to be consistent and asymptotically normal under appropriate regularity conditions. Its performance is demonstrated and compared to existing methods by means of Monte Carlo simulations.

In this paper, an extension of the indirect inference methodology to semiparametric estimation is explored in the context of censored regression. Motivated by weak small-sample performance of the censored regression quantile estimator proposed by Powell (J Econom 32:143-155, 1986a), two-and three-step estimation methods were introduced for estimation of the censored regression model under conditional quantile restriction. While those stepwise estimators have been proven to be consistent and asymptotically normal, their finite sample performance greatly depends on the specification of an initial estimator that selects the subsample to be used in subsequent steps. In this paper, an alternative semiparametric estimator is introduced that does not involve a selection procedure in the first step. The proposed estimator is based on the indirect inference principle and is shown to be consistent and asymptotically normal under appropriate regularity conditions. Its performance is demonstrated and compared to existing methods by means of Monte Carlo simulations.

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In this paper, a new class of semiparametric estimators for single-index binary-choice models is introduced. The proposed estimators are based on the semiparametric indirect inference that identifies and estimates the parameters of the model via possibly misspecified auxiliary criteria. A large class of considered auxiliary criteria includes the ordinary least squares, nonlinear least squares, and nonlinear least absolute deviations estimators. Besides deriving the consistency and asymptotic normality of the proposed methods, we demonstrate that the proposed indirect inference methodology – at least for selected auxiliary criteria – combines weak distributional assumptions, good estimation precision, and robustness to misclassification of responses. We conduct Monte Carlo experiments and an application study to compare the finite-sample performance of the proposed and existing estimators.

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