Robust kernel estimators for additive models with dependent observations

Robust nonparametric estimators for additive regression or autoregression models under an α-mixing condition are proposed. They are based on local M-estimators or local medians with kernel weights, and their asymptotic behaviour is studied. Moreover, diese local M-estimators achieve the same univariate rate of convergence as their linear relatives.     

L 1-estimation for varying coefficient models

The varying coefficient model is a useful extension of linear models and has many advantages in practical use. To estimate the unknown functions in the model, the kernel type with local linear least-squares (L ₂) estimation methods has been proposed by several authors. When the data contain outliers or come from population with heavy-tailed distributions, L ₁-estimation should yield better estimators. In this article, we present the local linear L ₁-estimation method and derive the asymptotic distributions of the L ₁-estimators. The simulation results for two examples, with outliers and heavy-tailed distribution, respectively, show that the L ₁-estimators outperform the L ₂-estimators.     

A variant of K nearest neighbor quantile regression

Compared with local polynomial quantile regression, K nearest neighbor quantile regression (KNNQR) has many advantages, such as not assuming smoothness of functions. The paper summarizes the research of KNNQR and has carried out further research on the selection of k, algorithm and Monte Carlo simulations. Additionally, simulated functions are Blocks, Bumps, HeaviSine and Doppler, which stand for jumping, volatility, mutagenicity slope and high frequency function. When function to be estimated has some jump points or catastrophe points, KNNQR is superior to local linear quantile regression in the sense of the mean squared error and mean absolute error criteria. To be mentioned, even high frequency, the superiority of KNNQR could be observed. A real data is analyzed as an illustration.     

Robust M-estimators of multivariate location and scatter in the presence of asymmetry

Robust estimation of location vectors and scatter matrices is studied under the assumption that the unknown error distribution is spherically symmetric in a central region and completely unknown in the tail region. A precise formulation of the model is given, an analysis of the identifiable parameters in the model is presented, and consistent initial estimators of the identifiable parameters are constructed. Consistent and asymptotically normal M-estimators are constructed (solved iteratively beginning with the initial estimates) based on “influence functions” which vanish outside specified compact sets. Finally M-estimators which are asymptotically minimax (in the sense of Huber) are derived.     

Consistency and Normality of M-Estimators in Partly Linear Models with Stochastic Adapted Errors

The robust M-estimators for the partly linear model under stochastic adapted errors are considered. It is shown that the M-estimator of parameter is asymptotically normal and the M-estimator of the nonparametric function achieves the optimal rate of convergence for nonparametric regression. Some known results are improved and generalized. Some simulations and a real data example are conducted to illustrate the proposed method.     

Robust estimation for functional coefficient regression models with spatial data

A global smoothing procedure is developed using B-spline function approximation for estimating the unknown functions of a functional coefficient regression model with spatial data. A general formulation is used to treat mean regression, median regression, quantile regression and robust mean regression in one setting. The global convergence rates of the estimators of unknown coefficient functions are established. Various applications of the main results, including estimating conditional quantile coefficient functions and robustifying the mean regression coefficient functions are given. Finite sample properties of our procedures are studied through Monte Carlo simulations. A housing data example is used to illustrate the proposed methodology.     

M-Estimators Based on Inverse Probability Weighted Estimating Equations with Response Missing at Random

Asymptotic properties of M-estimators with complete data are investigated extensively. In the presence of missing data, however, the standard inference procedures for complete data cannot be applied directly. In this article, the inverse probability weighted method is applied to missing response problem to define M-estimators. The existence of M-estimators is established under very general regularity conditions. Consistency and asymptotic normality of the M-estimators are proved, respectively. An iterative algorithm is applied to calculating the M-estimators. It is shown that one step iteration suffices and the resulting one-step M-estimate has the same limit distribution as in the fully iterated M-estimators.     

M-Estimator of a Generalized Linear Model with Measurement Errors

In this article, we propose a generalized linear model and estimate the unknown parameters using robust M-estimator. Under suitable conditions and by the strong law of large numbers and central limits theorem, the proposed M-estimators are proved to be consistent and asymptotically normal. We also evaluate the finite sample performance of our estimator through a Monte Carlo study.     

M-estimation in regression models for censored data

In this paper, we study M-estimators of regression parameters in semiparametric linear models for censored data. A class of consistent and asymptotically normal M-estimators is constructed. A resampling method is developed for the estimation of the asymptotic covariance matrix of the estimators.     

Efficient Robust Estimation for Linear Models with Missing Response at Random

Coefficient estimation in linear regression models with missing data is routinely carried out in the mean regression framework. However, the mean regression theory breaks down if the error variance is infinite. In addition, correct specification of the likelihood function for existing imputation approach is often challenging in practice, especially for skewed data. In this paper, we develop a novel composite quantile regression and a weighted quantile average estimation procedure for parameter estimation in linear regression models when some responses are missing at random. Instead of imputing the missing response by randomly drawing from its conditional distribution, we propose to impute both missing and observed responses by their estimated conditional quantiles given the observed data and to use the parametrically estimated propensity scores to weigh check functions that define a regression parameter. Both estimation procedures are resistant to heavy‐tailed errors or outliers in the response and can achieve nice robustness and efficiency. Moreover, we propose adaptive penalization methods to simultaneously select significant variables and estimate unknown parameters. Asymptotic properties of the proposed estimators are carefully investigated. An efficient algorithm is developed for fast implementation of the proposed methodologies. We also discuss a model selection criterion, which is based on an IC_Q ‐type statistic, to select the penalty parameters. The performance of the proposed methods is illustrated via simulated and real data sets.     

Comparisons of asymptotic biases and variances of m-estimators of scale under asymmetric contamination

We study robustness properties of two types of M-estimators of scale when both location and scale parameters are unknown: (i) the scale estimator arising from simultaneous M-estimation of location and scale; and (ii) its symmetrization about the sample median. The robustness criteria considered are maximal asymptotic bias and maximal asymptotic variance when the known symmetric unimodal error distribution is subject to unknown, possibly asymmetric, £-con-tamination. Influence functions and asymptotic variance functionals are derived, and computations of asymptotic biases and variances, under the normal distribution with ε-contamination at oo, are presented for the special subclass arising from Huber's Proposal 2 and its symmetrized version. Symmetrization is seen to reduce both asymptotic bias and variance. Some complementary theoretical results are obtained, and the tradeoff between asymptotic bias and variance is discussed.     

Asymptotic properties of one-step M-estimators

We study the asymptotic behavior of one-step M-estimators based on not necessarily independent identically distributed observations. In particular, we find conditions for asymptotic normality of these estimators. Asymptotic normality of one-step M-estimators is proven under a wide spectrum of constraints on the exactness of initial estimators. We discuss the question of minimal restrictions on the exactness of initial estimators. We also discuss the asymptotic behavior of the solution to an M-equation closest to the parameter under consideration. As an application, we consider some examples of one-step approximation of quasi-likelihood estimators in nonlinear regression.     

Automatic Bayesian quantile regression curve fitting

Quantile regression, including median regression, as a more completed statistical model than mean regression, is now well known with its wide spread applications. Bayesian inference on quantile regression or Bayesian quantile regression has attracted much interest recently. Most of the existing researches in Bayesian quantile regression focus on parametric quantile regression, though there are discussions on different ways of modeling the model error by a parametric distribution named asymmetric Laplace distribution or by a nonparametric alternative named scale mixture asymmetric Laplace distribution. This paper discusses Bayesian inference for nonparametric quantile regression. This general approach fits quantile regression curves using piecewise polynomial functions with an unknown number of knots at unknown locations, all treated as parameters to be inferred through reversible jump Markov chain Monte Carlo (RJMCMC) of Green (Biometrika 82:711–732, 1995). Instead of drawing samples from the posterior, we use regression quantiles to create Markov chains for the estimation of the quantile curves. We also use approximate Bayesian factor in the inference. This method extends the work in automatic Bayesian mean curve fitting to quantile regression. Numerical results show that this Bayesian quantile smoothing technique is competitive with quantile regression/smoothing splines of He and Ng (Comput. Stat. 14:315–337, 1999) and P-splines (penalized splines) of Eilers and de Menezes (Bioinformatics 21(7):1146–1153, 2005).     

On the trimmed mean and minimax-variance L-estimation in Kolmogorov neighbourhoods

We consider the properties of the trimmed mean, as regards minimax-variance L-estimation of a location parameter in a Kolmogorov neighbourhood K() of the normal distribution: We first review some results on the search for an L-minimax estimator in this neighbourhood, i.e. a linear combination of order statistics whose maximum variance in K_t() is a minimum in the class of L-estimators. The natural candidate – the L-estimate which is efficient for that member of K_t,() with minimum Fisher information – is known not to be a saddlepoint solution to the minimax problem. We show here that it is not a solution at all. We do this by showing that a smaller maximum variance is attained by an appropriately trimmed mean. We argue that this trimmed mean, as well as being computationally simple – much simpler than the efficient L-estimate referred to above, and simpler than the minimax M- and R-estimators – is at least “nearly” minimax.     

On Semiparametric Mode Regression Estimation

It has been found that, for a variety of probability distributions, there is a surprising linear relation between mode, mean, and median. In this article, the relation between mode, mean, and median regression functions is assumed to follow a simple parametric model. We propose a semiparametric conditional mode (mode regression) estimation for an unknown (unimodal) conditional distribution function in the context of regression model, so that any m-step-ahead mean and median forecasts can then be substituted into the resultant model to deliver m-step-ahead mode prediction. In the semiparametric model, Least Squared Estimator (LSEs) for the model parameters and the simultaneous estimation of the unknown mean and median regression functions by the local linear kernel method are combined to infer about the parametric and nonparametric components of the proposed model. The asymptotic normality of these estimators is derived, and the asymptotic distribution of the parameter estimates is also given and is shown to follow usual parametric rates in spite of the presence of the nonparametric component in the model. These results are applied to obtain a data-based test for the dependence of mode regression over mean and median regression under a regression model.     

The influence function of penalized regression estimators

To perform regression analysis in high dimensions, lasso or ridge estimation are a common choice. However, it has been shown that these methods are not robust to outliers. Therefore, alternatives as penalized M-estimation or the sparse least trimmed squares (LTS) estimator have been proposed. The robustness of these regression methods can be measured with the influence function. It quantifies the effect of infinitesimal perturbations in the data. Furthermore, it can be used to compute the asymptotic variance and the mean-squared error (MSE). In this paper we compute the influence function, the asymptotic variance and the MSE for penalized M-estimators and the sparse LTS estimator. The asymptotic biasedness of the estimators make the calculations non-standard. We show that only M-estimators with a loss function with a bounded derivative are robust against regression outliers. In particular, the lasso has an unbounded influence function.     

Multivariate τ-estimators for location and scatter

We discuss the robustness and asymptotic behaviour of τ-estimators for multivariate location and scatter. We show that τ-estimators correspond to multivariate M-estimators defined by a weighted average of redescending ψ-functions, where the weights are adaptive. We prove consistency and asymptotic normality under weak assumptions on the underlying distribution, show that τ-estimators have a high breakdown point, and obtain the influence function at general distributions. In the special case of a location-scatter family, τ-estimators are asymptotically equivalent to multivariate S-estimators defined by means of a weighted ψ-function. This enables us to combine a high breakdown point and bounded influence with good asymptotic efficiency for the location and covariance estimator.     

Bounded influence estimation for regression and scale

Bednarski and Müller [Optimal bounded influence regression and scale M-estimators in the context of experimental design, Statistics 35 (2001), pp. 349–369] introduced a class of bounded influence M estimates for the simultaneous estimation of regression and scale in the linear model with normal errors by solving the corresponding normal location and scale problem at each design point. This limits the proposal to regressor distributions with finite support. Based on their approach, we propose a slightly extended class of M estimates that is not restricted to finite support and is numerically easier to handle. Moreover, we employ the even more general class of asymptotically linear (AL) estimators which, in addition, is not restricted to normal errors. The superiority of AL estimates is demonstrated by numerical comparisons of the maximum asymptotic mean-squared error over infinitesimal contamination neighbourhoods.     

De l'unicité des estimateurs robustes en régression lorsque le paramètre d'échelle et le paramètre de la régression sont estimés simultanément

Several methods have been suggested to calculate robust M- and G-M -estimators of the regression parameter β and of the error scale parameter σ in a linear model. This paper shows that, for some data sets well known in robust statistics, the nonlinear systems of equations for the simultaneous estimation of β, with an M-estimate with a redescending ψ-function, and σ, with the residual median absolute deviation (MAD), have many solutions. This multiplicity is not caused by the possible lack of uniqueness, for redescending ψ-functions, of the solutions of the system defining β with known σ; rather, the simultaneous estimation of β and σ together creates the problem. A way to avoid these multiple solutions is to proceed in two steps. First take σ as the median absolute deviation of the residuals for a uniquely defined robust M-estimate such as Huber's Proposal 2 or the L₁-estimate. Then solve the nonlinear system for the M-estimate with σ equal to the value obtained at the first step to get the estimate of β. Analytical conditions for the uniqueness of M and G-M-estimates are also given.     

Quasi-likelihood from<Emphasis Type="Italic">M</Emphasis>-estimators: A numerical comparison with empirical likelihood

In this paper we compare two robust pseudo-likelihoods for a parameter of interest, also in the presence of nuisance parameters. These functions are obtained by computing quasi-likelihood and empirical likelihood from the estimating equations which define robustM-estimators. Application examples in the context of linear transformation models are considered. Monte Carlo studies are performed in order to assess the finite-sample performance of the inferential procedures based on quasi-and empirical likelihood, when the objective is the construction of robust confidence regions.