ESTIMATING THE ROOT OF A NONPARAMETRIC REGRESSION FUNCTION IN A ROBUST FASHION

For a nonparametric regression model y = m(x)+e with n independent observations, we analyze a robust method of finding the root of m(x) based on an M-estimation first discussed by Härdle & Gasser (1984). It is shown here that the robustness properties (minimaxity and breakdown function) of such an estimate are quite analogous to those of an M -estimator in the simple location model, but the rate of convergence is somewhat limited due to the nonparametric nature of the problem.     

Asymptotics of M-estimator in multivariate linear regression models for a class of random errors

It is known that linear regression models have immense applications in various areas such as engineering technology, economics and social sciences. In this paper, we investigate the asymptotic properties of M-estimator in multivariate linear regression model based on a class of random errors satisfying a generalised Bernstein-type inequality. By using the generalised Bernstein-type inequality, we obtain a general result on almost sure convergence for a class of random variables and then obtain the strong consistency for the M-estimator in multivariate linear regression models under some mild conditions. The result extends or improves some existing ones in the literature. Moreover, we also consider the case when the dimension $p$ tends to infinity by establishing the rate of almost sure convergence for a class of random variables satisfying generalised Bernstein-type inequality. Some numerical simulations are also provided to verify the validity of the theoretical results.     

M-Estimators of Scale with Minimum Gross Errors Sensitivity

The median absolute deviation (MAD) is known to be the M-estimator of scale with minimum gross errors sensitivity (GES) when the error distribution is known to be symmetric and strongly unimodal. The problem considered here is to find the Fisher consistent M-estimator with minimum GES when the error distribution is symmetric but not necessarily unimodal. Under some general conditions, the score function χ corresponding to the minimizing M-estimator has the form χ(x) = ?1 when |x| < a; χ(x) = c when a < |x| < b; χ(x) = 1 when |x| > b. An example is given in which the M-estimator with minimum GES does not correspond to the MAD.     

Minimum hellinger distance estimation for normal models

A robust estimator introduced by Beran (1977a, 1977b), which is based on the minimum Hellinger distance between a projection model density and a nonparametric sample density, is studied empirically. An extensive simulation provides an estimate of the small sample distribution and supplies empirical evidence of the estimator performance for a normal location-scale model. While the performance of the minimum Hellinger distance estimator is seen to be competitive with the maximum likelihood estimator at the true model, its robustness to deviations from normality is shown to be competitive in this setting with that obtained from the M-estimator and the Cramér-von Mises minimum distance estimator. Beran also introduced a goodness-of-fit statisticH ₂, based on the minimized Hellinger distance between a member of a parametric family of densities and a nonparametric density estimate. We investigate the statistic H (the square root of H ₂) as a test for normality when both location and scale are unspecified. Empirically derived critical values are given which do not require extensive tables. The power of the statistic H compares favorably with four other widely used tests for normality.     

New Results on the Small-Sample Properties of Some Robust Univariate Estimators of Location

This article compares eight estimators in terms of relative efficiencies with the univariate mean, some of which have not been compared previously. Four estimators, when testing hypotheses, are compared in terms of actual Type I errors. In terms of point estimation, the modified one-step M-estimator, one-step M-estimator, and rfch estimator are found to be the three best choices depending on the proportion of outliers. In terms of actual Type I errors, the modified one-step M estimator's and rfch estimator's level was between.045 and.055 in 5 out of 7 situations when real data were used in simulations.     

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.     

A Robust Variable Selection to t-type Joint Generalized Linear Models via Penalized t-type Pseudo-likelihood

Although the t-type estimator is a kind of M-estimator with scale optimization, it has some advantages over the M-estimator. In this article, we first propose a t-type joint generalized linear model as a robust extension to the classical joint generalized linear models for modeling data containing extreme or outlying observations. Next, we develop a t-type pseudo-likelihood (TPL) approach, which can be viewed as a robust version to the existing pseudo-likelihood (PL) approach. To determine which variables significantly affect the variance of the response variable, we then propose a unified penalized maximum TPL method to simultaneously select significant variables for the mean and dispersion models in t-type joint generalized linear models. Thus, the proposed variable selection method can simultaneously perform parameter estimation and variable selection in the mean and dispersion models. With appropriate selection of the tuning parameters, we establish the consistency and the oracle property of the regularized estimators. Simulation studies are conducted to illustrate the proposed methods.     

Minimax-variance L- and R-estimators of location

We consider the problem of minimax-variance, robust estimation of a location parameter, through the use of L- and R-estimators. We derive an easily checked necessary condition for L-estimation to be minimax, and a related sufficient condition for R-estimation to be minimax. Those cases in the literature in which L-estimation is known not to be minimax, and those in which R-estimation is minimax, are derived as consequences of these conditions. New classes of examples are given in each case. As well, we answer a question of Scholz (1974), who showed essentially that the asymptotic variance of an R-estimator never exceeds that of an L-estimator, if both are efficient at the same strongly unimodal distribution. Scholz raised the question of whether or not the assumption of strong unimodality could be dropped. We answer this question in the negative, theoretically and by examples. In the examples, the minimax property fails both for L-estimation and for R-estimation, but the variance of the L-estimator, as the distribution of the observation varies over the given neighbourhood, remains unbounded. That of the R-estimator is unbounded.     

Comparison of efficient L- and R-estimators of location

Scholz (1974) proved that the asymptotic variance of an R-estimator of location is no larger than that of an L-estimator when the observations come from a distribution G different from the distribution F for which the two estimators are efficient. This note extends this result to distributions F whose density has a first but no second derivative.     

Lp-estimators in ARCH models

For ergodic ARCH processes, we introduce a one-parameter family of L_p-estimators. The construction is based on the concept of weighted M-estimators. Under weak assumptions on the error distribution, the consistency is established. The asymptotic normality is proved for the special cases p=1 and 2. To prove the asymptotic normality of the L₁-estimator, one needs the existence of a density of the squares of the errors, whereas for the L₂-estimator the existence of fourth moments is assumed. The asymptotic covariance matrix of the estimator depends on the unknown parameter which can be substituted by consistent estimators. For the L₁-estimator we construct a kernel estimator for the unknown density of the square of the errors.     

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.     

One-step M-estimators: Jones and Faddy's skewed t-distribution

One-step M (OSM)-estimator needs some initial/preliminary estimates at the beginning of the calculation process. In this study, we propose to use new initial estimates for the calculation of the OSM-estimator. We consider simple location and simple linear regression models when the distribution of the error terms is Jones and Faddy's skewed t. Monte-Carlo simulation study shows that the OSM estimator(s) based on the proposed initial estimates is/are more efficient than the OSM estimator(s) based on the traditional initial estimates especially for the skewed cases. We also analyze some real data sets taken from the literature at the end of the paper.     

A consistent nonparametric density estimator for the deconvolution problem

The problem of nonparametric estimation of a probability density function when the sample observations are contaminated with random noise is studied. A particular estimator f?_n(x) is proposed which uses kernel-density and deconvolution techniques. The estimator f?_n(x) is shown to be uniformly consistent, and its appearance and properties are affected by constants M_n and h_n which the user may choose. The optimal choices of M_n and h_n depend on the sample size n, the noise distribution, and the true distribution which is being estimated. Particular selections for M_n and h_n which minimize upper-bound functions of the mean squared error for f?_n(x) are recommended.     

A mixture-based approach to robust analysis of generalised linear models

A method for robustness in linear models is to assume that there is a mixture of standard and outlier observations with a different error variance for each class. For generalised linear models (GLMs) the mixture model approach is more difficult as the error variance for many distributions has a fixed relationship to the mean. This model is extended to GLMs by changing the classes to one where the standard class is a standard GLM and the outlier class which is an overdispersed GLM achieved by including a random effect term in the linear predictor. The advantages of this method are it can be extended to any model with a linear predictor, and outlier observations can be easily identified. Using simulation the model is compared to an M-estimator, and found to have improved bias and coverage. The method is demonstrated on three examples.     

A New Test for New Better Than Used in Expectation Lifetimes

The mean residual life of a non negative random variable X with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. A popular nonparametric model of aging is new better than used in expectation (NBUE), when M(t) ? M(0) for all t ? 0. The exponential distribution lies at the boundary. There is a large literature on testing exponentiality against NBUE alternatives. However, comparisons of tests have been made only for alternatives much stronger than NBUE. We show that a new Kolmogorov-Smirnov type test is much more powerful than its competitors in most cases.     

ARFIMA processes and outliers: a weighted likelihood approach

In this paper, we consider the problem of robust estimation of the fractional parameter, d, in long memory autoregressive fractionally integrated moving average processes, when two types of outliers, i.e. additive and innovation, are taken into account without knowing their number, position or intensity. The proposed method is a weighted likelihood estimation (WLE) approach for which needed definitions and algorithm are given. By an extensive Monte Carlo simulation study, we compare the performance of the WLE method with the performance of both the approximated maximum likelihood estimation (MLE) and the robust M-estimator proposed by Beran (Statistics for Long-Memory Processes, Chapman & Hall, London, 1994). We find that robustness against the two types of considered outliers can be achieved without loss of efficiency. Moreover, as a byproduct of the procedure, we can classify the suspicious observations in different kinds of outliers. Finally, we apply the proposed methodology to the Nile River annual minima time series.     

Redescending M-estimators

In finite sample studies redescending M-estimators outperform bounded M-estimators (see for example, Andrews et al. [1972. Robust Estimates of Location. Princeton University Press, Princeton]). Even though redescenders arise naturally out of the maximum likelihood approach if one uses very heavy-tailed models, the commonly used redescenders have been derived from purely heuristic considerations. Using a recent approach proposed by Shurygin, we study the optimality of redescending M-estimators. We show that redescending M-estimator can be designed by applying a global minimax criterion to locally robust estimators, namely maximizing over a class of densities the minimum variance sensitivity over a class of estimators. As a particular result, we prove that Smith's estimator, which is a compromise between Huber's skipped mean and Tukey's biweight, provides a guaranteed level of an estimator's variance sensitivity over the class of densities with a bounded variance.     

Robust estimation and wavelet thresholding in partially linear models

This paper is concerned with a semiparametric partially linear regression model with unknown regression coefficients, an unknown nonparametric function for the non-linear component, and unobservable Gaussian distributed random errors. We present a wavelet thresholding based estimation procedure to estimate the components of the partial linear model by establishing a connection between an l ₁-penalty based wavelet estimator of the nonparametric component and Huber’s M-estimation of a standard linear model with outliers. Some general results on the large sample properties of the estimates of both the parametric and the nonparametric part of the model are established. Simulations are used to illustrate the general results and to compare the proposed methodology with other methods available in the recent literature.