Trimmed least squares estimator as best trimmed linear conditional estimator for linear regression model

A class of trimmed linear conditional estimators based on regression quantiles for the linear regression model is introduced. This class serves as a robust analogue of non-robust linear unbiased estimators. Asymptotic analysis then shows that the trimmed least squares estimator based on regression quantiles ( Koenker and Bassett ( 1978 ) ) is the best in this estimator class in terms of asymptotic covariance matrices. The class of trimmed linear conditional estimators contains the Mallows-type bounded influence trimmed means ( see De Jongh et al ( 1988 ) ) and trimmed instrumental variables estimators. A large sample methodology based on trimmed instrumental variables estimator for confidence ellipsoids and hypothesis testing is also provided.     

Bootstrap adaptive estimation: The trimmed-mean example

We consider the problem of choosing among a class of possible estimators by selecting the estimator with the smallest bootstrap estimate of finite sample variance. This is an alternative to using cross-validation to choose an estimator adaptively. The problem of a confidence interval based on such an adaptive estimator is considered. We illustrate the ideas by applying the method to the problem of choosing the trimming proportion of an adaptive trimmed mean. It is shown that a bootstrap adaptive trimmed mean is asymptotically normal with an asymptotic variance equal to the smallest among trimmed means. The asymptotic coverage probability of a bootstrap confidence interval based on such adaptive estimators is shown to have the nominal level. The intervals based on the asymptotic normality of the estimator share the same asymptotic result, but have poor small-sample properties compared to the bootstrap intervals. A small-sample simulation demonstrates that bootstrap adaptive trimmed means adapt themselves rather well even for samples of size 10.     

Linear trimmed means for the linear regression with AR(1) errors model

For the linear regression with AR(1) errors model, the robust generalized and feasible generalized estimators of Lai et al. (2003) of regression parameters are shown to have the desired property of a robust Gauss Markov theorem. This is done by showing that these two estimators are the best among classes of linear trimmed means. Monte Carlo and data analysis for this technique have been performed.     

Monte Carlo studies of some adaptive robust procedures for location

One linear and two nonlinear adaptive robust procedures have been developed in which preliminary statistics, based on tail lengths, attempt to identify distributions from which the samples arise so that a suitable robust estimator based on trimmed means can be used to estimate the location parameter. The efficiencies of the estimators based on the three proposed adaptive robust procedures have been obtained using Monte Carlo methods involving eight distributions and these efficiencies are compared with the efficiencies of nineteen other robust estimators.     

Trimmed likelihood estimators for lifetime experiments and their influence functions

We study the behaviour of trimmed likelihood estimators (TLEs) for lifetime models with exponential or lognormal distributions possessing a linear or nonlinear link function. In particular, we investigate the difference between two possible definitions for the TLE, one called original trimmed likelihood estimator (OTLE) and one called modified trimmed likelihood estimator (MTLE) which is the finite sample version of a form for location and linear regression used by Bednarski and Clarke [Trimmed likelihood estimation of location and scale of the normal distribution. Aust J Statist. 1993;35:141–153, Asymptotics for an adaptive trimmed likelihood location estimator. Statistics. 2002;36:1–8] and Bednarski et al. [Adaptive trimmed likelihood estimation in regression. Discuss Math Probab Stat. 2010;30:203–219]. The OTLE is always an MTLE but the MTLE may not be unique even in cases where the OLTE is unique. We compare especially the functional forms of both types of estimators, characterize the difference with the implicit function theorem and indicate situations where they coincide and where they do not coincide. Since the functional form of the MTLE has a simpler form, we use it then for deriving the influence function, again with the help of the implicit function theorem. The derivation of the influence function for the functional form of the OTLE is similar but more complicated.     

The Effect of Trimming and Normality Assumptions on the Efficiency of the O-BLUE Estimators

The seriousness of trimming samples, and considering the trimmed samples as if they are complete samples of the retained size, from a normal distribution is explored for the parameters of the simple linear regression model. The exact efficiencies of the O-BLUE estimators of the parameters under these conditions are investigated relative to the O-BLUE estimators based upon the trimmed samples from the actual distribution G. Five symmetric distributions are considered. It is found that the overall loss in relative efficiency is quite substantial especially when the true distribution has heavier tails than the normal distribution, and in particular for larger amounts of trimming.     

Computing least trimmed squares regression with the forward search

Least trimmed squares (LTS) provides a parametric family of high breakdown estimators in regression with better asymptotic properties than least median of squares (LMS) estimators. We adapt the forward search algorithm of Atkinson (1994) to LTS and provide methods for determining the amount of data to be trimmed. We examine the efficiency of different trimming proportions by simulation and demonstrate the increasing efficiency of parameter estimation as larger proportions of data are fitted using the LTS criterion. Some standard data examples are analysed. One shows that LTS provides more stable solutions than LMS.     

Trimmed Mean Isotonic Regression

The trimmed mean is well‐known in literature for being more robust and for having better efficiency than the sample mean when data is generated from heavy‐tailed distributions. In this article, the trimmed mean in the isotonic regression setup is proposed, and the asymptotic as well as the robustness properties of the estimator are studied. The usefulness of the proposed estimator is illustrated using different real and simulated data. Further, the performance of the estimator is compared with that of the mean and the median isotonic regression estimators.     

Trimmed mean and bounded influence estimators for the parameters of the ar(1) process

Trimmed mean type estimators are proposcd for estimating the parameters of an AR(1) process. Thcsc definitions are then extended to bounded influence trimmed mcans in analogy to those in the regression case. The behaviour of the estimators are studied numerically under two

outlicr generating models.     

Trimmed and Winsorized means based on a scaled deviation

Trimmed (and Winsorized) means based on a scaled deviation are introduced and studied. The influence functions of the estimators are derived and their limiting distributions are established via asymptotic representations. As a main focus of the paper, the performance of the estimators with respect to various robustness and efficiency criteria is evaluated and compared with leading competitors including the ordinary Tukey trimmed (and Winsorized) means. Unlike the Tukey trimming which always trims a fixed fraction of sample points at each end of data, the trimming scheme here only trims points at one or both ends that have a scaled deviation beyond some threshold. The resulting trimmed (and Winsorized) means are much more robust than their predecessors. Indeed they can share the best breakdown point robustness of the sample median for any common trimming thresholds. Furthermore, for appropriate trimming thresholds they are highly efficient at light-tailed symmetric models and more efficient than their predecessors at heavy-tailed or contaminated symmetric models. Detailed comparisons with leading competitors on various robustness and efficiency aspects reveal that the scaled deviation trimmed (Winsorized) means behave very well overall and consequently represent very favorable alternatives to the ordinary trimmed (Winsorized) means.     

Multivariate trimmed means based on the Tukey depth

In univariate statistics, the trimmed mean has long been regarded as a robust and efficient alternative to the sample mean. A multivariate analogue calls for a notion of trimmed region around the center of the sample. Using Tukey's depth to achieve this goal, this paper investigates two types of multivariate trimmed means obtained by averaging over the trimmed region in two different ways. For both trimmed means, conditions ensuring asymptotic normality are obtained; in this respect, one of the main features of the paper is the systematic use of Hadamard derivatives and empirical processes methods to derive the central limit theorems. Asymptotic efficiency relative to the sample mean as well as breakdown point are also studied. The results provide convincing evidence that these location estimators have nice asymptotic behavior and possess highly desirable finite-sample robustness properties; furthermore, relative to the sample mean, both of them can in some situations be highly efficient for dimensions between 2 and 10.     

A comparison of some robust,adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.

     

A comparison of some robust, adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.     

Robust test for means when population variances are unequal

We consider the problem of testing the equality of two population means when the population variances are not necessarily equal. We propose a Welch-type statistic, say T^* _c, based on Tiku!s ‘1967, 1980’ modified maximum likelihood estimators, and show that this statistic is robust to symmetric and moderately skew distributions. We investigate the power properties of the statistic T^* _c; T^* _c clearly seems to be more powerful than Yuen's ‘1974’ Welch-type robust statistic based on the trimmed sample means and the matching sample variances. We show that the analogous statistics based on the ‘adaptive’ robust estimators give misleading Type I errors. We generalize the results to testing linear contrasts among k population means     

Symmetric regression quantile and its application to robust estimation for the nonlinear regression model

Populational conditional quantiles in terms of percentage α are useful as indices for identifying outliers. We propose a class of symmetric quantiles for estimating unknown nonlinear regression conditional quantiles. In large samples, symmetric quantiles are more efficient than regression quantiles considered by Koenker and Bassett (Econometrica 46 (1978) 33) for small or large values of α, when the underlying distribution is symmetric, in the sense that they have smaller asymptotic variances. Symmetric quantiles play a useful role in identifying outliers. In estimating nonlinear regression parameters by symmetric trimmed means constructed by symmetric quantiles, we show that their asymptotic variances can be very close to (or can even attain) the Cramer–Rao lower bound under symmetric heavy-tailed error distributions, whereas the usual robust and nonrobust estimators cannot.     

Robust estimation methods for exponential data: a monte-carlo comparison

We compare the performance of seven robust estimators for the parameter of an exponential distribution. These include the debiased median and two optimally-weighted one-sided trimmed means. We also introduce four new estimators: the Transform, Bayes, Scaled and Bicube estimators. We make the Monte Carlo comparisons for three sample sizes and six situations. We evaluate the comparisons in terms of a new performance measure, Mean Absolute Differential Error (MADE), and a premium/protection interpretation of MADE. We organize the comparisons to enhance statistical power by making maximal use of common random deviates. The Transform estimator provides the best performance as judged by MADE. The singly-trimmed mean and Transform method define the efficient frontier of premium/protection.     

Inference under Heteroscedasticity of Unknown Form Using an Adaptive Estimator

The heteroscedasticity consistent covariance matrix estimators are commonly used for the testing of regression coefficients when error terms of regression model are heteroscedastic. These estimators are based on the residuals obtained from the method of ordinary least squares and this method yields inefficient estimators in the presence of heteroscedasticity. It is usual practice to use estimated weighted least squares method or some adaptive methods to find efficient estimates of the regression parameters when the form of heteroscedasticity is unknown. But HCCM estimators are seldom derived from such efficient estimators for testing purposes in the available literature. The current article addresses the same concern and presents the weighted versions of HCCM estimators. Our numerical work uncovers the performance of these estimators and their finite sample properties in terms of interval estimation and null rejection rate.     

Comparing J independent groups with a method based on trimmed means

The ANOVA-F test is the most popular and commonly used procedure for comparing J independent groups. However, it is well known that this method is very sensitive to non-normality, which has led to the derivation of alternative techniques based on robust estimators. In this work, ANOVA-F-test, trimmed mean Welch test, bootstrap-t trimmed mean Welch test, Schrader and Hettmansperger method with trimmed means, a percentile bootstrap method with trimmed means and a newly proposed method were compared in terms of both the Type I error probability and power. The proposed method compares well with ANOVA-F and other alternatives under various situations.     

Trimmed and Outer Means and Their Variances

Some properties of trimmed and outer means for the normal situation are considered; in particular, the mean of the outer half of the sample has the same variance as the mean of the inner half. An inequality involving variances of unbiased estimators of location and their complements is derived and some of its consequences are examined.     

Asymptotic efficiencies of estiamtors of location a comutational study

Asymptotic efficiencies of four classes of estimators of location are evaluated for a family of distributions consisting of t, lambda and contaminated normal densities. For the class of estimators derived from signed rank tests, maximin efficiencies between pairs of distributions in the family are computed using a formula of Gastwirth ( 1966 ). Asymptotic efficiencies are also evaluated for the scale dependent estimators of the form proposed by Hubcr ( 1964 ) and the efficiencies of procedures utilizing interquantiie ranges.are evaluated. Efficiencies of linear estimators such as trimmed means, BLUE's for the lambda family are computed for each density considered. Efficiencies of linear, polynomial and trigonometric approximations to BLUE weight functions are determined. Using the method of Birnbaum and Laska ( 1967 ) maximin efficiencies are computed using four linear or polynomial terms. On the basis of comparisons of these numerical results, suggestions for robust estimators are given