M-estimators In Linear Models Research Articles

Asymptotics for Redescending M-estimators in Linear Models with Increasing Dimension

Asymptotics for Redescending M-estimators in Linear Models with Increasing Dimension

The strong consistency of M-estimates in linear models with extended negatively dependent errors

ABSTRACTIn this paper, we first establish the strong convergence for weighted sums of extended negatively dependent (END) random variables. Based on the strong convergence and Bernstein inequality, we obtain the strong consistency of M-estimates of the regression parameters in a linear model for END random errors under some mild moment conditions. The results generalize and improve the ones obtained in the literature to the case of END random errors.

A note on the strong consistency of M-estimates in linear models

We improve a known result on the strong consistency of M-estimates of the regression parameters in a linear model for independent and identically distributed random errors under some mild conditions.

Asymptotic properties for M-estimators in linear models with dependent random errors

In this paper, we make use of the technique of martingales to establish the moderate deviations and strong Bahadur representations for M-estimators of the regression parameter in a linear model when the errors form a type of Markov chain. As an application, we obtain a law of the iterated logarithm for the estimators.

Moderate deviations for M-estimators in linear models with ϕ-mixing errors

In this paper, the moderate deviations for the M-estimators of regression parameter in a linear model are obtained when the errors form a strictly stationary ϕ-mixing sequence. The results are applied to study many different types of M-estimators such as Huber’s estimator, L p -regression estimator, least squares estimator and least absolute deviation estimator.

Approximating the Distribution of M-Estimators in Linear Models by Randomly Weighted Bootstrap

The asymptotic distribution of the M-estimators are generally related to quan- tities of the error distribution that can not be conveniently estimated.The randomly weighted bootstrap method provides a way of assessing the distribution of the M-estimators without estimating the nuisance quantities of the error distributions.In this paper,the distribution of M-estimators is approximated by the randomly weighted bootstrap method in linear models when the covariates are random.It is shown that the randomly weighted bootstrapping estima- tion of the distribution of the M-estimator is uniformly consistent.Also,the variance estimates is investigated by Monte Carlo simulations for different choices of the convex function,sample size and random weights.Poisson weighting is recommended for reducing the computational burden in the randomly weighted bootstrapping M-estimators.

Risk comparison of some shrinkage M-estimators in linear models

The problem of robust estimation of a (linear) regression parameter (vector), in the presence of nuisance scale parameter, is considered when it is a priori suspected that the regression could be restricted to a linear subspace. Asymptotic properties of variants of Stein-rule M-estimators (including the positive-rule shrinkage M-estimators) are studied. Under an asymptotic distributional quadratic risk criterion, their relative dominance picture is explored, analytically as well as by simulation. An extensive sampling experiment is used to examine the small sample characteristics of the proposed estimators over a wide-range of data sampling designs and distributions. Our simulation experiments have provided strong evidence that corroborates with the asymptotic theory. Two examples are provided to illustrate the performance of the estimators in real-life situations.

Strong consistency of M-estimates in linear models

The strong consistency of M-estimates of the regression coefficients in a linear model under some mild conditions is established, which is an essential improvement over the relevant results in the literature on the moment condition. Especially, in some important circumstances, only for some q ＞1 is needed, where is some score function of random error.

Strong consistency of M-estimates in linear models

Strong consistency of M-estimates in linear models

Asymptotic efficiency of randomly weighted bootstrap for linear models

This note discusses the method of randomly weighted bootstrap to derive the approximation of M-estimates in linear models, and shows that the approximation is asymptotically valid under some mild conditions.

Linear representation of M-estimates in linear models

Consider the linear regression model, yi = xiβ0 + ei, i = l,…,n, and an M-estimate β of βo obtained by minimizing Σρ(yi — xiβ), where ρ is a convex function. Let Sn = ΣXiXiXi and rn = Sn½ (β — β0) — Sn2 Σxih(ei), where, with a suitable choice of h(.), the expression Σ xix(e,) provides a linear representation of β. Bahadur (1966) obtained the order of rn as n ∞ when βo is a one-dimensional location parameter representing the median, and Babu (1989) proved a similar result for the general regression parameter estimated by the LAD (least absolute deviations) method. We obtain the stochastic order of rn as n ∞ for a general M-estimate as defined above, which agrees with the results of Bahadur and Babu in the special cases considered by them. Soient p une fonction convexe et β un M-estimateur de βo obtenu en minimisant Σρ(yi-xiβ)xiβ) dans le cadre du modele de regression lineaire yi=xiβ+ ei,i=[,…,n. En introduisant une fonction h(.) convenable, il est possible de reprdsenter β sous la forme ΣXih(ei(ei). Soient alors Sn =ΣXiXiet rn=Sn½(β — β0) — Sn2Σxih(ei)Σ xih(ei)Dans le cas particulier ou β0est un parametre de localisation unidimensionnel representant une mediane, l'ordre asymptotique de rna ete etabli par Bahadur (1966) et un resultat semblable a ete demontre par Babu (1989) pour un parametre de regression plus general estime par la methode des moindres ecarts absolus. Cet article a pour objectif d'etendre ces resultats en etablissant l'ordre stochastique de rnlorsque n ∞ dans le cadre abstrait decrit ci-haut.

M-estimators in linear models with long range dependent errors

This paper discusses the asymptotic behavior of a class of M-estimators in linear models when errors are Gaussian, or a function of Gaussian random variables, that are long range dependent. The asymptotics are discussed when the design variables are either i.i.d. or long range dependent, independent of the errors, or known constants. It is observed that in the latter two cases, the leading r.v.'s in the approximation of the class M-estimators of the regression parameter vector corresponding to the skew symmetric scores and symmetric errors is proportional to the least squares estimator in the Gaussian errors case. Moreover, if the design variables are either i.i.d. or the known constants then the limiting distributions are normal. But if the design variables are long range dependent then the limiting distributions are nonnormal.

Strong consistency of M-estimates in linear models

This article studies the strong consistency of M-estimates in linear regression models directly from the minimization problem ∑ i-1 n ρ(Y i−α−X i ′β):= min , where X 1. X 2, … can be random observations of a p-dimensional random vector X, or that they are simply known nonrandom p-vectors. It is shown that the solution (α̂ n, β̂′ n) of this minimization problem converges with probability one to the true parameter (α 0,β′ 0) under very general conditions on the function ϱ and the sequence {( X′ i, Y i )}.

The strong consistency of M-estimators in linear models

The strong consistency of M-estimators in linear models is considered. Under some conditions on the ratios of maximum and minimum eigenvalues of the information matrices the desired result is established.