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.     

Minimum distance estimation in linear regression with strong mixing errors

Abstract

Minimum distance estimation on the linear regression model with independent errors is known to yield an efficient and robust estimator. We extend the method to the model with strong mixing errors and obtain an estimator of the vector of the regression parameters. The goal of this article is to demonstrate the proposed estimator still retains efficiency and robustness. To that end, this article investigates asymptotic distributional properties of the proposed estimator and compares it with other estimators. The efficiency and the robustness of the proposed estimator are empirically shown, and its superiority over the other estimators is established.     

Wavelet-based estimation of regression function with strong mixing errors under fixed design

We consider wavelet-based non linear estimators, which are constructed by using the thresholding of the empirical wavelet coefficients, for the mean regression functions with strong mixing errors and investigate their asymptotic rates of convergence. We show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes B^s_{p, q}. The theory is illustrated with some numerical examples.

A new ingredient in our development is a Bernstein-type exponential inequality, for a sequence of random variables with certain mixing structure and are not necessarily bounded or sub-Gaussian. This moderate deviation inequality may be of independent interest.     

Strong consistency of non parametric kernel regression estimator for strong mixing samples

For α-mixing samples, we study Priestley–Chao kernel estimator for non parametric regression model. By using the moment inequality and the exponential inequality, the strong consistency and the uniformly strong consistency of the estimator are obtained for some weak conditions.     

Improved Liu estimator in a linear regression model

In this paper, we mainly aim to introduce the notion of improved Liu estimator (ILE) in the linear regression model y=Xβ+e. The selection of the biasing parameters is investigated under the PRESS criterion and the optimal selection is successfully derived. We make a simulation study to show the performance of ILE compared to the ordinary least squares estimator and the Liu estimator. Finally, the main results are applied to the Hald data.     

Liu estimator in partly linear regression models with correlated errors

This article is concerned with the parameter estimation in partly linear regression models when the errors are dependent. To overcome the multicollinearity problem, a generalized Liu estimator is proposed. The theoretical properties of the proposed estimator and its relationship with some existing methods designed for partly linear models are investigated. Finally, a hypothetical data is conducted to illustrate some of the theoretical results.     

A jackknifed difference-based ridge estimator in the partial linear model with correlated errors

Tabakan and Akdeniz [Difference-based ridge estimator of parameters in partial linear model. Statist Pap. 2010;51(2):357–368] proposed a difference-based ridge estimator (DBRE) in the partial linear model. In this paper, a new estimator is introduced by jackknifing the DBRE that Tabakan and Akdeniz presented. We investigate the performance of this new estimator over the DBRE and difference-based estimator introduced by Yatchew [An elementary estimator of the partial linear model. Econom Lett. 1997;57:135–143] in terms of mean-squared error and mean-squared error matrix and a numerical example is provided to demonstrate the performance of the estimators.     

Asymptotic properties of maximum quasi-likelihood estimator in quasi-likelihood non linear models with stochastic regression

In this paper, we establish the asymptotic properties of maximum quasi-likelihood estimator (MQLE) in quasi-likelihood non linear models (QLNMs) with stochastic regression under some mild regular conditions. We also investigate the existence, strong consistency, and asymptotic normality of MQLE in QLNMs with stochastic regression.     

Plug-in bandwidth choice for estimation of nonparametric part in partial linear regression models with strong mixing errors

Consider a regression model where the regression function is the sum of a linear and a nonparametric component. Assuming that the errors of the model follow a stationary strong mixing process with mean zero, the problem of bandwidth selection for a kernel estimator of the nonparametric component is addressed here. We obtain an asymptotic expression for an optimal band-width and we propose to use a plug-in methodology in order to estimate this bandwidth through preliminary estimates of the unknown quantities. Asymptotic optimality for the plug-in bandwidth is established.     

A non stochastic ridge regression estimator and comparison with the James-Stein estimator

Abstract

This article presents a non-stochastic version of the Generalized Ridge Regression estimator that arises from a discussion of the properties of a Generalized Ridge Regression estimator whose shrinkage parameters are found to be close to their upper bounds. The resulting estimator takes the form of a shrinkage estimator that is superior to both the Ordinary Least Squares estimator and the James-Stein estimator under certain conditions. A numerical study is provided to investigate the range of signal to noise ratio under which the new estimator dominates the James-Stein estimator with respect to the prediction mean square error.     

Variable selection in linear regression based on ridge estimator

In this article, we consider the problem of variable selection in linear regression when multicollinearity is present in the data. It is well known that in the presence of multicollinearity, performance of least square (LS) estimator of regression parameters is not satisfactory. Consequently, subset selection methods, such as Mallow's Cp, which are based on LS estimates lead to selection of inadequate subsets. To overcome the problem of multicollinearity in subset selection, a new subset selection algorithm based on the ridge estimator is proposed. It is shown that the new algorithm is a better alternative to Mallow's Cp when the data exhibit multicollinearity.     

On the restricted almost unbiased two-parameter estimator in linear regression model

Özkale and Kaçiranlar introduced the restricted two-parameter estimator (RTPE) to deal with the well-known multicollinearity problem in linear regression model. In this paper, the restricted almost unbiased two-parameter estimator (RAUTPE) based on the RTPE is presented. The quadratic bias and mean-squared error of the proposed estimator is discussed and compared with the corresponding competitors in literatures. Furthermore, a numerical example and a Monte Carlo simulation study are given to explain some of the theoretical results.     

Estimation in a linear regression model with stochastic linear restrictions: a new two-parameter-weighted mixed estimator

The present paper considers the weighted mixed regression estimation of the coefficient vector in a linear regression model with stochastic linear restrictions binding the regression coefficients. We introduce a new two-parameter-weighted mixed estimator (TPWME) by unifying the weighted mixed estimator of Schaffrin and Toutenburg [1] and the two-parameter estimator (TPE) of Özkale and Kaç?ranlar [2]. This new estimator is a general estimator which includes the weighted mixed estimator, the TPE and the restricted two-parameter estimator (RTPE) proposed by Özkale and Kaç?ranlar [2] as special cases. Furthermore, we compare the TPWME with the weighted mixed estimator and the TPE with respect to the matrix mean square error criterion. A numerical example and a Monte Carlo simulation experiment are presented by using different estimators of the biasing parameters to illustrate some of the theoretical results.     

The small sample properties of the restricted principal component regression estimator in linear regression model

In regression analysis, to deal with the problem of multicollinearity, the restricted principal components regression estimator is proposed. In this paper, we compared the restricted principal components regression estimator, the principal components regression estimator, and the ordinary least-squares estimator with each other under the Pitman's closeness criterion. We showed that the restricted principal components regression estimator is always superior to the principal components regression estimator, under certain conditions the restricted principal components regression estimator is superior to the ordinary least-squares estimator under the Pitman's closeness criterion and under certain conditions the principal components regression estimator is superior to the ordinary least-squares estimator under the Pitman's closeness criterion.     

On preliminary test almost unbiased two-parameter estimator in linear regression model with student's t errors

In this paper, the preliminary test approach to the estimation of the linear regression model with student's t errors is considered. The preliminary test almost unbiased two-parameter estimator is proposed, when it is suspected that the regression parameter may be restricted to a constraint. The quadratic biases and quadratic risks of the proposed estimators are derived and compared under both null and alternative hypotheses. The conditions of superiority of the proposed estimators for departure parameter and biasing parameters k and d are derived, respectively. Furthermore, a real data example and a Monte Carlo simulation study are provided to illustrate some of the theoretical results.     

Ridge regression methodology in partial linear models with correlated errors

In this article, a generalized restricted difference-based ridge estimator is defined for the vector parameter in a partial linear model when the errors are dependent. It is suspected that some additional linear constraints may hold on to the whole parameter space. The estimator is a generalization of the well-known restricted least-squares estimator and is confined to the (affine) subspace which is generated by the restrictions. The risk functions of the proposed estimators are derived under balanced loss function. Finally, the performance of the new estimators is evaluated by a simulated data set.     

New generalized regression estimator in the presence of non response under unequal probability sampling

In this paper, we propose a new generalized regression estimator for the problem of estimating the population total using unequal probability sampling without replacement. A modified automated linearization approach is applied in order to transform the proposed estimator to estimate variance of population total. The variance and estimated value of the variance of the proposed estimator is investigated under a reverse framework assuming that the sampling fraction is negligible and there are equal response probabilities for all units. We prove that the proposed estimator is an asymptotically unbiased estimator and that it does not require a known or estimated response probability to function.     

On the ridge regression estimator with sub-space restriction

In the linear regression model with elliptical errors, a shrinkage ridge estimator is proposed. In this regard, the restricted ridge regression estimator under sub-space restriction is improved by incorporating a general function which satisfies Taylor’s series expansion. Approximate quadratic risk function of the proposed shrinkage ridge estimator is evaluated in the elliptical regression model. A Monte Carlo simulation study and analysis based on a real data example are considered for performance analysis. It is evident from the numerical results that the shrinkage ridge estimator performs better than both unrestricted and restricted estimators in the multivariate t-regression model, for some specific cases.     

Performance of the stein-type two-parameter estimator in multiple linear regression model

In this article, we consider the Stein-type approach to the estimation of the regression parameter in a multiple regression model under a multicollinearity situation. The Stein-type two-parameter estimator is proposed when it is suspected that the regression parameter may be restricted to a subspace. The bias and the quadratic risk of the proposed estimator are derived and compared with the two-parameter estimator (TPE), the restricted TPE and the preliminary test TPE. The conditions of superiority of the proposed estimator are obtained. Finally, a real data example is provided to illustrate some of the theoretical results.