A Jackknife Method for Estimation of Variance Components

This paper concerns a method of estimation of variance components in a random effect linear model. It is mainly a resampling method and relies on the Jackknife principle. The derived estimators are presented as least squares estimators in an appropriate linear model, and one of them appears as a MINQUE (Minimum Norm Quadratic Unbiased Estimation) estimator. Our resampling method is illustrated by an example given by C. R. Rao [7] and some optimal properties of our estimator are derived for this example. In the last part, this method is used to derive an estimation of variance components in a random effect linear model when one of the components is assumed to be known.     

Outlier Resistant Estimation in Difference-Based Semiparametric Partially Linear Models

Partially linear models are extensions of linear models that include a nonparametric function of some covariate allowing an adequate and more flexible handling of explanatory variables than in linear models. The difference-based estimation in partially linear models is an approach designed to estimate parametric component by using the ordinary least squares estimator after removing the nonparametric component from the model by differencing. However, it is known that least squares estimates do not provide useful information for the majority of data when the error distribution is not normal, particularly when the errors are heavy-tailed and when outliers are present in the dataset. This paper aims to find an outlier-resistant fit that represents the information in the majority of the data by robustly estimating the parametric and the nonparametric components of the partially linear model. Simulations and a real data example are used to illustrate the feasibility of the proposed methodology and to compare it with the classical difference-based estimator when outliers exist.     

Estimating partially linear panel data models with one-way error components

We consider the problem of estimating a partially linear panel data model whenthe error follows an one-way error components structure. We propose a feasiblesemiparametric generalized least squares (GLS) type estimator for estimating the coefficient of the linear component and show that it is asymptotically more efficient than a semiparametric ordinary least squares (OLS) type estimator. We also discussed the case when the regressor of the parametric component is correlated with the error, and propose an instrumental variable GLS-type semiparametric estimator.     

Second-order least squares estimation of censored regression models

This paper proposes the second-order least squares estimation, which is an extension of the ordinary least squares method, for censored regression models where the error term has a general parametric distribution (not necessarily normal). The strong consistency and asymptotic normality of the estimator are derived under fairly general regularity conditions. We also propose a computationally simpler estimator which is consistent and asymptotically normal under the same regularity conditions. Finite sample behavior of the proposed estimators under both correctly and misspecified models are investigated through Monte Carlo simulations. The simulation results show that the proposed estimator using optimal weighting matrix performs very similar to the maximum likelihood estimator, and the estimator with the identity weight is more robust against the misspecification.     

Model selection by LASSO methods in a change-point model

The paper considers a linear regression model with multiple change-points occurring at unknown times. The LASSO technique is very interesting since it allows simultaneously the parametric estimation, including the change-points estimation, and the automatic variable selection. The asymptotic properties of the LASSO-type (which has as particular case the LASSO estimator) and of the adaptive LASSO estimators are studied. For this last estimator the Oracle properties are proved. In both cases, a model selection criterion is proposed. Numerical examples are provided showing the performances of the adaptive LASSO estimator compared to the least squares estimator.     

Score estimation in the monotone single‐index model

We consider estimation in the single‐index model where the link function is monotone. For this model, a profile least‐squares estimator has been proposed to estimate the unknown link function and index. Although it is natural to propose this procedure, it is still unknown whether it produces index estimates that converge at the parametric rate. We show that this holds if we solve a score equation corresponding to this least‐squares problem. Using a Lagrangian formulation, we show how one can solve this score equation without any reparametrization. This makes it easy to solve the score equations in high dimensions. We also compare our method with the effective dimension reduction and the penalized least‐squares estimator methods, both available on CRAN as R packages, and compare with link‐free methods, where the covariates are elliptically symmetric.     

Robust Linear Calibration

We regard the simple linear calibration problem where only the response y of the regression line y = β₀ + β₁ t is observed with errors. The experimental conditions t are observed without error. For the errors of the observations y we assume that there may be some gross errors providing outlying observations. This situation can be modeled by a conditionally contaminated regression model. In this model the classical calibration estimator based on the least squares estimator has an unbounded asymptotic bias. Therefore we introduce calibration estimators based on robust one-step-M-estimators which have a bounded asymptotic bias. For this class of estimators we discuss two problems: The optimal estimators and their corresponding optimal designs. We derive the locally optimal solutions and show that the maximin efficient designs for non-robust estimation and robust estimation coincide.     

Adaptation over parametric families of symmetric linear estimators

This paper treats an abstract parametric family of symmetric linear estimators for the mean vector of a standard linear model. The estimator in this family that has smallest estimated quadratic risk is shown to attain, asymptotically, the smallest risk achievable over all candidate estimators in the family. The asymptotic analysis is carried out under a strong Gauss–Markov form of the linear model in which the dimension of the regression space tends to infinity. Leading examples to which the results apply include: (a) penalized least squares fits constrained by multiple, weighted, quadratic penalties; and (b) running, symmetrically weighted, means. In both instances, the weights define a parameter vector whose natural domain is a continuum.     

Estimation of variance components in an unbalanced one-way classification

In the unbalanced one-way random effects model the weighted least squares approach with estimated weights is used to develop a relatively simple estimator of variance components. As the number of classes increases, the proposed estimator is seen not only to be best asymptotically normal but also to be asymptotically equivalent to the maximum likelihood estimator.     

More on the restricted ridge regression estimation

Several alternative methods for derivation of the restricted ridge regression estimator (RRRE) are provided. Theoretical comparison and relationship of RRRE with related methods for regression with the multicollinearity problem are described. We also find inter-connections among RRRE, ordinary ridge regression estimator (ORRE), restricted least squares estimator (RLSE), modified ridge regression estimator (MRRE) and restricted modified generalized ridge estimator (RMGRE). Finally, numerical comparison, in addition to theoretical derivation, is also conducted with a Monte Carlo simulation and a real data example.     

Penalized least absolute deviations estimation for nonlinear model with change-points

This paper studies the asymptotic properties of a smoothed least absolute deviations estimator in a nonlinear parametric model with multiple change-points occurring at the unknown times with independent and identically distributed errors. The model is nonlinear in the sense that between two successive change-points the regression function is nonlinear into respect to parameters. It is shown via Monte Carlo simulations that its performance is competitive with that of least absolute deviations estimator and it is more efficient than the least squares estimator, particularly in the presence of the outlier points. If the number of change-points is unknown, an estimation criterion for this number is proposed. Interest of this method is that the objective function is approximated by a differentiable function and if the model contains outliers, it detects correctly the location of the change-points.     

On generalized least squares estimation of the weibull distribution

A simple estimation procedure, based on the generalized least squares method, for the parameters of the Weibull distribution is described and investigated. Through a simulation study, this estimation technique is compared with maximum likelihood estimation, ordinary least squares estimation, and Menon's estimation procedure; this comparison is based on observed relative efficiencies (that is, the ratio of the Cramer-Rao lower bound to the observed mean squared error). Simulation results are presented for samples of size 25. Among the estimators considered in this simulation study, the generalized least squares estimator was found to be the "best" estimator for the shape parameter and a close competitor to the maximum likelihood estimator of the scale parameter.     

Robust Sparse Regression with High-Breakdown Value

Penalized least squares estimators are sensitive to the influence of outliers like the ordinary least squares estimator. We propose a sparse regression estimator for robust variable selection and estimation based on a robust initial estimator. It is proven that our estimator has at least the same breakdown value as the initial estimator. Numerical examples are presented to illustrate our method.     

A Note on the Kuks-Olman Estimator

In the linear regression model the unknown parameter vector θ is supposed to vary in a known ellipsoid. Under this parameter constraint Kuks and Olman derived an estimator by demanding a minimax property. Since sometimes the Kuks-Olman estimator takes values outside of the ellipsoid a modification is proposed in the paper. It is shown that this modified variant is a least squares estimator in the restricted model.     

Series Estimation in Partially Linear In‐Slide Regression Models

Abstract. The partially linear in‐slide model (PLIM) is a useful tool to make econometric analyses and to normalize microarray data. In this article, by using series approximations and a least squares procedure, we propose a semiparametric least squares estimator (SLSE) for the parametric component and a series estimator for the non‐parametric component. Under weaker conditions than those imposed in the literature, we show that the SLSE is asymptotically normal and that the series estimator attains the optimal convergence rate of non‐parametric regression. We also investigate the estimating problem of the error variance. In addition, we propose a wild block bootstrap‐based test for the form of the non‐parametric component. Some simulation studies are conducted to illustrate the finite sample performance of the proposed procedure. An example of application on a set of economical data is also illustrated.     

空间回归模型估计中的最小二乘法

空间回归模型由于引入了空间地理信息而使得其参数估计变得复杂,因为主要采用最大似然法,致使一般人认为在空间回归模型参数估计中不存在最小二乘法。通过分析空间回归模型的参数估计技术,研究发现,最小二乘法和最大似然法分别用于估计空间回归模型的不同的参数,只有将两者结合起来才能快速有效地完成全部的参数估计。数理论证结果表明,空间回归模型参数最小二乘估计量是最佳线性无偏估计量。空间回归模型的回归参数可以在估计量为正态性的条件下而实施显著性检验,而空间效应参数则不可以用此方法进行检验。     

Properties of the mixed regression estimator when disturbances are not necessarily normal

Small-disturbance approximations for the bias vector and mean squared error matrix of the mixed regression estimator for the coefficients in a linear regression model are derived and efficiency with respect to least squares estimator is examined.     

On inference for a semiparametric partially linear regression model with serially correlated errors

The authors consider a semiparametric partially linear regression model with serially correlated errors. They propose a new way of estimating the error structure which has the advantage that it does not involve any nonparametric estimation. This allows them to develop an inference procedure consisting of a bandwidth selection method, an efficient semiparametric generalized least squares estimator of the parametric component, a goodness‐of‐fit test based on the bootstrap, and a technique for selecting significant covariates in the parametric component. They assess their approach through simulation studies and illustrate it with a concrete application.     

Efficiency of least squares estimators in the presence of spatial autocorrelation

The effect of spatial autocorrelation on inferences made using ordinary least squares estimation is considered. It is found, in some cases, that ordinary least squares estimators provide a reasonable alternative to the estimated generalized least squares estimators recommended in the spatial statistics literature. One of the most serious problems in using ordinary least squares is that the usual variance estimators are severely biased when the errors are correlated. An alternative variance estimator that adjusts for any observed correlation is proposed. The need to take autocorrelation into account in variance estimation negates much of the advantage that ordinary least squares estimation has in terms of computational simplicity     

On equality of ordinary least squares estimator,best linear unbiased estimator and best linear unbiased predictor in the general linear model

The equality of ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE) and best linear unbiased predictor (BLUP) in the general linear model with new observations is investigated through matrix rank method, some new necessary and sufficient conditions are given.