Corrected proof of the result of 'A prediction error property of the Lasso estimator and its generalization' by Huang (2003)

The Lasso achieves variance reduction and variable selection by solving an ?₁‐regularized least squares problem. Huang (2003) claims that ‘there always exists an interval of regularization parameter values such that the corresponding mean squared prediction error for the Lasso estimator is smaller than for the ordinary least square estimator’. This result is correct. However, its proof in Huang (2003) is not. This paper presents a corrected proof of the claim, which exposes and uses some interesting fundamental properties of the Lasso.     

Adaptive Lasso Variable Selection for the Accelerated Failure Models

This article considers the adaptive lasso procedure for the accelerated failure time model with multiple covariates based on weighted least squares method, which uses Kaplan-Meier weights to account for censoring. The adaptive lasso method can complete the variable selection and model estimation simultaneously. Under some mild conditions, the estimator is shown to have sparse and oracle properties. We use Bayesian Information Criterion (BIC) for tuning parameter selection, and a bootstrap variance approach for standard error. Simulation studies and two real data examples are carried out to investigate the performance of the proposed method.     

On the equality of usual and amemiya's partially generalized least squares estimator

Equivalent conditions are derived for the equality of GLSE (generalized least squares estimator) and partially GLSE (PGLSE), the latter introduced by Amemiya (1983). By adopting a more general approach the ordinary least squares estimator (OLSE) can shown to be a special PGLSE. Furthcrmore, linearly restricted estimators proposed by Balestra (1983) are investigated in this context. To facilitate the comparison of estimators extensive use of oblique and orthogonal projectors is made.     

On the Principal Component Liu-type Estimator in Linear Regression

In this article, we present a principal component Liu-type estimator (LTE) by combining the principal component regression (PCR) and LTE to deal with the multicollinearity problem. The superiority of the new estimator over the PCR estimator, the ordinary least squares estimator (OLSE) and the LTE are studied under the mean squared error matrix. The selection of the tuning parameter in the proposed estimator is also discussed. Finally, a numerical example is given to explain our theoretical results.     

Estimation and Prediction in the Presence of Spatial Confounding for Spatial Linear Models

In studies that produce data with spatial structure, it is common that covariates of interest vary spatially in addition to the error. Because of this, the error and covariate are often correlated. When this occurs, it is difficult to distinguish the covariate effect from residual spatial variation. In an i.i.d. normal error setting, it is well known that this type of correlation produces biased coefficient estimates, but predictions remain unbiased. In a spatial setting, recent studies have shown that coefficient estimates remain biased, but spatial prediction has not been addressed. The purpose of this paper is to provide a more detailed study of coefficient estimation from spatial models when covariate and error are correlated and then begin a formal study regarding spatial prediction. This is carried out by investigating properties of the generalized least squares estimator and the best linear unbiased predictor when a spatial random effect and a covariate are jointly modelled. Under this setup, we demonstrate that the mean squared prediction error is possibly reduced when covariate and error are correlated.     

Two Stochastic Restricted Principal Components Regression Estimator in Linear Regression

In this article, we propose two stochastic restricted principal components regression estimator by combining the approach followed in obtaining the ordinary mixed estimator and the principal components regression estimator in linear regression model. The performance of the two new estimators in terms of matrix MSE criterion is studied. We also give an example and a Monte Carlo simulation to show the theoretical results.     

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.     

Prediction of response values in linear regression models from replicated experiments

This paper considers the problem of prediction in a linear regression model when data sets are available from replicated experiments. Pooling the data sets for the estimation of regression parameters, we present three predictors — one arising from the least squares method and two stemming from Stein-rule method. Efficiency properties of these predictors are discussed when they are used to predict actual and average values of response variable within/outside the sample. Received: November 17, 1999; revised version: August 10, 2000     

Linearized Ridge Regression Estimator in Linear Regression

In this article, we aim to study the linearized ridge regression (LRR) estimator in a linear regression model motivated by the work of Liu (1993). The LRR estimator and the two types of generalized Liu estimators are investigated under the PRESS criterion. The method of obtaining the optimal generalized ridge regression (GRR) estimator is derived from the optimal LRR estimator. We apply the Hald data as a numerical example and then make a simulation study to show the main results. It is concluded that the idea of transforming the GRR estimator as a complicated function of the biasing parameters to a linearized version should be paid more attention in the future.     

Least Trimmed Squares Estimator in the Errors-in-Variables Model

We propose a robust estimator in the errors-in-variables model using the least trimmed squares estimator. We call this estimator the orthogonal least trimmed squares (OLTS) estimator. We show that the OLTS estimator has the high breakdown point and appropriate equivariance properties. We develop an algorithm for the OLTS estimate. Simulations are performed to compare the efficiencies of the OLTS estimates with the total least squares (TLS) estimates and a numerical example is given to illustrate the effectiveness of the estimate.     

Confidence regions based on inefficient estimators

We consider an unbiased estimator of a function of mean value parameters, which is not efficient. This inefficient estimator is correlated with a residual vector. Thus, if a unit dispersion is unknown, it is impossible to determine the correct confidence region for a function of mean value parameters via a standard estimator of an unknown dispersion with the exception of the case when the ordinary least squares (OLS) estimator is considered in a model with a special covariance structure such that the OLS and the generalized least squares (GLS) estimator are the same, that is the OLS estimator is efficient. Two different estimators of a unit dispersion independent of an inefficient estimator are derived in a singular linear statistical model. Their quality was verified by simulations for several types of experimental designs. Two new estimators of the unit dispersion were compared with the standard estimators based on the GLS and the OLS estimators of the function of the mean value parameters. The OLS estimator was considered in the incorrect model with a different covariance matrix such that the originally inefficient estimator became efficient. The numerical examples led to a slightly surprising result which seems to be due to data behaviour. An example from geodetic practice is presented in the paper.     

Two-Stage Welsh's Trimmed Mean for the Simultaneous Equations Model

This paper discusses the large sample theory of the two-stage Welsh's trimmed mean for the limited information simultaneous equations model. Besides having asymptotic normality, this trimmed mean, as the two-stage least squares estimator, is a generalized least squares estimator. It also acts as a robust Aitken estimator for the simultaneous equations model. Examples illustrate real data analysis and large sample inferences based on this trimmed mean.     

The Equality of the Ordinary Least Squares Estimator and the Best Linear Unbiased Estimator

It is well known that the ordinary least squares estimator of Xβ in the general linear model E _y = Xβ, cov y = σ² V, can be the best linear unbiased estimator even if V is not a multiple of the identity matrix. This article presents, in a historical perspective, the development of the several conditions for the ordinary least squares estimator to be best linear unbiased. Various characterizations of these conditions, using generalized inverses and orthogonal projectors, along with several examples, are also given. In addition, a complete set of references is provided.     

Book Reviews

Books reviewed:
Philip Hans Franses & Dick van Dijk, Non-linear Time Series Models in Empirical Finance
Herbert Spirer, Louise Spirer & A.J. Jaffe, Misused Statistics
Deborah J. Bennett, Randomness
C.E. Linneborg, Data Analysis by Resampling: Concepts and Applications
I. Clark and W.V. Harper, Practical Geostatistics 2000     

放回抽样下HT估计量的性质及应用

放回抽样下传统的估计方法是采用Hansen-Hurwitz估计。而放回抽样下HH估计量并不是一致最小方差无偏估计,本文提出了另一种估计方法,即采用Horvitz-Thompson估计,并论证了放回抽样下HT估计量的三条定理,及与HH估计量的比较。然后以放回简单随机抽样和PPS抽样为例,通过理论公式、计算机模拟以及具体案例,进行更具体的分析。说明在一定条件下,HT估计量相对更优。在实际应用中,本文也提出了通过比较方差估计作为选取估计量的准则。     

An approximation of the distribution function of the LIML identifiability test statistic using the method of moments

This paper contains an application of the asymptotic expansion of a pFp() function to a problem encountered in econometrics. In particular we consider an approximation of the distribution function of the limited information maximum likelihood (LIML) identifiability test statistic using the method of moments. An expression for the Sth order asymptotic approximation of the moments of the LIML identifiability test statistic is derived and tabulated. The exact distribution function of the test statistic is approximated by a member of the class of F (variance ratio) distribution functions having the same first two integer moments. Some tabulations of the approximating distribution function are included.     

An asymptotic theory for semiparametric generalized least squares estimation in partially linear regression models

Consider a partially linear regression model with an unknown vector parameter β, an unknown functiong(·), and unknown heteroscedastic error variances. In this paper we develop an asymptotic semiparametric generalized least squares estimation theory under some weak moment conditions. These moment conditions are satisfied by many of the error distributions encountered in practice, and our theory does not require the number of replications to go to infinity.     

The General Expressions for the Moments of the Stochastic Shrinkage Parameters of the Liu Type Estimator

ABSTRACT

One of the problems with the Liu estimator is the appropriate value for the unknown biasing parameter d. In this article we consider the optimum value for d and give upper bound for the expected value of the estimator of this biasing parameter. We also derive the general expressions for the moments of the stochastic shrinkage parameters of the Liu estimator and the generalized Liu estimator. Numerical calculations are carried out to illustrate the behavior of the mean and variance of the biasing parameter. Also, a numerical example is given to illustrate the effect of the biasing parameter d, on the mean square error of the Liu estimator.