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.     

Another look at Huber's estimator: A new minimax estimator in regression with stochastically bounded noise

Huber's estimator has had a long lasting impact, particularly on robust statistics. It is well known that under certain conditions, Huber's estimator is asymptotically minimax. A moderate generalization in rederiving Huber's estimator shows that Huber's estimator is not the only choice. We develop an alternative asymptotic minimax estimator and name it regression with stochastically bounded noise (RSBN). Simulations demonstrate that RSBN is slightly better in performance, although it is unclear how to justify such an improvement theoretically. We propose two numerical solutions: an iterative numerical solution, which is extremely easy to implement and is based on the proximal point method; and a solution by applying state-of-the-art nonlinear optimization software packages, e.g., SNOPT. Contribution: the generalization of the variational approach is interesting and should be useful in deriving other asymptotic minimax estimators in other problems.     

Some General Comparative Points on Chao's and Zelterman's Estimators of the Population Size

Abstract. Two simple and frequently used capture–recapture estimates of the population size are compared: Chao's lower‐bound estimate and Zelterman's estimate allowing for contaminated distributions. In the Poisson case it is shown that if there are only counts of ones and twos, the estimator of Zelterman is always bounded above by Chao's estimator. If counts larger than two exist, the estimator of Zelterman is becoming larger than that of Chao's, if only the ratio of the frequencies of counts of twos and ones is small enough. A similar analysis is provided for the binomial case. For a two‐component mixture of Poisson distributions the asymptotic bias of both estimators is derived and it is shown that the Zelterman estimator can experience large overestimation bias. A modified Zelterman estimator is suggested and also the bias‐corrected version of Chao's estimator is considered. All four estimators are compared in a simulation study.     

Regularization through variable selection and conditional MLE with application to classification in high dimensions

It is often the case that high-dimensional data consist of only a few informative components. Standard statistical modeling and estimation in such a situation is prone to inaccuracies due to overfitting, unless regularization methods are practiced. In the context of classification, we propose a class of regularization methods through shrinkage estimators. The shrinkage is based on variable selection coupled with conditional maximum likelihood. Using Stein's unbiased estimator of the risk, we derive an estimator for the optimal shrinkage method within a certain class. A comparison of the optimal shrinkage methods in a classification context, with the optimal shrinkage method when estimating a mean vector under a squared loss, is given. The latter problem is extensively studied, but it seems that the results of those studies are not completely relevant for classification. We demonstrate and examine our method on simulated data and compare it to feature annealed independence rule and Fisher's rule.     

Penalized Empirical Likelihood via Bridge Estimator in Cox's Proportional Hazard Model

We propose the penalized empirical likelihood method via bridge estimator in Cox's proportional hazard model for parameter estimation and variable selection. Under reasonable conditions, we show that penalized empirical likelihood in Cox's proportional hazard model has oracle property. A penalized empirical likelihood ratio for the vector of regression coefficients is defined and its limiting distribution is a chi-square distributions. The advantage of penalized empirical likelihood as a nonparametric likelihood approach is illustrated in testing hypothesis and constructing confidence sets. The method is illustrated by extensive simulation studies and a real example.     

Performance of Kibria's Method for the Heteroscedastic Ridge Regression Model: Some Monte Carlo Evidence

It is common for a linear regression model that the error terms display some form of heteroscedasticity and at the same time, the regressors are also linearly correlated. Both of these problems have serious impact on the ordinary least squares (OLS) estimates. In the presence of heteroscedasticity, the OLS estimator becomes inefficient and the similar adverse impact can also be found on the ridge regression estimator that is alternatively used to cope with the problem of multicollinearity. In the available literature, the adaptive estimator has been established to be more efficient than the OLS estimator when there is heteroscedasticity of unknown form. The present article proposes the similar adaptation for the ridge regression setting with an attempt to have more efficient estimator. Our numerical results, based on the Monte Carlo simulations, provide very attractive performance of the proposed estimator in terms of efficiency. Three different existing methods have been used for the selection of biasing parameter. Moreover, three different distributions of the error term have been studied to evaluate the proposed estimator and these are normal, Student's t and F distribution.     

Proportion estimation in mixtures with covariates

Linear maps of a single unclassified observation are used to estimate the mixing proportion in a mixture of two populations with homogeneous variances in the presence of covariates. with complete knowledge of the parameters of the individual populations, the linear map for which the estimator is unbiased and has minimum variance amongst all similar estimators can be determined. Plug-in estimator based on independent training samples from the component populations can be constructed and is asymptotically equivalent to Cochran's classification statistic V* for covariate classification; see Memon and Okamoto (1970). Under normality assumptions, asymptotic expansion of the distribution of the plug-in estimator is available. In the absence of covariates, our estimator reduces to that suggested by Walker (1980) who has investigated the problem based on information on large unclassified samples from a mixture of two populations with heterogeneous variances. In contrast, distribution of Walker's estimator seems intractable in moderate sample sizes even with normality assumption.     

Bayesian estimation of renewal function for inverse Gaussian renewal process

Two approximation methods are used to obtain the Bayes estimate for the renewal function of inverse Gaussian renewal process. Both approximations use a gamma-type conditional prior for the location parameter, a non-informative marginal prior for the shape parameter, and a squared error loss function. Simulations compare the accuracy of the estimators and indicate that the Tieney and Kadane (T–K)-based estimator out performs Maximum Likelihood (ML)- and Lindley (L)-based estimator. Computations for the T–K-based Bayes estimate employ the generalized Newton's method as well as a recent modified Newton's method with cubic convergence to maximize modified likelihood functions. The program is available from the author.     

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.     

Liu-Type Logistic Estimator

It is known that multicollinearity inflates the variance of the maximum likelihood estimator in logistic regression. Especially, if the primary interest is in the coefficients, the impact of collinearity can be very serious. To deal with collinearity, a ridge estimator was proposed by Schaefer et al. The primary interest of this article is to introduce a Liu-type estimator that had a smaller total mean squared error (MSE) than the Schaefer's ridge estimator under certain conditions. Simulation studies were conducted that evaluated the performance of this estimator. Furthermore, the proposed estimator was applied to a real-life dataset.     

Grenander functionals and Cauchy's formula

Let ${\widehat{f}}_n$ be the nonparametric maximum likelihood estimator of a decreasing density. Grenander characterized this as the left‐continuous slope of the least concave majorant of the empirical distribution function. For a sample from the uniform distribution, the asymptotic distribution of the L₂‐distance of the Grenander estimator to the uniform density was derived in an article by Groeneboom and Pyke by using a representation of the Grenander estimator in terms of conditioned Poisson and gamma random variables. This representation was also used in an article by Groeneboom and Lopuhaä to prove a central limit result of Sparre Andersen on the number of jumps of the Grenander estimator. Here we extend this to the proof of the main result on the L₂‐distance of the Grenander estimator to the uniform density and also prove a similar asymptotic normality results for the entropy functional. Cauchy's formula and saddle point methods are the main tools in our development.     

Comparison of estimators of the location parameter using pitman's closeness criterion

Necessary and sufficient conditions for a linear estimator to dominate another linear estimator of a location parameter under the Pitman's criterion of comparison are discussed. Consequently it is demonstrated that a linear biased estimator can not dominate a linear unbiased estimator under Pitman's criterion and that the sample mean is the Closest Linear Unbiased Estimator (CLUE). It is also shown that the ridge regression estimator with a known biasing constant can not dominate the ordinary least squares estimator. If an estimator δdominates an estimator δin the average loss sense then sufficient conditions are obtained under which δis also preferred over δunder Pitman's criterion. Further we obtain sufficient conditions under which preference under the Pitman's criterion will lead to preference under the mean squared error sense.     

Empirical distribution function under heteroscedasticity

Neglecting heteroscedasticity of error terms may imply the wrong identification of a regression model (see appendix). Employment of (heteroscedasticity resistent) White's estimator of covariance matrix of estimates of regression coefficients may lead to the correct decision about the significance of individual explanatory variables under heteroscedasticity. However, White's estimator of covariance matrix was established for least squares (LS)-regression analysis (in the case when error terms are normally distributed, LS- and maximum likelihood (ML)-analysis coincide and hence then White's estimate of covariance matrix is available for ML-regression analysis, tool). To establish White's-type estimate for another estimator of regression coefficients requires Bahadur representation of the estimator in question, under heteroscedasticity of error terms. The derivation of Bahadur representation for other (robust) estimators requires some tools. As the key too proved to be a tight approximation of the empirical distribution function (d.f.) of residuals by the theoretical d.f. of the error terms of the regression model. We need the approximation to be uniform in the argument of d.f. as well as in regression coefficients. The present paper offers this approximation for the situation when the error terms are heteroscedastic.     

Adjustive Liu-Type Estimators in Linear Regression Models

In this article, we aim to put forward the notion of adjustive Liu-type estimator (ALTE) in the linear regression model. First, the explicit expression of the optimal selection of the adjustive factors is derived under the PRESS criterion through matrix techniques. Then, the results are applied to the dataset on Portland cement. Moreover, to select biasing parameters from the theoretical point of view, we extend ALTE to the generalized version (GALTE) and obtained the optimal ones. The results of the Portland cement data show that ALTE's and GALTE's can substantially improve the ordinary least squares estimator and Liu-type estimators.     

Estimation of a multivariate normal covariance matrix under a certain structure

We consider the estimation of Σ of the p-dimensional normal distribution N_p (0, Σ) when Σ?=?θ₀ I_p ?+?θ₁ aa′, where a is an unknown p-dimensional normalized vector and θ₀?>?0, θ₁?≥?0 are also unknown. First, we derive the restricted maximum likelihood (REML) estimator. Second, we propose a new estimator, which dominates the REML estimator with respect to Stein's loss function. Finally, we carry out Monte Carlo simulation to investigate the magnitude of the new estimator's superiority.     

Comparison of estimators for heteroskedastic models

Ordinary least squares (OLS) yield inefficient parameter estimates and inconsistent estimates of the covariance matrix in case of heteroskedastic errors. Robinson's adaptive estimator and the Cragg estimator avoid any explicit parameterization of heteroskedasticity, and reduce the danger of misspecification. A small Monte Carlo experiment is performed to compare the behavior of the adaptive estimator with the performance of the Cragg estimator. The Monte Carlo experiment includes simulations of the Generalized Least Squares (GLS) estimator. Indeed, an interesting question is how more sophisticated techniques, like the adaptive estimator, compare with GLS when the latter relies on an incorrect specification of the heteroskedastic process. It turns out that the regression parameters, when estimated adaptively, display small mean squared errors and great efficiency in case of medium or high heteroskedasticity. The covariance matrix, instead, is better estimated by the Cragg estimator or by GLS based on a misspecified error term, since the adaptive estimator overpredicts the standard errors of the regression parameters.     

Using Liu-Type Estimator to Combat Collinearity

Abstract

Linear regression model and least squares method are widely used in many fields of natural and social sciences. In the presence of collinearity, the least squares estimator is unstable and often gives misleading information. Ridge regression is the most common method to overcome this problem. We find that when there exists severe collinearity, the shrinkage parameter selected by existing methods for ridge regression may not fully address the ill conditioning problem. To solve this problem, we propose a new two-parameter estimator. We show using both theoretic results and simulation that our new estimator has two advantages over ridge regression. First, our estimator has less mean squared error (MSE). Second, our estimator can fully address the ill conditioning problem. A numerical example from literature is used to illustrate the results.     

Conditional inference in finite population sampling under a calibration model

Several estimators, including the classical and the regression estimators of finite population mean, are compared, both theoretically and empirically, under a calibration model, where the dependent variable(y), and not the independent variable(x), can be observed for all units of the finite population. It is shown asymptotically that when conditioned on x, the bias of the classical estimator may be much smaller than that of the regression estimators; whereas when conditioned on y, the regression estimator may have much smaller conditional bias than the classical estimator. Since all the y's(not x's) can be observed, it seems appropriate to make comparison under the conditional distribution of each estimator with y fixed. In this case, the regression estimator has smaller variance, smaller conditional bias, and the conditional coverage probability closer to its nominal level     

A Class of s–K Type Principal Components Estimators in the Linear Model

In this article, we introduce a new class of estimators called the s–K type principal components estimators to combat multicollinearity, which include the principal components regression (PCR) estimator, the r–k estimator and the s–K estimator as special cases. Necessary and sufficient conditions for the superiority of the new estimator over the PCR estimator, the r–k estimator and the s–K estimator are derived in the sense of the mean squared error matrix criterion. A Monte Carlo simulation study and a numerical example are given to illustrate the performance of the proposed estimator.