S. Ejaz Ahmed

Improving the performance of kurtosis estimator

In this communication, sample measures of kurtosis adapted by various software packages are compared for data from normal and non-normal populations. Further, two improved estimators of population kurtosis are proposed and their performance is compared with the currently used measures. The suggested estimators have considerably lower mean squared error (MSE) for various sampling designs in our simulation study. Two empirical examples are given to illustrate the usefulness of suggested estimators in practice.

Penalty, Shrinkage and Pretest Strategies

The objective of this book is to compare the statistical properties of penalty and non-penalty estimation strategies for some popular models. Specifically, it considers the full model, submodel, penalty, pretest and shrinkage estimation techniques for three regression models before presenting the asymptotic properties of the non-penalty estimators and their asymptotic distributional efficiency comparisons. Further, the risk properties of the non-penalty estimators and penalty estimators are explored through a Monte Carlo simulation study. Showcasing examples based on real datasets, the book will be useful for students and applied researchers in a host of applied fields.The books level of presentation and style make it accessible to a broad audience. It offers clear, succinct expositions of each estimation strategy. More importantly, it clearly describes how to use each estimation strategy for the problem at hand. The book is largely self-contained, as are the individual chapters, so that anyone interested in a particular topic or area of application may read only that specific chapter. The book is specially designed for graduate students who want to understand the foundations and concepts underlying penalty and non-penalty estimation and its applications. It is well-suited as a textbook for senior undergraduate and graduate courses surveying penalty and non-penalty estimation strategies, and can also be used as a reference book for a host of related subjects, including courses on meta-analysis. Professional statisticians will find this book to be a valuable reference work, since nearly all chapters are self-contained.

Absolute penalty and shrinkage estimation in partially linear models

In the context of a partially linear regression model, shrinkage semiparametric estimation is considered based on the Stein-rule. In this framework, the coefficient vector is partitioned into two sub-vectors: the first sub-vector gives the coefficients of interest, i.e., main effects (for example, treatment effects), and the second sub-vector is for variables that may or may not need to be controlled. When estimating the first sub-vector, the best estimate may be obtained using either the full model that includes both sub-vectors, or the reduced model which leaves out the second sub-vector. It is demonstrated that shrinkage estimators which combine two semiparametric estimators computed for the full model and the reduced model outperform the semiparametric estimator for the full model. Using the semiparametric estimate for the reduced model is best when the second sub-vector is the null vector, but this estimator suffers seriously from bias otherwise. The relative dominance picture of suggested estimators is investigated. In particular, suitability of estimating the nonparametric component based on the B-spline basis function is explored. Further, the performance of the proposed estimators is compared with an absolute penalty estimator through Monte Carlo simulation. Lasso and adaptive lasso were implemented for simultaneous model selection and parameter estimation. A real data example is given to compare the proposed estimators with lasso and adaptive lasso estimators.

Estimation of parameters in the growth curve model via an outer product least squares approach for covariance

In this paper, we propose a framework of outer product least squares for covariance (COPLS) to directly estimate covariance in the growth curve model based on an analogy, between the outer product of a data vector and covariance of a random vector, and the ordinary least squares technique. The COPLS estimator of covariance has an explicit expression and is shown to have the following properties: (1) following a linear transformation of two independent Wishart distribution for a normal error matrix; (2) having asymptotic normality for a nonnormal error matrix; and (3) having unbiasedness and invariance under a linear transformation group. And, a corresponding two-stage generalized least squares (GLS) estimator for the regression coefficient matrix in the model is obtained and its asymptotic normality is investigated. Simulation studies confirm that the COPLS estimator and the two-stage GLS estimator of the regression coefficient matrix are satisfying competitors with some evident merits to the existing maximum likelihood estimator in finite samples.

An application of shrinkage estimation to the nonlinear regression model

Various large sample estimation techniques in a nonlinear regression model are presented. These estimators are based around preliminary tests of significance, and the James-Stein rule. The properties of these estimators are studied when estimating regression coefficients in the multiple nonlinear regression model when it is a priori suspected that the coefficients may be restricted to a subspace. A simulation based on a demand for money model shows the superiority of the positive-part shrinkage estimator, in terms of standard measures of asymptotic distributional quadratic bias and risk measures, over a range of economically meaningful parameter values. Further work remains in analysing the use of these estimators in economic applications, relative to the inferential approach which is best to use in these circumstances.

Shrinkage and penalized estimation in semi-parametric models with multicollinear data

ABSTRACT In this paper, we consider estimation techniques based on ridge regression when the matrix appears to be ill-conditioned in the partially linear model using kernel smoothing. Furthermore, we consider that the coefficients can be partitioned as where is the coefficient vector for main effects, and is the vector for ‘nuisance’ effects. We are essentially interested in the estimation of when it is reasonable to suspect that is close to zero. We suggest ridge pretest, ridge shrinkage and ridge positive shrinkage estimators for the above semi-parametric model, and compare its performance with some penalty estimators. In particular, suitability of estimating the nonparametric component based on the kernel smoothing basis function is also explored. Monte Carlo simulation study is used to compare the relative efficiency of proposed estimators, and a real data example is presented to illustrate the usefulness of the suggested methods. Moreover, the asymptotic properties of the proposed estimators are obtained.

Model selection and parameter estimation of a multinomial logistic regression model

In the multinomial regression model, we consider the methodology for simultaneous model selection and parameter estimation by using the shrinkage and LASSO (least absolute shrinkage and selection operation) [R. Tibshirani, Regression shrinkage and selection via the LASSO, J. R. Statist. Soc. Ser. B 58 (1996), pp. 267–288] strategies. The shrinkage estimators (SEs) provide significant improvement over their classical counterparts in the case where some of the predictors may or may not be active for the response of interest. The asymptotic properties of the SEs are developed using the notion of asymptotic distributional risk. We then compare the relative performance of the LASSO estimator with two SEs in terms of simulated relative efficiency. A simulation study shows that the shrinkage and LASSO estimators dominate the full model estimator. Further, both SEs perform better than the LASSO estimators when there are many inactive predictors in the model. A real-life data set is used to illustrate the suggested shrinkage and LASSO estimators.

Shrinkage estimation for the regression parameter matrix in multivariate regression model

In this paper, we consider optimal shrinkage estimation strategies for the regression parameter matrix in multivariate multiple regression models. It uses the context of two competing models, where one model includes all predictors and the other restricts variable coefficients to a candidate linear subspace based on prior information. In this scenario, we suggest a shrinkage estimation strategy for the targeted regression parameter matrix. The goal of this paper is to critically examine the relative performances of the shrinkage estimators in the direction of the subspace and candidate subspace restricted-type estimators. We derive expression for bias and quadratic risk of the suggested estimators. Furthermore, we appraise the relative performance of the suggested estimators with the classical estimators. Our analytical and numerical results show that the proposed shrinkage estimators overall perform the best. The methods are also applied on a real data set for illustrative purposes.

Stabilizing the Performance of Kurtosis Estimator of Multivariate Data

The estimation of the kurtosis parameter of the underlying distribution plays a central role in many statistical applications. The central theme of the article is to improve the estimation of the kurtosis parameter using a priori information. More specifically, we consider the problem of estimating kurtosis parameter of a multivariate population when some prior information regarding the the parameter is available. The rationale is that the sample estimator of the kurtosis parameter has a large estimation error. In this situation we consider shrinkage and pretest estimation methodologies and reappraise their statistical properties. The estimation based on these strategies yield relatively smaller estimation error in comparison with the sample estimator in the candidate subspace. A large sample theory of the suggested estimators are developed and compared. The results demonstrate that suggested estimators outperform the estimator based on the sample data only in the candidate subspace. In an effort to appreciate the relative behavior of the estimators in a finite sample scenario, a Monte-carlo simulation study is planned and performed. The result of simulation study strongly corroborates the asymptotic result. To illustrate the application of the estimators, some example are showcased based on recently published data.

Data-Based Adaptive Estimation in an Investment Model

We consider improved estimation strategies for the parameters in a capital asset pricing model under a general linear constraint; we suggest candidate subspace, a preliminary test, and shrinkage estimators. We develop a large sample theory for the estimators that include derivation of asymptotic bias and asymptotic distributional risk of the suggested estimators. The asymptotic results demonstrate the superiority of the suggested estimation technique. A simulation study shows that the method suggested here has sound finite sample properties and strongly corroborates with the theoretical result of the article. A data example is also presented to illustrate the suggested estimation strategies.