Efficiency of the generalized difference-based Liu estimators in semiparametric regression models with correlated errors

In this paper, a generalized difference-based estimator is introduced for the vector parameter β in the semiparametric regression model when the errors are correlated. A generalized difference-based Liu estimator is defined for the vector parameter β in the semiparametric regression model. Under the linear nonstochastic constraint Rβ=r, the generalized restricted difference-based Liu estimator is given. The risk function for the β?_GRD(η) associated with weighted balanced loss function is presented. The performance of the proposed estimators is evaluated by a simulated data set.     

JAMES-STEIN RULE ESTIMATORS IN LINEAR REGRESSION MODELS WITH MULTIVARIATE-t DISTRIBUTED ERROR

This paper considers estimation of β in the regression model y =Xβ+μ, where the error components in μ have the jointly multivariate Student-t distribution. A family of James-Stein type estimators (characterised by nonstochastic scalars) is presented. Sufficient conditions involving only X are given, under which these estimators are better (with respect to the risk under a general quadratic loss function) than the usual minimum variance unbiased estimator (MVUE) of β. Approximate expressions for the bias, the risk, the mean square error matrix and the variance-covariance matrix for the estimators in this family are obtained. A necessary and sufficient condition for the dominance of this family over MVUE is also given.     

Regression model with elliptically contoured errors

For the regression model y=X β+ε where the errors follow the elliptically contoured distribution, we consider the least squares, restricted least squares, preliminary test, Stein-type shrinkage and positive-rule shrinkage estimators for the regression parameters, β.

We compare the quadratic risks of the estimators to determine the relative dominance properties of the five estimators.     

Minimum average risk estimators for coefficients in linear models

We consider the problem of estimating the coefficient vector β of a linear regression model with quadratic loss function. Some biased estimators which utilize the prior information about β are considered. Also studied is the problem of estimating the parameters of an over-identified structural equation from undersized samples.     

A restricted Liu estimator for binary regression models and its application to an applied demand system

In this article, we propose a restricted Liu regression estimator (RLRE) for estimating the parameter vector, β, in the presence of multicollinearity, when the dependent variable is binary and it is suspected that β may belong to a linear subspace defined by Rβ?=?r. First, we investigate the mean squared error (MSE) properties of the new estimator and compare them with those of the restricted maximum likelihood estimator (RMLE). Then we suggest some estimators of the shrinkage parameter, and a simulation study is conducted to compare the performance of the different estimators. Finally, we show the benefit of using RLRE instead of RMLE when estimating how changes in price affect consumer demand for a specific product.     

Developing a restricted two-parameter Liu-type estimator: A comparison of restricted estimators in the binary logistic regression model

In the context of estimating regression coefficients of an ill-conditioned binary logistic regression model, we develop a new biased estimator having two parameters for estimating the regression vector parameter β when it is subjected to lie in the linear subspace restriction Hβ = h. The matrix mean squared error and mean squared error (MSE) functions of these newly defined estimators are derived. Moreover, a method to choose the two parameters is proposed. Then, the performance of the proposed estimator is compared to that of the restricted maximum likelihood estimator and some other existing estimators in the sense of MSE via a Monte Carlo simulation study. According to the simulation results, the performance of the estimators depends on the sample size, number of explanatory variables, and degree of correlation. The superiority region of our proposed estimator is identified based on the biasing parameters, numerically. It is concluded that the new estimator is superior to the others in most of the situations considered and it is recommended to the researchers.     

Estimation of scale parameters in mixture distributions

Simultaneous estimation of scale parameters is considered in mixture distributions under squared-error loss. A general class of estimators is obtained which dominates the componentwise best multiple estimators and the moment estimators. As special cases, improved estimators are obtained for the multivariate t-distribution and the p-variate Lomax distribution.     

Comparison of improved regression estimators with and without moments

For estimating the coefficients in a linear regression model, the double k–class estimators are considered and the small disturbance asymptotic approximation for their density function is obtained. Then employing the criterion of concentration probability around the true parameter values, a comparison is made between the estimators possessing finite moments and the estimators having no finite moments.     

Maximum likelihood estimator in a multi-phase random regression model

We consider a random regression model with several-fold change-points. The results for one change-point are generalized. The maximum likelihood estimator of the parameters is shown to be consistent, and the asymptotic distribution for the estimators of the coefficients is shown to be Gaussian. The estimators of the change-points converge, with n ^?1 rate, to the vector whose components are the left end points of the maximizing interval with respect to each change-point. The likelihood process is asymptotically equivalent to the sum of independent compound Poisson processes.     

Improved pretest nonparametric estimation in a multivariate regression model

Shrinkage pretest nonparametric estimation of the location parameter vector in a multivariate regression model is considered when nonsample information (NSI) about the regression parameters is available. By using the quadratic risk criterion, the dominance of the pretest estimators over the usual estimators has been investigated. We demonstrate analytically and computationally that the proposed improved pretest estimator establishes a wider dominance range for the parameter under consideration than that of the usual pretest estimator in which it is superior over the unrestricted estimator.     

A semiparametric approach for modelling multivariate nonlinear time series

In this article, a semiparametric time‐varying nonlinear vector autoregressive (NVAR) model is proposed to model nonlinear vector time series data. We consider a combination of parametric and nonparametric estimation approaches to estimate the NVAR function for both independent and dependent errors. We use the multivariate Taylor series expansion of the link function up to the second order which has a parametric framework as a representation of the nonlinear vector regression function. After the unknown parameters are estimated by the maximum likelihood estimation procedure, the obtained NVAR function is adjusted by a nonparametric diagonal matrix, where the proposed adjusted matrix is estimated by the nonparametric kernel estimator. The asymptotic consistency properties of the proposed estimators are established. Simulation studies are conducted to evaluate the performance of the proposed semiparametric method. A real data example on short‐run interest rates and long‐run interest rates of United States Treasury securities is analyzed to demonstrate the application of the proposed approach. The Canadian Journal of Statistics 47: 668–687; 2019 © 2019 Statistical Society of Canada     

Improving the estimators of the parameters of a probit regression model: A ridge regression approach

This paper considered the estimation of the regression parameters of a general probit regression model. Accordingly, we proposed five ridge regression (RR) estimators for the probit regression models for estimating the parameters (β)$(\beta )$ when the weighted design matrix is ill-conditioned and it is suspected that the parameter β$\beta$ may belong to a linear subspace defined by Hβ=h$H\beta =h$. Asymptotic properties of the estimators are studied with respect to quadratic biases, MSE matrices and quadratic risks. The regions of optimality of the proposed estimators are determined based on the quadratic risks. Some relative efficiency tables and risk graphs are provided to illustrate the numerical comparison of the estimators. We conclude that when q≥3$q\ge 3$, one would uses PRRRE; otherwise one uses PTRRE with some optimum size α$\alpha$. We also discuss the performance of the proposed estimators compare to the alternative ridge regression method due to Liu (1993).     

Asymptotics in the β-model for networks with a differentially private degree sequence

Abstract

The β-model is a natural model for characterizing the degree heterogeneity that widely exists in the network data. The estimators of the model parameters in the differentially private β-model with the denoised process have been shown to be consistent and asymptotically normal. In this paper, we show that the moment estimators of the parameters based on the differentially private degree sequence without the denoised process is consistent and asymptotically normal.     

General linear estimators under the prediction error sum of squares criterion in a linear regression model

In this paper, the notion of the general linear estimator and its modified version are introduced using the singular value decomposition theorem in the linear regression model y=X β+e to improve some classical linear estimators. The optimal selections of the biasing parameters involved are theoretically given under the prediction error sum of squares criterion. A numerical example and a simulation study are finally conducted to illustrate the superiority of the proposed estimators.     

High breakdown point robust estimators with missing data

In this paper, we propose a new procedure to estimate the distribution of a variable y when there are missing data. To compensate the presence of missing responses, it is assumed that a covariate vector x is observed and that y and x are related by means of a semi-parametric regression model. Observed residuals are combined with predicted values to estimate the missing response distribution. Once the responses distribution is consistently estimated, we can estimate any parameter defined through a continuous functional T using a plug in procedure. We prove that the proposed estimators have high breakdown point.     

The superharmonic condition for simultaneous estimation of means in exponential familles

The mean vector associated with several independent variates from the exponential subclass of Hudson (1978) is estimated under weighted squared error loss. In particular, the formal Bayes and “Stein-like” estimators of the mean vector are given. Conditions are also given under which these estimators dominate any of the “natural estimators”. Our conditions for dominance are motivated by a result of Stein (1981), who treated the N_p (θ, I) case with p ≥ 3. Stein showed that formal Bayes estimators dominate the usual estimator if the marginal density of the data is superharmonic. Our present exponential class generalization entails an elliptic differential inequality in some natural variables. Actually, we assume that each component of the data vector has a probability density function which satisfies a certain differential equation. While the densities of Hudson (1978) are particular solutions of this equation, other solutions are not of the exponential class if certain parameters are unknown. Our approach allows for the possibility of extending the parametric Stein-theory to useful nonexponential cases, but the problem of nuisance parameters is not treated here.     

Combining the unrestricted estimators into a single estimator and a simulation study on the unrestricted estimators

The purpose of this paper is to combine several regression estimators (ordinary least squares (OLS), ridge, contraction, principal components regression (PCR), Liu, r?k and r?d class estimators) into a single estimator. The conditions for the superiority of this new estimator over the PCR, the r?k class, the r?d class, β?(k, d), OLS, ridge, Liu and contraction estimators are derived by the scalar mean square error criterion and the estimators of the biasing parameters for this new estimator are examined. Also, a numerical example based on Hald data and a simulation study are used to illustrate the results.     

Target estimation for the logistic regression model

The method of target estimation developed by Cabrera and Fernholz [(1999). Target estimation for bias and mean square error reduction. The Annals of Statistics, 27(3), 1080–1104.] to reduce bias and variance is applied to logistic regression models of several parameters. The expectation functions of the maximum likelihood estimators for the coefficients in the logistic regression models of one and two parameters are analyzed and simulations are given to show a reduction in both bias and variability after targeting the maximum likelihood estimators. In addition to bias and variance reduction, it is found that targeting can also correct the skewness of the original statistic. An example based on real data is given to show the advantage of using target estimators for obtaining better confidence intervals of the corresponding parameters. The notion of the target median is also presented with some applications to the logistic models.