A generalized inverse regression estimator in multi-univariate linear calibration

A multivariate linear calibration problem, in which response variable is multivariate and explanatory variable is univariate, is considered. In this paper a class of generalized inverse regression estimators is proposed in multi-univariate linear calibration. It includes the classical estimator and the inverse regression one (or Krutchkoff estimator). For the proposed estimator we derive the expressions of bias and mean square error (MSE). Furthermore the behavior of these characteristics is investigated through an analytical method. In addition through a numerical study we confirm the existence of a generalized inverse regression estimator to improve both the classical and the inverse regression estimators on the MSE criterion.     

Robust ridge and robust Liu estimator for regression based on the LTS estimator

In the multiple linear regression analysis, the ridge regression estimator and the Liu estimator are often used to address multicollinearity. Besides multicollinearity, outliers are also a problem in the multiple linear regression analysis. We propose new biased estimators based on the least trimmed squares (LTS) ridge estimator and the LTS Liu estimator in the case of the presence of both outliers and multicollinearity. For this purpose, a simulation study is conducted in order to see the difference between the robust ridge estimator and the robust Liu estimator in terms of their effectiveness; the mean square error. In our simulations, the behavior of the new biased estimators is examined for types of outliers: X-space outlier, Y-space outlier, and X-and Y-space outlier. The results for a number of different illustrative cases are presented. This paper also provides the results for the robust ridge regression and robust Liu estimators based on a real-life data set combining the problem of multicollinearity and outliers.     

On the ridge regression estimator with sub-space restriction

In the linear regression model with elliptical errors, a shrinkage ridge estimator is proposed. In this regard, the restricted ridge regression estimator under sub-space restriction is improved by incorporating a general function which satisfies Taylor’s series expansion. Approximate quadratic risk function of the proposed shrinkage ridge estimator is evaluated in the elliptical regression model. A Monte Carlo simulation study and analysis based on a real data example are considered for performance analysis. It is evident from the numerical results that the shrinkage ridge estimator performs better than both unrestricted and restricted estimators in the multivariate t-regression model, for some specific cases.     

Estimation in a linear regression model with stochastic linear restrictions: a new two-parameter-weighted mixed estimator

The present paper considers the weighted mixed regression estimation of the coefficient vector in a linear regression model with stochastic linear restrictions binding the regression coefficients. We introduce a new two-parameter-weighted mixed estimator (TPWME) by unifying the weighted mixed estimator of Schaffrin and Toutenburg [1] and the two-parameter estimator (TPE) of Özkale and Kaç?ranlar [2]. This new estimator is a general estimator which includes the weighted mixed estimator, the TPE and the restricted two-parameter estimator (RTPE) proposed by Özkale and Kaç?ranlar [2] as special cases. Furthermore, we compare the TPWME with the weighted mixed estimator and the TPE with respect to the matrix mean square error criterion. A numerical example and a Monte Carlo simulation experiment are presented by using different estimators of the biasing parameters to illustrate some of the theoretical results.     

The relative efficiency of the restricted estimators in linear regression models

This paper deals with the problem of multicollinearity in a multiple linear regression model with linear equality restrictions. The restricted two parameter estimator which was proposed in case of multicollinearity satisfies the restrictions. The performance of the restricted two parameter estimator over the restricted least squares (RLS) estimator and the ordinary least squares (OLS) estimator is examined under the mean square error (MSE) matrix criterion when the restrictions are correct and not correct. The necessary and sufficient conditions for the restricted ridge regression, restricted Liu and restricted shrunken estimators, which are the special cases of the restricted two parameter estimator, to have a smaller MSE matrix than the RLS and the OLS estimators are derived when the restrictions hold true and do not hold true. Theoretical results are illustrated with numerical examples based on Webster, Gunst and Mason data and Gorman and Toman data. We conduct a final demonstration of the performance of the estimators by running a Monte Carlo simulation which shows that when the variance of the error term and the correlation between the explanatory variables are large, the restricted two parameter estimator performs better than the RLS estimator and the OLS estimator under the configurations examined.     

A ridge-type estimator and good prior means

Swindel (1976) introduced a modified ridge regression estimator based on prior information. A necessary and sufficient condition is derived for Swindel's proposed estimator to have lower risk than the conventional ordinary ridge regression estimator when both estimators are computed using the same value of k.     

Feasible robust estimator in restricted semiparametric regression models based on the LTS approach

Under some nonstochastic linear restrictions based on either additional information or prior knowledge in a semiparametric regression model, a family of feasible generalized robust estimators for the regression parameter is proposed. The least trimmed squares (LTS) method proposed by Rousseeuw as a highly robust regression estimator is a statistical technique for fitting a regression model based on the subset of h observations (out of n) whose least-square fit possesses the smallest sum of squared residuals. The coverage h may be set between n/2 and n. The LTS estimator involves computing the hyperplane that minimizes the sum of the smallest h squared residuals. For practical purpose, it is assumed that the covariance matrix of the error term is unknown and thus feasible estimators are replaced. Then, we develop an algorithm for the LTS estimator based on feasible methods. Through the Monte Carlo simulation studies and a real data example, performance of the feasible type of robust estimators is compared with the classical ones in restricted semiparametric regression models.     

Understanding Estimators of Treatment Effects in Regression Discontinuity Designs

In this paper, we propose two new estimators of treatment effects in regression discontinuity designs. These estimators can aid understanding of the existing estimators such as the local polynomial estimator and the partially linear estimator. The first estimator is the partially polynomial estimator which extends the partially linear estimator by further incorporating derivative differences of the conditional mean of the outcome on the two sides of the discontinuity point. This estimator is related to the local polynomial estimator by a relocalization effect. Unlike the partially linear estimator, this estimator can achieve the optimal rate of convergence even under broader regularity conditions. The second estimator is an instrumental variable estimator in the fuzzy design. This estimator will reduce to the local polynomial estimator if higher order endogeneities are neglected. We study the asymptotic properties of these two estimators and conduct simulation studies to confirm the theoretical analysis.     

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.     

A new biased estimator based on ridge estimation

In this paper we introduce a new biased estimator for the vector of parameters in a linear regression model and discuss its properties. We show that our new biased estimator is superior, in the mean square error(mse) sense, to the ordinary least squares (OLS) estimator, the ordinary ridge regression (ORR) estimator and the Liu estimator. We also compare the performance of our new biased estimator with two other special Liu-type estimators proposed in Liu (2003). We illustrate our findings with a numerical example based on the widely analysed dataset on Portland cement.     

The optimal Lp norm estimator in linear regression models

The least squares estimator is usually applied when estimating the parameters in linear regression models. As this estimator is sensitive to departures from normality in the residual distribution, several alternatives have been proposed. The L_p norm estimators is one class of such alternatives. It has been proposed that the kurtosis of the residual distribution be taken into account when a choice of estimator in the L_p norm class is made (i.e. the choice of p). In this paper, the asymtotic variance of the estimators is used as the criterion in the choice of p. It is shown that when this criterion is applied, other characteristics of the residual distribution than the kurtosis (namely moments of order p-2 and 2p-2) are important.     

Setting confidence intervals by ratio estimator

For the survey population total of a variable y when values of an auxiliary variable x are available a popular procedure is to employ the ratio estimator on drawing a simple random sample without replacement (SRSWOR) especially when the size of the sample is large. To set up a confidence interval for the total, various variance estimators are available to pair with the ratio estimator. We add a few more variance estimators studded with asymptotic design-cum-model properties. The ratio estimator is traditionally known to be appropriate when the regression of y on x is linear through the origin and the conditional variance of y given x is proportional to x. But through a numerical exercise by simulation we find the confidence intervals to fare better if the regression line deviates from the origin or if the conditional variance is disproportionate with x. Also, comparing the confidence intervals using alternative variance estimators we find our newly proposed variance estimators to yield favourably competitive results.     

On the efficiency of ratio estimator over the regression estimator

It is well known that the ratio and product estimators have the limitation of having efficiency not exceeding that of the linear regression estimator. This paper develops a new approach to ratio estimation that produces a more precise and efficient ratio estimator that is superior to the regression estimator both in efficiency and biasedness. An empirical study is given.     

On the restricted r–k class estimator and the restricted r–d class estimator in linear regression

In this article, the restricted r–k class estimator and restricted r–d class estimator are introduced, which are general estimators of the r–k class estimator by Baye and Parker [Combining ridge and principal component regression: A money demand illustration, Commun. Stat. Theory Methods 13(2) (1984), pp. 197–205] and the r–d class estimator by Kaç?ranlar and Sakall?o?lu [Combining the Liu estimator and the principal component regression estimator, Commun. Stat. Theory Methods 30(12) (2001), pp. 2699–2705], respectively. For the two cases when the restrictions are true and not true, the superiority of the restricted r–k class estimator and r–d class estimator over the restricted ridge regression estimator by Sarkar [A new estimator combining the ridge regression and the restricted least squares methods of estimation, Commun. Stat. Theory Methods 21 (1992), pp. 1987–2000] and the restricted Liu estimator by Kaç?ranlar et al. [A new biased estimator in linear regression and a detailed analysis of the widely analysed dataset on Portland cement, Sankhya - Indian J. Stat. 61B(3) (1999), pp. 443–459] are discussed with respect to the mean squared error matrix criterion. Furthermore, a Monte Carlo evaluation of the estimators is given to illustrate some of the theoretical results.     

A modified Liu-type estimator with an intercept term under mixture experiments

We consider ridge regression with an intercept term under mixture experiments. We propose a new estimator which is shown to be a modified version of the Liu-type estimator. The so-called compound covariate estimator is applied to modify the Liu-type estimator. We then derive a formula of the total mean squared error (TMSE) of the proposed estimator. It is shown that the new estimator improves upon existing estimators in terms of the TMSE, and the performance of the new estimator is invariant under the change of the intercept term. We demonstrate the new estimator using a real dataset on mixture experiments.     

Principal components regression estimator of the parameters in partially linear models

ABSTRACT

As a compromise between parametric regression and non-parametric regression models, partially linear models are frequently used in statistical modelling. This paper is concerned with the estimation of partially linear regression model in the presence of multicollinearity. Based on the profile least-squares approach, we propose a novel principal components regression (PCR) estimator for the parametric component. When some additional linear restrictions on the parametric component are available, we construct a corresponding restricted PCR estimator. Some simulations are conducted to examine the performance of our proposed estimators and the results are satisfactory. Finally, a real data example is analysed.     

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.     

Superiority of the r-k class estimator over some estimators in a misspecified linear model

ABSTRACT

In this article, we discuss the superiority of r-k class estimator over some estimators in a misspecified linear model. We derive the necessary and sufficient conditions for the superiority of the r-k class estimator over each of these estimators under the Mahalanobis loss function by the average loss criterion in the misspecified linear model.     

Improvement of generalized difference-based mixed Liu estimator in partially linear model

In this paper, a generalized difference-based mixed Liu estimator in partially linear model is presented, when it is supposed that the regression parameter may be restricted to a subspace and compare the proposed estimators in the sense of matrix mean squared error criteria. Finally a simulation study is presented to show the performance of the estimators.