Superiority of the r–k Class Estimator Over Some Estimators In A Linear Model

In regression analysis, to overcome the problem of multicollinearity, the r ? k class estimator is proposed as an alternative to the ordinary least squares estimator which is a general estimator including the ordinary ridge regression estimator, the principal components regression estimator and the ordinary least squares estimator. In this article, 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. Then, we compare these estimators with each other using the same criterion. Also, we suggest to test to verify if these conditions are indeed satisfied. Finally, a numerical example and a Monte Carlo simulation are done to illustrate the theoretical results.     

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.     

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.     

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.     

MSE performance of the weighted average estimators consisting of shrinkage estimators when each individual regression coefficient is estimated

In this paper, we analytically derive the exact formula for the mean squared error (MSE) of two weighted average (WA) estimators for each individual regression coefficient. Further, we execute numerical evaluations to investigate small sample properties of the WA estimators, and compare the MSE performance of the WA estimators with the other shrinkage estimators and the usual OLS estimator. Our numerical results show that (1) the WA estimators have smaller MSE than the other shrinkage estimators and the OLS estimator over a wide region of parameter space; (2) the range where the relative MSE of the WA estimator is smaller than that of the OLS estimator gets narrower as the number of explanatory variables k increases.     

General procedure for estimating the population mean using ranked set sampling

In this paper, we suggest a class of estimators for estimating the population mean ? of the study variable Y using information on X?, the population mean of the auxiliary variable X using ranked set sampling envisaged by McIntyre [A method of unbiased selective sampling using ranked sets, Aust. J. Agric. Res. 3 (1952), pp. 385–390] and developed by Takahasi and Wakimoto [On unbiased estimates of the population mean based on the sample stratified by means of ordering, Ann. Inst. Statist. Math. 20 (1968), pp. 1–31]. The estimator reported by Kadilar et al. [Ratio estimator for the population mean using ranked set sampling, Statist. Papers 50 (2009), pp. 301–309] is identified as a member of the proposed class of estimators. The bias and the mean-squared error (MSE) of the proposed class of estimators are obtained. An asymptotically optimum estimator in the class is identified with its MSE formulae. To judge the merits of the suggested class of estimators over others, an empirical study is carried out.     

PMSE dominance of the positive-part shrinkage estimator in a regression model with proxy variables

Consider a linear regression model with some relevant regressors are unobservable. In such a situation, we estimate the model by using the proxy variables as regressors or by simply omitting the relevant regressors. In this paper, we derive the explicit formula of predictive mean squared error (PMSE) of a general family of shrinkage estimators of regression coefficients. It is shown analytically that the positive-part shrinkage estimator dominates the ordinary shrinkage estimator even when proxy variables are used in place of the unobserved variables. Also, as an example, our result is applied to the double k-class estimator proposed by Ullah and Ullah (Double k-class estimators of coefficients in linear regression. Econometrica. 1978;46:705–722). Our numerical results show that the positive-part double k-class estimator with proxy variables has preferable PMSE performance.     

Comparisons of the <Emphasis Type="Italic">r</Emphasis> − <Emphasis Type="Italic">k</Emphasis> class estimator to the ordinary least squares estimator under the Pitman’s closeness criterion

In the presence of multicollinearity, the r − k class estimator is proposed as an alternative to the ordinary least squares (OLS) estimator which is a general estimator including the ordinary ridge regression (ORR), the principal components regression (PCR) and the OLS estimators. Comparison of competing estimators of a parameter in the sense of mean square error (MSE) criterion is of central interest. An alternative criterion to the MSE criterion is the Pitman’s (1937) closeness (PC) criterion. In this paper, we compare the r − k class estimator to the OLS estimator in terms of PC criterion so that we can get the comparison of the ORR estimator to the OLS estimator under the PC criterion which was done by Mason et al. (1990) and also the comparison of the PCR estimator to the OLS estimator by means of the PC criterion which was done by Lin and Wei (2002).     

Estimators of location based on Kolmogorov-Smirnov-type statistics

Two classes of estimators of a location parameter ø₀ are proposed, based on a nonnegative functional H₁^* of the pair (D₁^ø_N, G^ø_N), where and where F_N is the sample distribution function. The estimators of the first class are defined as a value of ø minimizing H₁^*; the estimators of the second class are linearized versions of those of the first. The asymptotic distribution of the estimators is derived, and it is shown that the Kolmogorov-Smirnov statistic, the signed linear rank statistics, and the Cramérvon Mises statistics are special cases of such functionals H₁^*;. These estimators are closely related to the estimators of a shift in the two-sample case, proposed and studied by Boulanger in B2 (pp. 271–284).     

r-d Class Estimator Under Misspecification

Omission of some relevant explanatory variables and multicollinearity in regression models are very serious problems in applied works. There are some papers examining the multicollinearity and misspecification which is due to omission of some relevant explanatory variables, concurrently. To remedy the problem of multicollinearity, Kaç?ranlar and Sakall?o?lu (2001 Kaç?ranlar, S., Sakall?o?lu, S. (2001). Combining the Liu estimator and the principal component regression estimator. Commun. Stat. Theory Methods. 30:2699–2705.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) proposed the r-d class estimator that includes the ordinary least squares, principal components regression, and Liu estimators as special cases. The aim of this paper is to examine the performance of the r-d class estimator in misspecificied linear models.     

Optimal estimation for finite population mean in two-phase sampling

Using two-phase sampling scheme, we propose a general class of estimators for finite population mean. This class depends on the sample means and variances of two auxiliary variables. The minimum variance bound for any estimator in the class is provided (up to terms of ordern ⁻¹). It is also proved that there exists at least a chain regression type estimator which reaches this minimum. Finally, it is shown that other proposed estimators can reach the minimum variance bound, i.e. the optimal estimator is not unique.     

A REGRESSION APPROACH TO THE ESTIMATION OF THE FINITE POPULATION MEAN IN THE PRESENCE OF NON‐RESPONSE

This paper presents various estimators for estimating the population mean of the study variable y using information on the auxiliary variable x in the presence of non‐response. Properties of the suggested estimators are studied and compared with those of existing estimators. It is shown that the estimators suggested in this paper are among the best of all the estimators considered. An empirical study is carried out to demonstrate the performance of the suggested estimators and of others, and it is found that the empirical results support the theoretical study.     

New types of shrinkage estimators of Poisson means under the normalized squared error loss

In estimating p( ? 2) independent Poisson means, Clevenson and Zidek (1975) have proposed a class of estimators that shrink the unbiased estimator to the origin and dominate the unbiased one under the normalized squared error loss. This class of estimators was subsequently enlarged in several directions. This article deals with the problem and proposes new classes of dominating estimators using prior information pertinently. Dominance is shown by partitioning the sample space into disjoint subsets and averaging the loss difference over each subset. Estimation of several Poisson mean vectors is also discussed. Further, simultaneous estimation of Poisson means under order restriction is treated and estimators which dominate the isotonic regression estimator are proposed for some types of order restrictions.     

On preliminary test almost unbiased two-parameter estimator in linear regression model with student's t errors

In this paper, the preliminary test approach to the estimation of the linear regression model with student's t errors is considered. The preliminary test almost unbiased two-parameter estimator is proposed, when it is suspected that the regression parameter may be restricted to a constraint. The quadratic biases and quadratic risks of the proposed estimators are derived and compared under both null and alternative hypotheses. The conditions of superiority of the proposed estimators for departure parameter and biasing parameters k and d are derived, respectively. Furthermore, a real data example and a Monte Carlo simulation study are provided to illustrate some of the theoretical results.     

Three estimators for the poisson regression model with measurement errors

We consider two consistent estimators for the parameters of the linear predictor in the Poisson regression model, where the covariate is measured with errors. The measurement errors are assumed to be normally distributed with known error variance σ _u ² . The SQS estimator, based on a conditional mean-variance model, takes the distribution of the latent covariate into account, and this is here assumed to be a normal distribution. The CS estimator, based on a corrected score function, does not use the distribution of the latent covariate. Nevertheless, for small σ _u ² , both estimators have identical asymptotic covariance matrices up to the order of σ _u ² . We also compare the consistent estimators to the naive estimator, which is based on replacing the latent covariate with its (erroneously) measured counterpart. The naive estimator is biased, but has a smaller covariance matrix than the consistent estimators (at least up to the order of σ _u ² ).     

Comparison of estimation methods for the finite population mean in simple random sampling: symmetric super-populations

In this paper, a new estimator combined estimator (CE) is proposed for estimating the finite population mean ¯ Y _N in simple random sampling assuming a long-tailed symmetric super-population model. The efficiency and robustness properties of the CE is compared with the widely used and well-known estimators of the finite population mean ¯ Y _N by Monte Carlo simulation. The parameter estimators considered in this study are the classical least squares estimator, trimmed mean, winsorized mean, trimmed L-mean, modified maximum-likelihood estimator, Huber estimator (W24) and the non-parametric Hodges–Lehmann estimator. The mean square error criteria are used to compare the performance of the estimators. We show that the CE is overall more efficient than the other estimators. The CE is also shown to be more robust for estimating the finite population mean ¯ Y _N , since it is insensitive to outliers and to misspecification of the distribution. We give a real life example.     

Preliminary test almost unbiased two-parameter estimators with student’s t errors and conflicting test statistics

Abstract

In this paper, we consider the preliminary test approach to the estimation of the regression parameter in a multiple regression model under multicollinearity situation. The preliminary test almost unbiased two-parameter estimators based on the Wald, the Likelihood ratio, and the Lagrangian multiplier tests are given, when it is suspected that the regression parameter may be restricted to a subspace and the regression error is distributed with multivariate Student’s t errors. The bias and quadratic risk of the proposed estimators are derived and compared. Furthermore, a Monte Carlo simulation is provided to illustrate some of the theoretical results.     

Combining Unbiased Ridge and Principal Component Regression Estimators

In the presence of multicollinearity problem, ordinary least squares (OLS) estimation is inadequate. To circumvent this problem, two well-known estimation procedures often suggested are the unbiased ridge regression (URR) estimator given by Crouse et al. (1995 Crouse , R. , Jin , C. , Hanumara , R. ( 1995 ). Unbiased ridge estimation with prior information and ridge trace . Commun. Statist. Theor. Meth. 24 : 2341 – 2354 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and the (r, k) class estimator given by Baye and Parker (1984 Baye , M. , Parker , D. ( 1984 ). Combining ridge and principal component regression: a money demand illustration . Commun. Statist. Theor. Meth. 13 : 197 – 205 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). In this article, we proposed a new class of estimators, namely modified (r, k) class ridge regression (MCRR) which includes the OLS, the URR, the (r, k) class, and the principal components regression (PCR) estimators. It is based on a criterion that combines the ideas underlying the URR and the PCR estimators. The standard properties of this new class estimator have been investigated and a numerical illustration is done. The conditions under which the MCRR estimator is better than the other two estimators have been investigated.     

VARIANCE ESTIMATION IN TWO-PHASE SAMPLING

Two‐phase sampling is often used for estimating a population total or mean when the cost per unit of collecting auxiliary variables, x, is much smaller than the cost per unit of measuring a characteristic of interest, y. In the first phase, a large sample s₁ is drawn according to a specific sampling design p(s₁) , and auxiliary data x are observed for the units i∈s₁ . Given the first‐phase sample s₁ , a second‐phase sample s₂ is selected from s₁ according to a specified sampling design {p(s₂∣s₁) } , and (y, x) is observed for the units i∈s₂ . In some cases, the population totals of some components of x may also be known. Two‐phase sampling is used for stratification at the second phase or both phases and for regression estimation. Horvitz–Thompson‐type variance estimators are used for variance estimation. However, the Horvitz–Thompson ( Horvitz & Thompson, J. Amer. Statist. Assoc. 1952 ) variance estimator in uni‐phase sampling is known to be highly unstable and may take negative values when the units are selected with unequal probabilities. On the other hand, the Sen–Yates–Grundy variance estimator is relatively stable and non‐negative for several unequal probability sampling designs with fixed sample sizes. In this paper, we extend the Sen–Yates–Grundy ( Sen , J. Ind. Soc. Agric. Statist. 1953; Yates & Grundy , J. Roy. Statist. Soc. Ser. B 1953) variance estimator to two‐phase sampling, assuming fixed first‐phase sample size and fixed second‐phase sample size given the first‐phase sample. We apply the new variance estimators to two‐phase sampling designs with stratification at the second phase or both phases. We also develop Sen–Yates–Grundy‐type variance estimators of the two‐phase regression estimators that make use of the first‐phase auxiliary data and known population totals of some of the auxiliary variables.