A new estimator combining the ridge regression and the restricted least squares methods of estimation

It is well-known in the literature on multicollinearity that one of the major consequences of multicollinearity on the ordinary least squares estimator is that the estimator produces large sampling variances, which in turn might inappropriately lead to exclusion of otherwise significant coefficients from the model. To circumvent this problem, two accepted estimation procedures which are often suggested are the restricted least squares method and the ridge regression method. While the former leads to a reduction in the sampling variance of the estimator, the later ensures a smaller mean square error value for the estimator. In this paper we have proposed a new estimator which is based on a criterion that combines the ideas underlying these two estimators. The standard properties of this new estimator have been studied in the paper. It has also been shown that this estimator is superior to both the restricted least squares as well as the ordinary ridge regression estimators by the criterion of mean sauare error of the estimator of the regression coefficients when the restrictions are indeed correct. The conditions for superiority of this estimator over the other two have also been derived for the situation when the restrictions are not correct.     

Caliper pair-matching on a continuous variable in case-control studies

The use of matched pairs has been criticized as being less efficient than estimators based on random samples. This paper compares the mean square error of an analysis of covariance estimator based on random samples with two estimators based on caliper matched pairs. The first of these is a simple mean difference estimator and the second a regression estimator suggested by Rubin (1973b). Under conditions which commonly occur in epidemiologic case-control studies, both of the matched pair estimators can have smaller mean square errors than analysis o f covariance estimator. When there is a weak relationship between the matching and response variate, the mean difference estimator has a lower mean square error than the regression estimator.     

Two-parameter ridge estimator in the binary logistic regression

The binary logistic regression is a commonly used statistical method when the outcome variable is dichotomous or binary. The explanatory variables are correlated in some situations of the logit model. This problem is called multicollinearity. It is known that the variance of the maximum likelihood estimator (MLE) is inflated in the presence of multicollinearity. Therefore, in this study, we define a new two-parameter ridge estimator for the logistic regression model to decrease the variance and overcome multicollinearity problem. We compare the new estimator to the other well-known estimators by studying their mean squared error (MSE) properties. Moreover, a Monte Carlo simulation is designed to evaluate the performances of the estimators. Finally, a real data application is illustrated to show the applicability of the new method. According to the results of the simulation and real application, the new estimator outperforms the other estimators for all of the situations considered.     

Some Imputation Methods to Minimize the Effect of Non Response in Two-Occasion Rotation Patterns

Among modern strategies applied to cope with non response, a major problem faced by survey statisticians, imputation is one of the most common. Imputation is the filling up method of incomplete data for adapting the standard analytic model in statistics. The purpose of the present work is to study the use of imputation methods in dealing with non response which occur at both occasions in two-occasion successive (rotation) sampling. Chain-type regressions in ratio estimators have been proposed for estimating the population mean at current occasion. Expressions for optimum estimator and its mean square error have been derived. To study the effectiveness of the imputation methods, performances of the proposed estimators are compared in two different situations: with and without non response. Behavior of the proposed estimators is demonstrated through empirical studies.     

New Shrinkage Parameters for the Liu-type Logistic Estimators

The binary logistic regression is a widely used statistical method when the dependent variable has two categories. In most of the situations of logistic regression, independent variables are collinear which is called the multicollinearity problem. It is known that multicollinearity affects the variance of maximum likelihood estimator (MLE) negatively. Therefore, this article introduces new shrinkage parameters for the Liu-type estimators in the Liu (2003) in the logistic regression model defined by Huang (2012) in order to decrease the variance and overcome the problem of multicollinearity. A Monte Carlo study is designed to show the goodness of the proposed estimators over MLE in the sense of mean squared error (MSE) and mean absolute error (MAE). Moreover, a real data case is given to demonstrate the advantages of the new shrinkage parameters.     

Bias of maximum likelihood estimators of the response standard deviation in regression

The bias of maximum likelihood estimators of the standard deviation of the response in location/scale regression models is considered. Results are obtained for a very wide family of densities for the response variable. These are used to propose point estimators with improved mean square error properties and to demonstrate the importance of bias correction in statistical inference when samples are moderately small.     

On preliminary test ridge regression estimators for linear restrictions in a regression model with non-normal disturbances

In this paper, we study the properties of the preliminary test, restricted and unrestricted ridge regression estimators of the linear regression model with non-normal disturbances. We present the estimators of the regression coefficients combining the idea of preliminary test and ridge regression methodology, when it is suspected that the regression coefficients may be restricted to a subspace and the regression error is distributed as multivariate t. Accordingly we consider three estimators, namely the Unrestricted Ridge Regression Estimator (URRRE), the Restricted Ridge Regression Estimator (RRRE) and finally the Preliminary test Ridge Regression Estimator (PTRRE). The biases and the mean square error (MSE) of the estimators are derived under the null and alternative hypotheses and compared with the usual estimators. By studying the MSE criterion, the regions of optimahty of the estimators are determined.     

An improved class of estimators for the population mean

Starting from the Rao (Commun Stat Theory Methods 20:3325–3340, 1991) regression estimator, we propose a class of estimators for the unknown mean of a survey variable when auxiliary information is available. The bias and the mean square error of the estimators belonging to the class are obtained and the expressions for the optimum parameters minimizing the asymptotic mean square error are given in closed form. A simple condition allowing us to improve the classical regression estimator is worked out. Finally, in order to compare the performance of some estimators with the regression one, a simulation study is carried out when some population parameters are supposed to be unknown.     

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.     

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.     

Reliability estimation for the exponential distribution

This paper is concerned with classical statistical estimation of the reliability function for the exponential density with unknown mean failure time θ, and with a known and fixed mission time τ. The minimum variance unbiased (MVU) estimator and the maximum likelihood (ML) estimator are reviewed and their mean square errors compared for different sample sizes. These comparisons serve also to extend previous work, and reinforce further the nonexistence of a uniformly best estimator. A class of shrunken estimators is then defined, and it produces a shrunken quasi-estimator and a shrunken estimator. The mean square errors for both these estimators are compared to the mean square errors of the MVU and ML estimators, and the new estimators are found to perform very well. Unfortunately, these estimators are difficult to compute for practical applications. A second class of estimators, which is easy to compute is also developed. Its mean square error properties are compared to the other estimators, and it outperforms all the contending estimators over the high and low reliability parameter space. Since, for all the estimators, analytical mean square error comparisons are not tractable, extensive numerical analyses are done in obtaining both the exact small sample and large sample results.     

A simulation study of ridge and other regression estimators

We consider a number of estimators of regression coefficients, all of generalized ridge, or 'shrinkage' type. Results of a simulation study indicate that with respect to two commonly used mean square error criteria, two ordinary ridge estimators, one proposed by Hoerl, Kennard and Baldwin, and the other introduced here, perform substantially better than both least squares and the other estimators discussed here     

Optimal estimation of a finite population mean under generalized random permutation models

Employing certain generalized random permutation models and a general class of linear estimators of a finite population mean, it is shown that many of the conventional estimators are “optimal” in the sense of minimum average mean square error. Simple proofs are provided by using a well-known theorem on UMV estimation. The results also cover certain simple response error situations.     

Improved two-parameter estimators for the negative binomial and Poisson regression models

Negative binomial regression (NBR) and Poisson regression (PR) applications have become very popular in the analysis of count data in recent years. However, if there is a high degree of relationship between the independent variables, the problem of multicollinearity arises in these models. We introduce new two-parameter estimators (TPEs) for the NBR and the PR models by unifying the two-parameter estimator (TPE) of Özkale and Kaç?ranlar [The restricted and unrestricted two-parameter estimators. Commun Stat Theory Methods. 2007;36:2707–2725]. These new estimators are general estimators which include maximum likelihood (ML) estimator, ridge estimator (RE), Liu estimator (LE) and contraction estimator (CE) as special cases. Furthermore, biasing parameters of these estimators are given and a Monte Carlo simulation is done to evaluate the performance of these estimators using mean square error (MSE) criterion. The benefits of the new TPEs are also illustrated in an empirical application. The results show that the new proposed TPEs for the NBR and the PR models are better than the ML estimator, the RE and the LE.     

Efficient improved estimation of the parameters in two seemingly unrelated regression models

For a system of two seemingly unrelated regression equations, this paper proposes a two-stage covariance improved estimator of the regression coefficients. The new estimator is shown to uniformly dominate the present estimators in terms of generalized mean square error criterion. In addition, we also propose the exact generalized mean square error of new estimator.     

Some conditions for optimality of the almost unbiased estimators of regression coefficients

The purpose of this paper is to examine the asymptotic properties of the operational almost unbiased estimator of regression coefficients which includes almost unbiased ordinary ridge estimator a s a special case. The small distrubance approximations for the bias and mean square error matrix of the estimator are derived. As a consequence, it is proved that, under certain conditions, the estimator is more efficient than a general class of estimators given by Vinod and Ullah (1981). Also it is shown that, if the ordinary ridge estimator (ORE) dominates the ordinary least squares estimator then the almost unbiased ordinary ridge estimator does not dominate ORE under the mean square error criterion.     

Estimation of population mean in presence of random non response in two-stage cluster sampling

The present investigation addresses the problem of estimating a finite population mean in two-phase cluster sampling in presence of random non response situations. Utilizing information on an auxiliary variable, regression type estimators has been proposed. Effective imputation techniques have been suggested to deal with the random non response situations. The properties of the proposed estimation strategies have been studied for different cases of random non response situations in practical surveys. The superiority of the suggested methodology over the natural sample mean estimator of population mean has been established through empirical studies carried over the data sets of natural population and artificially generated population.     

Good ridge estimators based on prior information

Ridge regression is re-examined and ridge estimators based on prior information are introduced. A necessary and sufficient condition is given for such ridge estimators to yield estimators of every nonnull linear combination of the regression coefficients with smaller mean square error than that of the Gauss-Markov best linear unbiased estimator.     

Norm restricted maximum likelihood estimators for binary regression models with parametric link

Parametric link transformation families have shown to be useful in the analysis of binary regression data since they avoid th? problem of link misspecifaction. Inference for these models are commonly based on likelihood methods. Duffy and Santner (1988, 1989) however showed that ordinary logistic maximum likelihood estimators (MLE) have poor mean square error (MSE) behavior in small samples compared to alternative norm restricted estimators. This paper extends these alternative norm restricted estimators to binary regression models with any specified parametric link family. These extended norm restricted MLE's are strongly consistent and efficient under regularity conditions. Finally a simulation study shows that an empiric version of norm restricted MLE's exhibit superior MSE behavior in small samples compared to MLE's with fixed known link.     

A new modified Jackknifed estimator for the Poisson regression model

The Poisson regression is very popular in applied researches when analyzing the count data. However, multicollinearity problem arises for the Poisson regression model when the independent variables are highly intercorrelated. Shrinkage estimator is a commonly applied solution to the general problem caused by multicollinearity. Recently, the ridge regression (RR) estimators and some methods for estimating the ridge parameter k in the Poisson regression have been proposed. It has been found that some estimators are better than the commonly used maximum-likelihood (ML) estimator and some other RR estimators. In this study, the modified Jackknifed Poisson ridge regression (MJPR) estimator is proposed to remedy the multicollinearity. A simulation study and a real data example are provided to evaluate the performance of estimators. Both mean-squared error and the percentage relative error are considered as the performance criteria. The simulation study and the real data example results show that the proposed MJPR method outperforms the Poisson ridge regression, Jackknifed Poisson ridge regression and the ML in all of the different situations evaluated in this paper.