A New Proof of Murthy's Estimator which Applies to Sequential Sampling

Murthy's estimator has been used for constructing an unbiased estimator of a population total or mean from a sample of fixed size when there is unequal probability sampling without replacement. Traditionally, the estimator is derived by constructing an unordered version of Raj's ordered unbiased estimator. This paper presents an elementary proof of Murthy's estimator which applies the Rao–Blackwell theorem to a very simple estimator. This proof includes any sequential sampling scheme, thus extending the usefulness of Murthy's estimator. We demonstrate this extension by deriving unbiased estimators for inverse sampling.     

The exact mean squared error risks of preliminary test type estimators for the multivariate normal mean

The exact mean squared error risks of the preliminary test estimtor and the Sclove modified Stein rule estimator (Sclove, Morris and Radhakrishnan, 1972) for the multivariate normal mean are computed and their risks are compared with the risks of Stein estimators.     

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.     

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.     

An Improved Estimation in Regression Parameter Matrix in Multivariate Regression Model

We consider shrinkage and preliminary test estimation strategies for the matrix of regression parameters in multivariate multiple regression model in the presence of a natural linear constraint. We suggest a shrinkage and preliminary test estimation strategies for the parameter matrix. The goal of this article is to critically examine the relative performances of these estimators in the direction of the subspace and candidate subspace restricted type estimators. Our analytical and numerical results show that the proposed shrinkage and preliminary test estimators perform better than the benchmark estimator under candidate subspace and beyond. The methods are also applied on a real data set for illustrative purposes.     

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.     

Two Stochastic Restricted Principal Components Regression Estimator in Linear Regression

In this article, we propose two stochastic restricted principal components regression estimator by combining the approach followed in obtaining the ordinary mixed estimator and the principal components regression estimator in linear regression model. The performance of the two new estimators in terms of matrix MSE criterion is studied. We also give an example and a Monte Carlo simulation to show the theoretical results.     

New calibration estimator based on two auxiliary variables in stratified sampling

In recent years, calibration estimation has become an important field of research in survey sampling. This paper proposes a new calibration estimator for the population mean in the presence of two auxiliary variables in stratified sampling. The theory of new calibration estimator is given and optimum calibration weights are derived. A simulation study is carried out to performance of the proposed calibration estimator over other existing calibration estimators. The results reveal that the proposed calibration estimators are more efficient than other existing calibration estimators in stratified sampling.     

Combining poisson means

The problem of estimating the Poisson mean is considered based on the two samples in the presence of uncertain prior information (not in the form of distribution) that two independent random samples taken from two possibly identical Poisson populations. The parameter of interest is λ₁ from population I. Three estimators, i.e. the unrestricted estimator, restricted estimator and preliminary test estimator are proposed. Their asymptotic mean squared errors are derived and compared; parameter regions have been found for which restricted and preliminary test estimators are always asymptotically more efficient than the classical estimator. The relative dominance picture of the estimators is presented. Maximum and minimum asymptotic efficiencies of the estimators relative to the classical estimator are tabulated. A max-min rule for the size of the preliminary test is also discussed. A Monte Carlo study is presented to compare the performance of the estimator with that of Kale and Bancroft (1967).     

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.     

Estimating the proportion of true null hypotheses, with application to DNA microarray data

Summary.  We consider the problem of estimating the proportion of true null hypotheses, π ₀, in a multiple-hypothesis set-up. The tests are based on observed p -values. We first review published estimators based on the estimator that was suggested by Schweder and Spjøtvoll. Then we derive new estimators based on nonparametric maximum likelihood estimation of the p -value density, restricting to decreasing and convex decreasing densities. The estimators of π ₀ are all derived under the assumption of independent test statistics. Their performance under dependence is investigated in a simulation study. We find that the estimators are relatively robust with respect to the assumption of independence and work well also for test statistics with moderate dependence.     

多辅助变量线性组合的回归估计

以提高估计量的精度为目的,定义了一种新的多个辅助变量的回归估计法,从理论上研究了该方法下权的选取方法,并将该方法下的估计量与Raj多辅助变量回归估计量、Ghosh多元线性回归估计量在精度上进行了数值比较,结果表明：这种新的多辅助变量的回归估计法在精度上优于Raj多辅助变量回归估计和Ghosh多元线性回归估计。     

Robust estimation of the mean vector for high-dimensional data set using robust clustering

The first step in statistical analysis is the parameter estimation. In multivariate analysis, one of the parameters of interest to be estimated is the mean vector. In multivariate statistical analysis, it is usually assumed that the data come from a multivariate normal distribution. In this situation, the maximum likelihood estimator (MLE), that is, the sample mean vector, is the best estimator. However, when outliers exist in the data, the use of sample mean vector will result in poor estimation. So, other estimators which are robust to the existence of outliers should be used. The most popular robust multivariate estimator for estimating the mean vector is S-estimator with desirable properties. However, computing this estimator requires the use of a robust estimate of mean vector as a starting point. Usually minimum volume ellipsoid (MVE) is used as a starting point in computing S-estimator. For high-dimensional data computing, the MVE takes too much time. In some cases, this time is so large that the existing computers cannot perform the computation. In addition to the computation time, for high-dimensional data set the MVE method is not precise. In this paper, a robust starting point for S-estimator based on robust clustering is proposed which could be used for estimating the mean vector of the high-dimensional data. The performance of the proposed estimator in the presence of outliers is studied and the results indicate that the proposed estimator performs precisely and much better than some of the existing robust estimators for high-dimensional data.     

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.     

Wavelet threshold based on Stein's unbiased risk estimators of restricted location parameter in multivariate normal

In this paper, the problem of estimating the mean vector under non-negative constraints on location vector of the multivariate normal distribution is investigated. The value of the wavelet threshold based on Stein''s unbiased risk estimators is calculated for the shrinkage estimator in restricted parameter space. We suppose that covariance matrix is unknown and we find the dominant class of shrinkage estimators under Balance loss function. The performance evaluation of the proposed class of estimators is checked through a simulation study by using risk and average mean square error values.     

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.     

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.     

Second-order variance estimation in poststratified two-stage sampling

The linearization or Taylor series variance estimator and jackknife linearization variance estimator are popular for poststratified point estimators. In this note we propose a simple second-order linearization variance estimator for the poststratified estimator of the population total in two-stage sampling, using the second-order Taylor series expansion. We investigate the properties of the proposed variance estimator and its modified version and their empirical performance through some simulation studies in comparison to the standard and jackknife linearization variance estimators. Simulation studies are carried out on both artificially generated data and real data.     

Minimax Linear Smoothers

We consider the problem of estimating the mean of a multivariate distribution. As a general alternative to penalized least squares estimators, we consider minimax estimators for squared error over a restricted parameter space where the restriction is determined by the penalization term. For a quadratic penalty term, the minimax estimator among linear estimators can be found explicitly. It is shown that all symmetric linear smoothers with eigenvalues in the unit interval can be characterized as minimax linear estimators over a certain parameter space where the bias is bounded. The minimax linear estimator depends on smoothing parameters that must be estimated in practice. Using results in Kneip (1994), this can be done using Mallows' C _L -statistic and the resulting adaptive estimator is now asymptotically minimax linear. The minimax estimator is compared to the penalized least squares estimator both in finite samples and asymptotically.