Hartley-Ross Type Estimators for Population Mean Using Known Parameters of Auxiliary Variate

This article proposes Hartley-Ross type unbiased estimators of finite population mean using information on known parameters of auxiliary variate when the study variate and auxiliary variate are positively correlated. The variances of the proposed unbiased estimators are obtained. It has been shown that the proposed estimators are more efficient than the simple mean estimator, usual ratio estimator and estimators proposed by Sisodia and Dwivedi (1981 Sisodia , B. V. S. , Dwivedi , V. K. ( 1981 ). A modified ratio estimator using coefficient of variation of auxiliary variable . J. Indian Soc. Agricultural Statist. 33 ( 1 ): 13 – 18 . [Google Scholar]), Kadilar and Cingi (2006 Kadilar , C. , Cingi , H. ( 2006 ). A new ratio estimator using correlation coefficient . Int. Statist. 1 – 11 . [Google Scholar]), and Kadilar et al. (2007 Kadilar , C. , Candan , M. , Cingi , H. ( 2007 ). Ratio estimators using robust regression . Hacet. J. Math. Statist. 36 ( 2 ): 181 – 188 .[Web of Science ®] , [Google Scholar]) under certain realistic conditions. Empirical studies are also carried out to demonstrate the merits of the proposed unbiased estimators over other estimators considered in this article.     

Exponential-Type Estimators of the Mean of a Sensitive Variable in the Presence of Nonsensitive Auxiliary Information

Sousa et al. and Gupta et al. suggested ratio and regression-type estimators of the mean of a sensitive variable using nonsensitive auxiliary variable. This article proposes exponential-type estimators using one and two auxiliary variables to improve the efficiency of mean estimator based on a randomized response technique. The expressions for the mean squared errors (MSEs) and bias, up to first-order approximation, have been obtained. It is shown that the proposed exponential-type estimators are more efficient than the existing estimators. The gain in efficiency over the existing estimators has also been shown with a simulation study and by using real data.     

Selecting a test level for random effects

A method is presented for selecting an a-level to use when testing for group difference in a one-way classification random effects model. The a-level is chosen to make the power of the test equal to .5 when the parameters are such that between group mean square and total mean square are equally good minimum expected squared error estimators of the variance of y the estimator of the mean     

Improved estimation of variance components in mixed models

Taking Albert's (1976) formulation of a mixed model ANOVA, we consider improved estimation of the variance components for balanced designs under squared error loss. Two approaches are presented. One extends the ideas of Stein (1964), The other is developed from the fact that variance components can be expressed as linear combinations of chi-square scale parameters. Encouraging simulation results are presented.     

The Research on Two Kinds of Restricted Biased Estimators Based on Mean Squared Error Matrix

Sakall?oglu et al. (2001 Sakall?oglu , Kaç?ranlar , Akdeniz ( 2001 ). Mean squared error comparisons of some biased estimators . Commun. Statist. Theor. Meth. 30 : 347 – 361 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) dealt with the comparisons among the ridge estimator, Liu estimator, and iteration estimator. Akdeniz and Erol (2003 Akdeniz , F. , Erol , H. ( 2003 ). Mean squared error matrix comparisons of some biased estimators in linear regression . Commun. Statist. Theor. Meth. 32 : 2389 – 2413 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) have compared the (almost unbiased) generalized ridge regression estimator with the (almost unbiased) generalized Liu estimator in the matrix mean squared error sense. In this article, we study the ridge estimator and Liu estimator with respect to linear equality restriction, and establish some sufficient conditions for the superiority of the restricted ridge estimator over the restricted Liu estimator and the superiority of the restricted Liu estimator over the restricted ridge estimator under mean squared error matrix, respectively. Furthermore, we give a numerical example.     

Numerical Study of Small Sample Variances of Estimators of Variance Components in the Two-Way Factorial Model

Many estimation procedures have been proposed for estimating variance components in unbalanced factorial models. A large proportion of these are based on the solution to a system of linear equations obtained from a set of quadratic forms and their expected value. This paper will present a numerical study of the small sample variance of eight variance component estimators of this type. The variances will be compared to the Bhattacharyya lower bound for unbiased estimators.     

Alternative moment approximations for berkson's minimum logit chi-squared estimator

Expressions are derived for the bias to order J^-1 , the variance to order J^-2 and the mean squared error to order J^-2 of Berkson's minimum logit chi-squared estimator where J is the number of distinct design points. These moment approximations are numerically compared to Monte Carlo estimates of the true moments and the moment approximations of Amemiya (1980) which are appropriate when the “average” number of observations per design point is large. They are used to compare the mean squared error of the minimum logit chi-squared estimator to that of the maximum likelihood estimator and to investigate the effect of bias on confidence intenrals constructed using the minimum logit chi-squared estimator.     

Inference for Variance Components in a Mixed Model for Unbalanced Split Plot Design

Abstract

We consider the unbalanced split-plot design with the whole plot and the subplot effect from nonnormal universes. The three estimators for the whole plot effect variance component are obtained. An approximate test for significance of the whole plot effect variance component is presented.     

A Note on Model-Based Stratification

A slight improvement on the stratification methodology in Godfrey et al. (1984) is proposed and an example is advanced to demonstrate its usefulness.     

On a class of shrinkage estimators of vector of regression coefficients the

This paper studies a class of shrinkage estimators of the vector of regression coefficients. The small disturbance approximations for the bias and the mean squared error matrix of the estimator are derived. In the sense of mean squared error, these estimators dominate the least squares estimator and the generalized Stein estimator developed by Hosmane (1988).     

A study on the chain ratio-ratio-type exponential estimator for finite population variance

This paper considers the problem of estimating the population variance S²_y of the study variable y using the auxiliary information in sample surveys. We have suggested the (i) chain ratio-type estimator (on the lines of Kadilar and Cingi (2003)), (ii) chain ratio-ratio-type exponential estimator and their generalized version [on the lines of Singh and Pal (2015)] and studied their properties under large sample approximation. Conditions are obtained under which the proposed estimators are more efficient than usual unbiased estimator s²_y and Isaki (1893) ratio estimator. Improved version of the suggested class of estimators is also given along with its properties. An empirical study is carried out in support of the present study.     

A note on the estimation of variance of sample mean Using the knowledge of coefficient of variation in normal population

A class of estimators for the variance of sample mean is defined and its properties are studied in case of normal population. It is identified that the usual unbiased estimator, Singh, Pandey and Hirano (1973) -type estimator and Lee (1931) estimator are particular members of the proposed class of estimators. It is found that the minimum Mean Squared Error (MSE) of the proposed class of estimators is less than that of other estimators.     

Mean intergrated squared error properties and optimal kernels when estimating a diatribution function

Let X₁,X₂,… X_n be a sample of independent identically distributed (i.i.d)random variables having an unknown absolutely continuous distribution function f with density f the twofold aim of his paper consists in, firstly deriving asymptotic expressions of the mean intergrated squared error (MISE) of a kernel estimator of F when f is either assumed to be continuous everywhere or problem of finding optimal kernels in these two cases is studied in detail.     

Improved Ratio- and Product-Type Exponential Estimators for Population Mean in Case of Post-Stratification

This paper discusses the problem of estimation of population mean in case of post-stratification. Improved ratio- and product-type exponential estimators of finite population mean are suggested with their case of post-stratification. Bias and mean-squared error of the suggested estimators are obtained up to the first degree of approximation. Suggested estimators have been compared with unbiased estimator, ratio estimator, and product estimator in case of post-stratification. An empirical study has been carried out to demonstrate the performance of the suggested estimator.     

Numerical performance of block thresholded wavelet estimators

Usually, methods for thresholding wavelet estimators are implemented term by term, with empirical coefficients included or excluded depending on whether their absolute values exceed a level that reflects plausible moderate deviations of the noise. We argue that performance may be improved by pooling coefficients into groups and thresholding them together. This procedure exploits the information that coefficients convey about the sizes of their neighbours. In the present paper we show that in the context of moderate to low signal-to-noise ratios, this block thresholding approach does indeed improve performance, by allowing greater adaptivity and reducing mean squared error. Block thresholded estimators are less biased than term-by-term thresholded ones, and so react more rapidly to sudden changes in the frequency of the underlying signal. They also suffer less from spurious aberrations of Gibbs type, produced by excessive bias. On the other hand, they are more susceptible to spurious features produced by noise, and are more sensitive to selection of the truncation parameter.     

A class of exponential-type ratio estimators of a general parameter

This paper addresses the problem of estimating a general parameter using information on an auxiliary variable X. We have suggested a class of exponential-type ratio estimators for the parameter and its properties are studied. It is identified that the estimators due to Upadhyaya et al. [Journal of Statistical Theory and Practice (2011), 5(2), 285–302] and Yadav and Kadilar [Revista Columbiana de Estadistica, (2013), 36(1), 145–152] are members of the proposed estimator. We have also shown that the suggested estimator is more efficient than the estimators of Upadhyaya et al. (2011 Upadhyaya, L.N., Singh, H.P., Chatterjee, S., Yadav, R. (2011). Improved ratio and product exponential type estimators. J. Stat. Theo. Pract. 5 (2): 285–302.[Taylor &; Francis Online] , [Google Scholar]) and Yadav and Kadilar (2013 Yadav, S.K., Kadilar, C. (2013). Improved exponential type ratio estimator of population variance. Revis. Colum. de Estadist. 36(1): 145–152. [Google Scholar]). Numerical illustration is provided in support of the present study.     

Empirical Bayes Estimation of Small Area Means under a Nested Error Linear Regression Model with Measurement Errors in the Covariates

Abstract.  Previously, small area estimation under a nested error linear regression model was studied with area level covariates subject to measurement error. However, the information on observed covariates was not used in finding the Bayes predictor of a small area mean. In this paper, we first derive the fully efficient Bayes predictor by utilizing all the available data. We then estimate the regression and variance component parameters in the model to get an empirical Bayes (EB) predictor and show that the EB predictor is asymptotically optimal. In addition, we employ the jackknife method to obtain an estimator of mean squared prediction error (MSPE) of the EB predictor. Finally, we report the results of a simulation study on the performance of our EB predictor and associated jackknife MSPE estimators. Our results show that the proposed EB predictor can lead to significant gain in efficiency over the previously proposed EB predictor.     

Half-sample variance estimation

This paper studies an alternative to the jackknife variance estimator, the half-sample variance estimator. Both theoretical and Monte Carlo comparisons between the half-sample variance estimator and the jackknife variance estimator indicate that the former is better in some situations.     

Frequentist and Bayesian Interval Estimators for the Normal Mean

It is well known that a Bayesian credible interval for a parameter of interest is derived from a prior distribution that appropriately describes the prior information. However, it is less well known that there exists a frequentist approach developed by Pratt (1961 Pratt , J. W. ( 1961 ). Length of confidence intervals . J. Amer. Statist. Assoc. 56 : 549 – 657 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) that also utilizes prior information in the construction of frequentist confidence intervals. This frequentist approach produces confidence intervals that have minimum weighted average expected length, averaged according to some weight function that appropriately describes the prior information. We begin with a simple model as a starting point in comparing these two distinct procedures in interval estimation. Consider X ₁,…, X _n that are independent and identically N(μ, σ²) distributed random variables, where σ² is known, and the parameter of interest is μ. Suppose also that previous experience with similar data sets and/or specific background and expert opinion suggest that μ = 0. Our aim is to: (a) develop two types of Bayesian 1 ? α credible intervals for μ, derived from an appropriate prior cumulative distribution function F(μ) more importantly; (b) compare these Bayesian 1 ? α credible intervals for μ to the frequentist 1 ? α confidence interval for μ derived from Pratt's frequentist approach, in which the weight function corresponds to the prior cumulative distribution function F(μ). We show that the endpoints of the Bayesian 1 ? α credible intervals for μ are very different to the endpoints of the frequentist 1 ? α confidence interval for μ, when the prior information strongly suggests that μ = 0 and the data supports the uncertain prior information about μ. In addition, we assess the performance of these intervals by analyzing their coverage probability properties and expected lengths.