Combining the unrestricted estimators into a single estimator and a simulation study on the unrestricted estimators

The purpose of this paper is to combine several regression estimators (ordinary least squares (OLS), ridge, contraction, principal components regression (PCR), Liu, r?k and r?d class estimators) into a single estimator. The conditions for the superiority of this new estimator over the PCR, the r?k class, the r?d class, β?(k, d), OLS, ridge, Liu and contraction estimators are derived by the scalar mean square error criterion and the estimators of the biasing parameters for this new estimator are examined. Also, a numerical example based on Hald data and a simulation study are used to illustrate the results.     

On improvement in estimating the population mean in simple random sampling

Kadilar and Cingi [Ratio estimators in simple random sampling, Appl. Math. Comput. 151 (3) (2004), pp. 893–902] introduced some ratio-type estimators of finite population mean under simple random sampling. Recently, Kadilar and Cingi [New ratio estimators using correlation coefficient, Interstat 4 (2006), pp. 1–11] have suggested another form of ratio-type estimators by modifying the estimator developed by Singh and Tailor [Use of known correlation coefficient in estimating the finite population mean, Stat. Transit. 6 (2003), pp. 655–560]. Kadilar and Cingi [Improvement in estimating the population mean in simple random sampling, Appl. Math. Lett. 19 (1) (2006), pp. 75–79] have suggested yet another class of ratio-type estimators by taking a weighted average of the two known classes of estimators referenced above. In this article, we propose an alternative form of ratio-type estimators which are better than the competing ratio, regression, and other ratio-type estimators considered here. The results are also supported by the analysis of three real data sets that were considered by Kadilar and Cingi.     

A Note on Superiority Comparisons of Homogeneous Linear Estimators

This note extends some results on homogeneous linear estimators to the general, even nonlinear case.A Sufficient condition for the difference of mean square error matrices of minimum conditional mean square error estimator and minimum average risk linear estimator to be postive definite is derived.     

Approximations to the chi-square distribution

The chi-square distribution arises frequently in applied statistics.Associated with the chi-square random variable with v degrees of freedom are two interdependent variables: the probability integral and the percentage point.Given one of these variables,the other can be obtained from chi-square tables for selected values.In order to overcome the inconvenience of statistical tables and interpolation,many approximations have been suggested.The computational difficulty and accuracy of various approximations is compared.     

Estimation of a distribution function bounded by two known distribution functions

The problem of estimation of a cumulative distribution function (cdf), bounded by two known cdf's, is considered. An estimator satisfying the desired restriction has been obtained by suitably adjusting the empirical cdf. Consistency of the adjusted estimator has been established and its mean square error (MSE) has been shown to be smallerthan that of the empirical cdf. The new estimator has been comparedwith the empirical cdf for some special cases.     

On estimating the mean of the selected population with unknown variance

This paper compares four estimators of the mean of the selected population from two normal populations with unknown means and common but unknown variance. The selection procedure is that the population yielding the largest sample mean is selected. The four estimators considered are invariant under both location and scale transformations. The bias and mean square errors of the four estimators are computed and compared. The conclusions are close to those reported by Dahiya ‘1974’, even for small sample sizes     

Estimation of the mean of the exponential distribution with an outlying observation

Given a life testing experiment consisting of n items, n-1 of which have the expected life λ while one could have an expected life λ/α with 0 < α < 1 the problem is. to find a mean square error (MSE) minimizing estimation function. The standard estimators for the homogeneous case (α = 1) overestimate the expected life and their MSE tend to infinity when a tends to 0.

Looking at the estimation problem as an insurance (see Anscombe (1960)) two different “testimators” are compared with respect to their MSE, Numerical results show that an estimation function based on the “Epstein-statistic” x_(n)/[xbar] is the best one.     

Estimating Damages in a Class Action Litigation

In a class action litigation, actual damages are not known exactly and must be estimated. Various estimators are proposed and assessed by using a model that identifies possible sources of error. Estimators that have been used in practice are shown to be seriously biased. An empirical Bayes estimator and an empirical minimal mean squared error estimator are found to be more satisfactory methods for estimating damages.     

Minimax linear empirical bayes estimation of the binomial parameter

The minimax linear Empirical Bayes estimators for a binomial parameter are obtained, assuming some information about the moments of the prior. The form of these estimates is used to propose a criterion which may be helpful in determining whether Empirical Bayes estimation is Indicated for a given problem.     

Optimal minimax squared error risk estimation of the mean of a multivariate normal distribution

Minimax squared error risk estimators of the mean of a multivariate normal distribution are characterized which have smallest Bayes risk with respect to a spherically symmetric prior distribution for (i) squared error loss, and (ii) zero-one loss depending on whether or not estimates are consistent with the hypothesis that the mean is null. In (i), the optimal estimators are the usual Bayes estimators for prior distributions with special structure. In (ii), preliminary test estimators are optimal. The results are obtained by applying the theory of minimax-Bayes-compromise decision problems.     

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.     

Interval shrinkage estimators

This paper considers estimation of an unknown distribution parameter in situations where we believe that the parameter belongs to a finite interval. We propose for such situations an interval shrinkage approach which combines in a coherent way an unbiased conventional estimator and non-sample information about the range of plausible parameter values. The approach is based on an infeasible interval shrinkage estimator which uniformly dominates the underlying conventional estimator with respect to the mean square error criterion. This infeasible estimator allows us to obtain useful feasible counterparts. The properties of these feasible interval shrinkage estimators are illustrated both in a simulation study and in empirical examples.     

Optimal kernels when estimating non-smooth densities

The derivation of new kernel functions for the kernel estimator of an unknown density function is given. These kernels are shown to be optimal in some sense when the underlying density f is continuous but its derivative f′ is not, and consequently a solu tion is presented for an unsolved problem which was stated by van Eeden (1985). Other attractive features of these kernels are also discussed and a number of graphs are listed.     

Operational ridge regression estimators under the prediction goal

In this paper we study the Mean Square Error and Conditional Mean Forecasting of Operational Ordinary Ridge Estimator. We use the G( ) functions to provide both the exact and the approximate bias and Mean Square Error of ordinary ridge estimator (ORE), We show, among other things, that ORE dominates OLS up to a certain order of approximation under the conditional mean forecasting sense.     

Bootstrap for finite populations

The bootstrap method is compared with the classical (linearization) and jackknife procedures for estimating the mean square errors (MSEs) of the ratio estimator and the combined ratio estimator. The initial samples are considered to be selected without replacement, and different procedures for selecting the bootstrap samples with or without replacement from them are examined. The biases, stabilities, coverage probabilities and confidence widths of all the procedures are compared.     

Estimation and testing under exchangeable exponential model with a single outlier

Several estimators of mean of an exponential distribution, when an unidentified single outlier in a sample of size n is present* are discussed. It is assumed that n?1 of these observations have a mean σ, While one could have a mean σ/α. The estimation of σ has been considered in detail with some reference to the estimation of α. Finally, tests of hypotheses about σare briefly mentioned.     

An optimal property of the “(r,k ) class estimators”

The purpose of this work is to display optimal conditions, in terms of the Mean Square Error criterion of the (r,k) class estimators. This will be done with respect to the ordinary ridge repression, principal components and ordinary least squares estimate.     

A note on the distribution of mean square error of forecasting

Simulation has been a very important and widely used method in the study of misspecification or order determination in time series analysis. Mean square error of forecasting (MSEF) has been a major criterion for comparing the performance of different models. In simulation studies, standard deviations of MSEF's are calculated from the computed values of the MSEF's, In this note, the distribution of MSEF from simulation studies is established. Exact variance of the MSEF can be obtained from the prespecified values of the model selected for simulation. This variance should be a more appropriate criterion for evaluating the performance between models.