Mean square error comparison among variance estimators with known coefficient of variation

The improved estimators for the population parameters were considered by several statisticians under various conditions. Recently Laheetharan and Wijekoon (Improved estimation of the population parameters when some additional information is available. Stat Papers doi:, 2008) demonstrated a generalized procedure for obtaining optimal shrunken estimators, and derived such estimators for both population mean and variance when coefficient of variation is known. In this article the mean square errors of those estimators were compared, and a numerical illustration was done using the scaled mean square error loss as used by Kanefuji and Iwase (Stat Papers 39:377–388, 1998) to understand the efficiency of the estimators with increasing sample size.     

Improved estimation of the mean in one-parameter exponential families with known coefficient of variation

The value for which the mean square error of a biased estimatoraT for the mean μ is less than the variance of an unbiased estimatorT is derived by minimizingMSE(aT). The resulting optimal value is 1/[1+c(n)v ²], wherev=σ/μ, is the coefficient of variation. WhenT is the UMVUE , thenc(n)=1/n, and the optimal value becomes 1/(n+v ²) (Searls, 1964). Whenever prior information about the size ofv is available the shrinkage procedure is useful. In fact for some members of the one-parameter exponential families it is known that the variance is at most a quadratic function of the mean. If we identify the pertinent coefficients in the quadratic function, it becomes easy to determinev.     

Improvement of the Liu estimator in linear regression model

In the presence of stochastic prior information, in addition to the sample, Theil and Goldberger (1961) introduced a Mixed Estimator for the parameter vector β in the standard multiple linear regression model (T,Xβ,σ² I). Recently, the Liu estimator which is an alternative biased estimator for β has been proposed by Liu (1993). In this paper we introduce another new Liu type biased estimator called Stochastic restricted Liu estimator for β, and discuss its efficiency. The necessary and sufficient conditions for mean squared error matrix of the Stochastic restricted Liu estimator to exceed the mean squared error matrix of the mixed estimator will be derived for the two cases in which the parametric restrictions are correct and are not correct. In particular we show that this new biased estimator is superior in the mean squared error matrix sense to both the Mixed estimator and to the biased estimator introduced by Liu (1993).     

Improved estimation of the population parameters when some additional information is available

Estimation of population parameters is considered by several statisticians when additional information such as coefficient of variation, kurtosis or skewness is known. Recently Wencheko and Wijekoon (Stat Papers 46:101–115, 2005) have derived minimum mean square error estimators for the population mean in one parameter exponential families when coefficient of variation is known. In this paper the results presented by Gleser and Healy (J Am Stat Assoc 71:977–981, 1976) and Arnholt and Hebert (, 2001) were generalized by considering T (X) as a minimal sufficient estimator of the parametric function g(θ) when the ratio t²=[ g(q) ]^-2Var[ T(X ) ]{\tau^{2}=[ {g(\theta )} ]^{-2}{\rm Var}[ {T(\boldsymbol{X} )} ]} is independent of θ. Using these results the minimum mean square error estimator in a certain class for both population mean and variance can be obtained. When T (X) is complete and minimal sufficient, the ratio τ² is called “WIJLA” ratio, and a uniformly minimum mean square error estimator can be derived for the population mean and variance. Finally by applying these results, the improved estimators for the population mean and variance of some distributions are obtained.     

Stein-rule estimator under inclusion of superfluous variables in linear regression models

Stein-rule estimation is a well-known method to improve the unbiased OLSE in the sense of smaller Mean-Square-Error. The paper is investigating the behaviour of this efficiency relation in case of misspecification of the linear model caused by inclusion of superfluous variables     

Optimal generalized logistic estimator

In this paper, we propose a new efficient estimator namely Optimal Generalized Logistic Estimator (OGLE) for estimating the parameter in a logistic regression model when there exists multicollinearity among explanatory variables. Asymptotic properties of the proposed estimator are also derived. The performance of the proposed estimator over the other existing estimators in respect of Scalar Mean Square Error criterion is examined by conducting a Monte Carlo simulation.