Estimating the Mean with Known Coefficient of Variation

Searls in 1964 showed that when the coefficient of variation is known, the sample mean is dominated with respect to mean squared error by an improved estimator that makes use of that coefficient. In this article we illustrate that this is true for a general class of estimators. Expressions for the minimum mean squared error and the relative efficiency are given for general distributions. The improvement, as measured by relative efficiency, is seen to be independent of the form of the distribution.     

Choosing a robustness tuning parameter

A novel method is proposed for choosing the tuning parameter associated with a family of robust estimators. It consists of minimising estimated mean squared error, an approach that requires pilot estimation of model parameters. The method is explored for the family of minimum distance estimators proposed by [Basu, A., Harris, I.R., Hjort, N.L. and Jones, M.C., 1998, Robust and efficient estimation by minimising a density power divergence. Biometrika, 85, 549–559.] Our preference in that context is for a version of the method using the L ₂ distance estimator [Scott, D.W., 2001, Parametric statistical modeling by minimum integrated squared error. Technometrics, 43, 274–285.] as pilot estimator.     

An effective approach to linear calibration estimation with its applications

Abstract

The availability of some extra information, along with the actual variable of interest, may be easily accessible in different practical situations. A sensible use of the additional source may help to improve the properties of statistical techniques. In this study, we focus on the estimators for calibration and intend to propose a setup where we reply only on first two moments instead of modeling the whole distributional shape. We have proposed an estimator for linear calibration problems and investigated it under normal and skewed environments. We have partitioned its mean squared error into intrinsic and estimation components. We have observed that the bias and mean squared error of the proposed estimator are function of four dimensionless quantities. It is to be noticed that both the classical and the inverse estimators become the special cases of the proposed estimator. Moreover, the mean squared error of the proposed estimator and the exact mean squared error of the inverse estimator coincide. We have also observed that the proposed estimator performs quite well for skewed errors as well. The real data applications are also included in the study for practical considerations.     

Relative squared error prediction in the generalized linear regression model

In linear regression models, predictors based on least squares or on generalized least squares estimators are usually applied which, however, fail in case of multicollinearity. As an alternative biased estimators like ridge estimators, Kuks-Olman estimators, Bayes or minimax estimators are sometimes suggested. In our analysis the relative instead of the generally used absolute squared error enters the objective function. An explicit minimax solution is derived which, in an important special case, can be viewed as a predictor based on a Kuks-Olman estimator.     

Minimum mean squared error estimation of each individual coefficient in a linear regression model

In this paper, we derive the exact mean squared error (MSE) of the minimum MSE estimator for each individual coefficient in a linear regression model, and show a sufficient condition for the minimum MSE estimator for each individual coefficient to dominate the OLS estimator. Numerical results show that when the number of independent variables is 2 and 3, the minimum MSE estimator for each individual coefficient can be a good alternative to the OLS and Stein-rule estimators.     

Asymptotic Theory for the Multiscale Wavelet Density Derivative Estimator

Abstract

In this paper we consider the wavelet-based estimation of density derivatives. The multiscale density derivative estimator is proposed which is constructed by using a number of scaling functions. Asymptotic theory is developed in which asymptotic expressions for the bias, the variance and the mean integrated squared error are included. In addition, asymptotic normality of the proposed estimator is proved. Theoretical and numerical comparisons with the usual kernel-based estimators are also reported.     

An improved and efficient biased estimation technique in logistic regression model

Abstract

In this article, we propose a new improved and efficient biased estimation method which is a modified restricted Liu-type estimator satisfying some sub-space linear restrictions in the binary logistic regression model. We study the properties of the new estimator under the mean squared error matrix criterion and our results show that under certain conditions the new estimator is superior to some other estimators. Moreover, a Monte Carlo simulation study is conducted to show the performance of the new estimator in the simulated mean squared error and predictive median squared errors sense. Finally, a real application is considered.     

A comparison of estimators for the proportion in the tail of a normal distribution

Three estimators of the proportion in a tail of the normal distribution are compared using the criteria of mean squared error and mean absolute error. The estimators that we compare are the maximum likelihood estimator, the minimum variance unbiased estimator, and an intuitive estimator that is frequently used in practice. The intuitive estimator is similar to the MLE but uses the usual unbiased estimator of σ² rather than the MLE of σ². We show that the intuitive estimator has low efficiency, and for this reason it is not recommended. For very smallp and for largep the MVUE has the highest efficiency. The MLE is best for moderate values ofp.     

Minimum power-divergence estimator in three-way contingency tables

Cressie et al. (2000; 2003) introduced and studied a new family of statistics, based on the φ-divergence measure, for solving the problem of testing a nested sequence of loglinear models. In that family of test statistics the parameters are estimated using the minimum φ-divergence estimator which is a generalization of the maximum likelihood estimator. In this paper we study the minimum power-divergence estimator (the most important family of minimum φ-divergence estimator) for a nested sequence of loglinear models in three-way contingency tables under assumptions of multinomial sampling. A simulation study illustrates that the minimum chi-squared estimator is simultaneously the most robust and efficient estimator among the family of the minimum power-divergence estimator.     

An empirical investigation of different operating characteristics of several estimators of the intraclass correlation in the analysis of binary data

This paper is concerned with properties (bias, standard deviation, mean square error and efficiency) of twenty six estimators of the intraclass correlation in the analysis of binary data. Our main interest is to study these properties when data are generated from different distributions. For data generation we considered three over-dispersed binomial distributions, namely, the beta-binomial distribution, the probit normal binomial distribution and a mixture of two binomial distributions. The findings regarding bias, standard deviation and mean squared error of all these estimators, are that (a) in general, the distributions of biases of most of the estimators are negatively skewed. The biases are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution; (b) the standard deviations are smallest when data are generated from the beta-binomial distribution; and (c) the mean squared errors are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution. Of the 26, nine estimators including the maximum likelihood estimator, an estimator based on the optimal quadratic estimating equations of Crowder (1987), and an analysis of variance type estimator is found to have least amount of bias, standard deviation and mean squared error. Also, the distributions of the bias, standard deviation and mean squared error for each of these estimators are, in general, more symmetric than those of the other estimators. Our findings regarding efficiency are that the estimator based on the optimal quadratic estimating equations has consistently high efficiency and least variability in the efficiency results. In the important range in which the intraclass correlation is small (≤0 5), on the average, this estimator shows best efficiency performance. The analysis of variance type estimator seems to do well for larger values of the intraclass correlation. In general, the estimator based on the optimal quadratic estimating equations seems to show best efficiency performance for data from the beta-binomial distribution and the probit normal binomial distribution, and the analysis of variance type estimator seems to do well for data from the mixture distribution.     

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.     

On the use of the Stein variance estimator in the double <Emphasis Type="Italic">k</Emphasis>-class estimator when each individual regression coefficient is estimated

In this paper we consider the double k-class estimator which incorporates the Stein variance estimator. This estimator is called the SVKK estimator. We derive the explicit formula for the mean squared error (MSE) of the SVKK estimator for each individual regression coefficient. It is shown analytically that the MSE performance of the Stein-rule estimator for each individual regression coefficient can be improved by utilizing the Stein variance estimator. Also, MSE’s of several estimators included in a family of the SVKK estimators are compared by numerical evaluations.     

A minimum variance kernel estimator and a discrete frequency polygon estimator for ordinal contingency tables

This paper introduces two estimators, a boundary corrected minimum variance kernel estimator based on a uniform kernel and a discrete frequency polygon estimator, for the cell probabilities of ordinal contingency tables. Simulation results show that the minimum variance boundary kernel estimator has a smaller average sum of squared error than the existing boundary kernel estimators. The discrete frequency polygon estimator is simple and easy to interpret, and it is competitive with the minimum variance boundary kernel estimator. It is proved that both estimators have an optimal rate of convergence in terms of mean sum of squared error, The estimators are also defined for high-dimensional tables.     

Estimation of a Gamma Scale Parameter Under Asymmetric Squared Log Error Loss

Abstract

Estimation of scale parameter under the squared log error loss function is considered with restriction to the principle of invariance and risk unbiasedness. An explicit form of minimum risk scale-equivariant estimator under this loss is obtained. The admissibility and inadmissibility of a class of linear estimators of the form (cT + d) are considered, where T follows a gamma distribution with an unknown scale parameter η and a known shape parameter ν. This includes the admissibility of the minimum risk equivariant estimator on η (MRE).     

PMSE performance of two different types of preliminary test estimators under a multivariate t error term

Abstract

In this paper, assuming that the error terms follow a multivariate t distribution, we derive the exact formula for the predictive mean squared error (PMSE) of two different types of pretest estimators. It is shown analytically that one of the pretest estimator dominates the SR estimator if a critical value of the pretest is chosen appropriately. Also, we compare the PMSE of the pretest estimators with the MMSE, AMMSE, SR and PSR estimators by numerical evaluations. Our results show that the pretest estimators dominate the OLS estimator for all combinations when the degrees of freedom is not more than 5.     

A family of shrinkage estimators for Weibull shape parameter in censored sampling

In this paper a new class of shrinkage estimators has been introduced for the shape parameter in an independently identically distributed two-parameterWeibull model under censored sampling. The main idea is to incorporate the prior guessed value by correcting the standard estimator, which is essentially an unbiased estimator, with optimally weighted ratios of the guessed value and the standard estimator, instead of considering a convex combination of the standard estimator and the difference of the guessed value and the standard estimator. The resulting estimator dominates the standard estimator in a surprisingly large neighborhood of the guessed value. The suggested estimator has also been compared with the minimum mean squared error estimator and a class of estimators suggested by Singh and Shukla in IAPQR Trans 25(2), 107–118, 2000. It is found that the suggested class of estimators has lesser bias as well as lesser mean squared error than its competitors subject to certain conditions.      

On the stability of the jackknife variance estimator in ratio estimation

The stability of a slightly modified version of the usual jackknife variance estimator is evaluated exactly in small samples under a suitable linear regression model and compared with that of two different linearization variance estimators. Depending on the degree of heteroscedasticity of the error variance in the model, the stability of the jackknife variance estimator is found to be somewhat comparable to that of one or the other of the linearization variance estimators under conditions especially favorable to ratio estimation (i.e., regression approximately through the origin with a relatively small coefficient of variation in the x population). When these conditions do not hold, however, the jackknife variance estimator is found to be less stable than either of the linearization variance estimators.     

Developing a restricted two-parameter Liu-type estimator: A comparison of restricted estimators in the binary logistic regression model

In the context of estimating regression coefficients of an ill-conditioned binary logistic regression model, we develop a new biased estimator having two parameters for estimating the regression vector parameter β when it is subjected to lie in the linear subspace restriction Hβ = h. The matrix mean squared error and mean squared error (MSE) functions of these newly defined estimators are derived. Moreover, a method to choose the two parameters is proposed. Then, the performance of the proposed estimator is compared to that of the restricted maximum likelihood estimator and some other existing estimators in the sense of MSE via a Monte Carlo simulation study. According to the simulation results, the performance of the estimators depends on the sample size, number of explanatory variables, and degree of correlation. The superiority region of our proposed estimator is identified based on the biasing parameters, numerically. It is concluded that the new estimator is superior to the others in most of the situations considered and it is recommended to the researchers.     

A new stochastic mixed Liu estimator in linear regression model

Abstract

To overcome multicollinearity, a new stochastic mixed Liu estimator is presented and its efficiency is considered. We also compare the proposed estimators in the sense of matrix mean squared error criteria. Finally a numerical example and a simulation study are given to show the performance of the estimators.