Some improved estimators for a measures of dispersion of an inverse gaussian distribution

In this paper some improved estimators for the measure of dispersion of an inverse Gaussian distribution have been obtained. If some guessed value of λ is available in the form of a point esitmate λ₀ the shrikage technique has been applied and an estimator has been proposed which has smaller mean squared error than the usual estimator. Since the shrinkage estimator has better performance if the guessed value is in the vicinity of the true value, a shrinkage testimator has also been proposed and compared with the usual 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.     

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).     

Imputation by power transformation

In this paper, a new power transformation estimator of population mean in the presence of non-response has been suggested. The estimator of mean obtained from proposed technique remains better than the estimators obtained from ratio or mean methods of imputation. The mean squared error of the resultant estimator is less than that of the estimator obtained on the basis of ratio method of imputation for the optinum choice of parameters. An estimator for estimating a parameter involved in the process of new method of imputation has been discussed. The MSE expressions for the proposed estimators have been derived analytically and compared empirically. Product method of imputation for negatively correlated variables has also been introduced. The work has been extended to the case of multi-auxiliary information to be used for imputation.     

A Class of Estimators Using Auxiliary Information for Estimating Finite Population Variance in Presence of Measurement Errors

This article addresses the problem of estimating the population variance using auxiliary information in the presence of measurement errors. When the measurement error variance associated with study variable is known, a class of estimators of the population variance using auxiliary information has been proposed. We obtain the bias and mean squared errors of the suggested class of estimators upto the terms of order n ^?1, and also optimum estimators in asymptotic sense of the class with approximate mean squared error formula.     

Bias-adjustment and calibration of jackknife variance estimator in the presence of non-response

In this paper, bias-adjustment in the jackknife estimator of variance accredited to Rao and Sitter (1995) has been considered. Then the bias-adjusted Rao and Sitter (1995) estimator has been calibrated such that its expected value under the imputing superpopulation model remains the same as the expected value of the mean squared error of the ratio estimator in the presence of non-response. A simulation study has been performed to compare the six different estimators of variance: out of them four estimators belong to Rao and Sitter (1995) and the other two proposed estimators are named as bias-adjusted and bias-adjusted-cum-calibrated estimators. The empirical relative bias and empirical relative efficiency of the two proposed estimators with respect to the four existing estimators accredited to Rao and Sitter (1995) have been investigated through simulations. The bias-adjusted-cum-calibrated estimator has been found to be an efficient estimator in the case of heteroscadastic populations. The present paper considers the situation of simple random and without replacement sampling. The possibility of obtaining a negative estimate of variance by the estimator due to Kim et al. (2006) has been pointed out.     

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.     

Mem squared error estimation in finite populations under nonlinear models

Four estimators of the prediction mean squared error (MSB) of an estimated finite population total for a zero-one characteristic are examined. The characteristic associated with each population unit is modeled as the realization of a Bernoulli random variable whose expected value is a nonlinear function of a parameter vector and a set of known auxiliary variables. To compare the estimators, a simulation study is conducted using a population of hospitals. The MSB estimator Implied by the form of the assumed model underestimates the mean squared error in each of the cases studied and produces confidence lntervals with less than the nominal coverage probabilities. Of the three alternative MSE estimators presented, a linear approximation to the jackknife produces the best results and improves upon the model-specific estimator.     

A family of efficient estimators of the finite population mean in simple random sampling

In this paper an estimator of the finite population mean using auxiliary information in sample surveys has been proposed. The bias and mean squared error are obtained under large sample approximation. It has been shown that the proposed estimator performs better than some recently published estimators.     

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.     

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.     

A comparison of estimators for the mean in the inverse gaussian distribution with a known coefficient of variation

In this paper, we discuss an estimation problem of the mean in the inverse Gaussian distribution with a known coefficient of variation. Two types of linear estimators for the mean, the linear minimum variance unbiased estimator and the linear minimum mean squared error estimator, are constructed by using the squared error loss function and their properties are examined. It is observed that, for small samples the performance of the proposed estimators is better than that of the maximum likelihood estimator, when the coefficient of variation is large.     

Robust Mean Estimation Under a Possibly Incorrect Log-Normality Assumption

Nonparametric and parametric estimators are combined to minimize the mean squared error among their linear combinations. The combined estimator is consistent and for large sample sizes has a smaller mean squared error than the nonparametric estimator when the parametric assumption is violated. If the parametric assumption holds, the combined estimator has a smaller MSE than the parametric estimator. Our simulation examples focus on mean estimation when data may follow a lognormal distribution, or can be a mixture with an exponential or a uniform distribution. Motivating examples illustrate possible application areas.     

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.     

Improved Empirical Bayes Estimation in Group Testing Procedure for Small Proportions

Group testing has been long recognized as an efficient method to classify all the experimental units into two mutually exclusive categories: defective or not defective. In recent years, more attention has been brought to the estimation of the population prevalence rate p of a disease, or of some property, using group testing. In this article, we propose two scaled squared-error loss functions, which improve the Bayesian approach to estimating p in terms of minimizing the mean squared error (MSE) of the Bayes estimators of p for small p. We show that the new estimators are preferred over the estimator from the usual squared-error loss function and the maximum likelihood estimator (MLE) for small p.     

Caliper pair-matching on a continuous variable in case-control studies

The use of matched pairs has been criticized as being less efficient than estimators based on random samples. This paper compares the mean square error of an analysis of covariance estimator based on random samples with two estimators based on caliper matched pairs. The first of these is a simple mean difference estimator and the second a regression estimator suggested by Rubin (1973b). Under conditions which commonly occur in epidemiologic case-control studies, both of the matched pair estimators can have smaller mean square errors than analysis o f covariance estimator. When there is a weak relationship between the matching and response variate, the mean difference estimator has a lower mean square error than the regression estimator.     

One-sided trimming in small samples with asymmetric contamination

An estimator for location, given a sample of only four or five observations, is proposed. The underlying distribution on of the sample may (with probability p) be contaminated by an outlier from a rightly-skewed distribution. The estimator minimizes the maximum mean squared error over all values of p. In fact, there exists an estimator which is unbiased in both the outlier - free and extreme-outlier cases, but its mean square error is substantially higher than the mean squared error for the minimax estimator. Mean squared errors for various underlying distributional situations are calculated and compared with those of other location estimators such as the mean and the median.     

Second-order asymptotic comparison of estimators under φ-divergence loss

Our main concern is on the second-order asymptotic optimality problem of estimators. The φ-divergence loss is used as a criterion for evaluating the performance of estimators. In the comparison problem of any two estimators, the condition that one estimator dominates another estimator under the φ-divergence risk is given by evaluating the second-order term in the difference between the risks. As a result, it is proved that the condition is characterized by a peculiar value of the φ-divergence loss, which is called the divergence-loss coefficient. Furthermore, it is shown that the comparison based on the φ-divergence loss does not correspond with that based on any standard loss functions including the mean squared error, the absolute loss and the 0-1 loss. In addition, a necessary and sufficient condition for an estimator to be second-order admissible is derived.     

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.