Bounds for the variance of the graybill-deal estimator of the common mean of two normal distributions

In this note we derive sharp lower and upper bounds for the variance of the Graybill-Deal estimator of the common mean of two normal distributions with unknown variances when the sample sizes are not necessarily equal. We also derive similar bounds for the variance of the Brown-Cohen (1974) T _a(1) class of unbiased es-timators to which the Graybill-Deal estimator belongs. Further, we illustrate the sharpness of the bounds by numerical computations in the case of the Graybill-Deal estimator.     

A note on the common mean of two normal populations with order restricted variances

Consider the problem of estimating the common mean of two normal populations when the order of the unknown variances is known. In this article we have constructed a simple improved estimator which is better than the usual Graybill-Deal estimator in terms of stochastic domination.     

Unbiased estimation of the variance of the Graybill-Deal estimator of the common mean of several normal populations

An identity for the chi-squared distribution is used to derive an unbiased estimator of the variance of the familiar Graybill-Deal (1959) estimator of the common mean of several normal populations with possibly different unknown variances. This result appears to be new. It is observed that the unbiased estimator is a convergent series whose suitable truncation allows unbiased estimation up to any desired degree of accuracy.     

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.     

On estimating the common mean in two normal distributions after a preliminary test for equality of variances

Given two random samples of equal size from two normal distributions with common mean but possibly different variances, we examine the sampling performance of the pre-test estimator for the common mean after a preliminary test for equality of variances. It is shown that when the alternative in the pretest is one-sided, the Graybill-Deal estimator is dominated by the pre-test estimator if the critical value is chosen appropriately. It is also shown that all estimators, the grand mean, the Graybill-Deal estimator and the pre-test estimator, are admissible when the alternative in the pre-test is two-sided. The optimal critical values in the two-sided pre-test are sought based on the minimax regret and the minimum average risk criteria, and it is shown that the Graybill-Deal estimator is most preferable under the minimum average risk criterion when the alternative in the pre-test is two-sided.     

Classification into two normal populations with a common mean and unequal variances

The problem of classification into two univariate normal populations with a common mean is considered. Several classification rules are proposed based on efficient estimators of the common mean. Detailed numerical comparisons of probabilities of misclassifications using these rules have been carried out. It is shown that the classification rule based on the Graybill-Deal estimator of the common mean performs the best. Classification rules are also proposed for the case when variances are assumed to be ordered. Comparison of these rules with the rule based on the Graybill-Deal estimator has been done with respect to individual probabilities of misclassification.     

Smooth estimation of a monotone density

We investigate the interplay of smoothness and monotonicity assumptions when estimating a density from a sample of observations. The nonparametric maximum likelihood estimator of a decreasing density on the positive half line attains a rate of convergence of [Formula: See Text] at a fixed point t if the density has a negative derivative at t. The same rate is obtained by a kernel estimator of bandwidth [Formula: See Text], but the limit distributions are different. If the density is both differentiable at t and known to be monotone, then a third estimator is obtained by isotonization of a kernel estimator. We show that this again attains the rate of convergence [Formula: See Text], and compare the limit distributions of the three types of estimators. It is shown that both isotonization and smoothing lead to a more concentrated limit distribution and we study the dependence on the proportionality constant in the bandwidth. We also show that isotonization does not change the limit behaviour of a kernel estimator with a bandwidth larger than [Formula: See Text], in the case that the density is known to have more than one derivative.     

Estimation of the mean of an exponential distribution in the presence of an outlier

Kale and Sinha (1971) have found an estimator of the mean of an exponential distribution in the présence of an outlying observation with higher expected value. Here an alternative estimator of the mean is proposed and it is compared with the estimator of Kale and Sinha (1971) and the maximum likelihood estimator given by Kale (1975). The proposed estimator is found to be more efficient than the latter two estimators in some cases.     

Improving the Efficiency of the Nelson–Aalen Estimator: the Naive Local Constant Estimator

Abstract.  The Nelson–Aalen estimator is well known to be an asymptotically efficient estimator of the cumulative hazard function, see Andersen et al. ( Statistical models based on counting processes , Springer-Verlag, New York, 1993) among many others. In this paper, we show that the efficiency of the Nelson–Aalen estimator can be considerably improved by using more information in the estimation process than the traditional Nelson–Aalen estimator uses. While our approach results in a biased estimator, the variance improvement is substantial. By optimizing the balance between the bias loss and the variance improvement, we obtain results on the efficiency gain. Several examples for known failure time distributions are used to illustrate these ideas.     

Estimation of variance of mean using known coefficient of variation

An optimum unbiased estimator of the variance of mean is given It is defined as a function of the mean and itscustomary unbiased variance estimator, utilizing known coefficient of variation, skewness and kurtosis of the underlying distributions. Exact results are obtained. Normal and large sample cases receive particular treatment. The proposed variance estimator is generally more efficient than the customary variance estimator; its relative efficiency becomes appreciably higher for smaller coefficient of variation, smaller sample (in the normal case at least), higher negative skewness, or higher positive skewness with sufficiently large kurtosis. The empirical findings are reassuring and supportive.     

Estimation of finite population variance in successive sampling using multi-auxiliary variables

ABSTRACT

In this paper, a general class of estimators for estimating the finite population variance in successive sampling on two occasions using multi-auxiliary variables has been proposed. The expression of variance has also been derived. Further, it has been shown that the proposed general class of estimators is more efficient than the usual variance estimator and the class of variance estimators proposed by Singh et al. (2011) when we used more than one auxiliary variable. In addition, we support this with the aid of numerical illustration.     

Asymptotics for an Adaptive Trimmed Likelihood Location Estimator

An asymptotic normality result is given for an adaptive trimmed likelihood estimator of location, which parallels the asymptotic normality result for the adaptive trimmed mean. The new result comes out of studying the adaptive trimmed likelihood estimator modelled parametrically by a normal family but then examining the behavior when the underlying distribution is in fact some F different from normal. The asymptotic variance of the adaptive estimator is equal to the asymptotic variance of the trimmed likelihood estimator at the optimal trimming proportion for the distribution F, subject to that trimming proportion being positive and F being suitably smooth.     

Estimating variances of strata in ranked set sampling

In RSS, the variance of observations in each ranked set plays an important role in finding an optimal design for unbalanced RSS and in inferring the population mean. The empirical estimator (i.e., the sample variance in a given ranked set) is most commonly used for estimating the variance in the literature. However, the empirical estimator does not use the information in the entire data over different ranked sets. Further, it is highly variable when the sample size is not large enough, as is typical in RSS applications. In this paper, we propose a plug-in estimator for the variance of each set, which is more efficient than the empirical one. The estimator uses a result in order statistics which characterizes the cumulative distribution function (CDF) of the rth order statistics as a function of the population CDF. We analytically prove the asymptotic normality of the proposed estimator. We further apply it to estimate the standard error of the RSS mean estimator. Both our simulation and empirical study show that our estimators consistently outperform existing methods.     

Adaptive estimators for parameters of the autoregression function of a Markov chain

Suppose we observe an ergodic Markov chain on the real line, with a parametric model for the autoregression function, i.e. the conditional mean of the transition distribution. If one specifies, in addition, a parametric model for the conditional variance, one can define a simple estimator for the parameter, the maximum quasi-likelihood estimator. It is robust against misspecification of the conditional variance, but not efficient. We construct an estimator which is adaptive in the sense that it is efficient if the conditional variance is misspecified, and asymptotically as good as the maximum quasi-likelihood estimator if the conditional variance is correctly specified. The adaptive estimator is a weighted nonlinear least-squares estimator, with weights given by predictors for the conditional variance.     

A new ranked set sample estimator of variance

Summary. We develop an unbiased estimator of the variance of a population based on a ranked set sample. We show that this new estimator is better than estimating the variance based on a simple random sample and more efficient than the estimator based on a ranked set sample proposed by Stokes. Also, a test to determine the effectiveness of the judgment ordering process is proposed.     

Quantile Estimation Using Ranked Set Samples from a Population with Known Mean

Ranked set sampling (RSS) is a cost-efficient technique for data collection when the units in a population can be easily judgment ranked by any cheap method other than actual measurements. Using auxiliary information in developing statistical procedures for inference about different population characteristics is a well-known approach. In this work, we deal with quantile estimation from a population with known mean when data are obtained according to RSS scheme. Through the simple device of mean-correction (subtract off the sample mean and add on the known population mean), a modified estimator is constructed from the standard quantile estimator. Asymptotic normality of the new estimator and its asymptotic efficiency relative to the original estimator are derived. Simulation results for several underlying distributions show that the proposed estimator is more efficient than the traditional one.     

Trace of the Variance-Covariance Matrix in Natural Exponential Families

In this article, we introduce the notion of trace variance function which is the trace of the variance-covariance matrix. Under some conditions, we prove that this trace variance function characterizes the Natural Exponential Family (NEF). We apply this characterization in order to estimate the distribution which belongs to some NEFs. Therefore, we introduce the estimator of this trace variance function. We give the asymptotic properties of this estimator. Finally, we illustrate our results using a simulation study.     

Can efficiency be gained by correcting for misclassification?

This paper considers 2×2 tables arising from case–control studies in which the binary exposure may be misclassified. We found circumstances under which the inverse matrix method provides a more efficient odds ratio estimator than the naive estimator. We provide some intuition for the findings, and also provide a formula for obtaining the minimum size of a validation study such that the variance of the odds ratio estimator from the inverse matrix method is smaller than that of the naive estimator, thereby ensuring an advantage for the misclassification corrected result. As a corollary of this result, we show that correcting for misclassification does not necessarily lead to a widening of the confidence intervals, but, rather, in addition to producing a consistent estimate, can also produce one that is more efficient.     

Estimating a uniform distribution when data are measured with a normal additive error with unknown variance

The problem of estimating the width of a symmetric uniform distribution on the line together with the error variance, when data are measured with normal additive error, is considered. The main purpose is to analyse the maximum-likelihood (ML) estimator and to compare it with the moment-method estimator. It is shown that this two-parameter model is regular so that the ML estimator is asymptotically efficient. Necessary and sufficient conditions are given for the existence of the ML estimator. As numerical problems are known to frequently occur while computing the ML estimator in this model, useful suggestions for computing the ML estimator are also given.     

A note on calibration weightings for stratified double sampling with equal probability

This study proposes a more efficient calibration estimator for estimating population mean in stratified double sampling using new calibration weights. The variance of the proposed calibration estimator has been derived under large sample approximation. Calibration asymptotic optimum estimator and its approximate variance estimator are derived for the proposed calibration estimator and existing calibration estimators in stratified double sampling. Analytical results showed that the proposed calibration estimator is more efficient than existing members of its class in stratified double sampling. Analysis and evaluation are presented.