Improved estimators of common variance of <Emphasis Type="Italic">p</Emphasis>-populations when Kurtosis is known

The pooled variance of p samples presumed to have been obtained from p populations having common variance σ², has invariably been adopted as the default estimator for σ². In this paper, alternative estimators of the common population variance are developed. These estimators are biased and have lower mean-squared error values than . The comparative merit of these estimators over the unbiased estimator is explored using relative efficiency (a ratio of mean-squared error values).     

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.     

Estimation of the population mean when the coefficient of variation is known

Theory has been developed to provide an optimum estimator of the population mean based on a “mean per unit” estimator and the estimated standard deviation, assuming that the form of the distribution as well as its coefficient of variation (c.v.) are known. Theory has been extended to the case when an estimate of c.v. is available from an independent sample drawn in the past; the case when the form of the distribution is not known is also discussed. It is shown that the relative efficiency of the estimator with respect to “mean per unit estimator” is generally high for normal or near normal populations. For log-normal populations, an increase in efficiency of about 17 percent can be achieved. The results have been illustrated with data from biological populations.     

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.     

Estimation of parameters in heavy-tailed distribution when its second order tail parameter is known

Estimating parameters in heavy-tailed distribution plays a central role in extreme value theory. It is well known that classical estimators based on the first order asymptotics such as the Hill, rank-based and QQ estimators are seriously biased under finer second order regular variation framework. To reduce the bias, many authors proposed the so-called second order reduced bias estimators for both first and second order tail parameters. In this work, estimation of parameters in heavy-tailed distributions are studied under the second order regular variation framework when the second order parameter in the distribution tail is known. This is motivated in large part by a recent work by the authors showing that the second order tail parameter is known for a large class of popular random difference equations (for example, ARCH models). The focus is on least squares estimators that generalize rank-based and QQ estimators. Though other possible estimators are also briefly discussed, the least squares estimators are most simple to use and perform best for finite samples in Monte Carlo simulations.     

Comparison of the Stein and the usual estimators for the regression error variance under the Pitman nearness criterion when variables are omitted

This paper compares the Stein and the usual estimators of the error variance under the Pitman nearness (PN) criterion in a regression model which is mis-specified due to missing relevant explanatory variables. The exact expression of the PN-probability is derived and numerically evaluated. Contrary to the well-known result under mean squared errors (MSE), with the PN criterion the Stein variance estimator is uniformly dominated by the usual estimator when no relevant variables are excluded from the model. With an increased degree of model mis-specification, neither estimator strictly dominates the other. The authors are grateful to two anonymous referees for their valuable comments. Also, the first author is grateful to the Japan Society for the Promotion of Science for partial financial support.     

Estimation of variance of a normal population

In this paper we have suggested two estimators of variance of a normal population developed from the estimators of u² suggested by Govindarajulu and Sahai and Das. These have been shown to be more efficient than the usual estimator s².     

An approximation to the distribution of the sample variance

A simple three-moment approximation is introduced for the distribution of the sample variance. Comparisons are given with other approximations discussed by Tan and Wong (1977) and with an approximation developed very recently by Mudholkar and Trivedi (1981).     

Estimating powers of the generalized variance under the Pitman closeness criterion

For estimating powers of the generalized variance under a multivariate normal distribution with an unknown mean, the inadmissibility of the closest affine equivariant estimator is shown for the Pitman closeness criterion.     

A note on the estimation of variance of sample mean Using the knowledge of coefficient of variation in normal population

A class of estimators for the variance of sample mean is defined and its properties are studied in case of normal population. It is identified that the usual unbiased estimator, Singh, Pandey and Hirano (1973) -type estimator and Lee (1931) estimator are particular members of the proposed class of estimators. It is found that the minimum Mean Squared Error (MSE) of the proposed class of estimators is less than that of other estimators.     

On the optimality of the sample variance

The present paper explores the structure of linear exponential families for which the sample variance is a uniformly minimum variance unbiased estimator.     

Estimation of Long Memory in Integrated Variance

A stylized fact is that realized variance has long memory. We show that, when the instantaneous volatility is a long memory process of order d, the integrated variance is characterized by the same long-range dependence. We prove that the spectral density of realized variance is given by the sum of the spectral density of the integrated variance plus that of a measurement error, due to the sparse sampling and market microstructure noise. Hence, the realized volatility has the same degree of long memory as the integrated variance. The additional term in the spectral density induces a finite-sample bias in the semiparametric estimates of the long memory. A Monte Carlo simulation provides evidence that the corrected local Whittle estimator of Hurvich et al. (2005 Hurvich , C. M. , Moulines , E. , Soulier , P. ( 2005 ). Estimating long memory in volatility . Econometrica 73 ( 4 ): 1283 – 1328 .[Crossref], [Web of Science ®] , [Google Scholar]) is much less biased than the standard local Whittle estimator and the empirical application shows that it is robust to the choice of the sampling frequency used to compute the realized variance. Finally, the empirical results suggest that the volatility series are more likely to be generated by a nonstationary fractional process.     

Maximum likelihood estimation for the generalized poisson distribution when sample mean is larger than sample variance

The generalized Poisson distribution (GPD), studied by many researchers and containing two parameters θ and λ, has been found to fit very well data sets arising in biological, ecological, social and marketing fields. Consul and Shoukri (1985) have shown that for negative values of λ the GPD gets truncated and the model becomes deficient; however, the truncation error becomes less than 0.0005 if the minimum number of non-zero probability classes ≥ 4 for all values of θ and λ and the GPD model can be safely used in all such cases. The problem of admissible maximum likelihood (ML) estimation when the sample mean is larger than the sample variance is considered in this paper which complements the earlier work of Consul and Shoukri (1984) on the existence of unique ML estimators of θ and λ when the sample mean is smaller than or equal to the sample variance.     

A sufficient condition for the MSE dominance of the positive-part shrinkage estimator when each individual regression coefficient is estimated in a misspecified linear regression model

In this paper, assuming that there exist omitted explanatory variables in the specified model, we derive the exact formula for the mean squared error (MSE) of a general family of shrinkage estimators for each individual regression coefficient. It is shown analytically that when our concern is to estimate each individual regression coefficient, the positive-part shrinkage estimators have smaller MSE than the original shrinkage estimators under some conditions even when the relevant regressors are omitted. Also, by numerical evaluations, we showed the effects of our theorem for several specific cases. It is shown that the positive-part shrinkage estimators have smaller MSE than the original shrinkage estimators for wide region of parameter space even when there exist omitted variables in the specified model.     

Estimation of the parameters of Downton's bivariate exponential distribution using ranked set sampling scheme

The parameters of Downton's bivariate exponential distribution are estimated based on a ranked set sample. Parametric and nonparametric methods are considered. The suggested estimators are compared to the corresponding ones based on simple random sampling. It turns out that some of the suggested estimators are significantly more efficient than the ones based on simple random sampling.     

Efficient computation of the performance of bootstrap and jackknife estimators of the variance of L-statistics

Calculation of the bootstrap and the jackknife estimators of the variance of a statistic often relies on approximation techniques because the exact values are difficult if not impossible to obtain analytically. For the special case where the statistic is a linear combination of order statistics we propose to calculate the exact values combinatorically, thus completely eliminating the second-stage simulation error.     

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.     

Estimation of Population Variance from Multi-ranker Ranked Set Sampling Designs

In this article, we develop an estimator for a population variance based on a multi-ranker ranked set sampling design. In a multi-ranker design, the units are ranked by more than one ranker allowing ties whenever the confidence level of the rankers is low. The ranking information of all rankers is then combined in a meaningful way to create a single measure. This measure is used to construct the sampling design and a new estimator for the population variance. The article investigates the bias and relative efficiency of the proposed variance estimator. It is shown that the new estimator performs as good as or better than its competitors in the literature.     

The exact distribution and density functions of the stein-type estimator for normal variance

In this paper, we derive the exact distribution and density functions of the Stein-type estimator for the normal variance. It is shown by numerical evaluation that the density function of the Stein-type estimator is unimodal and concentrates around the mode more than that of the usual estimator.