Comments on the paper “Bias-adjustment and calibration of jackknife variance estimator in the presence of non-response”

Singh and Arnab (2010) presented a bias adjustment to the jackknife variance estimator of Rao and Sitter (1995) in the presence of non-response. In their paper, they obtained a second-order approximation of the bias of the Rao-Sitter variance estimator and then proposed a bias-adjusted estimator based on this approximation. To compare their proposed variance estimator to various other variance estimators, they performed a simulation study and showed that their variance estimator is superior to the Rao-Sitter variance estimator. In fact they showed that the Rao-Sitter variance estimator suffers from severe underestimation. These results contradict those in the literature, which indicate that the Rao-Sitter variance estimator suffers from a positive bias if the sampling fractions are not negligible; see Rao and Sitter (1995), Lee et al. (1995) and Haziza and Picard (2011). Because of this contradiction, we felt that a further investigation was warranted. In this paper, we attempt to recreate the results of Singh and Arnab (2010) and, in fact, show that their second order approximation to the bias of the Rao-Sitter variance estimator is incorrect and that their simulation results are also questionable.     

Deviation of order p for estimators of the variance in first-order stochastic differential equation (SDE)

In this work, we consider a non-parametric estimator of the variance in one-dimensional diffusion models or, more generally, in Itô processes with a deterministic diffusion term and a general non-anticipative drift. The estimation is based on the quadratic variation of discrete time observations over a finite interval. In particular, a central limit theorem (CLT) is proved for the deviation in L ^p norm (p≥; 1) between the variance and this estimator. The method of the proof consists in writing the L ^p norm of the deviation, when the drift term is equal to zero, as a sum of 4-dependent random variables. The moments are then computed by means of a Gaussian approximation and a CLT for m-dependent random variables is applied. The convergence is stable in law, this allows the result for processes with general drifts to be obtained, by using Girsanov's formula.     

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.     

On the UMVU estimator for the generalized variance of the multinomial family

Abstract

The generalized variance is an important statistical indicator which appears in a number of statistical topics. It is a successful measure for multivariate data concentration. In this article, we established, in a closed form, the bias of the generalized variance maximum likelihood estimator of the Multinomial family. We also derived, with a complete proof, the uniformly minimum variance unbiased estimator (UMVU) for the generalized variance of this family. These results rely on explicit calculations, the completeness of the exponential family and the Lehmann–Scheffé theorem.     

Second-order variance estimation in poststratified two-stage sampling

The linearization or Taylor series variance estimator and jackknife linearization variance estimator are popular for poststratified point estimators. In this note we propose a simple second-order linearization variance estimator for the poststratified estimator of the population total in two-stage sampling, using the second-order Taylor series expansion. We investigate the properties of the proposed variance estimator and its modified version and their empirical performance through some simulation studies in comparison to the standard and jackknife linearization variance estimators. Simulation studies are carried out on both artificially generated data and real data.     

A generalized improvement procedure for variance bounds for minimum variance unbiased estimator in inverse sampling

New bounds are obtained for the variance of the minimum variance unbiased estimator of p i n inverse sampling. A generalized procedure for further improving the bounds is also discussed.     

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.     

Nonparametric measures of intraclass correlation

Three nonparametric measures of intraclass correlation based on the notion of concordance are considered. Their unbiased estimators and nonparametric tests based on the estimators are studied and it is shown that an analogue of the Kendall's tau provides small variance estimator and relatively powerful test. Furthermore, the approximate variance of the estimator is given when the correlation is small in the normal model.     

A note on the greenwood variance estimator in cohort life tables: Isolating the effect of one source of bias

A modification of the Greenwood variance estimator is defined and shown to be free of bias whenever its constitu­ent interval estimators are conditionally unbiased, given the sample size at the start of the interval. Using the modified estimator as a standard of comparison, the original Greenwood estimator is seen to have an intrinsic positive bias.Under­estimation of variances through the use of Greenwood's formula must be due to bias in the constituent interval estimators and/or, with fixed interval bounds, due to disregarding the random character of the total number of life table intervals to exhaustion of ttje sample. Some easy to prove properties of the modified and the original Greenwood estimators are stated that apply in the absence of censoring. A suggest­ion is made for reducing the bias of the interval variance estimators.     

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 the variance when kurtosis is known

The unbiased estimator of a population variance σ², S ² has traditionally been overemphasized, regardless of sample size. In this paper, alternative estimators of population variance are developed. These estimators are biased and have the minimum possible mean-squared error [and we define them as the “minimum mean-squared error biased estimators” (MBBE)]. The comparative merit of these estimators over the unbiased estimator is explored using relative efficiency (RE) (a ratio of mean-squared error values). It is found that, across all population distributions investigated, the RE of the MBBE is much higher for small samples and progressively diminishes to 1 with increasing sample size. The paper gives two applications involving the normal and exponential distributions.     

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.     

Outlier detection using difference-based variance estimators in multiple regression

In this article, we propose an outlier detection approach in a multiple regression model using the properties of a difference-based variance estimator. This type of a difference-based variance estimator was originally used to estimate error variance in a non parametric regression model without estimating a non parametric function. This article first employed a difference-based error variance estimator to study the outlier detection problem in a multiple regression model. Our approach uses the leave-one-out type method based on difference-based error variance. The existing outlier detection approaches using the leave-one-out approach are highly affected by other outliers, while ours is not because our approach does not use the regression coefficient estimator. We compared our approach with several existing methods using a simulation study, suggesting the outperformance of our approach. The advantages of our approach are demonstrated using a real data application. Our approach can be extended to the non parametric regression model for outlier detection.     

A nonparametric measure of interclass correlation

A nonparametric measure of interclass correlation is considered and its unbiased estimator and a test based on the estimator are studied. Hie measure is an analogue of the Kendall's measure of dependence. It is shown that the variance of the estimator is small and the information loss of the test based on the estimator is not serious relative to a standard parametric test in the sense of the Pitman asymptotic relative efficiency. Furthermore, the approximate variance of the estimator is given in the normal model.     

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.     

Asymptotic properties of maximum quasi-likelihood estimator in quasi-likelihood nonlinear models with misspecified variance function

The quasi-likelihood function proposed by Wedderburn [Quasi-likelihood functions, generalized linear models, and the Gauss–Newton method. Biometrika. 1974;61:439–447] broadened the application scope of generalized linear models (GLM) by specifying the mean and variance function instead of the entire distribution. However, in many situations, complete specification of variance function in the quasi-likelihood approach may not be realistic. Following Fahrmeir's [Maximum likelihood estimation in misspecified generalized linear models. Statistics. 1990;21:487–502] treating with misspecified GLM, we define a quasi-likelihood nonlinear models (QLNM) with misspecified variance function by replacing the unknown variance function with a known function. In this paper, we propose some mild regularity conditions, under which the existence and the asymptotic normality of the maximum quasi-likelihood estimator (MQLE) are obtained in QLNM with misspecified variance function. We suggest computing MQLE of unknown parameter in QLNM with misspecified variance function by the Gauss–Newton iteration procedure and show it to work well in a simulation study.     

Difference-based variance estimator for nonparametric regression in complex surveys

A difference-based variance estimator is proposed for nonparametric regression in complex surveys. By using a combined inference framework, the estimator is shown to be asymptotically normal and to converge to the true variance at a parametric rate. Simulation studies show that the proposed variance estimator works well for complex survey data and also reveals some finite sample properties of the estimator.     

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.     

Recovery of inter-block information: extensions in a two variance component model

For a two variance component mixed linear model, it is shown that under suitable conditions there exists a nonlinear unbiased estimator that is better than a best linear unbiased estimator defined with respect to a given singular covariance matrix. It is also shown how this result applies to improving on intra-block estimators and on estimators like the unweighted means estimator in a random one-way model.     

On generalized variance of product of powered components and multiple stable Tweedie models

A flexible family of multivariate models, named multiple stable Tweedie (MST) models, is introduced and produces generalized variance functions which are products of powered components of the mean. These MST models are built from a fixed univariate stable Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are also real independent Tweedie variables, with the same dispersion parameter equal to the fixed component. In this huge family of MST models, generalized variance estimators are explicitly pointed out by maximum likelihood method and, moreover, computably presented for the uniform minimum variance and unbiased approach. The second estimator is brought from modified Lévy measures of MST which lead to some solutions of particular Monge–Ampère equations.