Robust estimation of variance components

New robust estimates for variance components are introduced. Two simple models are considered: the balanced one-way classification model with a random factor and the balanced mixed model with one random factor and one fixed factor. However, the method of estimation proposed can be extended to more complex models. The new method of estimation we propose is based on the relationship between the variance components and the coefficients of the least-mean-squared-error predictor between two observations of the same group. This relationship enables us to transform the problem of estimating the variance components into the problem of estimating the coefficients of a simple linear regression model. The variance-component estimators derived from the least-squares regression estimates are shown to coincide with the maximum-likelihood estimates. Robust estimates of the variance components can be obtained by replacing the least-squares estimates by robust regression estimates. In particular, a Monte Carlo study shows that for outlier-contaminated normal samples, the estimates of variance components derived from GM regression estimates and the derived test outperform other robust procedures.     

On shrinkage estimation of normal population variance

Some shrunken estimators of the normal population variance 2 are proposed and compared with the usual estimator, s², in terms of mean squared error.     

Improved robust Bayes estimators of the error variance in linear models

We consider the problem of estimating the error variance in a general linear model when the error distribution is assumed to be spherically symmetric, but not necessary Gaussian. In particular we study the case of a scale mixture of Gaussians including the particularly important case of the multivariate-t distribution. Under Stein's loss, we construct a class of estimators that improve on the usual best unbiased (and best equivariant) estimator. Our class has the interesting double robustness property of being simultaneously generalized Bayes (for the same generalized prior) and minimax over the entire class of scale mixture of Gaussian distributions.     

Minimum variance unbiased estimation for modified power series distribution

In this paper we study the problem of finding the minimum variance unbiased (MVU) estimators of the functions of the para-meters of the modified power series distributions (MPSD). A theorem giving the necessary and sufficient conditions for the existence of the MVU estimators has been proved. Also, the estimators for a number of estimable functions of a parameter are obtained. Two other theorems dealing with the MVU estimation of the left truncated MPSD with unknown truncation point are also given. The particular case of the Lagrangian Poisson, the Lagrangian binomial and the Borel-Tanner distributions are considered and tables are also provided for the MVU estimators for some functions of the parameters. The variances of the estimators are also given for some cases.     

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.     

Simple heterogeneity variance estimation for meta-analysis

Summary.  A simple method of estimating the heterogeneity variance in a random-effects model for meta-analysis is proposed. The estimator that is presented is simple and easy to calculate and has improved bias compared with the most common estimator used in random-effects meta-analysis, particularly when the heterogeneity variance is moderate to large. In addition, it always yields a non-negative estimate of the heterogeneity variance, unlike some existing estimators. We find that random-effects inference about the overall effect based on this heterogeneity variance estimator is more reliable than inference using the common estimator, in terms of coverage probability for an interval estimate.     

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.     

Half-sample variance estimation

This paper studies an alternative to the jackknife variance estimator, the half-sample variance estimator. Both theoretical and Monte Carlo comparisons between the half-sample variance estimator and the jackknife variance estimator indicate that the former is better in some situations.     

Linear models and convex geometry: aspects of non-negative variance estimation 1

Geometric aspects of linear model theory are surveyed as they bear on mean estimation, or variance covariance component estimation. It is outlined that notions associated with linear subspaces suffice for those of the customary procedures which are solely based on linear, or multilinear algebra. While conceptually simple, these methods do not always respect convexity constraints which naturally arise in variance component estimation.

Previous work on negative estimates of variance is reviewed, followed by a more detailed study of the non-negative definite analogue of the MINQUE procedure. Some characterizations are proposed which are based on convex duality theory. Optimal estimators now correspond to (non-linear) projections onto closed convex cones, they are easy to visualise, but hard to compute. No ultimate solution can be recommended, instead the paper concludes with a list of open problems.     

Secondary design considerations for minimum bias estimation

Several authors have suggested the method of minimum bias estimation for estimating response surfaces. The minimum bias estimation procedure achieves minimum average squared bias of the fitted model without depending on the values of the unknown parameters of the true surface. The only requirement is that the design satisfies a simple estimability condition. Subject to providing minimum average squared bias, the minimum bias estimator also provides minimum average variance of ?(x) where ?(x) is the estimate of the response at the point x.

To support the estimation of the parameters in the fitted model, very little has been suggested in the way of experimental designs except to say that a full rank matrix X of independent variables should be used. This paper presents a closer look at the estimability conditions that are required for minimum bias estimation, and from the form of the matrix X, a formula is derived which measures the amount of design flexibility available. The design flexibility is termed “the degrees of freedom” of the X matrix and it is shown how the degrees of freedom can be used to decide if other design optimality criteria might be considered along with minimum bias estimation. Several examples are provided.     

Improved estimation of variance components in mixed models

Taking Albert's (1976) formulation of a mixed model ANOVA, we consider improved estimation of the variance components for balanced designs under squared error loss. Two approaches are presented. One extends the ideas of Stein (1964), The other is developed from the fact that variance components can be expressed as linear combinations of chi-square scale parameters. Encouraging simulation results are presented.     

On the estimation of variance components in stability analysis

Chow and Shao (1989, 1991) indicated that the presence of batch-to-batch variation has an impact on the determination of drug shelf-life in stability studies. In this paper, we propose two unbiased estmators for batch-to-batch variation. The proposed estimators are compared in terms of their corresponding variances. An example concerning a stability study is discussed to illustrate the use of the proposed estimators.     

Error variance function estimation in nonparametric regression models

In this article, a new class of variance function estimators is proposed in the setting of heteroscedastic nonparametric regression models. To obtain a variance function estimator, the main proposal is to smooth the product of the response variable and residuals as opposed to the squared residuals. The asymptotic properties of the proposed methodology are investigated in order to compare its asymptotic behavior with that of the existing methods. The finite sample performance of the proposed estimator is studied through simulation studies. The effect of the curvature of the mean function on its finite sample behavior is also discussed.     

A unified approach to minimum variance unbiased estimation of a probability function belonging to an exponential family

The problem of minimum variance unbiased estimation of the probability density function of a random variable belonging to an exponential family is considered. The method of estimation proposed in this paper requires the solution of a certain integral equation. For many probability distributions the solution of this equation is given by a known result in integral transform theory.     

C293. A note on using a hypothesis-dependent variance estimate

Detecting the number of signals and estimating the parameters of the signals is an important problem in signal processing. Quite a number of papers appeared in the last twenty years regarding the estimation of the parameters of the sinusoidal components but not that much of attention has been given in estimating the number of terms present in a sinusoidal signal. Fuchs developed a criterion based on the perturbation analysis of the data auto correlation matrix to estimate the number of sinusoids, which is in some sense a subjective-based method. Recently Reddy and Biradar proposed two criteria based on AIC and MDL and developed an analytical framework for analyzing the performance of these criteria. In this paper we develop a method using the extended order modelling and singular value decomposition technique similar to that of Reddy and Biradar. We use penalty function technique but instead of using any fixed penalty function like AIC or MDL, a class of penalty functions satisfying some special properties has been used. We prove that any penalty function from that special class will give consistent estimate under the assumptions that the error random variables are independent and identically distributed with mean zero and finite variance. We also obtain the probabilities of wrong detection for any particular penalty function under somewhat weaker assumptions than that of Reddy and Biradar of Kaveh et al. It gives some idea to choose the proper penalty function for any particular model. Simulations are performed to verify the usefulness of the analysis and to compare our methods with the existing ones.     

Statistical estimation for a failure model with damage accumulation in a case of small samples

A failure model with damage accumulation is considered. Damages occur according to a Poisson process and they degenerate into failures in a random time. The rate of the Poisson process and the degeneration time distribution are unknown. Two sample populations are available: a sample of intervals between damages and a sample of degeneration times. The case of small samples is considered. The purpose is to estimate the expectation and the distribution of the number of damages and failures at time t. We consider the plug-in and resampling estimators of the above mentioned characteristics. The expectations and variances of the suggested estimators are investigated. The numerical examples show that the resampling estimator has some advantages.     

Maximum likelihood estimation for the beta distribution

A fast method of calculating the two-parameter maximum-likelihood estimates of the beta distribution is given which does not require starting values and is generally free from convergence problems.     

Conditional variance estimation in heteroscedastic regression models

First, we propose a new method for estimating the conditional variance in heteroscedasticity regression models. For heavy tailed innovations, this method is in general more efficient than either of the local linear and local likelihood estimators. Secondly, we apply a variance reduction technique to improve the inference for the conditional variance. The proposed methods are investigated through their asymptotic distributions and numerical performances.     

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.