An overview of variance component estimation

Variance components estimation originated with estimating error variance in analysis of variance by equating error mean square to its expected value. This equating procedure was then extended to random effects models, first for balanced data (for which minimum variance properties were subsequently established) and later for unbalanced data. Unfortunately, this ANOVA methodology yields no optimum properties (other than unbiasedness) for estimation from unbalanced data. Today it is being replaced by maximum likelihood (ML) and restricted maximum likelihood (REML) based on normality assumptions and involving nonlinear equations that have to be solved numerically. There is also minimum norm quadratic unbiased estimation (MINQUE) which is closely related to REML but with fewer advantages.An invited paper for the ProbaStat '94 conference, Smolenice, Slovakia, May 30–June 3, 1994 Paper number BU-677 in the Biometrics Unit. Cornell University Ithaca NY     

Realized range-based estimation of integrated variance

We provide a set of probabilistic laws for estimating the quadratic variation of continuous semimartingales with the realized range-based variance—a statistic that replaces every squared return of the realized variance with a normalized squared range. If the entire sample path of the process is available, and under a set of weak conditions, our statistic is consistent and has a mixed Gaussian limit, whose precision is five times greater than that of the realized variance. In practice, of course, inference is drawn from discrete data and true ranges are unobserved, leading to downward bias. We solve this problem to get a consistent, mixed normal estimator, irrespective of non-trading effects. This estimator has varying degrees of efficiency over realized variance, depending on how many observations that are used to construct the high–low. The methodology is applied to TAQ data and compared with realized variance. Our findings suggest that the empirical path of quadratic variation is also estimated better with the realized range-based variance.     

On estimation of variance components*

In this survey paper the estimation of variance components is given. The least squares approach in variance component estimation is a unifying principle which includes the analysis of variance estimators and the MINQUE. When normality is assumed the maximum likelihood estimators can be used. Many variance component estimators are not permissible because they are not non-negative. The development of non-negative variance component estimators is indicated.     

A theory of robust long-run variance estimation

Long-run variance estimation can typically be viewed as the problem of estimating the scale of a limiting continuous time Gaussian process on the unit interval. A natural benchmark model is given by a sample that consists of equally spaced observations of this limiting process. The paper analyzes the asymptotic robustness of long-run variance estimators to contaminations of this benchmark model. It is shown that any equivariant long-run variance estimator that is consistent in the benchmark model is highly fragile: there always exists a sequence of contaminated models with the same limiting behavior as the benchmark model for which the estimator converges in probability to an arbitrary positive value. A class of robust inconsistent long-run variance estimators is derived that optimally trades off asymptotic variance in the benchmark model against the largest asymptotic bias in a specific set of contaminated models.     

Model-based variance estimation under unequal probability sampling

This paper deals with a model-based variance estimation of the Horvitz–Thompson (HT) estimator when auxiliary information is available. A small simulation study is carried out to illustrate and establish some of the findings.     

Bayesian estimation of explained variance in ANOVA designs

We propose to use the squared multiple correlation coefficient as an effect size measure for experimental analysis‐of‐variance designs and to use Bayesian methods to estimate its posterior distribution. We provide the expressions for the squared multiple, semipartial, and partial correlation coefficients corresponding to four commonly used analysis‐of‐variance designs and illustrate our contribution with two worked examples.     

Stratification,midzuno-sen strategy and nonnegative unbiased variance estimation

When the ratio method is appropriate for estimating the population total one is faced with the problem of nonavailability of uniformly nonnegative unbiased variance estimators (nnuve). Here we highlight the twofold role of stratification in that it not only improves the efficiency of the ratio method of estimation but it also enhances the chances of getting uniformly nonnegative unbiased variance estimators.     

Sequential approach to simultaneous estimation of the mean and variance

The problem of simultaneous estimation of the mean and variance of a normal distribution has been studied. We propose a semi-circular region _n ={(a,b)':b>0} of radiusd, which has approximately a preassigned coverage probability. Asymptotic efficiency and asymptotic consistency (asd0) of our proposed sequential procedures have been proved.Research partially supported by U.S. Army Research Grant No. DAAG29-76-G-0038.     

Minimum variance unbiased estimation of the parameters of the Pareto distribution

Summary This paper presents a simpler proof of some results concerning estimation of the Pareto distribution due toLike [1969].     

Admissible unbiased variance estimation in finite population sampling under randomized response

LetP be the proportion of units in a finite population possessing a sensitive attribute. We prove the admissibility of (i) an unbiased estimator of the variance of a general homogeneous linear unbiased estimator ofP and (ii) an unbiased estimator of the population varianceP(1−P), based on an arbitrary but fixed sampling design, under the randomized response plans due to Warner (1965) and Eriksson (1973). Admissibility of an unbiased strategy for estimating the population variance is also established.     

Optimality of balanced designs for minimum norm quadratic unbiased estimation of variance components

This paper develops an approximate theory for the optimality of balanced designs under minimum norm quadratic unbiased estimation of variance components in one-way classified data.     

Sequential simultaneous estimation of the mean and variance: The rate of convergence

We reconsider the problem of simultaneous estimation of the mean and variance of a normal distribution along the lines of Mukhopadhyay (1981), and derive the rate of convergence of the appropriately normalized stopping time to normality. The Berry-Esseen type results for randomly stoppedU-statistics have been utilized repeatedly in this context.     

On uniformly minimum variance unbiased estimation when no complete sufficient statistics exist

Summary A simple generalization of the Lehmann-Scheffé theorem is given. It is used to find cases when UMVUE's exist but complete sufficient statistics do not. Another method to find such cases is also presented.     

A note on estimation of variance components in a multi-stage sampling design based on interpenetrating sub-samples

Summary In this note it is shown that unbiased estimators of the components of the sampling variance of an estimator in the case of a stratified multi-stage sampling design could easily be obtained by selecting the samples at the different stages in the form of two or more independent interpenetrating sub-samples.