# Improved estimation of the ratio of variance components for a balanced one-way random effects model

TL;DR: In this article, the ratio of variance components is considered for a balanced one-way random effects model, and two different paths are followed to cope with this problem: one way, a class of nonnegative estimates with improved MSE is considered by looking into the structure of the available estimates.

Abstract: Estimation of the ratio of variance components is considered for a balanced one-way random effects model. Problem arises as the readily available estimates can take negative values. Although there are some nonnegative estimates, they do not have any optimum property. Two different paths are followed to cope with this problem. In one way, a class of nonnegative estimates with improved MSE is considered by looking into the structure of the available estimates. In this context, a class of admissible nonnegative estimates is also characterized. In the other a class of nonanalytic estimates (truncated at zero) is formed.

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01 Jan 2005

TL;DR: In particular, the study of variance through a class of linear models known as random and mixed models is a central topic in statistics with wide ramifications in both theory and applications as discussed by the authors.

Abstract: The variability among data may have quite complex structure and the splitting of variation into its individual components is important in many areas of applications. In particular, the study of variation through a class of linear models known as random and mixed models is a central topic in statistics with wide ramifications in both theory and applications. Sometimes these models are called components of variance models since the interest in the model lies in investigating the variances of the effects of factor levels which are assumed to be randomly selected from a large population of levels. These variances are also known as variance components since the total variance can be expressed as the sum of these variances and the error variance. Knowledge of the variance components, i.e., the information on the relative importance and magnitude of the sources of variability in the data is of critical importance in many problems since the isolation of the total variance into the individual components of variance would result in the physical elimination of the sources of variability in question.

42 citations

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TL;DR: In this paper, a family of estimators for the intraclass correlation coefficient in the balanced one-way random effects model is presented, which includes the familiar REML and ML estimators, and under certain conditions, in terms of mean-squared error, most members of the family are admissible within the family.

Abstract: In this paper pivotal quantities are used to obtain estimators of parameters. As an application of this technique, a family of estimators is presented for the intraclass correlation coefficient in the balanced one-way random effects model. The family is derived by equating a pivotal quantity to different values of the pivoting distribution, and includes the familiar REML and ML estimators. It is shown that under certain conditions, in terms of mean-squared error, most members of the family of estimators are admissible within the family. The authors provide guidance concerning the choice of an individual member of the family for estimation purposes and indicate how the method can be extended to unbalanced designs.

8 citations

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01 Jan 2000TL;DR: In this article, the authors considered the random effect model involving only a single factor or variable in an experimental study involving a comparison of a set of treatments, where each of the treatments can be randomly assigned to experimental units.

Abstract: In this chapter, we consider the random effect model involving only a single factor or variable in an experimental study involving a comparison of a set of treatments, where each of the treatments can be randomly assigned to experimental units. Such a layout is commonly known as the one-way classification or the completely randomized design. The one-way classification is the simplest and most useful model in statistics. In a one-way random effects model, treatments, groups, or levels of a factor are regarded to be a random sample from a large population. It is the simplest nontrivial and widely used variance component model. Moreover, the statistical concepts and tools developed to handle a one-way random model can be adapted to provide solutions to more complex models. Models involving two or more factors will be considered in succeeding chapters.

6 citations

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26 Apr 1998

TL;DR: In this paper, a family of point estimators is presented for the intraclass correlation coefficient (or heritability) in the balanced one-way random effects model, and the family is obtained by equating a pivotal quantity to different values of the pivoting distribution.

Abstract: A family of point estimators is presented for the intraclass correlation coefficient (or heritability) in the balanced one-way random effects model. The family is obtained by equating a pivotal quantity to different values of the pivoting distribution, and includes the familiar ML and REML estimators. In terms of mean-squared error, most members of the family of estimators are admissible within the family. A sire model is used to illustrate the estimation of heritability. The authors provide guidance concerning the choice of an individual member of the family for estimation purposes and indicate how the method can be extended to unbalanced designs.

1 citations

### Cites background from "Improved estimation of the ratio of..."

...On a closely related topic, Loh (1986), Das, Meneghini and Giri (1990), Das (1992), and Ye (1994) examine inferences of a ratio of variance components in a balanced oneway random effects model....

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...On a closely related topic, Loh (1986), Das, Meneghini and Giri (1990), Das (1992), and Ye (1994) examine inferences of a ratio of variance components in a balanced oneway random effects model. In particular, Loh (1986) develops a point estimator that has uniformly smaller mean-squared error than the maximum likelihood, restricted maximum Conference on Applied Statistics in Agriculture Kansas State University...

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TL;DR: In this article, a sample space, orbit-by-orbit analysis of the conditional expected loss given the orbit is used to obtain the limit of a certain sequence of testimator-like estimators.

Abstract: Techniques for improving on equivariant estimators are described. They may be applied, although without assurance of success, whatever be the family of underlying distributions. The loss function is required to satisfy an intuitively reasonable condition but is otherwise arbitrary. One of these techniques amounts to a sample space, orbit-by-orbit analysis of the conditional expected loss given the orbit. It yields, when successful, a "testimator". A second technique obtains the limit of a certain sequence of "testimator-like" estimators. The result is "smoother" than a testimator and often identical to a generalized Bayes estimator over much of its domain. Applications are presented. In the first we extend results of Stein (1964) and obtain a minimax estimator which is generalized Bayes, and in a univariate subcase, admissible within the class of scale-equivariant estimators. In the second, we extend a result of Srivastava and Bancroft (1967).

260 citations

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TL;DR: In this paper, various estimators of components of variance for the balanced one way layout are compared using mean squared error as a measure of performance and several modifications of the maximum likelihood estimator and several formal Bayes estimators are compared and mean square error relationships are given.

Abstract: Various estimators of components of variance for the balanced one way layout are compared using mean squared error as a measure of performance. Several modifications of the maximum likelihood estimator and several formal Bayes estimators are compared and mean square error relationships are given. Inadmissibility is established for a large class of translation and scale invariant estimators. Numerical results are tabled and graphed to exhibit the relative efficiency of the estimators considered.

80 citations

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TL;DR: In this paper, the problem of estimating a ratio of variance components in the balanced one-way random effects model is considered, and it is shown that in terms of mean squared error, the ML, REML (or truncated ANOVA), and Bayes modal estimators (using the noninformative prior) are inadmissible.

Abstract: The problem of estimating a ratio of variance components in the balanced one-way random effects model is considered. It is shown that in terms of mean squared error, the ML, REML (or truncated ANOVA), and Bayes modal estimators (using the noninformative prior) are inadmissible. An estimator that dominates all three is derived. Two other estimators that are adaptive in nature are also introduced. The new estimators are shown to possess much-improved mean squared error properties. The results easily extend to balanced higher-way random or mixed effects models.

17 citations

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TL;DR: In this paper, the problem of estimating the ratio of variance components is considered from the invariance point of view, and a class of shrinkage estimators is also proposed, which dominate the ML, REML and even in some cases, Bayes modal estimators, although this is not true for the optimal estimator.

Abstract: The problem of estimation of the ratio of variance components is considered from the invariance point of view. This approach actually paves the way for proving the inadmissibility of the optimal estimator developed by Loh (1986). A class of shrinkage estimators is also proposed. These estimators dominate the ML, REML and even in some cases, Bayes modal estimators, although this is not true for the optimal estimator. Numerical comparisons are made to establish the appealing behaviour of the proposed estimators.

6 citations