Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems:RejoinderAuthor(s): David A. HarvilleSource: Journal of the American Statistical Association, Vol. 72, No. 358 (Jun., 1977), pp. 339-340Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2286798 .

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Variance Component Estimation 339

promising since it avoids negative solutions if the starting values are nonnegative. However, the specification of "good" starting values which are nonnegative may be difficult (e.g., if the ANOVA estimates are employed as starting values, some of them may turn out to be negative).

4. Harville seems to prefer the restricted likelihood function, Li, obtained from a set of linearly independent contrasts, but LI is not minimal sufficient unlike the full likelihood function L. Criteria such as the marginal sufficiency and the argument that restricted ML estimates reduce to the usual ANOVA estimates in the balanced cases are not adequate justifications for preferring re- stricted ML estimates over ML estimates. We need ex- tensive results on the relative efficiencies and also on the relative powers of tests based on these estimates.

5. Unlike the ANOVA methods, ML and MINQUE tech- niques both provide estimates for the general mixed model in a unified way. We therefore need further work on the comparison of the latter methods from the view- point of computations as well as efficiency. Recently, Liu and Senturia (1976) simplified the computation of MINQUE of variance components along the lines of Hemmerle and Hartley (1973) for the ML estimates.

6. As mentioned earlier, this paper is largely con- cerned with point estimation under the assumption of normality, even though the asymptotic standard errors are also given. Hartley and Rao (1967) provide likelihood ratio tests and exact small sample probability statements in the form of exact confidence regions for the variance components. Broemeling (1969) provides a simple method of finding conservative confidence regions, but it assumes that the analysis of variance produces mutually inde- pendent mean squares distributed as chi-square variates.

However, the large-sample standard errors and the above-mentioned tests and confidence regions are likely to be sensitive to departures from normality even for large samples, as evidenced by the investigations of Arvesen and Schmitz (1970) and Arvesen and Layard (1975) for the one-way balanced and unbalanced layouts. We need robust tests and robust confidence regions; perhaps the "jackknife" tests, which appear to be robust at least in the case of a one-way layout, could be extended to cover the general mixed model.

REFERENCES

Arvesen, James N., and Layard, Maxwell W.J. (1975), "Asymptot- ically Robust Tests in Unbalanced Variance Components," Annals of Statistics, 3, 1122-34.

and Schmitz, Thomas H. (1970), "Robust Procedures for Variance Component Problems Using the Jackknife," Biometrics, 26, 677-86.

Broemeling, Lyle D. (1969), "Confidence Regions for Variance Ratios of Random Models," Journal of the American Statistical Association, 64, 660-4.

Hartley, Herman O., and Rao, J.N.K. (1967), "Maximum Likeli- hood Estimation for Mixed Analysis of Variance Model," Bio- metrika, 54, 93-108.

Hemmerle, William J., and Hartley, Herman 0. (1973), "Computing Maximum Likelihood Estimates for the Mixed A.O.V. Model Using the W Transform," Technometrics, 15, 819-31.

Klotz, Jerome, and Putter, Joseph (1970), "Remarks on Variance Components: Likelihood Summits and Flats," Technical Report No. 247, Department of Statistics, University of Wisconsin, Madison.

Liu, Lon-Mu, and Senturia, Jerome (1976), "Computation of MINQUE Variance Component Estimates," Technical Report No. 408, Department of Statistics, University of Wisconsin, Madison.

Miller, John J. (1973), "Asymptotic Properties and Computation of Maximum Likelihood Estimates in the Mixed Model of the Analysis of Variance," Technical Report No. 12, Department of Statistics, Stanford University.

Re joinder DAVID A. HARVILLE

J.N.K. Rao has made a number of interesting points in his discussion. I agree with most of his comments and will limit my response to the relatively few areas where there may be some disagreement.

Rao suggests that I prefer the REML estimator of 0 to the ML estimator. I do think that, at worst, REML is a worthy competitor to ML, though I am in sympathy with Rao's position that further comparisons of the two procedures are needed. Corbeil and Searle's (1976) results indicate that which estimator is "best" depends heavily on the setting and, in general, on the underlying value of 0. While it seems unlikely that comparisons between REML and ML will lead to a clear-cut "winner," it would

seem to be a point in REML'S favor that it "never" pro- duces a "ridiculous" estimator as ML sometimes does, e.g., as in the problem of estimating the residual variance in the ordinary fixed ANOVA and regression models when n- p* is small relative to n. Also, criticism of the REML

estimator of 0 on the basis, "Li is not minimal sufficient unlike the full likelihood function L," seems irrelevant, since, as noted in the paper, the ML estimator turns out to depend on the data vector y only through the error contrasts.

Rao is quite correct in his assessment that the scope of the paper is largely limited to point estimation. The point estimation of 0 was emphasized, relative to test and

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340 Journal of the American Statistical Association, June 1977

interval procedures, because this made the review task manageable and because the estimation problem seems to be much closer to a satisfactory resolution than the test and interval problems. I am more optimistic than Rao about the possibility of using large-sample variances and covariances and asymptotic normality to construct approximate tests and confidence regions. Perhaps, parameterizations for 8 can be found such that these approximations will be satisfactory for "samples of reasonable size."

In many applications, interest centers on various linear combinations of the elements of a and 5 with 0 being regarded as something of a nuisance parameter. Satisfactory estimates and confidence regions for these linear combinations can be obtained by first estimating

6 and then proceeding as though the estimate were a true value, provided the data are extensive enough to give the estimate of 6 considerable precision (Harville 1976). Unfortunately, in instances where 0 cannot be estimated with a high degree of precision, this approach is apt to lead to confidence regions that are much too small. Modified procedures need to be developed that take into account the fact that an estimate of 0 is being used in place of its true value.

REFERENCES Corbeil, Robert R., and Searle, Shayle R. (1976), "A Comparison of

Variance Component Estimators," Biometrics, 32, 779-91. Harville, David A. (1976), "Confidence Intervals and Sets for Linear

Combinations of Fixed and Random Effects," Biometrics, 32, 403-7.

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