Monte Carlo and Applications

Thisted (1976) compared ridge 2, modified ridge 2, ridge 3, and generalized ridge 1 by the Monte Carlo method and found the somewhat paradoxical result that ridge 2, which is minimax for the smallest subset of Л, performs best in general.

Gunst and Mason (1977) compared by the Monte Carlo method the esti­mators (1) least squares, (2) principal components, (3) Stein’s, and (4) ridge 2. Their conclusion was that although (3) and (4) are frequently better than (1) and (2), the improvement is not large enough to offset the advantages of (1) and (2), namely, the known distribution and the ability to select regressors.

Dempster, SchatzofF, and Wermuth (1977) compared 57 estimators, be­longing to groups such as selection of regressors, principal components, Stein’s and ridge, in 160 normal linear models with factorial designs using both E(j} — — P) and E(j} — Д)’Х’Х(Д — p) as the risk function. The

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winner was their version of ridge based on the empirical Bayes estimation of у defined by

The fact that their ridge beat Stein’s estimator even with respect to the risk function E(fi — — p) casts some doubt on their design of Monte

Carlo experiments, as pointed out by Efron, Morris, and Thisted in the dis­cussion following the article.

For an application of Stein’s estimator (pulling toward the overall mean), see Efron and Morris (1975), who considered two problems, one of which is the prediction of the end-of-season batting average from the averages of the first forty at-bats. For applications of ridge estimators to the estimation of production functions, see Brown and Beattie (1975) and Vinod (1976). These authors determined у by modifications of the Hoerl and Kennard ridge trace analysis. A ridge estimator with a constant у is used by Brown and Payne (1975) in the study of election night forecasts. Aigner and Judge (1977) used generalized ridge estimators on economic data (see Section 2.2.8).

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