This article presents a new empirical Bayes estimator (EBE) and a shrinkage estimator for determining the relative potency from several multivariate bioassays by incorporating prior information on the model parameters based on Jeffreys' rules. The EBE can account for any extra variability among the bioassays, and if this extra variability is 0, then the EBE reduces to the maximum likelihood estimator for combinations of multivariate bioassays. The shrinkage estimator turns out to be a compromise of the prior information and the estimator from each multivariate bioassay, with the weights depending on the prior variance.

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Robust S-estimation is proposed for multivariate Gaussian mixture models generalizing the work of Hastie and Tibshirani (J. Roy. Statist. Soc. Ser. B 58 (1996) 155). In the case of Gaussian Mixture models, the unknown location and scale parameters are estimated by the EM algorithm. In the presence of outliers, the maximum likelihood estimators of the unknown parameters are affected, resulting in the misclassification of the observations. The robust Sestimators of the unknown parameters replace the non-robust estimators from M-step of the EM algorithm. The results were compared with the standard mixture discriminant analysis approach using the probability of misclassification criterion. This comparison showed a slight reduction in the average probability of misclassification using robust S-estimators as compared to the standard maximum likelihood estimators. r

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