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**Michael J. Daniels and Robert E. Kass**

Our investigation involves three alternative hierarchical priors. The first works with the spectral decomposition of the covariance matrix and produces both shrinkage of the eigenvalues toward each other and shrinkage of the rotation matrix toward the identity. The second produces shrinkage of the correlations toward zero, and the third uses a conjugate Wishart distribution to shrink toward diagonality. A simulation study shows that such hierarchical priors, especially the first, can be very effective in reducing small-sample risk. We evaluate the computational algorithm in the context of a Normal nonlinear random-effects model and illustrate the methodology with a Poisson random-effects model.

* Keywords:* hierarchical prior distributions, variance
estimation, importance sampling, Givens angles

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