Non-existence of empirical Bayes estimators for broader conjugate-prior families

Determine whether non-existence of empirical Bayes estimators analogous to the non-existence established for the Binomial–Beta and Poisson–Gamma models also occurs for other conjugate priors with more hyperparameters and under different loss functions.

Background

The paper shows that Type-II maximum likelihood fails to produce finite or meaningful hyperparameter estimates in the Binomial model with Beta priors and in the Poisson model with Gamma priors. In both settings, the marginal likelihood either approaches its supremum at the boundary or along diverging hyperparameter sequences, preventing the construction of a distinct, finite empirical Bayes estimator.

The authors explicitly propose investigating whether this failure is a broader phenomenon affecting other conjugate-prior families with higher-dimensional hyperparameters and whether it persists when alternative loss functions are used. This question is unresolved within the paper.