A mixture of a normal distribution with random mean and variance -- Examples of inconsistency of maximum likelihood estimates

We consider the estimation of the mixing distribution of a normal distribution where both the shift and scale are unobserved random variables. We argue that in general, the model is not identifiable. We give an elegant non-constructive proof that the model is identifiable if the shift parameter is bounded by a known value. However, we argue that the generalized maximum likelihood estimator is inconsistent even if the shift parameter is bounded and the shift and scale parameters are independent. The mixing distribution, however, is identifiable if we have more than one observations per any realization of the latent shift and scale.