=+15.16. Recall the open question given at the end of Section 15.4 regarding the consistency of MLE in a very simple case of the GLMM. The conjectured answer to the question is yes. Here is evidence (and you can surely find more).

Let m = m1∧m2. Consider a subset of the data Xk = ykk, k = 1,...,m. Note that X1,...,Xm are i.i.d. Let ˜μ denote the MLE of μ based on X1,...,Xm.

Show that ˜μ is a consistent estimator of μ; that is, ˜μ

P

−→ μ as m → ∞. The point is that even the MLE based on a subset of the data is consistent. If one uses the MLE based on the full data (and therefore more information), one should do better, right? The challenge (not necessarily for this exercise) is to provide a rigorous justification.