I have the following setting: $$ x_k \sim N(\mu,\sigma^2 + \hat{\delta}^2_k),k=1,\dots,K, $$
where $\{x_k,k=1,\dots,K\}$ - observed data, $\{\hat{\delta}^2_k\,k=1,\dots,K \}$ are known parameters (just consider them fixed), while $(\mu,\sigma^2)$ are unknown. My primary goal - making inference about $\mu$. Does anyone know of any results for this particular case in terms of consistency and asymptotic behavior of MLE estimator for $\mu$ as $K\rightarrow \infty$?
For the time being I am simply using a $\chi^2$ approximation for the likelihood ratio test, but I don't have solid theoretical argument for that as the data is not identically distributed. It also complicates things that I fail to get a closed-form solution for $\hat{\mu}_{mle}$
I have tried Hoadley's paper on "Asymptotic Properties of MLE for Independent Non-Identically distributed case" http://projecteuclid.org/download/pdf_1/euclid.aoms/1177693066, but verifying their general conditions is not coming to me easy as of yet.
Please let me know if anyone encountered this particular kind of a problem and knows what regularity conditions are needed for nice classic MLE properties. Obviously there should be a certain upper bound on values $\{ \hat{\delta}^2_k,\ k=1,\dots, \}$, but what kind of bound, and even with that bound - how to prove consistency/asymptotic normality.