Let $X_1,\cdots,X_n \stackrel{i.i.d}{\sim} \mathcal{N}(\theta,\tau\theta)$, where both parameters $\tau,\theta$ are positive.
$(a)$ Derive the likelihood ratio test of $H_0 : \tau = 1$, $\theta$ unknown, versus $H_a : \tau = 1$, $\theta$ unknown and simplify your test statistic as much as possible.
$(b)$ Assume $\tau=1$. Is the MLE of $\theta$ consistent? Justify your answer.
This is a past qual question and I did part $(a)$ but I can't find the Expectation of $\theta^*=\dfrac{-1+\sqrt{1+4(\sigma^2+\bar{X}^2)}}2$
For consistency it's sufficient to show that the MLE converges in probability to the parameter you are estimating. The MLE is given by: $\hat{\theta} = -1/2 + \sqrt{1+1/4(1+\bar{X^2})}$. The weak law of large numbers tells us that $\bar{X^2} \rightarrow^p \mathbb{E}[X^2]$. Then, by the the continuous mapping theorem we also know:
$$-1/2 + \sqrt{1+1/4(1+\bar{X^2})} \rightarrow^p \theta$$