Checking the MLE is consistent or not in $mathcal{N}(theta,tautheta)$.

Question

Let $X_1,cdots,X_n stackrel{i.i.d}{sim} mathcal{N}(theta,tautheta)$, 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 $theta^*=dfrac{-1+sqrt{1+4(sigma^2+bar{X}^2)}}2$

dmh · Answer

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:
$$-1/2 + sqrt{1+1/4(1+bar{X^2})} rightarrow^p theta$$

Checking the MLE is consistent or not in $mathcal{N}(theta,tautheta)$.

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