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

Mathematics Asked by Confuse_d on January 5, 2022

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 $$theta^*=dfrac{-1+sqrt{1+4(sigma^2+bar{X}^2)}}2$$

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:

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

Answered by dmh on January 5, 2022

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