Consistency and asymptotic normality of two-dimensional parameter

Question

In a textbook exercise, for a sequence of iid variables, I have calculated the score function to be $$\begin{bmatrix} - \frac{n}{2\lambda} + \sum_{i=1}^n \frac{( x_i - \mu)^2}{2\mu^2 x_i}& \sum_{i=1}^n \frac{\mu - x_i}{\mu^3} \end{bmatrix}$$ From this, the information matrix is readily calculated. I wish to show that the MLE estimators are consistent and asymptotically normal.

It is easy to see that the MLE estimator for $\mu$ is $\sum x_i/n$, and therefore the MLE estimator for $\lambda$ is $$ \frac{n}{2} \left(\sum_{i=1}^n \frac{( x_i - \hat{\mu})^2}{2\ \hat{\mu}^2 x_i}\right)^{-1}$$ where we plug in the MLE estimator $\mu$. The information matrix is positive-definite in these values.

It is hinted that $EX = \mu$, $E(1/X) = 1/\mu + 1/\lambda$, $VARX = \mu^3/\lambda$, $VAR1/X = \frac{1}{\mu \lambda} + \frac{2}{\lambda^2},$ and $COV(X,1/X) = -\mu/\lambda$.

I am not sure how to continue. I guess asymptotic normality of both follows by using CLT on $\hat{\mu}$ and then use the delta-method? The consistency of $\hat{\mu}$ follows from Chebyshev's inequality, but I have no clue how to show it for $\hat{\lambda}$.

Indeed my problem is that I am not sure how those hints of variances of $1/X$ and so on are supposed to come into play. Any answer or hint is appreciated.

Answer 1

$$\frac{n}{2} \left(\sum_{i=1}^n \frac{( x_i - \hat{\mu})^2}{2\ \hat{\mu}^2 x_i}\right)^{-1} = \hat{\mu}^2 \cdot \left(\frac 1n\sum_{i=1}^n \frac{( x_i - \hat{\mu})^2}{\ x_i}\right)^{-1} =$$

$$=\hat{\mu}^2 \cdot \left(\frac 1n\sum_{i=1}^n \left (x_i -2\hat {\mu} + \frac{\hat{\mu}^2}{x_i}\right) \right)^{-1}$$

So you are looking at sample means of an ergodic sample. Apply the probability limit and its properties (including the continuous mapping theorem), use the consistency for $\mu$, and Law of Large Numbers, as well as the hint about $E(1/X)$ (which importantly also implies that the expected value of the reciprocal exists in the first place).

These will lead you to consistency for $\lambda$.