How to simplify the variance of $\sqrt{1+\frac{4\sum_{i=1}^{n}X_i^2}{n}}$?

Question

Let $\{X_i\}_{i=1}^{n}$ be $n \ $ i.i.d random variables such that $X_i \sim N(\theta,\theta)$ where $\theta > 0$.

Now, I am trying to simplify $$\text{var}\bigg(\sqrt{1+\frac{4\sum_{i=1}^{n}X_i^2}{n}}\bigg).$$ I got this expression in some calculation that I am doing.

Only, the square root is creating problems. I can't get my head around this. Please help.

Thanks!

EDIT : I have to show that $$\text{var}\bigg(\sqrt{1+\frac{4\sum_{i=1}^{n}X_i^2}{n}}\bigg) \to 0$$ as $n \to \infty.$

EDIT 2 : MLE of $\theta$ is $$\hat{\theta}_{MLE} = \frac{-1 + \sqrt{1+\frac{4\sum_{i=1}^{n}X_i^2}{n}}}{2}$$.

I am trying to prove the consistency of $\hat{\theta}_{MLE}$.

Answer 1

I do not know your purpose but surely you can approximate

$$\mathbb{V}\Bigg[\sqrt{1+4\overline{X}_n}\Bigg]$$

using Delta Method.

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EDIT: your estimator is

$$T=\sqrt{1+4\frac{\Sigma_i X_i^2}{n}}$$

After usual calculations,

$$\hat{\theta}_{ML}=\frac{T-1}{2}$$

The asymptotic distribution of MLE is known...easy solved problem (as an approximation, obviously)

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EDIT: at the end, the problem was different.

Reading the complete original text

Let $X_1,...,X_n$ be $n$ i.i.d. random variables with ditribution $N(\theta;\theta)$ for some unknown $\theta>0$. Compute the MLE of $\theta$ and show it is Consistent

After calculating $\hat{\theta}_{ML}$ that you did correctly let's go back to the Score that is

$$l^*(\theta)=-\frac{n}{2\theta}-\frac{n}{2}+\frac{\Sigma_i X_i^2}{2\theta^2}$$

The derivative of the score is

$$l^{**}(\theta)=\frac{n}{2\theta^2}-\frac{\Sigma_i X_i^2}{\theta^3}$$

Thus the Fischer information is

$$I_n(\theta)=-\mathbb{E}[l^{**}(\theta)]=-\frac{n}{2\theta^2}+\frac{(\theta+\theta^2)n}{\theta^3}=\frac{(1+2\theta)n}{2\theta^2}$$

Concluding:

The asymptotic distribution is the following

$$\hat{\theta}_{ML}\dot{\sim} N\Bigg[\theta;\frac{2\theta^2}{(1+2\theta)n}\Bigg]$$

This means that

$$\lim\limits_{n \to\infty}\mathbb{E}[\hat{\theta}_{ML}]=\theta$$

and

$$\lim\limits_{n \to\infty}\mathbb{V}[\hat{\theta}_{ML}]=0$$

Which is a necessary and sufficient condition for consistency