What is the maximum likelihood estimator of a the normal distribution $N(\theta,\theta)$?

Question

Consider the normal distribution when $\theta=\sigma^2$: $N(\theta,\theta)$ with probability density function:

• $f(x;\theta)=\frac{1}{\sqrt{2\pi\theta}}e^{-\frac{1}{2\theta}(x-\theta)^2}$

I was asked to find the maximum likelihood estimator of the law. In other words, how to find the maximum value of:

• $L(\theta)=\frac{1}{(2\pi\theta)^{n/2}}e^{-\frac{1}{2\theta}\sum(X_i-\theta)^2}$

where $X_1,...,X_n$ are iid following $N(\theta,\theta)$.

Answer 1

HINT:

It would be much easier if you just used the log-likelihood $l(\theta)$ giving $$l(\theta)=-\frac n2\log2\pi\theta-\frac1{2\theta}\sum_i(X_i-\theta)^2$$ Then solve $$\frac{\partial l}{\partial \hat{\theta}}=0.$$