consistency of MLE estimator example Given the density function $f(x;\theta)=\theta x^{\theta -1}$ with $\theta \gt 0$ and $x \in (0,1)$, once found the MLE estimator $\hat{\theta}=\frac{-n}{\sum{\log{X_i}}}$ i want to show the consistency of such an estimator. I was suggested to apply the strong law of large numbers, although i am not sure how. Should I first obtain the expected value of the random variable $\log(X_i)$ and then apply the SLLN on them or is there another way?
 A: You can show that $Y = -\ln X \sim Exp(\theta)$, hence $\mathbb{E}[Y]=1/\theta$, thus by the SLLN and the continuous mapping theorem
$$
\hat{\theta}_n \xrightarrow{a.s.} (1/\theta)^{-1} = \theta, 
$$
as $n \to \infty$.
A: If you are not familiar with SLLN, there are several other ways to approach the  problem,

*

*The simplest: a property of ML Estimators is that they are consistent.


*Consistency you have to prove is $\hat{\theta}\xrightarrow{\mathcal{P}}\theta$
So first let's calculate the density of the estimator.
Observe that (it is very easy to prove this with the fundamental transformation theorem)
$$Y=-logX\sim Exp(\theta)$$
Thus $W=\Sigma_iY_i\sim Gamma(n;\theta)$ and $\frac{1}{W}\sim \text{Inverse Gamma}$ and consequently
$$\mathbb{E}[\hat{\theta}]=\frac{n}{n-1}\theta$$
$$\mathbb{V}[\hat{\theta}]=\frac{n^2}{(n-1)^2(n-2)}\theta$$
This grant the following are both true
$$\lim\limits_{n\to \infty}\mathbb{E}[\hat{\theta}]=\theta$$
$$\lim\limits_{n\to \infty}\mathbb{V}[\hat{\theta}]=0$$
This is an IFF condition for the convergence in $L^2$ which implies also the convergence in probability (the consistency of your estimator)


*Calculate the convergence in probability by the definition

Using 2 is a better way because you proved also the convergence in quadratic mean, that it is stronger

Trying all the method we showed is an useful exercise (IMHO)
