I need to show that given an i.i.d sample $X_1,\dots X_n$ arising from the model:
$$\{f(x,\theta)=\theta x^{\theta-1}exp\{-x^{\theta}\},x>0,\theta\in (0,\infty)\}$$
that the MLE exists with probability one and is consistent.
I have been instructed to use the following lemma(which I have proved):
Let $S_n$ be a sequence of random real-valued continuous functions defined on $\Theta$ such that, as $n \xrightarrow{}\infty, S_n(\theta) \xrightarrow{P}S(\theta) \forall\theta \in \Theta$ where $S:\Theta \xrightarrow{} R$ is nonrandom. Suppose for some $\theta_0$ in the interior of $\Theta$ and every $\epsilon >0$ small enough we have $S(\theta_0 \pm \epsilon)<0<S(\theta_0 \mp \epsilon)$ and that $S_n$ has exactly one zero $\hat{\theta}_n$ for every natural number n. Then we must have $\hat{\theta}_n\xrightarrow{P} \theta_0$.
I have also been permitted to interchange differentiation $\frac{d}{d\theta}$ and dx-integration without justification.
I am not entirely sure how to begin to prove the existence of the MLE I have tried taking log-likelihoods and manipulating the derivative but to no avail.
Assuming its existence I thought of letting $S_n(\theta)=\frac{1}{n}\sum_{i=1}^n \frac{d}{d\theta}(log(f(X_i,\theta))$ so that we can use the law of large numbers but this has not proved successful. I can see that if I can find $S_n$ and $S$ to match the conditions in 1. then consistency would be immediate.
Any help would be appreciated