Maximum Likelihood Estimations examples

Question

I am trying to find the MLE of the following functions but I'm getting stuck. I know the method and steps to follow but Pi notation is confusing for me.

1) f(x) = øx^(ø-1), 0 < x < 1 and 0 < ∞. Let X1, X2, ... Xn be a random sample. What is the MLE of ø?

So I can get it into pi notation as multiple from i=1 to n, øxi^(ø-1). I figured I can pull out a ø^n, but I'm stuck after that. Is there some type of trick here?

2) f(x) = (1/ø)x^((1-ø)/ø), 0 < x < 1 and 0 < ∞. Let X1, X2, ... Xn be a random sample. What is the MLE of ø?

I'm not even sure how to approach this one. How can I get this into a sensible form?

Answer 1

Here's a general hint for MLEs: we almost never directly optimize the product (i.e., the actual sample likelihood) $L$:

$$L(\theta;x)=\prod f(\theta;x_i)$$

Instead, we take the natural log of the likelihood (called the log-likelihood) (sometimes shown as $\mathcal{L}$, and then proceed to try to maximize that:

$$\ln L(\theta;x) := \mathcal{L}(\theta;x) = \sum \ln(f(\theta;x_i))$$