Help with the Maximum Likelihood Estimator?

Question

I'm really struggling to understand this and am trying to learn it for my upcoming exam.

The question I'm trying to do is Write down the likelihood function and then find the Maximum likelihood estimator of $\theta$.

$$f_{x}(x)= \theta x^{\theta - 1} \qquad 0 \leq x \leq 1$$

So what i have is $$L(\theta : X) = \prod_{i=1}^{n} \theta x^{\theta - 1} = \theta ^{n} \bigg( \prod_{i=1}^{n} x_{i} \bigg)^{\theta - 1}$$

I don't really understand this and have just got to this through looking at previous examples. The bit i am stuck on is then taking the logs of this.

If anyone can help this would be greatly appreciated.

Answer 1

\begin{align} \ell(\theta) = \log L(\theta) = \log \left( \theta^n \bigg( \prod_{i=1}^n x_i \bigg)^{\theta - 1} \right) & = \log(\theta^n) + \log \left( \left( \prod_{i=1}^n x_i \right)^{\theta-1} \right) \\[10pt] & = n\log\theta + (\theta - 1) \log\prod_{i=1}^n x_i. \end{align}

You could then go on to write $\displaystyle\log\prod_{i=1}^n x_i$ as $\displaystyle\sum_{i=1}^n \log x_i$, but that won't actually help with finding the MLE, since you're concerned with this whole expression as a function of $\theta$ rather than as a function of $x_1,\ldots,x_n$.