Suppose $X = \theta{Y}$ for some $\theta>0$ with $Y∼Beta(8,1)$. What is the maximum likelihood estimator for $\hat{\theta}=\hat{\theta}(x)$?
The distribution function I got is $f_\theta(x) = \frac{8x^7}{\theta^8}$ on range$[0,\frac{1}{\theta}]$, but by taking the derivative of the log likelihood function for this distribution by $\theta$ does not work.
In problems like these you can still use ordinary calculus methods; you have a monotonic likelihood function with the MLE occurring at a boundary point. This gives rise to a biased estimator of the parameter value, which can then be adjusted to obtain a non-biased estimator. In your particular problem (with a single observation $x$) you have the log-likelihood function:
$$\ell_x (\theta) = -8 \ln \theta <0 \quad \quad \text{for all }\theta \geqslant x.$$
Since the log-likelihood is a decreasing function, the MLE occurs at the boundary point $\hat{\theta} = x$. It can easily be shown that this is a biased estimator with mean $\mathbb{E}(\hat{\theta}) = \tfrac{8}{9} \cdot \theta$, and so you can obtain an unbiased adjusted-MLE using the estimator $\tilde{\theta} = \tfrac{9}{8} \cdot x$.
