Let $X_1, \dots, X_n$ denote a random sample from the PDF $$f_{\varphi}(x)= \begin{cases} \varphi x^{\varphi - 1} &\text{if}\, 0 < x < 1, \varphi > 0\\ 0 &\text{otherwise} \end{cases}$$ This density function is a member of the one-parameter exponential family. Let $A_i = - \log(X_i)$, wher...