= Estimation of a beta shape parameter
{title2=$\widehat\theta=-n/\sum_i\log X_i$}
For an independent <statistical sample> from the <beta distribution> with parameters $(\theta,1)$, $\theta>0$, the <probability density function> is $\theta x^{\theta-1}$ on $(0,1)$. Its <raw moments> are $\mathbb E X^r=\theta/(\theta+r)$ for $r> -\theta$, by direct <integration>. Consequently the <method of moments (statistics)> gives $T_1=\bar X/(1-\bar X)$, whereas the <maximum-likelihood estimator> is $T_2=-n/\sum_i\log X_i$.
The second formula follows by setting the <score function> $n/\theta+\sum_i\log X_i$ to zero; its <derivative> is negative. Also $-\log X_i$ has an <exponential distribution> of rate $\theta$, since $\mathbb P(-\log X_i>u)=e^{-\theta u}$. The <central limit theorem> and <delta method> give
$$
\sqrt n(T_1-\theta)\xrightarrow dN\left(0,\frac{\theta(\theta+1)^2}{\theta+2}\right),\qquad
\sqrt n(T_2-\theta)\xrightarrow dN(0,\theta^2).
$$
Thus the <asymptotic relative efficiency> of $T_1$ relative to $T_2$ is $1-(\theta+1)^{-2}$. The transformed sum has a <gamma distribution> of shape $n$ and rate $\theta$, which also gives $\mathbb E T_2=n\theta/(n-1)$ for $n>1$ and $\operatorname{Var}(T_2)=n^2\theta^2/[(n-1)^2(n-2)]$ for $n>2$.
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