Estimation of a beta shape parameter ($\hat{\theta }=-n/{\sum }_i\log X_i$)

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 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 giveThus 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 $\mathrm{Var}(T_2)=n^2{\theta }^2/[(n-1{)}^2(n-2)]$ for $n>2$.