= Estimating Bernoulli variance
{title2=$\phi=\theta(1-\theta)$}
For $n\geq2$ <independent> Bernoulli observations with sum $K$, the extended maximum-likelihood <estimator> is $\widehat\phi=(K/n)(1-K/n)$. Its unbiased improvement is $K(n-K)/[n(n-1)]=n\widehat\phi/(n-1)$, obtained by conditioning $X_1(1-X_2)$ on $K$. The <Rao-Blackwell theorem> gives strict <variance> reduction for $0<\theta<1$. With a uniform prior on $\theta$ and <squared-error loss> for $\phi$, the Bayes <estimator> is $(K+1)(n-K+1)/[(n+2)(n+3)]$. For <parameter space> strictly $(0,1)$, <statistical sample>[samples] with $K=0$ or $n$ have only a boundary likelihood supremum, not an attained maximum.
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