Estimating Bernoulli variance ($\varphi =\theta (1-\theta )$)

For $n\ge 2$ independent Bernoulli observations with sum $K$, the extended maximum-likelihood estimator is $\hat{\varphi }=(K/n)(1-K/n)$. Its unbiased improvement is $K(n-K)/[n(n-1)]=n\hat{\varphi }/(n-1)$, 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 $\varphi$, the Bayes estimator is $(K+1)(n-K+1)/[(n+2)(n+3)]$. For parameter space strictly $(0,1)$, samples with $K=0$ or $n$ have only a boundary likelihood supremum, not an attained maximum.