Cramér–Rao bound for a Bernoulli proportion

Question

Let X1, \dots, Xn be i.i.d. Bernoulli(p). State the Cramér–Rao lower bound, compute the Fisher information for p, and determine whether \hat{p} = \bar{X} is efficient.

Accepted Answer

The bound. Under regularity conditions (support not depending on p, differentiability), any unbiased estimator T of p satisfies \operatorname{Var}(T) \geq \frac{1}{n\,I(p)}, where I(p) is the Fisher information of a single observation. An unbiased estimator achieving the bound is called efficient. Fisher information for Bernoulli. The log-density of one observation is \log f(x; p) = x\log p + (1-x)\log(1-p), so the score is s(x; p) = \frac{x}{p} - \frac{1-x}{1-p} = \frac{x - p}{p(1-p)}. (Sanity check: \mathbb{E}[s] = 0, as a score must be.) The information is the variance of the score: I(p) =…

Cramér–Rao bound for a Bernoulli proportion

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