The variance floor for estimating the odds of heads

Question

You flip a coin n times with unknown head probability p and want to estimate the odds of heads, \omega = g(p) = \dfrac{p}{1-p}. State the Cramer-Rao lower bound for an unbiased estimator of \omega.

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The bound for a function. When the target is a smooth function g(p) of the parameter, the Cramer-Rao lower bound is the usual floor scaled by the squared derivative: \operatorname{Var}(T) \ge \frac{[g'(p)]^2}{n\,I(p)}. The intuition: if \omega changes quickly with p (large g'), then errors in estimating p are magnified when converted to \omega, raising the floor. Plug in the Bernoulli pieces. The Fisher information for one flip is I(p) = \frac{1}{p(1-p)}. The target and its derivative are g(p) = \frac{p}{1-p}, \qquad g'(p) = \frac{1}{(1-p)^2}, \qquad [g'(p)]^2 = \frac{1}{(1-p)^4}. Assemble. \…

The variance floor for estimating the odds of heads

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