A signal fires on a fraction p of days. From n independent days, show that the sample frequency attains the Cramer-Rao bound for p, then compute the smallest n that puts its standard error below 0.01 when p is near 0.2.

A signal fires on a fraction p of days. From n independent days, show that the sample frequency attains the Cramer-Rao bound for p, then compute the smallest n that puts its standard error below 0.01 when p is near 0.2.

Approach: Write the Bernoulli log-likelihood, get the information from the variance of the score, then compare with the exact variance of the sample proportion.

1600. The log-likelihood for one observation is x*log(p) + (1 - x)*log(1 - p), whose score is (x - p)/(p*(1 - p)). Its variance is the Fisher information I(p) = 1/(p*(1 - p)), so the Cramer-Rao bound on the variance of any unbiased estimator of p is p*(1 - p)/n. The sample frequency p_hat = x/n is unbiased with Var(p_hat) = p*(1 - p)/n, so it attains the bound exactly and is efficient at every n. Setting the standard error sqrt(0.2*0.8/n) <= 0.01 gives n >= 0.16/0.0001 = 1600 days, which is more than six years of trading days for a single parameter.

Follow-up: How does the required sample size change if you instead need the standard error of the log-odds below 0.05?

Key concepts: Cramer-Rao bound, Fisher information, efficiency, standard error.