The naive estimator is biased. If there are $N$ trials and $i$ success, a Rao-Blackwellisation of the naive estimator gives the unbiased estimator $\frac{2l}{d}\left(\frac{n}{i}+\frac{1}{i}\right)$ (to be fair this hides an assumption for a uniform prior for the probability of crossing, which induces a weird prior on $\pi$).
One can look at the variance of the estimator conditional on obtaining one success. The strategy is then to set $l=d$. Intuitively this makes sense, we want the term in $1/i$ to be as small as possible. (Though for n $n$ small enough, it looks like the minimum may prior dominates and the optimum is actually be achieved for a low small value of $\frac{2l}{d\pi}$ ).2l/(d\pi)$. 3 added 45 characters in body; added 72 characters in body The naive estimator is biased. If there are$N$trials and$i$success, a Rao-Blackwellisation of the naive estimator gives the unbiased estimator$\frac{2l}{d}\left(\frac{n}{i}+\frac{1}{i}\right)$(to be fair this hides an assumption for a uniform prior for the probability of p).crossing, which induces a weird prior on$\pi$). One can look at the variance of the estimator conditional on obtaining one success. The strategy is then to set$l=d$l=d$. Intuitively this makes sense, we want the term in $1/i$ to be as small as possible. (Though for n small enough, there it looks like the minimum may actually be a local minimum achieved for a low value of $\frac{2l}{d\pi}$ but the formula seems intractable)).
The naive estimator is biased. If there are $N$ trials and $i$ success, a Rao-Blackwellisation of the naive estimator gives the unbiased estimator $\frac{2l}{t}\left(\frac{n}{i}+\frac{1}{i}\right)$. \frac{2l}{d}\left(\frac{n}{i}+\frac{1}{i}\right)$(to be fair this hides an assumption for a uniform prior for the probability of p). One can look at the variance of the estimator conditional on obtaining one success. The strategy is then to set$l=d$(for n small enough, there may actually be a local minimum for a low value of$\frac{2l}{d\pi}\$ but the formula seems intractable).