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Arthur B
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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. (ThoughThough for n$n$ small enough, it looks like the minimum mayprior dominates and the optimum is actually be achieved for a lowsmall value of $\frac{2l}{d\pi}$ )$2l/(d\pi)$.

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 small enough, it looks like the minimum may actually be achieved for a low value of $\frac{2l}{d\pi}$ ).

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 $n$ small enough, the prior dominates and the optimum is actually achieved for a small value of $2l/(d\pi)$.

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Arthur B
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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 pcrossing, 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. (forThough for n small enough, thereit looks like the minimum may actually be a local minimumachieved 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 $\frac{2l}{d}\left(\frac{n}{i}+\frac{1}{i}\right)$ (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).

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 small enough, it looks like the minimum may actually be achieved for a low value of $\frac{2l}{d\pi}$ ).

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Arthur B
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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).

The naive estimator is biased. If there are $N$ trials and $i$ success, a Rao-Blackwellisation of the naive estimator gives the unbiased estimator $\frac{2l}{t}\left(\frac{n}{i}+\frac{1}{i}\right)$.

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).

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).

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Arthur B
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