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From Wikipedia:

In the early 1940s Alexander Steven Corbet spent 2 years in British Malaya trapping butterflies.He kept track of how many species he observed, and how many members of each species were captured. For example, there were 74 different species of which he captured only 2 individual butterflies.

When Corbet returned to the United Kingdom, he approached biostatistician Ronald Fisher and asked how many new species of butterflies he could expect to catch if he went trapping for another two years;in essence, Corbet was asking how many species he observed zero times.

Fisher responded with a simple estimation: for an additional 2 years of trapping, Corbet could expect to capture 75 new species. He did this using a simple summation:

Take the number of species you found an odd number of, and subtract from that the number of species you found an even number of.

The wikipedia article cites this paper, which doesn't provide a proof. That paper cites this 1943 one by Fisher, Corbet, and Williams, but I don't see anything resembling the formula in there. The Wikipedia article also cites this paper by Good and Toulmin as confirming Fisher's sum, but it's not there either.

I am utterly flummoxed by this estimator.

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Fisher's formula is in the Good and Toulmin paper, though not very obviously. It's the estimator in equation 24 with $\lambda=2$, ie, the case where the second sample is the same size as the first sample.

The weird $\pm$ in the formula comes originally from equation 8, though it also shows up in equations 4 and 5, which represent an alternative approach that wasn't developed. Equation 8 is developed in equation 13.

I'm not sure there is a really good heuristic explanation -- there doesn't seem to be any simple inclusion/exclusion argument -- sometimes the maths just comes out that way

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Another source is "Computer Age Statistical Inference" by Efron and Hastie. I believe that section 6.2, entitled 'The Missing-Species Problem', contains the derivation you want.

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    $\begingroup$ See also Peter McCullagh's Ten Projects in Applied Statistics. $\endgroup$
    – Nick Cox
    Commented Aug 27 at 21:22
  • $\begingroup$ Thank you so much, that was excellent! $\endgroup$
    – goopy
    Commented Sep 11 at 20:53

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