Hey, there's this thing I can't wrap my head around.(adsbygoogle = window.adsbygoogle || []).push({});

Let's say we have a negative binomial variable x, with parameters p and r. That is, x is the number of failures we get before the rth sucess, while looking at random bernolli variables with sucsess rate p.

It can be shown that (r-1)/(x+r-1) is an unbiased estimator for p. So let's say before you start the experiment you want 5 sucesses, then the failures x is the variable. Let's say you get SFFSSFSFFS then the estimation for p=4/9=0,444444

Here comes the tricky part. One intuitive way the estimate the sucsessrate is to use r/(x+r), which I think is more logical(it is also actually the maximum-lilelyhood estimator, but biased). I mean, if you have 10 trials and 5 sucsesses 5/10 wouldn't be that bad would it? However this is not correct, and on avarage in the long run, since this is not a binomial experiment, but a negative binomial, it will tend to overestimate p.

But why does this tend to overestimate p, and why does subtracting 1 from r, give the correct answer? I know it can be shown from calculating the expected value of (r-1)/(x+r-1) that it is unbiased, but I am looking for a more intuitive answer. Which properties does the negative binomal model have that gives you the correct estimator, on avarage in the long run, if you subtract 1 from r.

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# Estimation, negative binomial variable

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