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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Estimation, negative binomial variable

**Physics Forums | Science Articles, Homework Help, Discussion**