       Re: fit a BinomialDistribution to exptl data?

• To: mathgroup at smc.vnet.net
• Subject: [mg80780] Re: fit a BinomialDistribution to exptl data?
• From: Ray Koopman <koopman at sfu.ca>
• Date: Sat, 1 Sep 2007 00:33:18 -0400 (EDT)
• References: <11420600.1188458042685.JavaMail.root@m35>

```On Aug 30, 9:03 pm, DrMajorBob <drmajor... at bigfoot.com> wrote:
> It is very instructive to see how the results vary from sample to
> sample. Yes, results vary wildly (for all the methods). In just a
> few trials of 200 draws from BinomialDistribution[20, 0.2], I got
> n anywhere from 13 to 96.
>
> In more extensive testing (300 trials), the simple moment estimator
> (setting theoretical mean and variance equal to the sample mean and
> variance, then solving for n and p) returned n from -1447 to +521,
> and the other methods were not much better. Darren Glosemeyer's
> FindFit solver failed to converge on one sample, returning n
> greater than 4000. And so forth.
>
> Statistics is more art than science.
>
> Bobby

The sample variance "ought" to be less than the sample mean, but
there is nothing to stop if from being bigger, in which case the
moment estimates will be inadmissible. Also, it is possible that
the ML estimate of n may not exist because the likelihood, maximized
w.r.t. p given n, may be monotone increasing in n, as is the case
with the following 200 samples from a Binomial[20,.2] distribution:

{x = {0,  1,  2,  3,  4,  5,  6, 7, 8, 9, 10, 11},
f = {3, 21, 28, 37, 43, 28, 22, 8, 6, 2,  1,  1}};

which has {mean, variance} = {3.875, 4.02952}. It is not obvious
(except perhaps to a Bayesian) what should be done in cases such as
this.

```

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