A Likelihood Problem:                    Torsten Sillke

Some years ago I analyzed an equipment to measure
the dust concentration of the air.
All boils down to an occupation problem, where an 
unknown number of particles are collected in n=400
cells. It is only possible to distinguish between 
empty and non empty cells.

This analysis lead me to the conjecture given below.


1. Occupation problem (Feller Vol 1. Exercise XII.6.4, C. Domb)
---------------------------------------------------------------

Let N have a Poisson distribution with mean lambda, and let
N balls be placed randomly into n cells. Show without calculation
that the probability of finding exactly m cells empty is:

(*)   Binomial(n,m) * exp(-lambda*m/n) * (1-exp(-lambda/n))^(n-m).

If n and m are given, what will be the most likely lambda?
The likelihood value of lambda (for m>0) (using (*)) is

   Likelihood(n,m) = n * ln(n/m)

Question: How good is the Likelihood-Estimater?
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2. Occupation problem (C. Domb)
-------------------------------

Show that when a fixed number r of balls is placed randomly into 
n cells the probability of finding exactly m cells empty equals
the coefficient of exp(-lambda)*lambda^r/r! in (*).


3. Maximum Likelihood
---------------------

Show that when a fixed number r of balls is placed randomly into 
n cells the probability of finding exactly k cells not empty equals

(**)    (n)_k * Stirling2nd(r,k) / n^r

Let m = n-k the number of empty cells a recursion can
be found in [Feller, Vol 1, II.11 equation 11.8]

                       n-m                m+1
 p_m[r+1,n] = p_m[r,n] ---  +  p_m+1[r,n] ---
                        n                  n 

What is the maximum likelihood value of r?
That means for fixed n and k find the parameter r which maximizes (**).
Let MaxLikelihood(n,k) be one r for which (**) is maximal.

Conjecture (Sillke):
====================

 | MaxLikelihood(n,k)  -  ln(n/(n-k))/ln(n/(n-1)) | < 1  and 
 | MaxLikelihood(n,k)  -  (n-1/2)*ln(n/(n-k))     | < 1

upto n = 600 I found

 -0.9807 < MaxLikelihood(n,k)  -  ln(n/(n-k))/ln(n/(n-1)) < 0.9482
 -0.9807 < MaxLikelihood(n,k)  -  (n-1/2)*ln(n/(n-k))     < 0.9470

with the minimum for n=595 and the maximum for n=439.

References:

- W. Feller, An Introduction to Probability Theory and its Applications,
     Vol 1., Wiley, 1968, 3rd Ed.

- C. Domb, On the use of a random parameter in combinatorial problems,
     Proc. Physical Society, Ser. A, 65 (1952) 305-309

- N.M. Temme, Asymptotic estimates of Stirling numbers,
     Studies in Applied Math., 89 (1993), 233--243.