[EM] Bias-Free (BF) is the unbiased divisor method for uniform probability-density

[Michael Ossipoff]

I defined and described BF in my most recent posting in the thread
called Sainte-Lague and bias, or something like that.

I mentioned that, in December 2006, I posted here that BF is the
unbiased divisor-method for allocation, if one is equally likely to
find a state anywhere on the population-scale.

Then, a few months later, I, for some reason, posted that that was
mistaken, and that Webster is the unbiased one after-all.

Looking at the problem now, I have no idea why I said that. Looking at
the problem now, it's obvious that it's BF.   ...for the reason that I
describe in my most recent post to the Sainte-Lague & Bias thread.

For a quotient, q,  (result of dividing a state's population by some
common divisor) between the consecutive integers a and b, BF's
rounding point, R, is:

R = (b^b/a^a)(1/e).

For the assumption of uniform probability distribution, BF is the
completely unbiased divisor method.

With uniform distribution, Webster is about 1.9% large-biased.

By that I mean;

Say there's a small state whose quotient is between 1 and 2, and is
equally likely to be anywhere in that interval.

Say there's a large state whose quotient is between 53 and 54, and is
equally likely to be anywhere in that interval.

The large state's expected s/q is 1.9% greater, that's 1.019 times
greater, than that of the small state.

Those integers (1,2,53 & 54) were chosen to cover the range between
the largest and smallest states for which s/q should be equal, so as
to give the greatest factor by which s/q can differ with whatever
method is being discussed.

But Webster's round-off point is closer to that of BF than Hill's
round-off point is. Between 1 and 2, Webster's round-off point is
about twice as close to that of BF as Hill's is.

For a non-uniform distribution, Weighted-BF is the unbiased divisor method.

In the other posting, my most recent one at the Sainte-Lague & bias
thread,I described how one could find the round-off point for
Weighted-BF (for a non-uniform distribution).

Mike Ossipoff