[EM] The Sainte-Lague index and proportionality

[Michael Ossipoff]

>> So either approach would be proposable, if that one overall
>> exponential is a good approximation. But Warren himself admitted that,
>> at the low-population end, it isn't accurate, because the states, at
>> some point, stop getting smaller. But Warren said that his single
>> exponential function worked pretty well in his tests.
>
>
> Gibrat's law ( https://en.wikipedia.org/wiki/Gibrat%27s_law ) suggests a
> log-normal distribution, and indeed the tail of such a distribution is
> exponential.

Thanks for that information.

I'd been skeptical about Warren's simple exponential, because there
aren't increasingly many small parties, as one looks at smaller and
smaller state populations.

>
> After reading about it, I did some tests with US state populations (of
> latest census), and a kernel density estimate of the logarithms of the
> populations show a distribution that looks a lot like a sum of Gaussians. So
> log-normal is a reasonable first approximation. To get a better
> approximation, use the exponential of a fitted sum of Gaussians, which would
> be log-normals multiplied -- I think.
>
> I haven't tested on past populations, so perhaps my sample size is
> insufficient. Still, if it is not, that would explain why Warren's
> exponential function worked well -- and presumably, a log-normal fit would
> work better yet.

Interesting. It suggests that there's no point in wasting time working
with an exponential approximation, when there are more accurate,
though more complicated approximations proposed. Thanks for finding
and reporting that information from Gibrat, and for your own
examination of the distribution.

If someone wanted to actually find the round-up point for
Weighted-Bias-Free, one should only use the better distribution
approximations. Maybe, for that purpose, one could start with the
log-normal, and then, afterwards, try the more accurate exponential of
a fitted sum of Gaussians.

But, as these distribution approximations get more complicated, the
chance of an analytical solution, for the roundoff-point R, gets
slimmer and slimmer.

WBF requires integrating 1/q times the probability-density. And even
if that could be done analytically, it isn't obvious to me that
solving the resulting equation for R can necessarily always be done as
an exact solution without a trial-&-error method.

Based on your information, for accurate WBF, finding the
rounding-point will be a numerically-solved problem.

Mike Ossipoff