[EM] Finding the constants for the approximating function

[Michael Ossipoff]

Some days ago I posted a suggestion for finding the values of the constants 
A & B in the approximating function B/(q+A), by least-squares. The same 
method could be used for B*exp(-A*q).

That's probably the most accurate way to evaluate the constants.

I'd noticed that Warren finds the constants based on the fact that the U.S. 
population and the number of states are know. Yes, those two known 
quantities make it possible to solve two equations to find the constants A & 
B.

The integral, with respect to q,  of the frequency distribution 
approximating function, from 0 to Q
represents the number of states from population zero out to a population of 
Q Hare quotas.

For the least-squares solution, I'd suggested fitting that integral function 
to the data points consisting of the states's Hare quotas and their 
cumulative numbers (The smallest state's cumulative number is 1. The 2nd 
smallest state's cumulative number is 2...etc), to get the values of the two 
constants.

But, for the two equations solution, you could set that integral, from 0 out 
to 52 Hare quotas, equal to the number of states. That's one of the two 
equations.

That integral function gives the cumulative number of states as a function 
of q. Its inverse gives q as a function of the cumulative number of states. 
Write the integral of that inverse from 0 to 50, and set it equal to 435 
(the U.S. population in Hare quotas). That's the 2nd equation. Solve those 
two equations for A and B.

Least squares is almost surely more accurate. If the distribution function 
really is the exponential function, then both methods are completely 
accurate. But if it isn't, then the least squares approximation gives the 
exponential function that gives the best approximation.

Another way to make two equations would be set the integral of the 
approximatng function equal to 34 when q has the value of the number of Hare 
quotas possessed by the 34th state. And to set the integral of the 
approximating function equal to 17 when q has the value of the number of 
Hare quotas possessed by the 17th state.

I'd considered that, and preferred least-squares, because, as I said, it's 
the closest B*exp(-A*q) approximation to the distribution when the 
distribution  isn't really that function.

Mike Ossipoff