The German Tank Problem dates back to World War II when the Allies used a
statistical approach to estimate the number of enemy tanks produced or on the
field from observed serial numbers after battles. Assuming that the tanks are
labeled consecutively starting from 1, if we observe k tanks from a total of
N tanks with the maximum observed tank being m, then the best estimate for
N is m(1+1/k)−1. We explore many generalizations. We looked at the
discrete and continuous one dimensional case. We explored different estimators
such as the L\textsuperscript{th} largest tank, and applied motivation from
portfolio theory and studied a weighted average; however, the original formula
was the best. We generalized the problem in two dimensions, with pairs instead
of points, studying the discrete and continuous square and circle variants.
There were complications from curvature issues and that not every number is
representable as a sum of two squares. We often concentrated on the large N
limit. For the discrete and continuous square, we tested various statistics,
finding the largest observed component did best; the scaling factor for both
cases is (2k+1)/2k. The discrete case was especially involved because we had
to use approximation formulas that gave us the number of lattice points inside
the circle. Interestingly, the scaling factors were different for the cases.
Lastly, we generalized the problem into L dimensional squares and circles.
The discrete and continuous square proved similar to the two dimensional square
problem. However, for the L\textsuperscript{th} dimensional circle, we had to
use formulas for the volume of the L-ball, and had to approximate the number
of lattice points inside it. The formulas for the discrete circle were
particularly interesting, as there was no L dependence in the formula.Comment: Version 1.0, 47 page
