A Test for the Herfindahl Index

Abstract

Regional scientists have adopted two rather different empirical views of agglomeration. One view, favored by economists, examines the distribution of some property across a class of entities (agents, regions). The other view, favored by geographers, examines the spatial correlation of that property based on the arrangement of those entities. Adopting the first approach, this paper develops new statistical properties for the Herfindahl–Hirschman concentration index. The methodology is based on the standard occupancy problem in physics, where r particles (points) are distributed across n cells (quadrats). Both r and n are random variables so the Herfindahl index itself is considered a random variable H(r, n). Including all the equi-probable states of this random variable, expected values for both the mean and standard deviation are specified. A test compares the Herfindahl score for a specific state (sample) to the score based on all possible states, allowing inferences to be made about whether the observed score conforms to a random process. A few applications follow and then a short discussion concludes the chapter.

Appendix

The formula for integers developed by James Stirling (1692–1770) has been given much attention by many prominent mathematicians, especially since the advances made in research on complex numbers by Abraham de Moivre (1667–1754). The formula provides an approximate solution for integer factorials where the absolute errors diverge, but the relative or proportional errors converge, as the integers become larger in size. In short, the formula is.

$$ n!\sim {\left(2\uppi \right)}^{0.5}\ {n}^{n+0.5}{e}^{-n} $$

where the sign ~ indicates that the ratio of the formula’s two sides approaches unity as the size of the integer n approaches infinity (Feller 1957). Note, when n = 10, the LHS of the formula is 3,628,800, while the RHS is 3,598,695.6, and the approximation generates a relative error of less than 1%. In fact, when n = 8 the error is 1.03%, when n = 10, the error is 0.83%, and when n = 12 the error is 0.63%.

As Parzen (1960) points out this factorial approximation for integers is a special case of the gamma function for real numbers, which continues to play an important role in various aspects of probability theory. This function also can be tied to the Riemann Zeta function, which has key applications to the study of prime numbers in mathematics and string theory in physics. It is worth noting that, in general, the gamma function Γ(n + 1) = n!, where Γ(1) = 1. When addressing increasingly larger integers, Stirling’s formula provides solutions that asymptotically approach the values for the real numbers that are calculated in the various integral solutions of the gamma function.