Distribution Theory

Abstract

The inverse Gaussian distribution has a history dating back to 1915 when Schrödinger and Smoluchowski presented independent derivations of the density of the first passage time distribution of Brownian motion with positive drift. The drift free case had already been published by Bachelier in 1900 in his doctoral thesis on the theory of speculation. Among the early advocates of this distribution one should single out Hadwiger (1940a, 1940b, 1941, 1942) and Halphen. The first of Hadwiger’s expositions dealt with the inverse Gaussian law as a solution to a functional equation in renewal theory. The other papers dealt with applications of the distribution to the reproduction functions arising in the study of population growth. Halphen is credited with the first formulation of what is now known as the generalized inverse Gaussian distribution. His discovery arose from the need to model hydrologic data whose behaviour was subject to decay for both large and small values. A general discussion of the early history can be found in Seshadri (1993) and Chhikara and Folks (1988). The modern day statistical community became aware of this law through the pioneering work of Tweedie (1941, 1945, 1946, 1947, 1956, 1957a, 1957b). The very name “inverse Gaussian” is Tweedie’s creation and is based on his observation that the cumulant function of this law is the inverse of the cumulant function of the normal law. From the point of view of probability and mathematical statistics the distribution can be regarded as a natural exponential family generated by the one-sided stable law with index $\frac{1}{2}$. Thus if λ > 0 and

$$\mu \left( {dx} \right) = \sqrt {\frac{1}{{2\pi {x^3}}}} \exp \left( { - \frac{1}{2}} \right){1_R}+ \left( x \right)dx$$

then the Laplace transform of µ(dх) is

$$\smallint _0^\infty {e^{\theta x}}\mu \left( {dx} \right) = - \sqrt { - 2\lambda \theta } for\theta \in \left( { - \infty ,0} \right]$$

.