A Characterization of the Inverse Gaussian Distribution

Abstract

M. C. K. Tweedie [2] defined the inverse Gaussian distributions via the density functions \begin{equation*} ag{1}f(x; m, \lambda)= \lbrack\lambda/(2\pi x^3)\rbrack^{\frac{1}{2}} \exp \lbrack -\lambda(x - m)^2/(2m^2x)\rbrack\quad ext{for}\quad x > 0 = 0\quad ext{for} x \leqq 0\end{equation*} The parameters λ and m are positive. The corresponding densities reflected about the origin, and with λ and m negative, may also be considered as in the Inverse Gaussian family. The characteristic function of the Inverse Gaussian distribution with parameters λ, m is \begin{equation*} ag{2}\phi(t) = \exp \lbrack\lambda\{1 - (1 - 2im^2t\lambda^{-1})^{\frac{1}{2}}\}/m\rbrack,\quad i = \sqrt{-1}\end{equation*} for all real values of t. If x1, x2, ⋯, xn are n independent observations from (1), then y = ∑n j=1 xj and z = ∑n j=1 x-1 j - n2y -1 are independently distributed. The distribution of y is f(y, nm, n2λ) and that of λ z is Chi-Square with (n - 1) degrees of freedom. In this note, we prove that, if x1, x2, ⋯, xn are independently and identically distributed variates, with the existence of certain moments (different from zero), and if y and z are independently distributed, then the distribution of xj is Inverse Gaussian.