Note on a Characterization of the Inverse Gaussian Distribution

Abstract

A random variable $x$ is said to have the inverse Gaussian [3], or Wald's ([4] pages 192-193) distribution function when its density function is given by \begin{equation*}\tag{1} f(x; m, \lambda) = (\lambda/2\pi x^3)^{\frac{1}{2}} \exp\{-\lambda(x - m)^2/2m^2x\},\quad 0 < x < \infty,\end{equation*} where the constants $\lambda$ and $m$ are assumed to be positive. If $x_1, x_2,\cdots, x_N$ are $N$ independent observations from (1), then it is proved by Tweedie [3] that the densities of the variates $y = \sum^N_{j=1} x_j$ and $z = \sum^N_{j=1} x_j^{-1} - N^2y^{-1}$ are independent; the distribution of $y$ is $f(y; Nm, N^2\lambda)$ and that of $\lambda z$ is $\chi^2$ with $N - 1$ degrees of freedom (df). By proving the converse of Tweedie's result, namely, if $x_1,\cdots, x_N$ are independently and identically distributed random variables and if $y$ and $z$ are independently distributed, then $x_1, x_2,\cdots, x_N$ each have inverse Gaussian distribution, Khatri [2] gave a characterization of the inverse Gaussian distribution. Now Wald ([4] pages 192-193) has proved that if $x$ has the distribution (1), then $t^2 = (x - m)^2/m^2x$ has a $\chi^2$ distribution with one df, or $t$ has a normal distribution with mean zero and variance $\lambda^{-1}$. It thus appears that certain properties of the inverse Gaussian distribution may be studied via the known properties of the normal distribution. Khatri's result on the characterization of the inverse Gaussian distribution is one such result. The following known result for the normal distribution, namely, if $t_1, t_2,\cdots, t_N$ are independent and identically distributed random variables and if a linear form $w$ in $t$ variates and a quadratic form $z$ of rank $N - 1$ in $t$ variates, where $\sum^N_1 t^2_i = w^2 + z$, are independently distributed, then the $t$ variates are normally distributed. Now this result when translated for the inverse Gaussian distribution is the result stated by Khatri, who proves the result by an involved method of characteristic functions. Perhaps, the simpler proof of Khatri's result established in the present paper might be of pedagogical interest.