Introduction

Abstract

The subject of the present study is the generalized inverse Gaussian distribution whose probability density function is given by

$$ {\frac{{(\Psi /X)}}{{2{K_{\lambda }}\;(\sqrt {{X\psi }} )}}^{{\frac{\lambda }{2}}}}\;{X^{{\lambda - 1}}}{e^{{ - \frac{1}{2}({X^{{{X^{{ - 1}}}}}} + \psi X)}}}\quad (x > 0), $$

(1.1)

where K_λ is the modified Bessel function of the third kind and with index λ. Special cases of (1.1) are the gamma distribution (^χ = 0, λ > 0)> the distribution of a reciprocal gamma variate (ψ = 0, λ < 0) (in the following denoted the reciprocal gamma distribution), the inverse Gaussian distribution (λ =-1/2) and the distribution of a reciprocal inverse Gaussian variate (λ = 1/2). Other important cases are λ = 0 (the hyperbola distribution) and λ = 1.