On the mixture of the inverse Gaussian distribution with its complementary reciprocal

By generalizing the inverse Gaussian distribution function, we obtain a new three- parameter family of distributions which includes as special cases the inverse Gaussian and the reciprocal inverse Gaussian distributions, while preserving some of the interesting properties of the inverse Gaussian distribution. We derive two representations of the new distribution, one as the mixture of an inverse Gaussian distribution with its complementary reciprocal, and the second as a sum of an inverse Gaussian variable and an independent compound Bernoulli variable. The family is a two-parameter exponential model for known value of the third parameter, and is intimately related with exponential dispersion model theory. We also consider estimation and inference properties for the family, and show that it may have applications for positive right-skewed unimodal data and, in particular, duration or failure-time data.