Characterizations for the Power Inverse Gaussian and Generalized Inverse Gaussian Distributions Based on Conditional Moments

In this study, we characterize the power inverse Gaussian and generalized inverse Gaussian distributions based on recurrence relations for conditional moments in terms of hazard rate functions. Relations for conditional moments of the largest order statistic (upper record value) given the second largest order statistic (upper record value) are derived from these recurrence relations. Characterizations of the inverse Gaussian, reciprocal inverse Gaussian, gamma and reciprocal gamma distributions are obtained as special cases of both the power inverse Gaussian and the generalized inverse Gaussian distributions.