Statistics and Its Interface

Volume 12 (2019)

Number 3

The inverse gamma-difference distribution and its first moment in the Cauchy principal value sense

Pages: 467 – 478



Aaron Hendrickson (Atlantic Test Ranges, Patuxent River, Maryland, U.S.A.)


In this paper, the probability density and distribution functions for the reciprocal-difference of independent gamma random variables with unequal shape parameters are derived. A theorem is developed and applied to evaluate the first moment of this distribution in the sense of the Cauchy principal value, which addresses the inverse chi-squared and inverse exponential-difference distributions as special cases. These results are used to find the first moment and an approximation to the centralized inverse-Fano distribution, which models the sampling distribution of the photon transfer conversion gain measurement of electro-optical imaging sensors. A Monte Carlo simulation is performed to show how the first moment of the inverse gamma-difference distribution can be utilized to control the bias of the conversion gain measurement in a live experiment. The low illumination problem of conversion gain measurement is introduced with a discussion motivating future application of the theoretical results derived.


Cauchy principal value, centralized inverse-Fano distribution, conversion gain, first negative moment, gamma-difference distribution, generalized central limit theorem, hypergeometric function, photon transfer

2010 Mathematics Subject Classification

Primary 26A03, 30E20, 78M99. Secondary 33C20, 46N30, 62E17.

Received 25 June 2018

Published 4 June 2019