Saddlepoint Inference for Nonlinear Statistics from Inverse Gaussian Models: Applications to Clinical, Engineering, and Environmental Data

This paper applies established saddlepoint approximation techniques to nonlinear statistics arising from inverse Gaussian models. In particular, we consider the product of independent inverse Gaussian random variables and the ratio of weighted linear combinations of inverse Gaussian random variables, for which exact distributions are generally unavailable in closed form. For the product statistic, a logarithmic transformation converts the problem into one involving the cumulant generating function of a sum of log-transformed variables. This cumulant generating function is expressed in terms of fractional moments including modified Bessel functions of the second kind. For the ratio statistic, the event including the ratio is reformulated in terms of a linear statistic, which enables the use of saddlepoint density and Lugannani-Rice distribution approximations. The proposed formulation accommodates heterogeneous model parameters, overlapping numerator and denominator components, and flexible coefficient structures subject to positivity of the denominator. Simulation studies show that the proposed approximations provide accurate results across a range of sample sizes, skewness regimes, and parameter configurations. Furthermore, simulation results indicate that the saddlepoint approximation is more accurate than the normal approximation. Sensitivity analysis confirms that the proposed approximations are reasonably stable under moderate inverse Gaussian parameter misspecification. Three real data applications including clinical illness scores, engineering repair times, and environmental runoff measurements illustrate the practical usefulness of the approach. Overall, the results indicate that the saddlepoint approximation provides an accurate and computationally efficient tool for inference on nonlinear statistics from inverse Gaussian models when exact distributions are not available.