The information loss in the inverse Gaussian model

In the inverse Gaussian model, the sample mean and sample reciprocal mean are minimum sufficient, and the distribution of sample reciprocal mean depends only on the dispersion parameter. Traditional inference about the dispersion parameter considers only the sample reciprocal mean instead of the whole sufficient statistic. This causes information loss, especially when the sample size is small. The purpose of this paper is to utilize the information of the dispersion parameter contained in the sample mean, and to improve the estimation of the dispersion parameter.