Statistical Properties of Inverse Gaussian Distributions. I

A report is presented on some statistical properties of the family of probability density functions - (x -) ²/2²x/2 x³^1/2 for a variate x and parameters and, with x, , each confined to (0, ). The expectation of x is, while is a measure of relative precision. The chief result is that the ml estimators of and have stochastically independent distributions, and are of a nature which permits of the construction of an analogue of the analysis of variance for nested classifications. The ml estimator of is the sample mean, and for a fixed sample size n its distribution is of the same family as x, with the same but with replaced by n. The distribution of the ml estimator of the reciprocal of is of the chi-square type. The probability distribution of 1/x, and the estimation of certain functions of the parameters in heterogeneous data, are also considered.