A note on the harmonic law: A two-parameter family of distributions for ratios

Consider two-parameter statistical models for positive continuous observations. Suppose that these models are closed under change of scale and under reciprocals, properties that are very appropriate when the observations are ratios of positive magnitudes. Additionally, suppose that the maximum likelihood estimator of the population mean is the sample mean (Gauss's principle). Surprisingly, only one statistical model satisfies these three properties and this is a special case of the generalized inverse gaussian family of distributions known as Harmonic Law.