The Ratio of the Geometric Mean to the Arithmetic Mean for a Random Sample from a Gamma Distribution

Let X 1, …, Xn denote a random sample from an unknown member of the family of two-parameter gamma densities x > 0, α > 0, k > 0. Define U to be the ratio of the geometric mean to the arithmetic mean, U = (Π Xi )1/n /(Σ Xi/n). The density of U is derived in a useable form which is exact for the interval e -2π/n < u < 1. Large-sample properties of U are also considered.