Approximation of the expected value of the harmonic mean and some applications

Although the harmonic mean (HM) is mentioned in textbooks along with the arithmetic mean (AM) and the geometric mean (GM) as three possible ways of summarizing the information in a set of observations, its appropriateness in some statistical applications is not mentioned in textbooks. During the last 10 y a number of papers were published giving some statistical applications where HM is appropriate and provides a better performance than AM. In the present paper some additional applications of HM are considered. The key result is to find a good approximation to [Formula: see text], the expectation of the harmonic mean of n observations from a probability distribution. In this paper a second-order approximation to [Formula: see text] is derived and applied to a number of problems.