Approximation of the expected value of the harmonic mean and some applications
Calyampudi Radhakrishna Rao, Xiaoping Shi, Yuehua Wu · Proceedings of the National Academy of Sciences · 2014
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.