title: Approximation of the expected value of the harmonic mean and some applications.
abstract: 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.
authors.affiliation: Department of Biostatistics, University at Buffalo, The State University of New York, Buffalo, NY 14221-3000; CRRAO Advanced Institute of Mathematics, Statistics And Computer Science, Hyderabad-500046, India; and crr1@psu.edu.
authors.forename: Calyampudi Radhakrishna
authors.initials: CR
authors.lastname: Rao
authors.affiliation: Department of Mathematics and Statistics, York University, Toronto, ON, Canada M3J 1P3.
authors.forename: Xiaoping
authors.initials: X
authors.lastname: Shi
authors.affiliation: Department of Mathematics and Statistics, York University, Toronto, ON, Canada M3J 1P3.
authors.forename: Yuehua
authors.initials: Y
authors.lastname: Wu
keywords: arithmetic mean; harmonic mean; image denoising; marginal likelihood; second-order approximation