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 E(H_n), the expectation of the harmonic mean of n observations from a probability distribution. In this paper a second-order approximation to E(H_n) is derived and applied to a number of problems.The harmonic mean H_n of n observations Z₁, …, Z_n drawn from a population is defined byHn=n∑i=1n1Zi.[1]There have been a number of applications of the harmonic mean in recent papers. A more general version of H_n with weights w₁, …, w_n isHn(w)=∑i=1nwi∑i=1nwiZi.[2]where w = (w₁,…,w_n)^T. The harmonic mean H_n is used to provide the average rate in physics and to measure the price ratio in finance as well as the program execution rate in computer engineering. Some statistical applications of the harmonic mean are given in refs. 1–4, among others. H_n(w) has been used in evaluation of the portfolio price-to-earnings ratio value (ref. 5, p. 339) and the signal-to-interference-and-noise ratio (6) among others. The asymptotic properties of H_n including the asymptotic expansion of E(H_n) are investigated in refs. 7 and 8 by either assuming that some moments of 1/Z_i are finite or that Z_i s follow the Poisson distribution. It is noted that recent papers (9, 10) enable one to use saddle-point approximation to give the asymptotic expansion of E(H_n) to any given order of 1/n for some constants c₀, c₁, c₂, …, i.e.,E(Hn)=c0+c1n+c2n2+⋯.[3]However, such methods are not applicable for obtaining the asymptotic expansion of H_n when the first moment of 1/Z_i is infinite. In ref. 3, Z_i s are assumed to follow a uniform distribution in the interval (0,1), i.e., U(0,1), motivated by learning theory. Using the property that the inverse of H_n converges to the stable law, ref. 3 showed thatE(Hn)∼1log(n),[4]where the symbol “∼” means asymptotic equivalence as n → ∞. Our interest in this paper is to determine the second term in the asymptotic expansion of E(H_n) or the general version E(H_n(w)) under more general assumptions on distributions of Z_i s. We show that under mild assumptions,E(Hn)∼1log(n){1+c1log(n)},[5]where the constant c₁ will be given. In addition, we use the approach for obtaining [5] to the case that the first moment of 1/Z_i is finite, motivated by evaluation of the marginal likelihood in ref. 11.