Estimation of Harmonic Mean of a Lognormal Variable

The harmonic mean has numerous engineering applications including characterization of the large-scale effective permeability in layered porous media, characterization of petrochemical properties of heterogeneous media, and the design of declining rate filter beds. The U.S. EPA also recommends the use of the harmonic mean daily streamflow as a design streamflow for the protection of human health against lifetime exposure to suspected carcinogens. The sampling properties of various estimators of the harmonic mean are derived and compared for observations arising from a lognormal distribution. Previous applications have recommended the use of a moment estimator of the harmonic mean. We document that the moment estimator of the harmonic mean exhibits significant upward bias and large root-mean-square error, particularly for large skews. A maximum likelihood estimator of the harmonic mean is generally preferred because it is nearly unbiased and can provide dramatic reductions in the root-mean-square error, compared with the moment estimator. In addition, a maximum likelihood estimator of the generalized mean (or p-norm) of a lognormal distribution is introduced.