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The usual moment estimator of the harmonic mean is 1/mean(1/x_i), the reciprocal of the mean of the reciprocals of the x_i. This is a nonparametric estimator and is produced by the HARMEAN function in the DATA step. Limbrunner et. al. (2000) indicate that this moment estimator can be significantly biased especially for data with large coefficient of variation. They show that the maximum likelihood (ML) estimator "is nearly unbiased and [is] generally more efficient (lower RMSE) than the alternative estimators for lognormal observations."

The ML estimator is easily produced using the ESTIMATE statement in PROC NLMIXED. In addition to a point estimate, the PREDICT statement provides a standard error, confidence interval, and test of the harmonic mean as shown in the following example.

These statements use the HARMEAN function to produce the moment estimate of the harmonic mean of the values 1, 2, and 4.

      data a; 
        input y @@; 
        harmean = harmean(1, 2, 4);
        datalines;
      1 2 4
      ;
      
      proc print data=a(obs=1) noobs;
        var harmean;
        title "Moment estimator of Harmonic Mean";
        run;

PROC NLMIXED is used to fit a lognormal model to the data. The variable LL is the log likelihood for the lognormal. The mean, MU, and variance, V, are estimated by PROC NLMIXED. The maximum likelihood estimator, exp(μ - ½σ²), is computed in the ESTIMATE statement.

      proc nlmixed data=a;
        ll=-((log(y)-mu)**2)/v - log(v);
        model y~general(ll);
        estimate "ML estimator of Harmonic Mean" exp(mu-.5*v);
        run;

_____

Limbrunner J.F., Vogel R.M., and Brown L.C. (2000), "Estimation of Harmonic Mean of a Lognormal Variable", J. Hydrologic Engineering, 5(1), 59-66.

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