Editing Harmonic mean (section)

==Statistics==

For a random sample, the harmonic mean is calculated as above. Both the [[Expected value|mean]] and the [[variance]] may be [[Infinity|infinite]] (if it includes at least one term of the form 1/0).

===Sample distributions of mean and variance===

The mean of the sample ''m'' is  asymptotically distributed normally with variance ''s''<sup>2</sup>.
:<math>s^2 = \frac{m \left[\operatorname{E}\left(\frac{1}{x} - 1\right)\right]}{m^2 n}</math>

The variance of the mean itself is<ref name=Zelen1972>Zelen M (1972) Length-biased sampling and biomedical problems. In: Biometric Society Meeting, Dallas, Texas</ref>
: <math>\operatorname{Var}\left(\frac{1}{x}\right) = \frac{m \left[\operatorname{E}\left(\frac{1}{x} - 1\right)\right]}{n m^2}</math>

where ''m'' is the arithmetic mean of the reciprocals, ''x'' are the variates, ''n'' is the population size and ''E'' is the expectation operator.

===Delta method===
{{Unreferenced section|date=December 2019}}

Assuming that the variance is not infinite and that the [[central limit theorem]] applies to the sample then using the [[delta method]], the variance is

: <math>\operatorname{Var}(H) = \frac{1}{n}\frac{s^2}{m^4}</math>

where ''H'' is the harmonic mean, ''m'' is the arithmetic mean of the reciprocals

: <math>m = \frac{1}{n} \sum{ \frac{1}{x} }.</math>

''s''<sup>2</sup> is the variance of the reciprocals of the data

: <math>s^2 = \operatorname{Var}\left( \frac{1}{x} \right) </math>

and ''n'' is the number of data points in the sample.

===Jackknife method===

A [[resampling (statistics)#Jackknife|jackknife]] method of estimating the variance is possible if the mean is known.<ref name=Lam1985>Lam FC (1985) Estimate of variance for harmonic mean half lives. J Pharm Sci 74(2) 229-231</ref> This method is the usual 'delete 1' rather than the 'delete m' version.

This method first requires the computation of the mean of the sample (''m'')
: <math>m = \frac{n}{ \sum{ \frac{1}{x} } }</math>

where ''x'' are the sample values.

A series of value ''w<sub>i</sub>'' is then computed where
: <math>w_i = \frac{n - 1}{ \sum_{j \neq i} \frac{1}{x} }.</math>

The mean (''h'') of the ''w''<sub>i</sub> is then taken:
: <math>h = \frac{1}{n} \sum{w_i}</math>

The variance of the mean is
: <math>\frac{n - 1}{n} \sum{(m - w_i)}^2.</math>

Significance testing and [[confidence interval]]s for the mean can then be estimated with the [[t test]].

===Size biased sampling===

Assume a random variate has a distribution ''f''( ''x'' ). Assume also that the likelihood of a variate being chosen is proportional to its value. This is known as length based or size biased sampling.

Let ''μ'' be the mean of the population. Then the [[probability density function]] ''f''*( ''x'' ) of the size biased population is
: <math>f^*(x) = \frac{x f(x)}{\mu}</math>

The expectation of this length biased distribution E<sup>*</sup>( ''x'' ) is<ref name="Zelen1972"/>
: <math>\operatorname{E}^*(x) = \mu \left[ 1 + \frac{\sigma^2}{\mu^2} \right]</math>

where ''σ''<sup>2</sup> is the variance.

The expectation of the harmonic mean is the same as the non-length biased version E( ''x'' )
:  <math> E^*( x^{ -1 } ) = E( x )^{ -1 } </math>

The problem of length biased sampling arises in a number of areas including textile manufacture<ref name=Cox1969>Cox DR (1969) Some sampling problems in technology. In: New developments in survey sampling. U.L. Johnson, H Smith eds. New York: Wiley Interscience</ref> pedigree analysis<ref name=Davidov2001>Davidov O, Zelen M (2001) Referent sampling, family history and relative risk: the role of length-biased sampling. Biostat 2(2): 173-181 {{doi|10.1093/biostatistics/2.2.173}}</ref> and survival analysis<ref name=Zelen1969>Zelen M, Feinleib M (1969) On the theory of screening for chronic diseases. Biometrika 56: 601-614</ref>

Akman ''et al.'' have developed a test for the detection of length based bias in samples.<ref name=Akman2007>Akman O, Gamage J, Jannot J, Juliano S, Thurman A, Whitman D (2007) A simple test for detection of length-biased sampling. J Biostats 1 (2) 189-195</ref>

===Shifted variables===

If ''X'' is a positive random variable and ''q'' > 0 then for all ''ε'' > 0<ref name=Chuen-Teck2008>Chuen-Teck See, Chen J (2008) Convex functions of random variables. J Inequal Pure Appl Math 9 (3) Art 80</ref>
: <math>\operatorname{Var} \left[\frac{1}{(X + \epsilon)^q}\right] < \operatorname{Var} \left(\frac{1}{X^q}\right) .</math>

===Moments===

Assuming that ''X'' and E(''X'') are > 0 then<ref name="Chuen-Teck2008"/>
: <math>\operatorname{E}\left[ \frac{1}{X} \right] \ge \frac{1}{ \operatorname{E}(X) }</math>

This follows from [[Jensen's inequality]].

Gurland has shown that<ref name=Gurland1967>Gurland J (1967) An inequality satisfied by the expectation of the reciprocal of a random variable. The American Statistician. 21 (2) 24</ref> for a distribution that takes only positive values, for any ''n'' > 0
: <math>\operatorname{E} \left(X^{-1}\right) \ge \frac{\operatorname{E} \left(X^{n-1}\right)}{\operatorname{E}\left(X^n\right)} .</math>

Under some conditions<ref name=Sung2010>Sung SH (2010) On inverse moments for a class of nonnegative random variables. J Inequal Applic {{doi|10.1155/2010/823767}}</ref>
: <math>\operatorname{E}(a + X)^{-n} \sim \operatorname{E}\left(a + X^{-n}\right)</math>

where ~ means approximately equal to.

===Sampling properties===

Assuming that the variates (''x'') are drawn from a lognormal distribution there are several possible estimators for ''H'':

: <math>\begin{align}
  H_1 &= \frac{n}{ \sum\left(\frac{1}{x}\right) } \\
  H_2 &= \frac{\left( \exp\left[ \frac{1}{n} \sum \log_e(x) \right] \right)^2}{ \frac{1}{n} \sum(x) } \\
  H_3 &= \exp \left(m - \frac{1}{2} s^2 \right)
\end{align}</math>

where
: <math>m = \frac{1}{n} \sum \log_e (x)</math>
: <math>s^2 = \frac{1}{n} \sum \left(\log_e (x) - m\right)^2</math>

Of these ''H''<sub>3</sub> is probably the best estimator for samples of 25 or more.<ref name=Stedinger1980>Stedinger JR (1980) Fitting lognormal distributions to hydrologic data. Water Resour Res 16(3) 481–490</ref>

===Bias and variance estimators===

A first order approximation to the [[bias]] and variance of ''H''<sub>1</sub> are<ref name=Limbrunner2000>Limbrunner JF, Vogel RM, Brown LC (2000) Estimation of harmonic mean of a lognormal variable. J Hydrol Eng 5(1) 59-66 {{cite web |url=http://engineering.tufts.edu/cee/people/vogel/publications/estimation-harmonic.pdf |title=Archived copy |access-date=2012-09-16 |url-status=dead |archive-url=https://web.archive.org/web/20100611205528/http://engineering.tufts.edu/cee/people/vogel/publications/estimation-harmonic.pdf |archive-date=2010-06-11 }}</ref>

: <math>\begin{align}
  \operatorname{bias}\left[ H_1 \right] &= \frac{H C_v}{n} \\
   \operatorname{Var}\left[ H_1 \right] &= \frac{H^2 C_v}{n}
\end{align}</math>

where ''C''<sub>v</sub> is the coefficient of variation.

Similarly a first order approximation to the bias and variance of ''H''<sub>3</sub> are<ref name="Limbrunner2000"/>

: <math>\begin{align}
  \frac{H \log_e \left(1 + C_v\right)}{2n} \left[ 1 + \frac{1 + C_v^2}{2} \right] \\
  \frac{H \log_e \left(1 + C_v\right)}{n} \left[ 1 + \frac{1 + C_v^2}{4} \right]
\end{align}</math>

In numerical experiments ''H''<sub>3</sub> is generally a superior estimator of the harmonic mean than ''H''<sub>1</sub>.<ref name="Limbrunner2000"/> ''H''<sub>2</sub> produces estimates that are largely similar to ''H''<sub>1</sub>.