Contraharmonic Mean - Uses in Statistics

Uses in Statistics

The contraharmonic mean of a random variable is equal to the sum of the (arithmetic) mean and the variance/mean. Since the variance is always >0 the contraharmonic mean is always greater than the arithmetic mean.

The problem of a size biased sample was discussed by Cox in 1969 on a problem of sampling fibres. The expectation of size biased sample is equal to its contraharmonic mean.

The probability of a fibre being sampled is proportional to its length. Because of this the usual sample mean (arithmetic mean) is a biased estimator of the true mean. To see this consider

where f(x) is the true population distribution, g(x) is the length weighted distribution and m is the sample mean. Taking the usual expectation of the mean here here gives the contraharmonic mean rather than the usual (arthimetic) mean of the sample. This problem can be overcome by taking instead the expectation of the harmonic mean ( 1 / x ). The expectation and variance of 1 / x are

and has variance

where E is the expectation operator. Asymptotically E( 1 / x ) is distributed normally.

The asymptotic efficiency of length biased sampling depends compared to random sampling on the underlying distribution. if f(x) is log normal the efficiency is 1 while if the population is gamma distributed with index b, the efficiency is b /( b - 1 ).

This distribution has been has been used in several areas.