Beta Distribution
The harmonic mean of a beta distribution with shape parameters α and β is:
The harmonic mean with α < 1 is undefined because its defining expression is not bounded in .
Letting α = β
showing that for α = β the harmonic mean ranges from 0 for α = β = 1, to 1/2 for α = β → ∞.
The following are the limits with one parameter finite (non zero) and the other parameter approaching these limits:
With the geometric mean the harmonic mean may be useful in maximum likelihood estimation in the four parameter case.
A second harmonic mean (H1 - X) also exists for this distribution
This harmonic mean with β < 1 is undefined because its defining expression is not bounded in .
Letting α = β in the above expression
showing that for α = β the harmonic mean ranges from 0, for α = β = 1, to 1/2, for α = β → ∞.
The following are the limits with one parameter finite (non zero) and the other approaching these limits:
Although both harmonic means are asymmetric, when α = β the two means are equal.
Read more about this topic: Harmonic Mean
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