In probability theory and statistics, the harmonic distribution is a continuous probability distribution. It was discovered by Étienne Halphen, who had become interested in the statistical modeling of natural events. His practical experience in data analysis motivated him to pioneer a new system of distributions that provided sufficient flexibility to fit a large variety of data sets. Halphen restricted his search to distributions whose parameters could be estimated using simple statistical approaches. Then, Halphen introduced for the first time what he called the harmonic distribution or harmonic law. The harmonic law is a special case of the generalized inverse Gaussian distribution family when γ = 0 {\displaystyle \gamma =0} .
History One of Halphen's tasks, while working as statistician for Electricité de France, was the modeling of the monthly flow of water in hydroelectric stations. Halphen realized that the Pearson system of probability distributions could not be solved; it was inadequate for his purpose despite its remarkable properties. Therefore, Halphen's objective was to obtain a probability distribution with two parameters, subject to an exponential decay both for large and small flows. In 1941, Halphen decided that, in suitably scaled units, the density of X should be the same as that of 1/X. Taken this consideration, Halphen found the harmonic density function. Nowadays known as a hyperbolic distribution, has been studied by Rukhin (1974) and Barndorff-Nielsen (1978). The harmonic law is the only one two-parameter family of distributions that is closed under change of scale and under reciprocals, such that the maximum likelihood estimator of the population mean is the sample mean (Gauss' principle). In 1946, Halphen realized that introducing an additional parameter, flexibility could be improved. His efforts led him to generalize the harmonic law to obtain the generalized inverse Gaussian distribution density.
Definition
Notation The harmonic distribution will be denoted by θ ( m , a ) {\displaystyle \theta (m,a)} . As a result, when a random variable X is distributed following a harmonic law, the parameter of scale m is the population median and a is the parameter of shape.
X ∼ Harm ( m , a ) {\displaystyle X\ \sim \operatorname {Harm} (m,a)\,}
Probability density function The density function of the harmonic law, which depends on two parameters, has the form,
f ( x ; m , a ) = 1 2 x K 0 ( a ) exp ( − a 2 ( x m + m x ) ) {\displaystyle f(x;m,a)={\frac {1}{2xK_{0}(a)}}\exp \left(-{\frac {a}{2}}\left({\frac {x}{m}}+{\frac {m}{x}}\right)\right)}
where
K 0 ( a ) {\displaystyle K_{0}(a)} denotes the third kind of the modified Bessel function with index 0,
m ≥ 0 , {\displaystyle m\geq 0,}
a ≥ 0. {\displaystyle a\geq 0.}
Properties
Moments To derive an expression for the non-central moment of order r, the integral representation of the Bessel function can be used.
μ r ′ = ∫ 0 ∞ x r f ( x ; m , a ) d x = m r K r ( a ) K 0 ( a ) {\displaystyle \mu '_{r}=\int _{0}^{\infty }x^{r}f(x;m,a)\,dx=m^{r}{\frac {K_{r}(a)}{K_{0}(a)}}}
where:
r denotes the order of the moment. Hence the mean and the succeeding three moments about it are
Skewness Skewness is the third standardized moment around the mean divided by the 3/2 power of the standard deviation, we work with,
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