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Harmonic distribution

Harmonic

Notation	$\mathrm {Harm} (m,a)\,$
Parameters	m ≥ 0, a ≥ 0
PMF	${\frac {1}{2xK_{0}(a)}}\exp(-{\frac {a}{2}}({\frac {x}{m}}+{\frac {m}{x}}))$
CDF	$\int _{X}^{\infty }{\frac {1}{2xK_{0}(a)}}\exp(-{\frac {a}{2}}({\frac {x}{m}}+{\frac {m}{x}}))dx$
Mean	$m{\frac {K_{1}(a)}{K_{0}(a)}}$
Variance	$m^{2}(1+{\frac {2K_{1}(a)}{K_{0}(a)a}}-{\frac {K_{1}^{2}(a)}{K_{0}^{2}(a)}})$
Skewness	${\frac {K_{0}^{2}(a)K_{3}(a)-3K_{0}(a)K_{1}(a)K_{2}(a)+2K_{1}^{3}(a)}{(K_{0}(a)K_{2}(a)-K_{1}^{2}(a))^{\frac {3}{2}}}}$
Excess kurtosis	${\frac {\mu _{4}}{\mu _{2}^{2}}}={\frac {K_{0}^{3}(a)K_{4}(a)-4K_{0}^{2}(a)K_{1}(a)K_{3}(a)+6K_{0}(a)K_{1}^{2}(a)K_{2}(a)-3K_{1}^{4}(a)}{(K_{0}(a)K_{2}(a)-K_{1}^{2}(a))^{2}}}$

In probability theory and statistics, the Harmonic distribution is a continuous probability distribution. Discovered by Étienne Halphen, who was searching a probability distributions with two parameters, the Harmonic Law is a special case of the Generalized Inverse Gaussian family of distribution when $\gamma =0$.

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 distribution could not be solved, it was inadequate for his purpose despite its remarkable properties. Halphen objective was to obtain a probability distribution with two parameters, subject a 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 1/X. Taken this consideration, Halphen found the density function. Nowadays known as an hyperbolic distribution, has been studied by Rukhin (1974) and Barndorff-Nielsen (1978).

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 density.

Definition

Notation

The Harmonic distribution is denoted by ${\theta }(m,a)$. As a result, when a random variable X is distributed by Harmonic Law, the parameter of scale m is the population median and a is the parameter of shape.

$X\ \sim \ \mathrm {Harm} (m,a)\,$

Probability Density Function

Being f be a two-parameter statistical model. Then, the probability density function with two parameters is,

$f(x;m,a)={\frac {1}{2xK_{0}(a)}}\exp(-{\frac {a}{2}}({\frac {x}{m}}+{\frac {m}{x}}))$

where:

• $K_{0}(a)$ denotes the third kind of the modified Bessel function with index 0.
• m ≥ 0.
• a ≥ 0.

Cumulative Density Function

The Cumulative distribution function for the Harmonic Law does not exist in closed form, it is not possible to derive an explicit expression. The Cumulative distribution function must be determined by numeric solving methods,

$\int _{X}^{\infty }{\frac {1}{2xK_{0}(a)}}\exp(-{\frac {a}{2}}({\frac {x}{m}}+{\frac {m}{x}}))dx={\frac {1}{T}}$

Quantiles

The Quantiles in Harmonic Law are calculate with the Cumulative Distribution but does not exist form and only we can see the extend expression for any quantile. To solve the quantiles, only we can get numerically.

• Q1

The first quantile, when T = 4 can be obtained from the integral of probability density function,

$Q1=\int _{X}^{\infty }{\frac {1}{2xK_{0}(a)}}\exp(-{\frac {a}{2}}({\frac {x}{m}}+{\frac {m}{x}}))dx={\frac {1}{4}}$

• Q2 = Median

Median, or second quantile, when T = 2 we can be reached from the integral of probability density function,

$Med=Q2=\int _{X}^{\infty }{\frac {1}{2xK_{0}(a)}}\exp(-{\frac {a}{2}}({\frac {x}{m}}+{\frac {m}{x}}))dx={\frac {1}{2}}$

• Q3

Finally, the third quantile, when T = 4/3 we comes from the integral of probability density function,

$Q3=\int _{X}^{\infty }{\frac {1}{2xK_{0}(a)}}\exp(-{\frac {a}{2}}({\frac {x}{m}}+{\frac {m}{x}}))dx={\frac {3}{4}}$

Properties

Moments

To derive an expression for the non-central moment of order r, it can be used the integral representation of the Bessel function. Its easy to show that,

$\mu _{r}=\int _{0}^{\infty }x^{r}f(x)dx=m^{r}{\frac {K_{r}(a)}{K_{0}(a)}}$

Where:

• r denotes the moment.

Hence the mean and the succeeding three moments about it are

Order	Moment	Cumulant
1	$\mu _{1}=m{\frac {K_{1}(a)}{K_{0}(a)}}$	$\mu$
2	$\mu _{2}=m^{2}({\frac {K_{2}(a)}{K_{0}(a)}}-{\frac {K_{1}^{2}(a)}{K_{0}^{2}(a)}})$	$\sigma ^{2}$
3	$\mu _{3}=m^{3}({\frac {K_{3}(a)}{K_{0}(a)}}-3{\frac {K_{1}(a)K_{2}(a)}{K_{0}^{2}(a)}}+2{\frac {K_{1}^{2}(a)}{K_{0}^{3}(a)}})$	$k_{3}$
4	$\mu _{4}=m^{4}({\frac {K_{4}(a)}{K_{0}(a)}}-4{\frac {K_{1}(a)K_{3}(a)}{K_{0}^{2}(a)}}+6{\frac {K_{1}^{2}(a)K_{2}(a)}{K_{0}^{3}(a)}}-3{\frac {K_{1}^{4}(a)}{K_{0}^{4}(a)}})$	$k_{4}$

Skewness

Skewness is the third moment centered, when it comes to Harmonic distribution, we work with,

$\gamma _{1}={\frac {\mu _{3}}{\mu _{2}^{\frac {3}{2}}}}={\frac {K_{0}^{2}(a)K_{3}(a)-3K_{0}(a)K_{1}(a)K_{2}(a)+2K_{1}^{3}(a)}{(K_{0}(a)K_{2}(a)-K_{1}^{2}(a))^{\frac {3}{2}}}}$

• Always $\gamma _{1}>0$, so the mass of the distribution is concentrated on the left.

Kurtosis

Kurtosis is the fourth moment centered, for Harmonic distribution it is known as,

$\gamma _{2}={\frac {\mu _{4}}{\mu _{2}^{2}}}={\frac {K_{0}^{3}(a)K_{4}(a)-4K_{0}^{2}(a)K_{1}(a)K_{3}(a)+6K_{0}(a)K_{1}^{2}(a)K_{2}(a)-3K_{1}^{4}(a)}{(K_{0}(a)K_{2}(a)-K_{1}^{2}(a))^{2}}}$

• Always $\gamma _{2}>0$ the distribution has a high acute peak around the mean and fatter tails.