Pearson distribution

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The Pearson distribution is a family of probability distributions that are a generalisation of the normal distribution.

These models are used in financial markets, given their ability to be parametrised in a way that has intuitive meaning for market traders. A number of models are in current use that capture the stochastic nature of the volatility of rates, stocks etc. and this family of distributions may prove to be one of the more important.

The Pearson distributions were given by Karl Pearson in 1895 in one of a series of articles on the mathematical theory of evolution (see references below). Pearson described five types of continuous distributions:

  • I. Limited range in both directions, with skewness.
  • II. Limited range in both directions, symmetric.
  • III. Limited range in one direction only (must be skew).
  • IV. Unlimited range in both directions, with skewness.
  • V. Unlimited range in both directions, symmetric.

Generalizing the hypergeometric distribution, Pearson proposed a probability density proportional to:

(1+x/a_1)^{\nu a_1} (1-x/a_2)^{\nu a_2}

for − a1 < x < a2, and by taking various limits of this derived forms for his Types I, II, III, and V. For Type IV he derived a related form:

exp( − νarctan(x / a)) / (1 + x2 / a2)m

These can be shifted to the desired location. Some of his forms correspond to other named distributions.

[edit] Pearson type III distribution

Pearson type III
Probability density function
Cumulative distribution function
Parameters \alpha\! location (real)
\beta>0\! scale (real)
p>0\! shape (real)
Support x\in [a;\infty)\!
Probability density function (pdf) \frac{1}{\beta\,\Gamma(p)} \left(\frac{x-\alpha}{\beta}\right)^{p-1} \!\!e^{-(x-\alpha)/\beta}\!
Cumulative distribution function (cdf) 1-\frac{\Gamma\left(p,\frac{x-\alpha}{\beta}\right)}{\Gamma(p)\!}
Mean \alpha + p\,\beta\!
Median
Mode
Variance p\,\beta^2\!
Skewness \frac{2}{\sqrt{p}}\!
Excess Kurtosis \frac{6}{p}\!
Entropy
mgf
Char. func. e^{i\alpha t}(1 - i\beta t)^{-p}\!

The Pearson Type III distribution is given by the probability density function

f(x) = \frac{1}{\beta\,\Gamma(p)} \left(\frac{x-\alpha}{\beta}\right)^{p-1} e^{-(x-\alpha)/\beta}, \!

where x ∈ [α,∞) and α, β and p are parameters of the distribution with β > 0 and p > 0 (Abramowitz and Stegun 1954, p. 930). Here, Γ denotes the Gamma function.

eiαt(1 − iβt) − p
α + pβ
pβ2
\frac{2}{\sqrt{p}}
\frac{6}{p}

When α=0, β=2, and p is half-integer, the Pearson Type III distribution becomes the chi squared distribution of 2p degrees of freedom.


[edit] References

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Continuous: Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square • inverse Gaussian • inverse gamma • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda Dirichlet • Kent • matrix normal • multivariate normal • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling • singular
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