Wakeby distribution

The Wakeby distribution is a five-parameter probability distribution defined by the transformation

X =\xi + \frac{\alpha}{\beta} (1 - (1-U)^{\beta}) - \frac{\gamma}{\delta} (1 - (1-U)^{-\delta})

where U is a standard uniform random variable. That is, the above equation defines the quantile function for the Wakeby distribution.^[1] The parameters β, γ and δ are shape parameters. The parameters ξ and α are location parameters.

The Wakeby distribution has been used for modelling flood flows.^[2] This distribution was first proposed by Harold A. Thomas Jr., who named it after Wakeby Pond in Cape Cod.

The following restrictions apply to the parameters of this distribution:

$\beta + \delta \ge 0$
Either $\beta + \delta > 0$ or $\beta = \gamma = \delta = 0$
If $\gamma > 0$ , then $\delta > 0$
$\gamma \ge 0$
$\alpha + \gamma \ge 0$

The domain of the Wakeby distribution is

$\xi$ to $\infty$ , if $\delta \ge 0$ and $\gamma > 0$
$\xi$ to $\xi + (\alpha/ \beta) - (\gamma/ \delta)$ , if $\delta < 0$ or $\gamma = 0$

With three shape parameters, the Wakeby distribution can model a wide variety of shapes.

The cumulative distribution function is computed by numerically inverting the quantile function given above. The probability density function is then found by using the following relation (given on page 46 of Johnson, Kotz, and Balakrishnan):

f(x) = \frac{(1 - F(x))^{(\delta+1)}}{\alpha t + \gamma}

where F is the cumulative distribution function and

t = (1 - F(x))^{(\beta + \delta)}

An implementation that computes the probability density function of the Wakeby distribution is included in the Dataplot scientific computation library, as routine WAKPDF.^[1]

References

1 2 "Dataplot reference manual: WAKPDF". NIST. Retrieved 20 August 2015.
↑ John C. Houghton (October 14, 1977). "Birth of a Parent: The Wakeby Distribution for Modeling Flood Flows; Working Paper No. MIT-EL77-033WP" (PDF). MIT.

External links

Discussion of the naming of the distribution on Stack Exchange

Some common univariate probability distributions

Continuous	beta Cauchy chi-squared exponential F gamma Laplace Asymmetric Laplace log-normal Maxwell–Boltzmann normal Pareto Rayleigh Student's t uniform Wakeby Weibull

Discrete	Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson

List of probability distributions

Note: this work is based on a NIST document that is in the public domain as a work of the U.S. federal government

This article is issued from Wikipedia - version of the Wednesday, October 07, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Wakeby distribution

See also

References

External links