Benini Distribution¶

The Benini distribution has the probability density function

\[f(x, \alpha, \beta, \sigma) = e^{\theta(x, \alpha, \beta, \sigma)} \left( \frac{\alpha}{x} + \frac{2\beta\log\left(\frac{x}{\sigma}\right)}{x} \right)\]

and cumulative distribution function

\[F(x, \alpha, \beta, \sigma) = 1 - e^{\theta(x, \alpha, \beta, \sigma)}\]

where

\[\theta(x, \alpha, \beta, \sigma) = -\alpha \log\left(\frac{x}{\sigma}\right) - \beta \log\left(\frac{x}{\sigma}\right)^2\]

\(\alpha\) and \(\beta\) are shape parameters; \(\sigma\) is a scale parameter.

The support is \(x > \sigma\).

See

Kleiber, Christian; Kotz, Samuel (2003). “Chapter 7.1: Benini Distribution”, Statistical Size Distributions in Economics and Actuarial Sciences. Wiley. ISBN 978-0-471-15064-0.
“Benini distribution”, Wikipedia, https://en.wikipedia.org/wiki/Benini_distribution

mpsci.distributions.benini.cdf(x, alpha, beta, scale)¶: Cumulative distribution function of the Benini distribution.

mpsci.distributions.benini.invcdf(p, alpha, beta, scale)¶: Inverse CDF of the Benini distribution.

mpsci.distributions.benini.invsf(p, alpha, beta, scale)¶: Inverse survival function of the Benini distribution.

mpsci.distributions.benini.logcdf(x, alpha, beta, scale)¶: Natural logarithm of the CDF of the Benini distribution.

mpsci.distributions.benini.logpdf(x, alpha, beta, scale)¶: Natural logarithm of the PDF of the Benini distribution.

mpsci.distributions.benini.logsf(x, alpha, beta, scale)¶: Natural logarithm of the survival function of the Benini distribution.

mpsci.distributions.benini.mean(alpha, beta, scale)¶: Mean of the Benini distribution.

mpsci.distributions.benini.pdf(x, alpha, beta, scale)¶: Probability density function of the Benini distribution.

mpsci.distributions.benini.sf(x, alpha, beta, scale)¶: Cumulative distribution function of the Benini distribution.

mpsci.distributions.benini.var(alpha, beta, scale)¶: Variance of the Benini distribution.