User:Jheald/scratch/Meixner

Convolution
Considering the (scaled) sum of $$r$$ independent and identically distributed hyperbolic secant random variables:
 * $$X = \frac{1}{\sqrt{r}}\; (X_1 + X_2 + \;...\; + X_r)$$

then in the limit $$r\to\infty$$ the distribution of $$X$$ will tend to the normal distribution $$N(0,1)$$, in accordance with the central limit theorem.

This allows a convenient family of distributions to be defined with properties intermediate between the hyperbolic secant and the normal distribution, controlled by the shape parameter $$r$$, which can be extended to non-integer values via the characteristic function
 * $$\varphi(t) = \big(\operatorname{sech}(t /\sqrt{r})\big)^r$$

Moments can be readily calculated from the characteristic function. The excess kurtosis is found to be $$2/r$$.

Skew
A skewed generalisation of the distribution can be obtained by multiplying by the exponential $$e^{\theta x}$$ and normalising, to give the distribution
 * $$f(x) = \cos \theta \; \frac{e^{\theta x}}{2 \operatorname{cosh}(\frac{\pi x}{2})} $$

where the parameter value $$\theta = 0$$ corresponds to the original distribution.

This is proportional to a horizontal scaling of the asymmetric curve $h(x)=\frac{1}{\exp(-a x) + \exp(b x)}$, considered in a shifted from by Losev (1989).

Location and scale
The distribution (and its generalisations) can also trivially be shifted and scaled in the usual way to give a corresponding location-scale family

All of the above
Allowing all four of the adjustments above gives distribution with four parameters, controlling shape, skew, location, and scale respectively, called either the Meixner distribution after Josef Meixner who first investigated the family, or the NEF-GHS distribution (Natural exponential family - Generalised Hyperbolic Secant distribution).

In financial mathematics the Meixner distribution has been used to model non-Gaussian movement of stock-prices, with applications including the pricing of options.