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Normal Inverse Gaussian Distribution

docs/tutorial:stats:continuous_norminvgauss

The probability density function is given by:

 \begin{eqnarray*}
	        f(x; a, b) = \frac{a \exp\left(\sqrt{a^2 - b^2} + b x \right)}{\pi \sqrt{1 + x^2}} \, K_1\left(a * \sqrt{1 + x^2}\right),
	\end{eqnarray*}

where $x$ is a real number, the parameter $a$ is the tail heaviness and $b$ is the asymmetry parameter satisfying $a > 0$ and $∣ b ∣ \leq a$ . $K_{1}$ is the modified Bessel function of second kind (scipy.special.k1).

A normal inverse Gaussian random variable with parameters $a$ and $b$ can be expressed as $X = bV + (V) X$ where $X$ is norm(0,1) and $V$ is invgauss(mu=1/sqrt(a**2 - b**2)). Hence, the normal inverse Gaussian distribution is a special case of normal variance-mean mixtures.

Another common parametrization of the distribution is given by the following expression of the pdf:

 \begin{eqnarray*}
        g(x, \alpha, \beta, \delta, \mu) = \frac{\alpha\delta K_1 \left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)}{\pi \sqrt{\delta^2 + (x - \mu)^2}} \,
        e^{\delta \sqrt{\alpha^2 - \beta^2} + \beta (x - \mu)}
	\end{eqnarray*}

In SciPy, this corresponds to $a = α δ, b = β δ, loc = μ, scale = δ$ .

Implementation: scipy.stats.norminvgauss