scipy.stats.norminvgauss = <scipy.stats._continuous_distns.norminvgauss_gen object>[source]

A Normal Inverse Gaussian continuous random variable.

As an instance of the rv_continuous class, norminvgauss object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.


The probability density function for norminvgauss is:

\[f(x; a, b) = (a \exp(\sqrt{a^2 - b^2} + b x)) / (\pi \sqrt{1 + x^2} \, K_1(a * \sqrt{1 + x^2}))\]

where x is a real number, the parameter a is the tail heaviness and b is the asymmetry parameter satisfying a > 0 and abs(b) <= a. K_1 is the modified Bessel function of second kind (scipy.special.k1).

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, norminvgauss.pdf(x, a, b, loc, scale) is identically equivalent to norminvgauss.pdf(y, a, b) / scale with y = (x - loc) / scale.

A normal inverse Gaussian random variable with parameters a and b can be expressed as X = b * V + sqrt(V) * X where X is norm(0,1) and V is invgauss(mu=1/sqrt(a**2 - b**2)). This representation is used to generate random variates.


O. Barndorff-Nielsen, “Hyperbolic Distributions and Distributions on Hyperbolae”, Scandinavian Journal of Statistics, Vol. 5(3), pp. 151-157, 1978.

O. Barndorff-Nielsen, “Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling”, Scandinavian Journal of Statistics, Vol. 24, pp. 1–13, 1997.


>>> from scipy.stats import norminvgauss
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate a few first moments:

>>> a, b = 1, 0.5
>>> mean, var, skew, kurt = norminvgauss.stats(a, b, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(norminvgauss.ppf(0.01, a, b),
...                 norminvgauss.ppf(0.99, a, b), 100)
>>> ax.plot(x, norminvgauss.pdf(x, a, b),
...        'r-', lw=5, alpha=0.6, label='norminvgauss pdf')

Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.

Freeze the distribution and display the frozen pdf:

>>> rv = norminvgauss(a, b)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = norminvgauss.ppf([0.001, 0.5, 0.999], a, b)
>>> np.allclose([0.001, 0.5, 0.999], norminvgauss.cdf(vals, a, b))

Generate random numbers:

>>> r = norminvgauss.rvs(a, b, size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)


rvs(a, b, loc=0, scale=1, size=1, random_state=None) Random variates.
pdf(x, a, b, loc=0, scale=1) Probability density function.
logpdf(x, a, b, loc=0, scale=1) Log of the probability density function.
cdf(x, a, b, loc=0, scale=1) Cumulative distribution function.
logcdf(x, a, b, loc=0, scale=1) Log of the cumulative distribution function.
sf(x, a, b, loc=0, scale=1) Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).
logsf(x, a, b, loc=0, scale=1) Log of the survival function.
ppf(q, a, b, loc=0, scale=1) Percent point function (inverse of cdf — percentiles).
isf(q, a, b, loc=0, scale=1) Inverse survival function (inverse of sf).
moment(n, a, b, loc=0, scale=1) Non-central moment of order n
stats(a, b, loc=0, scale=1, moments=’mv’) Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(a, b, loc=0, scale=1) (Differential) entropy of the RV.
fit(data, a, b, loc=0, scale=1) Parameter estimates for generic data.
expect(func, args=(a, b), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds) Expected value of a function (of one argument) with respect to the distribution.
median(a, b, loc=0, scale=1) Median of the distribution.
mean(a, b, loc=0, scale=1) Mean of the distribution.
var(a, b, loc=0, scale=1) Variance of the distribution.
std(a, b, loc=0, scale=1) Standard deviation of the distribution.
interval(alpha, a, b, loc=0, scale=1) Endpoints of the range that contains alpha percent of the distribution

Previous topic


Next topic