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# scipy.stats.norminvgauss

scipy.stats.norminvgauss = <scipy.stats._continuous_distns.norminvgauss_gen object>

  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.

  Methods

  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(order, a, b, loc=0, scale=1) | Non-central moment of the specified order.
  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) | Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments.
  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(confidence, a, b, loc=0, scale=1) | Confidence interval with equal areas around the median.

  Notes

  The probability density function for `norminvgauss` is:

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

  where \(x\) is a real number, the parameter \(a\) is the tail heaviness and \(b\) is the asymmetry parameter satisfying \(a > 0\) and \(|b| <= a\). \(K_1\) 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`. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.

  A normal inverse Gaussian random variable Y with parameters a and b can be expressed as a normal mean-variance mixture: `Y = 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.

  Another common parametrization of the distribution (see Equation 2.1 in [2]) is given by the following expression of the pdf:

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

  In SciPy, this corresponds to a = alpha * delta, b = beta * delta, loc = mu, scale=delta.

  References

  [1]

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

  [2]

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

  Examples

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

  Get the support:

      >>> a, b = 1.25, 0.5
      >>> lb, ub = norminvgauss.support(a, b)

  Calculate the first four moments:

      >>> 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))
      True

  Generate random numbers:

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

  And compare the histogram:

      >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
      >>> ax.set_xlim([x[0], x[-1]])
      >>> ax.legend(loc='best', frameon=False)
      >>> plt.show()