3.3.2.2.1. statsmodels.distributions.edgeworth.ExpandedNormal¶

class statsmodels.distributions.edgeworth.ExpandedNormal(cum, name='Edgeworth expanded normal', **kwds)[source]¶

Construct the Edgeworth expansion pdf given cumulants.

Parameters:	cum: array_like cum[j] contains (j+1)-th cumulant: cum[0] is the mean, cum[1] is the variance and so on.

Notes

This is actually an asymptotic rather than convergent series, hence higher orders of the expansion may or may not improve the result. In a strongly non-Gaussian case, it is possible that the density becomes negative, especially far out in the tails.

References

[R19]

E.A. Cornish and R.A. Fisher, Moments and cumulants in the specification of distributions, Revue de l’Institut Internat. de Statistique. 5: 307 (1938), reprinted in R.A. Fisher, Contributions to Mathematical Statistics. Wiley, 1950.

[R20]

http://en.wikipedia.org/wiki/Edgeworth_series

[R21]

S. Blinnikov and R. Moessner, Expansions for nearly Gaussian distributions, Astron. Astrophys. Suppl. Ser. 130, 193 (1998)

Examples

Construct the 4th order expansion for the chi-square distribution using the known values of the cumulants:

>>> from scipy.misc import factorial
>>> df = 12
>>> chi2_c = [2**(j-1) * factorial(j-1) * df for j in range(1, 5)]
>>> edgw_chi2 = ExpandedNormal(chi2_c, name='edgw_chi2', momtype=0)

Calculate several moments: >>> m, v = edgw_chi2.stats(moments=’mv’) >>> np.allclose([m, v], [df, 2 * df]) True

Plot the density function: >>> mu, sigma = df, np.sqrt(2*df) >>> x = np.linspace(mu - 3*sigma, mu + 3*sigma) >>> plt.plot(x, stats.chi2.pdf(x, df=df), ‘g-‘, lw=4, alpha=0.5) >>> plt.plot(x, stats.norm.pdf(x, mu, sigma), ‘b–’, lw=4, alpha=0.5) >>> plt.plot(x, edgw_chi2.pdf(x), ‘r-‘, lw=2) >>> plt.show()

__init__(cum, name='Edgeworth expanded normal', **kwds)[source]¶