scipy.stats.jf_skew_t

scipy.stats.jf_skew_t = <scipy.stats._continuous_distns.jf_skew_t_gen object>

Jones and Faddy skew-t distribution.

As an instance of the rv_continuous class, jf_skew_t 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.

Notes

The probability density function for jf_skew_t is:

\[f(x; a, b) = C_{a,b}^{-1} \left(1+\frac{x}{\left(a+b+x^2\right)^{1/2}}\right)^{a+1/2} \left(1-\frac{x}{\left(a+b+x^2\right)^{1/2}}\right)^{b+1/2}\]

for real numbers \(a>0\) and \(b>0\), where \(C_{a,b} = 2^{a+b-1}B(a,b)(a+b)^{1/2}\), and \(B\) denotes the beta function (scipy.special.beta).

When \(a<b\), the distribution is negatively skewed, and when \(a>b\), the distribution is positively skewed. If \(a=b\), then we recover the t distribution with \(2a\) degrees of freedom.

jf_skew_t takes \(a\) and \(b\) as shape parameters.

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

References

[1]

M.C. Jones and M.J. Faddy. “A skew extension of the t distribution, with applications” Journal of the Royal Statistical Society. Series B (Statistical Methodology) 65, no. 1 (2003): 159-174. DOI:10.1111/1467-9868.00378

Examples

Try it in your browser!

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

Calculate the first four moments:

>>> a, b = 8, 4
>>> mean, var, skew, kurt = jf_skew_t.stats(a, b, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(jf_skew_t.ppf(0.01, a, b),
...                 jf_skew_t.ppf(0.99, a, b), 100)
>>> ax.plot(x, jf_skew_t.pdf(x, a, b),
...        'r-', lw=5, alpha=0.6, label='jf_skew_t 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 = jf_skew_t(a, b)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = jf_skew_t.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()

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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.