scipy.stats.skewcauchy

scipy.stats.skewcauchy = <scipy.stats._continuous_distns.skewcauchy_gen object>

A skewed Cauchy random variable.

As an instance of the rv_continuous class, skewcauchy 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.

See also

cauchy

Cauchy distribution

Notes

The probability density function for skewcauchy is:

\[f(x) = \frac{1}{\pi \left(\frac{x^2}{\left(a\, \text{sign}(x) + 1 \right)^2} + 1 \right)}\]

for a real number \(x\) and skewness parameter \(-1 < a < 1\).

When \(a=0\), the distribution reduces to the usual Cauchy distribution.

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

“Skewed generalized t distribution”, Wikipedia https://en.wikipedia.org/wiki/Skewed_generalized_t_distribution#Skewed_Cauchy_distribution

Examples

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

Calculate the first four moments:

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

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = skewcauchy.rvs(a, size=1000)

And compare the histogram:

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

Methods

rvs(a, loc=0, scale=1, size=1, random_state=None)	Random variates.
pdf(x, a, loc=0, scale=1)	Probability density function.
logpdf(x, a, loc=0, scale=1)	Log of the probability density function.
cdf(x, a, loc=0, scale=1)	Cumulative distribution function.
logcdf(x, a, loc=0, scale=1)	Log of the cumulative distribution function.
sf(x, a, loc=0, scale=1)	Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).
logsf(x, a, loc=0, scale=1)	Log of the survival function.
ppf(q, a, loc=0, scale=1)	Percent point function (inverse of cdf — percentiles).
isf(q, a, loc=0, scale=1)	Inverse survival function (inverse of sf).
moment(n, a, loc=0, scale=1)	Non-central moment of order n
stats(a, loc=0, scale=1, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(a, 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,), 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, loc=0, scale=1)	Median of the distribution.
mean(a, loc=0, scale=1)	Mean of the distribution.
var(a, loc=0, scale=1)	Variance of the distribution.
std(a, loc=0, scale=1)	Standard deviation of the distribution.
interval(alpha, a, loc=0, scale=1)	Endpoints of the range that contains fraction alpha [0, 1] of the distribution

scipy.stats.semicircular scipy.stats.skewnorm