scipy.stats.argus = <scipy.stats._continuous_distns.argus_gen object>[source]#

Argus distribution

As an instance of the rv_continuous class, argus 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 argus is:

\[f(x, \chi) = \frac{\chi^3}{\sqrt{2\pi} \Psi(\chi)} x \sqrt{1-x^2} \exp(-\chi^2 (1 - x^2)/2)\]

for \(0 < x < 1\) and \(\chi > 0\), where

\[\Psi(\chi) = \Phi(\chi) - \chi \phi(\chi) - 1/2\]

with \(\Phi\) and \(\phi\) being the CDF and PDF of a standard normal distribution, respectively.

argus takes \(\chi\) as shape a parameter. Details about sampling from the ARGUS distribution can be found in [2].

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



“ARGUS distribution”,


Christoph Baumgarten “Random variate generation by fast numerical inversion in the varying parameter case.” Research in Statistics, vol. 1, 2023, doi:10.1080/27684520.2023.2279060.

Added in version 0.19.0.


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

Calculate the first four moments:

>>> chi = 1
>>> mean, var, skew, kurt = argus.stats(chi, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

>>> vals = argus.ppf([0.001, 0.5, 0.999], chi)
>>> np.allclose([0.001, 0.5, 0.999], argus.cdf(vals, chi))

Generate random numbers:

>>> r = argus.rvs(chi, 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)


rvs(chi, loc=0, scale=1, size=1, random_state=None)

Random variates.

pdf(x, chi, loc=0, scale=1)

Probability density function.

logpdf(x, chi, loc=0, scale=1)

Log of the probability density function.

cdf(x, chi, loc=0, scale=1)

Cumulative distribution function.

logcdf(x, chi, loc=0, scale=1)

Log of the cumulative distribution function.

sf(x, chi, loc=0, scale=1)

Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).

logsf(x, chi, loc=0, scale=1)

Log of the survival function.

ppf(q, chi, loc=0, scale=1)

Percent point function (inverse of cdf — percentiles).

isf(q, chi, loc=0, scale=1)

Inverse survival function (inverse of sf).

moment(order, chi, loc=0, scale=1)

Non-central moment of the specified order.

stats(chi, loc=0, scale=1, moments=’mv’)

Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).

entropy(chi, loc=0, scale=1)

(Differential) entropy of the RV.


Parameter estimates for generic data. See for detailed documentation of the keyword arguments.

expect(func, args=(chi,), 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(chi, loc=0, scale=1)

Median of the distribution.

mean(chi, loc=0, scale=1)

Mean of the distribution.

var(chi, loc=0, scale=1)

Variance of the distribution.

std(chi, loc=0, scale=1)

Standard deviation of the distribution.

interval(confidence, chi, loc=0, scale=1)

Confidence interval with equal areas around the median.