Example

>>> from scipy.stats import norm, uniform
>>> from skgof import ks_test, cvm_test, ad_test

>>> ks_test((1, 2, 3), uniform(0, 4))
GofResult(statistic=0.25, pvalue=0.97...)

>>> cvm_test((1, 2, 3), uniform(0, 4))
GofResult(statistic=0.04..., pvalue=0.95...)

>>> data = norm(0, 1).rvs(random_state=1, size=100)
>>> ad_test(data, norm(0, 1))
GofResult(statistic=0.75..., pvalue=0.51...)
>>> ad_test(data, norm(.3, 1))
GofResult(statistic=3.52..., pvalue=0.01...)

Simple tests

Scikit-gof currently only offers three nonparametric tests that let you compare a sample with a reference probability distribution. These are:

ks_test()

Kolmogorov-Smirnov supremum statistic; almost the same as scipy.stats.kstest() with alternative='two-sided' but with (hopefully) somewhat more precise p-value calculation;

cvm_test()

Cramer-von Mises L2 statistic, with a rather crude estimation of the statistic distribution (but seemingly the best available);

ad_test()

Anderson-Darling statistic with a fair approximation of its distribution; unlike the composite scipy.stats.anderson() this one needs a fully specified hypothesized distribution.

Simple test functions use a common interface, taking as the first argument the data (sample) to be compared and as the second argument a frozen scipy.stats distribution. They return a named tuple with two fields: statistic and pvalue.

For a simple example consider the hypothesis that the sample (.4, .1, .7) comes from the uniform distribution on [0, 1]:

if ks_test((.4, .1, .7), unif(0, 1)).pvalue < .05:
    print("Hypothesis rejected with 5% significance.")

If your samples are very large and you have them sorted ahead of time, pass assume_sorted=True to save some time that would be wasted resorting.

Extending

Simple tests are composed of two phases: calculating the test statistic and determining how likely is the resulting value (under the hypothesis). New tests may be defined by providing a new statistic calculation routine or an alternative distribution for a statistic.

Functions calculating statistics are given evaluations of the reference cumulative distribution function on sorted data and are expected to return a single number. For a simple test, if the sample indeed comes from the hypothesized (continuous) distribution, the values passed to the function should be uniformly distributed over [0, 1].

Here is a simplistic example of how a statistic function might look like:

def ex_stat(data):
    return abs(data.sum() - data.size / 2)

Statistic functions for the provided tests, ks_stat(), cvm_stat(), and ad_stat(), can be imported from skgof.ecdfgof.

Statistic distributions should derive from rv_continuous and implement at least one of the abstract _cdf() or _pdf() methods (you might also consider directly coding _sf() for increased precision of results close to 1). For example:

from numpy import sqrt
from scipy.stats import norm, rv_continuous

class ex_unif_gen(rv_continuous):
    def _cdf(self, statistic, samples):
        return 1 - 2 * norm.cdf(-statistic, scale=sqrt(samples / 12))

ex_unif = ex_unif_gen(a=0, name='ex-unif', shapes='samples')

The provided distributions live in separate modules, respectively ksdist, cvmdist, and addist.

Once you have a statistic calculation function and a statistic distribution the two parts can be combined using simple_test:

from functools import partial
from skgof.ecdfgof import simple_test

ex_test = partial(simple_test, stat=ex_stat, pdist=ex_unif)

Exercise: The example test has a fundamental flaw. Can you point it out?

Installation

pip install scikit-gof

Requires recent versions of Python (> 3), NumPy (>= 1.10) and SciPy.

Please fix or point out any errors, inaccuracies or typos you notice.