.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/statistics/likelihood/plot_02_wilks_likelihood_ratio.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_statistics_likelihood_plot_02_wilks_likelihood_ratio.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_statistics_likelihood_plot_02_wilks_likelihood_ratio.py:


Wilks's theorem and the likelihood-ratio test
===================================================

Repeatedly tests the true null hypothesis "the normal mean is 0" with
the likelihood-ratio statistic 2(l1 - l0). Wilks's theorem says the
statistic's null distribution approaches chi-square with one degree of
freedom, the number of parameters the null hypothesis fixes.

.. GENERATED FROM PYTHON SOURCE LINES 12-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy import stats

    from mathematicskit.statistics import likelihood_ratio_test, maximum_likelihood_fit








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Simulating the statistic under the null
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-38

.. code-block:: Python


    rng = np.random.default_rng(0)
    statistics = []
    for _ in range(1000):
        data = rng.normal(loc=0.0, scale=1.0, size=50)
        null = maximum_likelihood_fit(data, "norm", floc=0.0)
        alt = maximum_likelihood_fit(data, "norm")
        statistics.append(likelihood_ratio_test(null.log_likelihood, alt.log_likelihood, df=1).statistic)
    statistics = np.array(statistics)
    print(f"fraction above the chi2(1) 95% point: {np.mean(statistics > stats.chi2.ppf(0.95, 1)):.3f} (nominal 0.05)")

    fig, ax = plt.subplots()
    ax.hist(statistics, bins=40, density=True, alpha=0.6, label="simulated 2(l1 - l0)")
    grid = np.linspace(0.05, 10, 200)
    ax.plot(grid, stats.chi2.pdf(grid, 1), label=r"$\chi^2_1$ density")
    ax.set_ylim(0, 1.5)
    ax.legend()



.. image-sg:: /api/gallery/statistics/likelihood/images/sphx_glr_plot_02_wilks_likelihood_ratio_001.png
   :alt: plot 02 wilks likelihood ratio
   :srcset: /api/gallery/statistics/likelihood/images/sphx_glr_plot_02_wilks_likelihood_ratio_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    fraction above the chi2(1) 95% point: 0.066 (nominal 0.05)

    <matplotlib.legend.Legend object at 0x11a99dbe0>




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.082 seconds)


.. _sphx_glr_download_api_gallery_statistics_likelihood_plot_02_wilks_likelihood_ratio.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_wilks_likelihood_ratio.ipynb <plot_02_wilks_likelihood_ratio.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_wilks_likelihood_ratio.py <plot_02_wilks_likelihood_ratio.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_wilks_likelihood_ratio.zip <plot_02_wilks_likelihood_ratio.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_