.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/statistics/likelihood/plot_01_maximum_likelihood.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_01_maximum_likelihood.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_statistics_likelihood_plot_01_maximum_likelihood.py:


Fisher's maximum likelihood
================================

Fits an exponential and a gamma distribution to the same skewed data
by maximum likelihood, and plots the exponential model's log-likelihood
as a function of its scale: the curve peaks exactly at the sample mean,
the closed-form MLE.

.. 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 maximum_likelihood_fit








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Fitting two candidate models
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-30

.. code-block:: Python


    rng = np.random.default_rng(0)
    data = rng.gamma(shape=2.0, scale=1.5, size=300)

    expon_fit = maximum_likelihood_fit(data, "expon", floc=0)
    gamma_fit = maximum_likelihood_fit(data, "gamma", floc=0)
    print(f"exponential: scale={expon_fit.params[1]:.3f} (sample mean {data.mean():.3f}), AIC={expon_fit.aic():.1f}")
    print(f"gamma:       shape={gamma_fit.params[0]:.3f}, scale={gamma_fit.params[2]:.3f}, AIC={gamma_fit.aic():.1f}")





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

 .. code-block:: none

    exponential: scale=2.908 (sample mean 2.908), AIC=1242.4
    gamma:       shape=2.243, scale=1.296, AIC=1154.0




.. GENERATED FROM PYTHON SOURCE LINES 31-33

The log-likelihood curve
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 33-43

.. code-block:: Python


    scales = np.linspace(1.0, 6.0, 200)
    loglik = [np.sum(stats.expon.logpdf(data, scale=s)) for s in scales]

    fig, ax = plt.subplots()
    ax.plot(scales, loglik)
    ax.axvline(expon_fit.params[1], color="C1", ls="--", label="MLE = sample mean")
    ax.set_xlabel("exponential scale")
    ax.set_ylabel("log-likelihood")
    ax.legend()



.. image-sg:: /api/gallery/statistics/likelihood/images/sphx_glr_plot_01_maximum_likelihood_001.png
   :alt: plot 01 maximum likelihood
   :srcset: /api/gallery/statistics/likelihood/images/sphx_glr_plot_01_maximum_likelihood_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1192f8ec0>




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

   **Total running time of the script:** (0 minutes 0.024 seconds)


.. _sphx_glr_download_api_gallery_statistics_likelihood_plot_01_maximum_likelihood.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_maximum_likelihood.ipynb <plot_01_maximum_likelihood.ipynb>`

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

      :download:`Download Python source code: plot_01_maximum_likelihood.py <plot_01_maximum_likelihood.py>`

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

      :download:`Download zipped: plot_01_maximum_likelihood.zip <plot_01_maximum_likelihood.zip>`


.. only:: html

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

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