.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/statistics/bootstrap/plot_02_jackknife.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_bootstrap_plot_02_jackknife.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_statistics_bootstrap_plot_02_jackknife.py:


The jackknife
==================

Leaves out one observation at a time and recomputes the statistic.
For the plug-in variance, the jackknife's bias correction recovers the
unbiased n - 1 variance exactly, and for the mean its standard error is
exactly the classical s / sqrt(n). For a non-smooth statistic such as
the midrange, the jackknife standard error is unreliable and disagrees
with the bootstrap's -- one of the shortcomings Efron's bootstrap was
designed to overcome.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.statistics import bootstrap_confidence_interval, jackknife








.. GENERATED FROM PYTHON SOURCE LINES 21-23

Bias correction of the plug-in variance
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 23-31

.. code-block:: Python


    rng = np.random.default_rng(0)
    data = rng.normal(loc=10.0, scale=3.0, size=20)
    var_result = jackknife(data, np.var)
    print(f"plug-in variance    {var_result.estimate:.4f}")
    print(f"jackknife-corrected {var_result.bias_corrected:.4f}")
    print(f"unbiased (ddof=1)   {np.var(data, ddof=1):.4f}")





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

 .. code-block:: none

    plug-in variance    6.5159
    jackknife-corrected 6.8588
    unbiased (ddof=1)   6.8588




.. GENERATED FROM PYTHON SOURCE LINES 32-34

Standard error of the mean and of a non-smooth statistic
--------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-51

.. code-block:: Python



    def midrange(x):
        return 0.5 * (np.min(x) + np.max(x))


    mean_result = jackknife(data, np.mean)
    print(f"\nmean: jackknife SE {mean_result.std_error:.4f}, s/sqrt(n) {np.std(data, ddof=1) / np.sqrt(data.size):.4f}")
    mid_result = jackknife(data, midrange)
    mid_boot = bootstrap_confidence_interval(data, statistic=midrange, n_resamples=4000, method="percentile", seed=0)
    print(f"midrange: jackknife SE {mid_result.std_error:.4f}, bootstrap SE {mid_boot.std_error:.4f}")

    fig, ax = plt.subplots()
    ax.plot(mean_result.replicates, "o", label="leave-one-out means")
    ax.axhline(mean_result.estimate, color="k", lw=1, label="full-sample mean")
    ax.set_xlabel("observation left out")
    ax.legend()



.. image-sg:: /api/gallery/statistics/bootstrap/images/sphx_glr_plot_02_jackknife_001.png
   :alt: plot 02 jackknife
   :srcset: /api/gallery/statistics/bootstrap/images/sphx_glr_plot_02_jackknife_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    mean: jackknife SE 0.5856, s/sqrt(n) 0.5856
    midrange: jackknife SE 1.5742, bootstrap SE 0.8834

    <matplotlib.legend.Legend object at 0x11b8fbe00>




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

   **Total running time of the script:** (0 minutes 0.037 seconds)


.. _sphx_glr_download_api_gallery_statistics_bootstrap_plot_02_jackknife.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_jackknife.ipynb <plot_02_jackknife.ipynb>`

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

      :download:`Download Python source code: plot_02_jackknife.py <plot_02_jackknife.py>`

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

      :download:`Download zipped: plot_02_jackknife.zip <plot_02_jackknife.zip>`


.. only:: html

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

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