.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/statistics/shrinkage/plot_01_james_stein.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_shrinkage_plot_01_james_stein.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_statistics_shrinkage_plot_01_james_stein.py:


Stein's paradox
====================

Estimates p unrelated normal means from one observation each. Shrinking
all the observations toward zero by the James-Stein factor lowers the
total squared error for every p >= 3, even though no single
observation carries information about the others.

.. GENERATED FROM PYTHON SOURCE LINES 12-17

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.statistics import james_stein_estimator








.. GENERATED FROM PYTHON SOURCE LINES 18-20

Risk as a function of dimension
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-39

.. code-block:: Python


    rng = np.random.default_rng(0)
    dims = np.arange(3, 31)
    raw_risk, js_risk = [], []
    for p in dims:
        theta = rng.normal(scale=1.5, size=p)
        x = theta + rng.normal(size=(2000, p))
        shrunk = np.array([james_stein_estimator(row) for row in x])
        raw_risk.append(np.mean(np.sum((x - theta) ** 2, axis=1)))
        js_risk.append(np.mean(np.sum((shrunk - theta) ** 2, axis=1)))
        if p in (3, 10, 30):
            print(f"p={p:2d}: raw risk {raw_risk[-1]:6.2f}, James-Stein risk {js_risk[-1]:6.2f}")

    fig, ax = plt.subplots()
    ax.plot(dims, raw_risk, "o-", label="x (risk = p)")
    ax.plot(dims, js_risk, "s-", label="James-Stein")
    ax.set_xlabel("dimension p")
    ax.set_ylabel("mean total squared error")
    ax.legend()



.. image-sg:: /api/gallery/statistics/shrinkage/images/sphx_glr_plot_01_james_stein_001.png
   :alt: plot 01 james stein
   :srcset: /api/gallery/statistics/shrinkage/images/sphx_glr_plot_01_james_stein_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    p= 3: raw risk   2.97, James-Stein risk   1.95
    p=10: raw risk  10.03, James-Stein risk   7.10
    p=30: raw risk  29.92, James-Stein risk  21.74

    <matplotlib.legend.Legend object at 0x1192f8ec0>




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

   **Total running time of the script:** (0 minutes 0.102 seconds)


.. _sphx_glr_download_api_gallery_statistics_shrinkage_plot_01_james_stein.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_james_stein.ipynb <plot_01_james_stein.ipynb>`

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

      :download:`Download Python source code: plot_01_james_stein.py <plot_01_james_stein.py>`

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

      :download:`Download zipped: plot_01_james_stein.zip <plot_01_james_stein.zip>`


.. only:: html

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

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