scipy.stats.

order_statistic#

scipy.stats.order_statistic(X, /, *, r, n)[source]#

Probability distribution of an order statistic

Returns a random variable that follows the distribution underlying the \(r^{\text{th}}\) order statistic of a sample of \(n\) observations of a random variable \(X\).

Parameters:
XContinuousDistribution

The random variable \(X\)

rarray_like

The (positive integer) rank of the order statistic \(r\)

narray_like

The (positive integer) sample size \(n\)

Returns:
YContinuousDistribution

A random variable that follows the distribution of the prescribed order statistic.

Notes

If we make \(n\) observations of a continuous random variable \(X\) and sort them in increasing order \(X_{(1)}, \dots, X_{(r)}, \dots, X_{(n)}\), \(X_{(r)}\) is known as the \(r^{\text{th}}\) order statistic.

If the PDF, CDF, and CCDF underlying math:X are denoted \(f\), \(F\), and \(F'\), respectively, then the PDF underlying math:X_{(r)} is given by:

\[f_r(x) = \frac{n!}{(r-1)! (n-r)!} f(x) F(x)^{r-1} F'(x)^{n - r}\]

The CDF and other methods of the distribution underlying \(X_{(r)}\) are calculated using the fact that \(X = F^{-1}(U)\), where \(U\) is a standard uniform random variable, and that the order statistics of observations of U follow a beta distribution, \(B(r, n - r + 1)\).

Array API Standard Support

order_statistic has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library

CPU

GPU

NumPy

✅

n/a

CuPy

n/a

⛔

PyTorch

⛔

⛔

JAX

⛔

⛔

Dask

⛔

n/a

See Support for the array API standard for more information.

References

Examples

Suppose we are interested in order statistics of samples of size five drawn from the standard normal distribution. Plot the PDF underlying each order statistic and compare with a normalized histogram from simulation.

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>>
>>> X = stats.Normal()
>>> data = X.sample(shape=(10000, 5))
>>> sorted = np.sort(data, axis=1)
>>> Y = stats.order_statistic(X, r=[1, 2, 3, 4, 5], n=5)
>>>
>>> ax = plt.gca()
>>> colors = plt.rcParams['axes.prop_cycle'].by_key()['color']
>>> for i in range(5):
...     y = sorted[:, i]
...     ax.hist(y, density=True, bins=30, alpha=0.1, color=colors[i])
>>> Y.plot(ax=ax)
>>> plt.show()
../../_images/scipy-stats-order_statistic-1.png
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