`scipy.stats._distribution_infrastructure:order_statistic`

source: /scipy/stats/_distribution_infrastructure.py:4953

Signature

def order_statistic(X, /, r, n)

Summary

Probability distribution of an order statistic.

Extended Summary

Returns a random variable that follows the distribution of the $r^{th}$ order statistic of a sample of $n$ observations of a random variable $X$ .

Parameters

X : `ContinuousDistribution`: The random variable $X$ .
r : array_like: The (positive integer) rank of the order statistic $r$ , satisfying 1 <= r <= n.
n : array_like: The (positive integer) sample size $n$ .

Returns

Y : `ContinuousDistribution`: 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 $X_{(1)}, \dots, X_{(r)}, \dots, X_{(n)}$ , then $X_{(r)}$ is known as the $r^{th}$ order statistic.

If the PDF, CDF, and CCDF of $X$ are denoted by $f$ , $F$ , and $G = 1 - F$ , respectively, then the PDF of $X_{(r)}$ is given by:

f_{r} (x) = \frac{n !}{( r - 1 )! ( n - r )!} f (x) F (x)^{r - 1} G (x)^{n - r}

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

Examples

Suppose we are interested in order statistics of samples of size five drawn from the standard normal distribution. Plot the PDF of 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_data = 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_data[:, i]
    ax.hist(y, density=True, bins=30, alpha=0.1, color=colors[i])
Y.plot(ax=ax)

✗

<Axes: title={'center': 'order_statistic(StandardNormal(), r=[1. 2. 3. 4. 5.], n=[5. 5. 5. 5. 5.])'}, xlabel='$x$', ylabel='$pdf$'>

plt.show()

✓