`scipy.stats._distribution_infrastructure:OrderStatisticDistribution`

source: /scipy/stats/_distribution_infrastructure.py:4812

Signature

def OrderStatisticDistribution(dist, /, *args, r, n, **kwargs)

Members

Summary

Probability distribution of an order statistic.

Extended Summary

An instance of this class represents 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

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

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 the fourth order statistic and compare with a normalized histogram from simulation.

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
from scipy.stats._distribution_infrastructure import OrderStatisticDistribution
X = stats.Normal()
data = X.sample(shape=(10000, 5))
sorted_data = np.sort(data, axis=1)
Y = OrderStatisticDistribution(X, r=4, n=5)
ax = plt.gca()

✓

Y.plot(ax=ax)
ax.hist(sorted_data[:, 3], density=True, bins=30)

✗

(array([0.00215, 0.00358, 0.00788, 0.02435, 0.03223, 0.07306, 0.12248,
       0.17333, 0.27217, 0.38677, 0.48059, 0.55938, 0.69332, 0.72626,
       0.69045, 0.67684, 0.59161, 0.50495, 0.35525, 0.28291, 0.19195,
       0.11961, 0.08308, 0.04584, 0.0308 , 0.01791, 0.01003, 0.00215,
       0.     , 0.00143]), array([-1.47076, -1.33114, -1.19152, -1.0519 , -0.91228, -0.77267,
       -0.63305, -0.49343, -0.35381, -0.21419, -0.07457,  0.06505,
        0.20466,  0.34428,  0.4839 ,  0.62352,  0.76314,  0.90276,
        1.04238,  1.18199,  1.32161,  1.46123,  1.60085,  1.74047,
        1.88009,  2.01971,  2.15932,  2.29894,  2.43856,  2.57818,
        2.7178 ]), <BarContainer object of 30 artists>)

plt.show()

✓

Aliases

scipy.stats._distribution_infrastructure.OrderStatisticDistribution