`scipy.stats._distribution_infrastructure:OrderStatisticDistribution`

source: /scipy/stats/_distribution_infrastructure.py :4874

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 underlying 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 (integer) sample size $n$
r : array_like: The (integer) rank of the order statistic $r$

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 $r^{th}$ 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, $B (r, n - r + 1)$ .

Examples

Suppose we are interested in order statistics of samples of size five drawn from the standard normal distribution. Plot the PDF underlying 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))
ranks = np.sort(data, axis=1)
Y = OrderStatisticDistribution(X, r=4, n=5)
ax = plt.gca()

✓

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

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plt.show()

✓

Aliases

scipy.stats._distribution_infrastructure.OrderStatisticDistribution