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, \(B(r, n - r + 1)\).
References
[1]Order statistic. Wikipedia. https://en.wikipedia.org/wiki/Order_statistic
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()
