`scipy.stats._stats_py:expectile`

source: /scipy/stats/_stats_py.py :10100

Signature

def expectile ( a , alpha = 0.5 , * , weights = None )

Summary

Compute the expectile at the specified level.

Extended Summary

Expectiles are a generalization of the expectation in the same way as quantiles are a generalization of the median. The expectile at level alpha = 0.5 is the mean (average). See Notes for more details.

Parameters

a : array_like: Array containing numbers whose expectile is desired.
alpha : float, default: 0.5: The level of the expectile; e.g., alpha=0.5 gives the mean.
weights : array_like, optional: An array of weights associated with the values in a. The weights must be broadcastable to the same shape as a. Default is None, which gives each value a weight of 1.0. An integer valued weight element acts like repeating the corresponding observation in a that many times. See Notes for more details.

Returns

expectile : ndarray: The empirical expectile at level alpha.

Notes

In general, the expectile at level $α$ of a random variable $X$ with cumulative distribution function (CDF) $F$ is given by the unique solution $t$ of:

α E ((X - t)_{+}) = (1 - α) E ((t - X)_{+}) .

Here, $(x)_{+} = max (0, x)$ is the positive part of $x$ . This equation can be equivalently written as:

α \int_{t}^{\infty} (x - t) d F (x) = (1 - α) \int_{- \infty}^{t} (t - x) d F (x) .

The empirical expectile at level $α$ (alpha) of a sample $a_{i}$ (the array a) is defined by plugging in the empirical CDF of a. Given sample or case weights $w$ (the array weights), it reads $F_{a} (x) = \frac{1}{\sum _{i} w _{i}} \sum_{i} w_{i} 1_{a_{i} \leq x}$ with indicator function $1_{A}$ . This leads to the definition of the empirical expectile at level alpha as the unique solution $t$ of:

α i = 1 \sum n w_{i} (a_{i} - t)_{+} = (1 - α) i = 1 \sum n w_{i} (t - a_{i})_{+} .

For $α = 0.5$ , this simplifies to the weighted average. Furthermore, the larger $α$ , the larger the value of the expectile.

As a final remark, the expectile at level $α$ can also be written as a minimization problem. One often used choice is

argmin_{t} E (∣ 1_{t \geq X} - α ∣ (t - X)^{2}) .

Array API Standard Support

expectile 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 dev-arrayapi for more information.

Examples

import numpy as np
from scipy.stats import expectile
a = [1, 4, 2, -1]

✓

expectile(a, alpha=0.5) == np.mean(a)
expectile(a, alpha=0.2)
expectile(a, alpha=0.8)

✗

weights = [1, 3, 1, 1]

✓

expectile(a, alpha=0.8, weights=weights)

✗

Aliases

scipy.stats.expectile

Referenced by

This package

release:1.10.0-notes