scipy.stats.

expectile#

scipy.stats.expectile(a, alpha=0.5, *, weights=None)[source]#

Compute the expectile at the specified level.

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:

aarray_like: Array containing numbers whose expectile is desired.
alphafloat, default: 0.5: The level of the expectile; e.g., alpha=0.5 gives the mean.
weightsarray_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:

expectilendarray: The empirical expectile at level alpha.

See also

numpy.mean: Arithmetic average
numpy.quantile: Quantile

Notes

In general, the expectile at level $\alpha$ 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)_{+}) .

$\alpha E((X - t)_+) = (1 - \alpha) E((t - X)_+) \,.$

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) .

$\alpha \int_t^\infty (x - t)\mathrm{d}F(x) = (1 - \alpha) \int_{-\infty}^t (t - x)\mathrm{d}F(x) \,.$

The empirical expectile at level $\alpha$ (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 \sum_{i = 1}^{n} w_{i} (a_{i} - t)_{+} = (1 - α) \sum_{i = 1}^{n} w_{i} (t - a_{i})_{+} .

$\alpha \sum_{i=1}^n w_i (a_i - t)_+ = (1 - \alpha) \sum_{i=1}^n w_i (t - a_i)_+ \,.$

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

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

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

$\operatorname{argmin}_t E(\lvert 1_{t\geq X} - \alpha\rvert(t - X)^2) \,.$

References

[1]

W. K. Newey and J. L. Powell (1987), “Asymmetric Least Squares Estimation and Testing,” Econometrica, 55, 819-847.

[2]

T. Gneiting (2009). “Making and Evaluating Point Forecasts,” Journal of the American Statistical Association, 106, 746 - 762. DOI:10.48550/arXiv.0912.0902

Examples

>>> import numpy as np
>>> from scipy.stats import expectile
>>> a = [1, 4, 2, -1]
>>> expectile(a, alpha=0.5) == np.mean(a)
True
>>> expectile(a, alpha=0.2)
0.42857142857142855
>>> expectile(a, alpha=0.8)
2.5714285714285716
>>> weights = [1, 3, 1, 1]