scipy.stats.expectile

scipy.stats.expectile(a, alpha=0.5, *, weights=None)

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:

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

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

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

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 \(F_a(x) = \frac{1}{\sum_i a_i} \sum_i w_i 1_{a_i \leq x}\) with indicator function \(1_{A}\). This leads to the definition of the empirical expectile at level alpha as the unique solution \(t\) of:

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

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

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

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]