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:

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

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]