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=0.5gives 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 \(F_a(x) = \frac{1}{\sum_i w_i} \sum_i w_i 1_{a_i \leq x}\) with indicator function \(1_{A}\). This leads to the definition of the empirical expectile at level- alphaas 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]