Expectiles are configurably-optimistic expectations

Wednesday, June 11^th, 2025

Expectiles are a class of summary statistics generalising the well-known expected value [1, 2]. They have been relatively neglected since their introduction [3]. Perhaps this is because they lack a well-known, immediate interpretation like that of the expected value, or that of quantiles?

This note reviews the definition of expectiles, outlines an interpretation of expectiles as “expectations at varying degrees of positive outlook,” and compares them to the more well-known asymmetric summary statistic—quantiles.

Defining expectiles

It is well known that the expected value $\mu_X$ of a scalar random variable $X$ with finite second moment is the scalar that minimises the expected squared distance from the random variable. That is, $$\mu_X = \mathop{\mathrm{arg\,min}}_\mu \mathbb{E}\left[ (\mu-X)^2\right ].$$

Expectiles are a class of summary statistics generalising the expected value [1,2]. Given an asymmetry parameter $\tau \in (0, 1)$, the $\tau$-expectile of $X$, $\epsilon_{X}{(\tau)}$, is the minimiser of an asymmetric version of the expected squared distance, weighting squared positive distances by $\tau$ and squared negative distances by $1-\tau$:

$$\epsilon_{X}{(\tau)} = \mathop{\mathrm{arg\,min}}_ \epsilon \mathbb{E}\left[ [\hspace{-1.5pt}[X > \epsilon]\hspace{-1.5pt}]^{\tau}_ {1-\tau} \cdot (\epsilon-X)^2 \right].$$

Here, $[\hspace{-1.5pt}[P]\hspace{-1.5pt}]^a_b$ is a generalised Iverson bracket, evaluating to $a$ if $P$ is a true proposition, or to $b$ otherwise.

The expectiles can also be defined for $\tau \in \{0, 1\}$. It makes sense to define $\epsilon_{X}{(0)}$ as the infimum of the support of $X$, and $\epsilon_{X}{(1)}$ as the supremum.

Interpreting expectiles

The expectile varies with the asymmetry parameter, $\tau \in (0,1)$, as follows.

• The expected value $\mu_X$ is recovered as the $0.5$-expectile.

• Expectiles with $\tau > 0.5$ are more sensitive to “higher-than-expected” outcomes than they are to “lower-than-expected” outcomes. In this sense, these expectiles can be viewed as “optimistic” expectation values.

• Likewise, expectiles with $\tau < 0.5$ weight can be viewed as “pessimistic” expectations.

We can thus understand $\tau$ as capturing the degree of “positive outlook” of an expectile, on a scale from 0 to 1.

Calling it “positive outlook” suggests that higher values of the random variable are better. This is the case when measuring value or return. The interpretation could be flipped when measuring bad things like risk, cost, loss, or regret.

Expectiles and quantiles

Expectiles are closely related to the more well-known summary statistic, quantiles.

Quantiles minimise an asymmetric absolute distance from a random variable. Formally, given an asymmetry parameter $p \in (0,1)$, The $p$-quantile of $X$, $Q_X(p)$, is the minimiser of an asymmetric version of the expected absolute distance, weighting positive distances by $p$ and squared negative distances by $1-p$: $$Q_X(p) = \mathop{\mathrm{arg\,min}}_q \mathbb{E}\left[ [\hspace{-1.5pt}[X > q]\hspace{-1.5pt}]_ {1-p}^{p} \cdot |q - X| \right].$$ The symmetric version of the quantile, $Q_X(0.5)$, is the well-known median statistic. Thus, expectiles generalise the expected value of a random variable in direct analogy to how quantiles generalise the median value of a random variable.

It turns out that expectiles possess many properties similar to quantiles. Indeed, [4] showed that the expectiles of a random variable $X$ are the quantiles of a suitably transformed $X$.

References

Works cited above:

1. Dennis J. Aigner, Takeshi Amemiya, and Dale J. Poirier. “On the estimation of production frontiers: Maximum likelihood estimation of the parameters of a discontinuous density function.” International Economic Review, pages 377–396, 1976.

2. Whitney K. Newey and James L. Powell. “Asymmetric least squares estimation and testing.” Econometrica: Journal of the Econometric Society, pages 819–847, 1987.

3. Linda Schulze Waltrup, Fabian Sobotka, Thomas Kneib, and Göran Kauermann. “Expectile and quantile regression—David and Goliath?” Statistical Modelling, 15(5):433–456, 2015.

4. M. Chris Jones. “Expectiles and m-quantiles are quantiles.” Statistics & Probability Letters, 20(2):149–153, 1994.