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date: Wednesday, June 11^th^, 2025
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title: Expectiles are configurably-optimistic expectations
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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,](https://en.wikipedia.org/wiki/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.