- Source: Expectile
In the mathematical theory of probability, the expectiles of a probability distribution are related to the expected value of the distribution in a way analogous to that in which the quantiles of the distribution are related to the median.
For
τ
∈
(
0
,
1
)
{\textstyle \tau \in (0,1)}
expectile of the probability distribution with cumulative distribution function
F
{\textstyle F}
is characterized by any of the following equivalent conditions:
(
1
−
τ
)
∫
−
∞
t
(
t
−
x
)
d
F
(
x
)
=
τ
∫
t
∞
(
x
−
t
)
d
F
(
x
)
∫
−
∞
t
|
t
−
x
|
d
F
(
x
)
=
τ
∫
−
∞
∞
|
x
−
t
|
d
F
(
x
)
t
−
E
[
X
]
=
2
τ
−
1
1
−
τ
∫
t
∞
(
x
−
t
)
d
F
(
x
)
{\displaystyle {\begin{aligned}&(1-\tau )\int _{-\infty }^{t}(t-x)\,dF(x)=\tau \int _{t}^{\infty }(x-t)\,dF(x)\\[5pt]&\int _{-\infty }^{t}|t-x|\,dF(x)=\tau \int _{-\infty }^{\infty }|x-t|\,dF(x)\\[5pt]&t-\operatorname {E} [X]={\frac {2\tau -1}{1-\tau }}\int _{t}^{\infty }(x-t)\,dF(x)\end{aligned}}}
Quantile regression minimizes an asymmetric
L
1
{\displaystyle L_{1}}
loss (see least absolute deviations).
Analogously, expectile regression minimizes an asymmetric
L
2
{\displaystyle L_{2}}
loss (see ordinary least squares):
quantile
(
τ
)
∈
argmin
t
∈
R
E
[
|
X
−
t
|
|
τ
−
H
(
t
−
X
)
|
]
expectile
(
τ
)
∈
argmin
t
∈
R
E
[
|
X
−
t
|
2
|
τ
−
H
(
t
−
X
)
|
]
{\displaystyle {\begin{aligned}\operatorname {quantile} (\tau )&\in \operatorname {argmin} _{t\in \mathbb {R} }\operatorname {E} [|X-t||\tau -H(t-X)|]\\\operatorname {expectile} (\tau )&\in \operatorname {argmin} _{t\in \mathbb {R} }\operatorname {E} [|X-t|^{2}|\tau -H(t-X)|]\end{aligned}}}
where
H
{\displaystyle H}
is the Heaviside step function.
