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In-Depth Information
q
;i
D
ˇ
;0
C
ˇ
;1
x
i1
C
:::
C
ˇ
;p
x
ip
;
that is, the quantile q
of the response distribution is a linear combination of the
covariates as in the multiple linear regression model. Generalizations to structured
additive predictors are conceptually straightforward (although estimation is truly
a challenge). The response distribution is implicitly determined by the estimated
quantiles q
provided that quantiles for a reasonable dense grid of -values are esti-
mated. In contrast to the GAMLSS framework, a specific
parametric
distribution is
not specified a priori which makes quantile regression a distribution-free approach.
Estimation of the quantile-specific regression coefficients
ˇ
is achieved by
minimizing the asymmetrically weighted absolute error criterion
n
X
ˇ
D
argmin
w
.y
i
;
i
/
j
y
i
i
j
(5.7)
ˇ
i D1
where
i
D
x
i
ˇ
and
8
<
1
y
i
<
i
0
w
.y
i
;
i
/
D
y
i
D
i
:
y
i
>
i
:
Frequentist quantile regression as outlined above is extensively treated in
Koenker (
2005
), see also Fahrmeir et al. (
2013
).
Bayesian quantile regression has been developed utilizing the equivalence
between posterior mode and maximum likelihood estimation under noninformative
priors
ˇ
/
const; see Yu and Moyeed (
2001
) and Yue and Rue (
2011
). Therefore,
we have to define a specific distributional assumption for the error terms (or
equivalently the responses) to make the Bayesian standard machinery work. If we
start with the model,
D
x
i
ˇ
y
i
C
"
i
;
i
D
1;:::;n;
we will assume independent and identically distributed errors following an asym-
metric Laplace distribution, that is, "
i
j
2
i.i.d. ALD.0;
2
;/with density
exp
:
.1
/
2
w
."
i
;0/
j
"
i
j
2
p."
i
j
2
/
D
For the responses, the error distribution induces y
i
j ˇ
;
2
ALD.
x
i
ˇ
;
2
;/,
such that the density of the responses is given by
exp
:
.1
/
2
w
.y
i
;
x
i
ˇ
/
j
y
i
x
i
ˇ
j
2
p.y
i
j ˇ
;
2
/
D
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