blq {bsamGP}  R Documentation 
This function fits a Bayesian quantile regression model.
blq(formula, data = NULL, p, mcmc = list(), prior = list(), marginal.likelihood = TRUE)
formula 
an object of class “ 
data 
an optional data frame. 
p 
quantile of interest (default=0.5). 
mcmc 
a list giving the MCMC parameters.
The list includes the following integers (with default values in parentheses):

prior 
a list giving the prior information. The list includes the following parameters
(default values specify the noninformative prior):

marginal.likelihood 
a logical variable indicating whether the log marginal likelihood is calculated. The methods of Gelfand and Dey (1994) is used. 
This generic function fits a Bayesian quantile regression model.
Let y_i
and w_i
be the response and the vector of parametric predictors, respectively.
Further, let x_{i,k}
be the covariate related to the response, linearly.
The model is as follows.
y_i = w_i^T\beta + \epsilon_i, ~ i=1,\ldots,n,
where the error terms \{\epsilon_i\}
are a random sample from an asymmetric Laplace distribution, ALD_p(0,\sigma^2)
,
which has the following probability density function:
ALD_p(\epsilon; \mu, \sigma^2) = \frac{p(1p)}{\sigma^2}\exp\Big(\frac{(x\mu)[p  I(x \le \mu)]}{\sigma^2}\Big),
where 0 < p < 1
is the skew parameter, \sigma^2 > 0
is the scale parameter, \infty < \mu < \infty
is
the location parameter, and I(\cdot)
is the indication function.
The conjugate priors are assumed for \beta
and \sigma
:
\beta  \sigma \sim N(m_{0,\beta}, \sigma^2V_{0,\beta}), \quad \sigma^2 \sim IG\Big(\frac{r_{0,\sigma}}{2}, \frac{s_{0,\sigma}}{2}\Big)
An object of class blm
representing the Bayesian parametric linear model fit.
Generic functions such as print
and fitted
have methods to show the results of the fit.
The MCMC samples of the parameters in the model are stored in the list mcmc.draws
,
the posterior samples of the fitted values are stored in the list fit.draws
, and
the MCMC samples for the log marginal likelihood are saved in the list loglik.draws
.
The output list also includes the following objects:
post.est 
posterior estimates for all parameters in the model. 
lmarg 
log marginal likelihood using GelfandDey method. 
rsquarey 
correlation between 
call 
the matched call. 
mcmctime 
running time of Markov chain from 
Gelfand, A. E. and Dey, K. K. (1994) Bayesian model choice: asymptotics and exact calculations. Journal of the Royal Statistical Society. Series B  Statistical Methodology, 56, 501514.
Kozumi, H. and Kobayashi, G. (2011) Gibbs sampling methods for Bayesian quantile regression. Journal of Statistical Computation and Simulation, 81(11), 15651578.
#####################
# Simulated example #
#####################
# Simulate data
set.seed(1)
n < 100
w < runif(n)
y < 3 + 2*w + rald(n, scale = 0.8, p = 0.5)
# Fit median regression
fout < blq(y ~ w, p = 0.5)
# Summary
print(fout); summary(fout)
# fitted values
fit < fitted(fout)
# Plots
plot(fout)