This function fits quantile regression models under Bayesian inference. The function samples from the posterior distribution using Gibbs sampling with data augmentation. A multivariate normal prior is assumed for \(\beta\). The user supplies the prior parameters. A sample of the posterior distribution is returned as an mcmc object, which can then be analysed by functions in the coda package.
MCMCquantreg(
formula,
data = NULL,
tau = 0.5,
burnin = 1000,
mcmc = 10000,
thin = 1,
verbose = 0,
seed = sample(1:1e+06, 1),
beta.start = NA,
b0 = 0,
B0 = 0,
...
)Model formula.
Data frame.
The quantile of interest. Must be between 0 and 1. The default value of 0.5 corresponds to median regression.
The number of burn-in iterations for the sampler.
The number of MCMC iterations after burnin.
The thinning interval used in the simulation. The number of MCMC iterations must be divisible by this value.
A switch which determines whether or not the progress of the
sampler is printed to the screen. If verbose is greater than 0 the
iteration number and the most recently sampled values of \(\beta\)
and \(\sigma\) are printed to the screen every verboseth
iteration.
The seed for the random number generator. If NA, the Mersenne
Twister generator is used with default seed 12345; if an integer is passed
it is used to seed the Mersenne twister. The default value for this argument
is a random integer between 1 and 1,000,000. This default value ensures that
if the function is used again with a different value of \(\tau\), it
is extremely unlikely that the seed will be identical. The user can also
pass a list of length two to use the L'Ecuyer random number generator, which
is suitable for parallel computation. The first element of the list is the
L'Ecuyer seed, which is a vector of length six or NA (if NA a default seed
of rep(12345,6) is used). The second element of list is a positive
substream number. See the MCMCpack specification for more details.
The starting values for \(\beta\). This can either be a scalar or a column vector with dimension equal to the dimension of \(\beta\). The default value of NA will use the OLS estimate \(\hat{\beta}\) with \(\hat{\sigma}\Phi^{-1}(\tau)\) added on to the first element of \(\hat{\beta}\) as the starting value. (\(\hat{\sigma}^2\) denotes the usual unbiased estimator of \(\sigma^2\) under ordinary mean regression and \(\Phi^{-1}(\tau)\) denotes the inverse of the cumulative density function of the standard normal distribution.) Note that the default value assume that an intercept is included in the model. If a scalar is given, that value will serve as the starting value for all \(\beta\).
The prior mean of \(\beta\). This can either be a scalar or a column vector with dimension equal to the dimension of
\(\beta\). If this takes a scalar value, then that value will serve as the prior mean for all \(\beta\).
The prior precision of \(\beta\). This can either be a scalar or a square matrix with dimensions equal to the number of betas. If this takes a scalar value, then that value times an identity matrix serves as the prior precision of \(\beta\). Default value of 0 is equivalent to an improper uniform prior for \(\beta\).
further arguments to be passed
An mcmc object that contains the posterior sample. This object can be summarised by functions provided by the coda package.
MCMCquantreg simulates from the posterior distribution using Gibbs
sampling with data augmentation (see
http://people.brunel.ac.uk/~mastkky/). \(\beta\) are drawn
from a multivariate normal distribution. The augmented data are drawn
conditionally from the inverse Gaussian distribution. The simulation is
carried out in compiled C++ code to maximise efficiency. Please consult the
coda documentation for a comprehensive list of functions that can be used to
analyse the posterior sample.
We assume the model
$$Q_{\tau}(y_i|x_i) = x_i'\beta$$
where \(Q_{\tau}(y_i|x_i)\) denotes the conditional \(\tau\)th quantile of \(y_i\) given \(x_i\), and \(\beta=\beta(\tau)\) are the regression parameters possibly dependent on \(\tau\). The likelihood is formed based on assuming independent Asymmetric Laplace distributions on the \(y_i\) with skewness parameter \(\tau\) and location parameters \(x_i'\beta\). This assumption ensures that the likelihood function is maximised by the \(\tau\)th conditional quantile of the response variable. We assume standard, semi-conjugate priors on \(\beta\):
$$\beta \sim \mathcal{N}(b_0,B_0^{-1})$$
Only starting values for \(\beta\) are allowed for this sampler.
Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.2. http://scythe.lsa.umich.edu.
Craig Reed and Keming Yu. 2009. ``An Efficient Gibbs Sampler for Bayesian Quantile Regression.'' Technical Report.
Keming Yu and Jin Zhang. 2005. ``A Three Parameter Asymmetric Laplace Distribution and it's extensions.'' Communications in Statistics - Theory and Methods, 34, 1867-1879.
Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2006. ``Output Analysis and Diagnostics for MCMC (CODA)'', R News. 6(1): 7-11. https://CRAN.R-project.org/doc/Rnews/Rnews_2006-1.pdf.
# NOT RUN {
# }
# NOT RUN {
x<-rep(1:10,5)
y<-rnorm(50,mean=x)
posterior_50 <- MCMCquantreg(y~x)
posterior_95 <- MCMCquantreg(y~x, tau=0.95, verbose=10000,
mcmc=50000, thin=10, seed=2)
plot(posterior_50)
plot(posterior_95)
raftery.diag(posterior_50)
autocorr.plot(posterior_95)
summary(posterior_50)
summary(posterior_95)
# }
# NOT RUN {
# }
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