blq {bsamGP}

Bayesian Quantile Regression

Description

This function fits a Bayesian quantile regression model.

Usage

blq(formula, data = NULL, p, mcmc = list(), prior = list(), marginal.likelihood = TRUE)

Arguments

formula

an object of class “formula”

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): nblow (1000) giving the number of MCMC in transition period, nskip (1) giving the thinning interval, smcmc (1000) giving the number of MCMC for analysis.

prior

a list giving the prior information. The list includes the following parameters (default values specify the non-informative prior): beta_m0 and beta_v0 giving the hyperparameters of the multivariate normal distribution for parametric part including intercept, sigma2_m0 and sigma2_v0 giving the prior mean and variance of the inverse gamma prior for the scale parameter of response.

marginal.likelihood

a logical variable indicating whether the log marginal likelihood is calculated. The methods of Gelfand and Dey (1994) is used.

Details

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(1-p)}{\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)

Value

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 Gelfand-Dey method.

rsquarey

correlation between y and \hat{y}.

call

the matched call.

mcmctime

running time of Markov chain from system.time().

References

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, 501-514.

Kozumi, H. and Kobayashi, G. (2011) Gibbs sampling methods for Bayesian quantile regression. Journal of Statistical Computation and Simulation, 81(11), 1565-1578.

See Also

blr, gblr

Examples

#####################
# 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)