MCMC sampling for Bayesian quantile regression with an unknown (nonparametric) transformation. Like in traditional Bayesian quantile regression, an asymmetric Laplace distribution is assumed for the errors, so the regression models targets the specified quantile. However, these models are often woefully inadequate for describing observed data. We introduce a nonparametric transformation to improve model adequacy while still providing inference for the regression coefficients and the specified quantile. A g-prior is assumed for the regression coefficients.

## Usage

```
sbqr(
y,
X,
tau = 0.5,
X_test = X,
psi = length(y),
laplace_approx = TRUE,
approx_g = FALSE,
nsave = 1000,
nburn = 100,
ngrid = 100,
verbose = TRUE
)
```

## Arguments

- y
`n x 1`

response vector- X
`n x p`

matrix of predictors- tau
the target quantile (between zero and one)

- X_test
`n_test x p`

matrix of predictors for test data; default is the observed covariates`X`

- psi
prior variance (g-prior)

- laplace_approx
logical; if TRUE, use a normal approximation to the posterior in the definition of the transformation; otherwise the prior is used

- approx_g
logical; if TRUE, apply large-sample approximation for the transformation

- nsave
number of MCMC iterations to save

- nburn
number of MCMC iterations to discard

- ngrid
number of grid points for inverse approximations

- verbose
logical; if TRUE, print time remaining

## Value

a list with the following elements:

`coefficients`

the posterior mean of the regression coefficients`fitted.values`

the estimated`tau`

th quantile at test points`X_test`

`post_theta`

:`nsave x p`

samples from the posterior distribution of the regression coefficients`post_ypred`

:`nsave x n_test`

samples from the posterior predictive distribution at test points`X_test`

`post_qtau`

:`nsave x n_test`

samples of the`tau`

th conditional quantile at test points`X_test`

`post_g`

:`nsave`

posterior samples of the transformation evaluated at the unique`y`

values`model`

: the model fit (here,`sbqr`

)

as well as the arguments passed in.

## Details

This function provides fully Bayesian inference for a
transformed quantile linear model.
The transformation is modeled as unknown and learned jointly
with the regression coefficients (unless `approx_g`

= TRUE, which then uses
a point approximation). This model applies for real-valued data, positive data, and
compactly-supported data (the support is automatically deduced from the observed `y`

values).
The results are typically unchanged whether `laplace_approx`

is TRUE/FALSE;
setting it to TRUE may reduce sensitivity to the prior, while setting it to FALSE
may speed up computations for very large datasets.

## Examples

```
# \donttest{
# Simulate some heteroskedastic data (no transformation):
dat = simulate_tlm(n = 200, p = 10, g_type = 'box-cox', heterosked = TRUE, lambda = 1)
y = dat$y; X = dat$X # training data
y_test = dat$y_test; X_test = dat$X_test # testing data
# Target this quantile:
tau = 0.05
# Fit the semiparametric Bayesian quantile regression model:
fit = sbqr(y = y, X = X, tau = tau, X_test = X_test)
#> [1] "27 seconds remaining"
#> [1] "31 seconds remaining"
#> [1] "16 seconds remaining"
#> [1] "0 seconds remaining"
#> [1] "Total time: 56 seconds"
names(fit) # what is returned
#> [1] "coefficients" "fitted.values" "post_theta" "post_ypred"
#> [5] "post_qtau" "post_g" "model" "y"
#> [9] "X" "X_test" "psi" "approx_g"
#> [13] "tau"
# Posterior predictive checks on testing data: empirical CDF
y0 = sort(unique(y_test))
plot(y0, y0, type='n', ylim = c(0,1),
xlab='y', ylab='F_y', main = 'Posterior predictive ECDF')
temp = sapply(1:nrow(fit$post_ypred), function(s)
lines(y0, ecdf(fit$post_ypred[s,])(y0), # ECDF of posterior predictive draws
col='gray', type ='s'))
lines(y0, ecdf(y_test)(y0), # ECDF of testing data
col='black', type = 's', lwd = 3)
# }
```