Asymmetric Laplace Quantile Regression

Description

Asymmetric Laplace family for quantile regression and a wrapper for estimating one or more conditional quantile models.

Usage

ALD(tau = 0.5)

fit_ALD(formula, data, tau, anchor = 0.5, max_step = 0.05,
  restarts = c("always", "on_failure", "never"), start = NULL,
  monotone = FALSE, verbose = interactive(), ...)

Arguments

tau	For ALD, one finite quantile probability strictly between zero and one. For fit_ALD, a numeric vector of such probabilities.
formula	A model formula accepted by gamlss2.
data	A data frame containing the variables in formula.
anchor	Quantile probability from which the sequence of fits starts.
max_step	Maximum distance between successive quantiles along the fitting path. Intermediate bridge models are fitted when necessary. Use Inf to disable bridge models.
restarts	Character. With “always”, each warm-started fit is compared with a cold fit; “on_failure” uses a cold fit only when the warm fit fails or does not converge; and “never” uses continuation only.
start	Optional starting values for the anchor model, as accepted by gamlss2.
monotone	Logical. If TRUE, returned fitted values and predictions are rearranged pointwise to be nondecreasing in tau.
verbose	Logical. Should progress information be printed?
…	Further arguments passed to gamlss2.

Details

The asymmetric Laplace density is proportional to \(\exp\{-\rho_\tau((y-\mu)/\sigma)\}\), where \(\rho_\tau(u) = u\{\tau-I(u<0)\}\). Thus, \(\mu\) is the conditional \(\tau\)-quantile and \(\sigma>0\) is a scale parameter.

fit_ALD first estimates the anchor quantile and then proceeds toward the lower and upper requested quantiles. Coefficients and smoothing parameters are transferred between successive fits. Converged candidate fits are preferred and compared using their asymmetric Laplace log-likelihood. If no candidate converges, up to two retries halve and quarter the step length, retaining the smoothing criterion, iteration limits, and convergence tolerances. These retries use cold starts unless restarts = “never”. If no candidate converges after these retries, a finite candidate is returned with its convergence flag unchanged; callers should check the individual models or the diagnostics.

Monotone rearrangement affects only the values returned by fitted() and predict(); the individual fitted models are retained unchanged.

The ALD can be used as a working likelihood when the response distribution is not asymmetric Laplace. With constant scale, its data likelihood targets pinball loss for the requested quantile. The location score is nonsmooth, and working-likelihood intervals are not automatically calibrated under misspecification. Ordinary observed-information Gaussian inference for smooth likelihoods should not be interpreted as calibrated quantile confidence intervals here; a suitable calibration or resampling procedure requires separate justification.

Value

ALD returns an object of class “gamlss2.family”.

fit_ALD returns an object of class “ALDfit”, containing the individual models, fitted values, requested quantiles, and fitting diagnostics.

References

Yu K, Moyeed RA (2001). “Bayesian Quantile Regression”. Statistics & Probability Letters, 54(4), 437–447. doi:10.1016/S0167-7152(01)00124-9

Chernozhukov V, Fernandez-Val I, Galichon A (2010). “Quantile and Probability Curves Without Crossing”. Econometrica, 78(3), 1093–1125. doi:10.3982/ECTA7880

Sriram K (2015). “A Sandwich Likelihood Correction for Bayesian Quantile Regression Based on the Misspecified Asymmetric Laplace Density”. Statistics & Probability Letters, 107, 18–26. doi:10.1016/j.spl.2015.07.035

See Also

gamlss2, predict, mcmc.

Examples

library("gamlss2")

data("SpirometryUS", package = "gamlss2")

## subset for females
d <- subset(SpirometryUS, gender == "female")

## fit one conditional median model
m50 <- gamlss2(fev1 ~ s(age), data = d, family = ALD(tau = 0.5))

## estimate several conditional quantiles
qu <- c(0.025, 0.5, 0.975)
m <- fit_ALD(fev1 ~ s(age), data = d, tau = qu,
  max_step = 0.1, monotone = TRUE, verbose = FALSE)

## plot fitted quantile curves
fit <- predict(m)
i <- order(d$age)
plot(fev1 ~ age, data = d)
matlines(d$age[i], fit[i, ],
  lty = 1, lwd = 2, col = 4)

## estimate lower limits model using full MCMC
set.seed(1328)
m2.5 <- mcmc(m$models[["tau=0.025"]],
  n.iter = 1200, burnin = 200, thin = 1)

summary(m2.5)

## predict credible intervals
nd <- data.frame(age = seq(6, 80, length = 300))
nd$p <- predict(m2.5, parameter = "mu", newdata = nd,
  FUN = function(x) quantile(x, probs = c(0.025, 0.5, 0.975)))

## plot
plot(fev1 ~ age, data = d, col = adjustcolor(1, 0.3))
matlines(nd$age, nd$p, lty = c(2, 1, 2), lwd = 2, col = 2)