conquer.process: Convolution-Type Smoothed Quantile Regression Process

Description

Fit a smoothed quantile regression process over a quantile range. The algorithm is essentially the same as conquer.

Usage

conquer.process(
  X,
  Y,
  tauSeq = seq(0.1, 0.9, by = 0.05),
  kernel = c("Gaussian", "logistic", "uniform", "parabolic", "triangular"),
  h = 0,
  checkSing = FALSE,
  tol = 1e-04,
  iteMax = 5000,
  stepBounded = TRUE,
  stepMax = 100
)

Value

An object containing the following items will be returned:

coeff: A \((p + 1)\) by \(m\) matrix of estimated quantile regression process coefficients, including the intercept. m is the length of tauSeq.
bandwidth: Bandwidth value.
tauSeq: The sequence of quantile levels.
kernel: The choice of kernel function.
n: Sample size.
p: Number the covariates.

Arguments

X: An \(n\) by \(p\) design matrix. Each row is a vector of observations with \(p\) covariates. Number of observations \(n\) must be greater than number of covariates \(p\).
Y: An \(n\)-dimensional response vector.
tauSeq: (optional) A sequence of quantile values (between 0 and 1). Default is \(\{0.1, 0.15, 0.2, ..., 0.85, 0.9\}\).
kernel: (optional) A character string specifying the choice of kernel function. Default is "Gaussian". Choices are "Gaussian", "logistic", "uniform", "parabolic" and "triangular".
h: (optional) The bandwidth/smoothing parameter. Default is \(\max\{((log(n) + p) / n)^{0.4}, 0.05\}\). The default will be used if the input value is less than or equal to 0.
checkSing: (optional) A logical flag. Default is FALSE. If checkSing = TRUE, then it will check if the design matrix is singular before running conquer.
tol: (optional) Tolerance level of the gradient descent algorithm. The iteration will stop when the maximum magnitude of all the elements of the gradient is less than tol. Default is 1e-04.
iteMax: (optional) Maximum number of iterations. Default is 5000.
stepBounded: (optional) A logical flag. Default is TRUE. If stepBounded = TRUE, then the step size of gradient descent is upper bounded by stepMax. If stepBounded = FALSE, then the step size is unbounded.
stepMax: (optional) Maximum bound for the gradient descent step size. Default is 100.

References

Barzilai, J. and Borwein, J. M. (1988). Two-point step size gradient methods. IMA J. Numer. Anal., 8, 141–148.

Fernandes, M., Guerre, E. and Horta, E. (2021). Smoothing quantile regressions. J. Bus. Econ. Statist., 39, 338-357.

He, X., Pan, X., Tan, K. M., and Zhou, W.-X. (2022+). Smoothed quantile regression for large-scale inference. J. Econometrics, in press.

Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33-50.

Examples

Run this code

n = 500; p = 10
beta = rep(1, p)
X = matrix(rnorm(n * p), n, p)
Y = X %*% beta + rt(n, 2)

## Smoothed quantile regression process with Gaussian kernel
fit.Gauss = conquer.process(X, Y, tauSeq = seq(0.2, 0.8, by = 0.05), kernel = "Gaussian")
beta.hat.Gauss = fit.Gauss$coeff

## Smoothe quantile regression with uniform kernel
fit.unif = conquer.process(X, Y, tauSeq = seq(0.2, 0.8, by = 0.05), kernel = "uniform")
beta.hat.unif = fit.unif$coeff