R: Convolution-Type Smoothed Quantile Regression

conquer {conquer}

R Documentation

Convolution-Type Smoothed Quantile Regression

Description

Fit a smoothed quantile regression via convolution-type smoothing method. The solution is computed using gradient descent with Barzilai-Borwein step size. Constructs (1-alpha) confidence intervals with multiplier bootstrap.

Usage

conquer(
  X,
  Y,
  tau = 0.5,
  kernel = c("Gaussian", "uniform", "parabolic", "triangular"),
  h = 0,
  checkSing = FALSE,
  tol = 1e-04,
  iteMax = 5000,
  ci = FALSE,
  alpha = 0.05,
  B = 1000
)

Arguments

`X`	A n by p design matrix. Each row is a vector of observation with p covariates. Number of observations n must be greater than number of covariates p.
`Y`	An n-dimensional response vector.
`tau`	(optional) The desired quantile level. Default is 0.5. Value must be between 0 and 1.
`kernel`	(optional) A character string specifying the choice of kernel function. Default is "Gaussian". Other choices are "Gaussian", "uniform", "parabolic" or "triangular".
`h`	(optional) The bandwidth parameter for kernel smoothing. Default is max(((log(n) + p) / n)^{0.4}, 0.05). The default will be used if the input value is less than 0.05.
`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 gradient descent algorithm terminates when the maximal entry of the gradient is less than `tol`. Default is 1e-04.
`iteMax`	(optional) Maximum number of iterations. Default is 5000.
`ci`	(optional) A logical flag. Default is FALSE. If `ci = TRUE`, then three types of confidence intervals (percentile, pivotal and normal) will be constructed via multiplier bootstrap.
`alpha`	(optional) The nominal level for (1-alpha)-confidence intervals. Default is 0.05. The input value must be in (0, 1).
`B`	(optional) The size of bootstrap samples. Default is 1000.

Value

An object containing the following items will be returned:

coeff: A (p + 1)-vector of estimated quantile regression coefficients, including the intercept.
ite: The number of iterations of the gradient descent algorithm for convergence.
residual: The residuals of the quantile regression fit.
bandwidth: The value of smoothing bandwidth.
tau: The desired quantile level.
kernel: The choice of kernel function.
n: The sample size.
p: The dimension of the covariates.
perCI: The percentile confidence intervals for regression coefficients. Not available if ci = FALSE.
pivCI: The pivotal confidence intervals for regression coefficients. Not available if ci = FALSE
normCI: The normal-based confidence intervals for regression coefficients. Not available if ci = FALSE

Author(s)

Xuming He <xmhe@umich.edu>, Xiaoou Pan <xip024@ucsd.edu>, Kean Ming Tan <keanming@umich.edu>, and Wen-Xin Zhou <wez243@ucsd.edu>

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. (2019). Smoothing quantile regressions. J. Bus. Econ. Statist., in press.

He, X., Pan, X., Tan, K. M., and Zhou, W.-X. (2020). Smoothed quantile regression for large-scale inference. Preprint.

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

Examples

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

## Smoothed quantile regression with Gaussian kernel
fit.Gauss = conquer(X, Y, tau = 0.5, kernel = "Gaussian")
beta.hat.Gauss = fit.Gauss$coeff

## Smoothe quantile regression with uniform kernel
fit.unif = conquer(X, Y, tau = 0.5, kernel = "uniform")
beta.hat.unif = fit.unif$coeff

## Construct three types of confidence intervals via multiplier bootstrap
fit = conquer(X, Y, tau = 0.5, kernel = "Gaussian", ci = TRUE)
ci.per = fit$perCI
ci.piv = fit$pivCI
ci.norm = fit$normCI

[Package conquer version 1.0.2 Index]