sgdi_qr {SGDinference}

Averaged S-subGD and its Inference via Random Scaling in Linear Quantile Regression

Description

Compute the averaged S-subGD (stochastic subgradient) estimator for the coefficients in linear quantile regression and conduct inference via random scaling method.

Usage

sgdi_qr(
  formula,
  data,
  gamma_0 = NULL,
  alpha = 0.501,
  burn = 1,
  inference = "rs",
  bt_start = NULL,
  qt = 0.5,
  studentize = TRUE,
  no_studentize = 100L,
  intercept = TRUE,
  rss_idx = c(1),
  level = 0.95,
  path = FALSE,
  path_index = c(1)
)

Arguments

formula

formula. The response is on the left of a ~ operator. The terms are on the right of a ~ operator, separated by a + operator.

data

an optional data frame containing variables in the model.

gamma_0

numeric. A tuning parameter for the learning rate (gamma_0 x t ^ alpha). Default is NULL and it is determined by the adaptive method in Lee et al. (2023).

alpha

numeric. A tuning parameter for the learning rate (gamma_0 x t ^ alpha). Default is 0.501.

burn

numeric. A tuning parameter for "burn-in" observations. We burn-in up to (burn-1) observations and use observations from (burn) for estimation. Default is 1, i.e. no burn-in.

inference

character. Specifying the inference method. Default is "rs" (random scaling matrix for joint inference using all the parameters). "rss" is for ransom scaling subset inference. This option requires that "rss_indx" should be provided. "rsd" is for the diagonal elements of the random scaling matrix, excluding one for the intercept term.

bt_start

numeric. (p x 1) vector, excluding the intercept term. User-provided starting value. Default is NULL. Then, it is estimated by conquer.

qt

numeric. Quantile. Default is 0.5.

studentize

logical. Studentize regressors. Default is TRUE.

no_studentize

numeric. The number of observations to compute the mean and std error for studentization. Default is 100.

intercept

logical. Use the intercept term for regressors. Default is TRUE. If this option is TRUE, the first element of the parameter vector is the intercept term.

rss_idx

numeric. Index of x for random scaling subset inference. Default is 1, the first regressor of x. For example, if we want to focus on the 1st and 3rd covariates of x, then set it to be c(1,3).

level

numeric. The confidence level required. Default is 0.95. Can choose 0.90 and 0.80.

path

logical. The whole path of estimation results is out. Default is FALSE.

path_index

numeric. A vector of indices to print out the path. Default is 1.

Value

An object of class "sgdi", which is a list containing the following

coefficients

a vector of estimated parameter values

V

a random scaling matrix depending on the inference method

ci.lower

a vector of lower confidence limits

ci.upper

a vector of upper confidence limits

inference

character that specifies the inference method

level

The confidence level required. Default is 0.95.

path_coefficients

The path of coefficients.

Note

The dimension of coefficients is (p+1) if intercept=TRUE or p otherwise. The random scaling matrix V is a full matrix if "rs" is chosen; it is a scalar or smaller matrix, depending on the specification of "rss_indx" if "rss" is selected; it is a vector of diagonal elements of the full matrix if "rsd" is selected. In this case, the first element is missing if the intercept is included. The confidence intervals may contain NA under "rss" and "rsd".

Examples

n = 1e05
p = 5
bt0 = rep(5,p)
x = matrix(rnorm(n*(p-1)), n, (p-1))
y = cbind(1,x) %*% bt0 + rnorm(n)
my.dat = data.frame(y=y, x=x)
sgdi.out = sgdi_qr(y~., data=my.dat)