sgdi_lm {SGDinference}

Averaged SGD and its Inference via Random Scaling

Description

Compute the averaged SGD estimator and conduct inference via random scaling method.

Usage

sgdi_lm(
  formula,
  data,
  gamma_0 = NULL,
  alpha = 0.501,
  burn = 1,
  inference = "rs",
  bt_start = NULL,
  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: 1/sd(y).

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. User-provided starting value Default is NULL.

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

coefficient

A (p + 1)-vector of estimated parameter values including the intercept.

var

A (p+1)x (p+1) variance-covariance matrix of coefficient

ci.lower

The lower part of the 95% confidence interval

ci.upper

The upper part of the 95% confidence interval

level

The confidence level required. Default is 0.95.

path_coefficients

The path of coefficients.

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_lm(y~., data=my.dat)