QF: Inference for quadratic forms of the regression vector in...

In SIHR: Statistical Inference in High Dimensional Regression

Inference for quadratic forms of the regression vector in high dimensional generalized linear regressions

Description

Inference for quadratic forms of the regression vector in high dimensional generalized linear regressions

Usage

QF(
  X,
  y,
  G,
  A = NULL,
  model = c("linear", "logistic", "logistic_alter"),
  intercept = TRUE,
  beta.init = NULL,
  split = TRUE,
  lambda = NULL,
  mu = NULL,
  prob.filter = 0.05,
  rescale = 1.1,
  tau = c(0.25, 0.5, 1),
  alpha = 0.05,
  verbose = FALSE
)

Arguments

X	Design matrix, of dimension n x p
y	Outcome vector, of length n
G	The set of indices, G in the quadratic form
A	The matrix A in the quadratic form, of dimension |G|\times|G|. If NULL A would be set as the |G|\times|G| submatrix of the population covariance matrix corresponding to the index set G (default = NULL)
model	The high dimensional regression model, either "linear" or "logistic" or "logistic_alter"
intercept	Should intercept be fitted for the initial estimator (default = TRUE)
beta.init	The initial estimator of the regression vector (default = NULL)
split	Sampling splitting or not for computing the initial estimator. It take effects only when beta.init = NULL. (default = TRUE)
lambda	The tuning parameter in fitting initial model. If NULL, it will be picked by cross-validation. (default = NULL)
mu	The dual tuning parameter used in the construction of the projection direction. If NULL it will be searched automatically. (default = NULL)
prob.filter	The threshold of estimated probabilities for filtering observations in logistic regression. (default = 0.05)
rescale	The factor to enlarge the standard error to account for the finite sample bias. (default = 1.1)
tau	The enlargement factor for asymptotic variance of the bias-corrected estimator to handle super-efficiency. It allows for a scalar or vector. (default = c(0.25,0.5,1))
alpha	Level of significance to construct two-sided confidence interval (default = 0.05)
verbose	Should intermediate message(s) be printed. (default = FALSE)

Value

est.plugin	The plugin(biased) estimator for the quadratic form of the regression vector restricted to G
est.debias	The bias-corrected estimator of the quadratic form of the regression vector
se	Standard errors of the bias-corrected estimator, length of tau; corrsponding to different values of tau
ci.mat	The matrix of two.sided confidence interval for the quadratic form of the regression vector; row corresponds to different values of tau

Examples

X = matrix(rnorm(100*5), nrow=100, ncol=5)
y = X[,1] * 0.5 + X[,2] * 1 + rnorm(100)
G = c(1,2)
A = matrix(c(1.5, 0.8, 0.8, 1.5), nrow=2, ncol=2)
Est = QF(X, y, G, A, model="linear")
## compute confidence intervals
ci(Est, alpha=0.05, alternative="two.sided")

## summary statistics
summary(Est)

SIHR documentation built on April 9, 2023, 5:08 p.m.