# QF: Inference for quadratic forms of the regression vector in... In SIHR: Statistical Inference in High Dimensional Regression

## Description

Computes the bias-corrected estimator of the quadratic form of the regression vector, restricted to the set of indices G for the high dimensional linear regression and the corresponding standard error. It also constructs the confidence interval for the quadratic form and test whether it is above zero or not.

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 QF( X, y, G, Cov.weight = TRUE, A = NULL, tau.vec = c(1), init.coef = NULL, lambda = NULL, mu = NULL, step = NULL, resol = 1.5, maxiter = 6, alpha = 0.05, verbose = TRUE ) 

## 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 Cov.weight Logical, if set to TRUE then A  is the |G|\times|G| submatrix of the population covariance matrix corresponding to the index set G, else need to provide an A (default = TRUE) A The matrix A in the quadratic form, of dimension |G|\times|G| (default = NULL) tau.vec The vector of enlargement factors for asymptotic variance of the bias-corrected estimator to handle super-efficiency (default = 1) init.coef Initial estimator for the regression vector (default = NULL) lambda The tuning parameter used in the construction of init.coef (default = NULL) mu The dual tuning parameter used in the construction of the projection direction (default = NULL) step The step size used to compute mu; if set to NULL it is computed to be the number of steps (< maxiter) to obtain the smallest mu such that the dual optimization problem for constructing the projection direction converges (default = NULL) resol Resolution or the factor by which mu is increased/decreased to obtain the smallest mu such that the dual optimization problem for constructing the projection direction converges (default = 1.5) maxiter Maximum number of steps along which mu is increased/decreased to obtain the smallest mu such that the dual optimization problem for constructing the projection direction converges (default = 6) alpha Level of significance to test the null hypothesis which claims that the quadratic form of the regression vector is equal to 0 (default = 0.05) verbose Should inetrmediate message(s) be printed (default = TRUE)

## Value

 prop.est The bias-corrected estimator of the quadratic form of the regression vector se The standard error of the bias-corrected estimator CI The matrix of confidence interval for the quadratic form of the regression vector; row corresponds to different values of tau.vec decision decision=1 implies the quadratic form of the regression vector is above zero \newline decision=0 implies the quadratic form of the regression vector is zero \newline row corresponds to different values of tau.vec proj The projection direction, of length p plug.in The plug-in estimator for the quadratic form of the regression vector restricted to G

\insertRef

grouplinSIHR

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 n = 100 p = 200 A1gen <- function(rho,p){ A1=matrix(0,p,p) for(i in 1:p){ for(j in 1:p){ A1[i,j]<-rho^(abs(i-j)) } } A1 } mu <- rep(0,p) mu[1:5] <- c(1:5)/5 rho = 0.5 Cov <- (A1gen(rho,p))/2 beta <- rep(0,p) beta[1:10] <- c(1:10)/5 X <- MASS::mvrnorm(n,mu,Cov) y = X%*%beta + rnorm(n) test.set =c(30:50) Est <-SIHR::QF(X = X, y = y, G = test.set) 

SIHR documentation built on Oct. 7, 2021, 9:08 a.m.