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.