constrInResponseLasso: Compute the truncation set in the response after outlier...

In shuxiaoc/outference: Valid Inference in Linear Regression Corrected for Outlier Removal

Description

This function computes a matrix A and a vector b, such that outlier detection using lasso is equivalent to A y ≥ b.

Usage

constrInResponseLasso(n, p, PXperp, outlier.det, outlier.det.sign, cutoff)

Arguments

n,	the number of observations.
p,	the number of variables, including the intercept.
PXperp,	the projection matrix onto the orthogonal complement of the column space of the design matrix X.
outlier.det,	indexes of detected outliers, can be empty.
outlier.det.sign,	the sign of the active variable estimated by lasso.
cutoff,	the cutoff λ (see details).

Details

Consider solving the following program

minimize ||y-Xβ-u||_2^2/(2n) + λ ||u||_1.

The i-th observation is considered as an outlier if and only if \hat u_i \neq 0. This is equivalent to solving

minimize ||P_X^\perp (y-u)||_2^2/(2n) + λ ||u||_1.

Then the variable selection can be characterized by a set of affine constraints Ay ≥ b. In essence, this function is equivalent to selectiveInference:::fixedLasso.poly, but adapted to our notations and up to some scaling factors and linear transformations.

Value

This function returns a list (A, b).

References

Lee, Jason D., et al. "Exact post-selection inference, with application to the lasso." The Annals of Statistics 44.3 (2016): 907-927.

Tibshirani, R., et al. "selectiveInference: Tools for Post-Selection Inference." R package version 1.3 (2016).

shuxiaoc/outference documentation built on July 8, 2019, 8:30 p.m.