Ridge-Regularized Largest Root Tests for General Linear Hypotheses

Tests general linear hypotheses in high-dimensional multivariate regression using a ridge-regularized largest root statistic. Estimates the population spectral distribution by linear programming and computes Tracy-Widom centering and scaling constants. Supports fixed, Bayesian, and minimax regularization choices. Implements the methods of Li (2025) .

RidgeLRGLH

Ridge-regularized largest root tests for high-dimensional general linear hypotheses, implementing Haoran Li's article accepted by the Annals of Statistics. The public preprint is available at https://arxiv.org/abs/2504.15510.

Install

From the directory containing the source archive:

install.packages(c("lpSolve", "RMTstat"))
install.packages("RidgeLRGLH_0.1.0.tar.gz", repos = NULL, type = "source")
library(RidgeLRGLH)

Multivariate regression

Rows of Y are response variables; columns are independent observations. Rows of X are predictors. Include the intercept explicitly. The hypothesis is B %*% C = 0 in Y = B %*% X + E.

set.seed(8)
Y <- matrix(rnorm(80 * 150), nrow = 80)
X <- rbind(intercept = 1, matrix(rnorm(20 * 150), nrow = 20))
C <- diag(21)[, -1, drop = FALSE]

fixed <- ridge_glht(Y, X, C)  # ridge defaults to mean residual eigenvalue
adaptive <- ridge_glht(Y, X, C, selection = "minimax")
adaptive
adaptive$tuning
adaptive$spectrum[c("sigma", "weights", "loss")]

Other adaptive choices are "bayes_identity" and "bayes_covariance". Set lambda for a fixed ridge or lambda_grid for adaptive selection. Do not select the smallest p-value across methods and interpret it as a single calibrated test. The reported p-values use an asymptotic Tracy-Widom approximation under the article's assumptions.

Precomputed covariance matrices

set.seed(17)
Z1 <- matrix(rnorm(40 * 30), 40)
Z2 <- matrix(rnorm(40 * 80), 40)
W1 <- tcrossprod(Z1) / 30
W2 <- tcrossprod(Z2) / 80
ridge_root(W1, W2, n1 = 30, n2 = 80, lambda = 1)

The matrices must be normalized by their respective degrees of freedom. ridge_root() is a double Wishart test, not a general equality-of-covariance matrices test for arbitrary pairs of sample covariance matrices.

Numerical controls

Defaults use 500 transform points and 500 population-spectrum masses with first- and second-order spectral constraints. Use control = list(size = 100, grid_size = 150) for exploratory computations. Compare results with larger grids before drawing conclusions. Inspect fit$spectrum$loss; the package does not assign a universal statistical cutoff to this approximation loss.

Supply spectrum = list(sigma = ..., weights = ...) to use a known population spectrum, for example in oracle experiments. ridge_calibration() exposes centering, scaling, the regular edge and transform derivatives independently. Read vignette("RidgeLRGLH", package = "RidgeLRGLH") for a worked introduction.

Development checks

R CMD build RidgeLRGLH
R CMD check --as-cran RidgeLRGLH_0.1.0.tar.gz

Tests cover analytic limits, an independent ODE implementation, GLHT projection identities, singular covariance matrices, adaptive criteria and invalid inputs. The original manuscript and simulation files remain outside this package.

Finite-sample validation

Seeded Gaussian identity-covariance checks found 5 rejections in 100 null replicates for default minimax selection at p = 200, n1 = 100, n2 = 400. At the smaller setting p = 60, n1 = 40, n2 = 120, it rejected 26 of 200 null replicates (13%) at nominal 5%, using the full default grids. Increasing the spectral grid alone did not resolve that finite-sample inflation. These limited experiments do not establish accuracy across the model class. Use particular care with smaller dimensions and validate calibration for your intended setting. The package implements the published asymptotic method without an additional finite-sample correction.