Package {RidgeLRGLH}

Estimate a Population Covariance Spectrum by Linear Programming

Description

Fits nonnegative spectral masses using the complex-valued relative discrepancy constraints in Algorithm 1 of Li (2025).

Usage

estimate_spectrum(eigenvalues, gamma2, lambda = NULL,
  size = 500L, grid_size = 500L, sigma_grid = NULL,
  z_grid = NULL, order = 2L, truncation = 2)

Arguments

Details

The default grid is obtained by inverting the empirical companion Stieltjes transform. Its real parts run from the transform at 1.05 times the largest sample eigenvalue to the transform at minus the fitting ridge. The imaginary transform component is 0.01 divided by the largest eigenvalue. The default population grid follows Supplement S.2. A bounded Newton iteration with step halving performs complex inversion; failed inversion or optimization raises an error. The returned loss is measured before truncation. Finite grids are approximations; inspect sensitivity to grid size, fitting ridge and constraint order.

Value

A list containing positive weights, their sigma locations, pre-truncation loss, solver status (zero on success), fitting lambda, z_grid, and constraint order.

References

Li, H. (2025). Ridge-Regularized Largest Root Test For High-Dimensional General Linear Hypotheses. doi:10.48550/arXiv.2504.15510. Accepted by the Annals of Statistics.

Examples

set.seed(12)
Z <- matrix(rnorm(20 * 60), 20)
ev <- eigen(tcrossprod(Z) / 60, symmetric = TRUE, only.values = TRUE)$values
spectrum <- estimate_spectrum(ev, 20 / 60, size = 40, grid_size = 60)
sum(spectrum$weights)

Compute Tracy-Widom Centering and Scaling Constants

Description

Computes the spectral edge and calibration constants for a specified population covariance spectrum.

Usage

ridge_calibration(sigma, gamma1, gamma2, lambda,
  weights = rep(1 / length(sigma), length(sigma)))

Arguments

Details

Uses the scalar parameterization x(h)=h+[1+\gamma_2 H_1(h)]^{-1} from Section 3. It solves for the regular branch and \beta^2s'(\beta)=1/\gamma_1 by bracketed root finding. Derivatives follow by implicit differentiation. This is equivalent to the manuscript's ODE characterization, and avoids evaluating a dense ODE trajectory near the edge. Root tolerances are applied after normalization of the population spectrum. The Tracy-Widom approximation assumes the high-dimensional regime and spectral regularity conditions in the article; it is not an exact finite-sample calibration.

Value

A named numeric vector: center, scale, beta, rho, s, s_prime, and s_double_prime, with derivatives evaluated at beta. The normalized statistic is p^{2/3}(l_{max}-center)/scale.

References

Li, H. (2025). Ridge-Regularized Largest Root Test For High-Dimensional General Linear Hypotheses. doi:10.48550/arXiv.2504.15510. Accepted by the Annals of Statistics.

Examples

ridge_calibration(sigma = 1, gamma1 = 2, gamma2 = 0.5, lambda = 1)

Test a General Linear Hypothesis in Multivariate Regression

Description

Tests the hypothesis BC = 0 in the model Y = BX + E using a ridge-regularized largest root statistic.

Usage

ridge_glht(Y, X, C, lambda = NULL,
  selection = c("fixed", "bayes_identity", "bayes_covariance", "minimax"),
  lambda_grid = NULL, spectrum = NULL, control = list())

Arguments

Details

QR factorizations construct the tested and residual subspaces, without forming observation-by-observation projection matrices. Numerator and denominator degrees of freedom are q and n - m. Nonzero right-hand sides and rank-deficient designs or contrasts are not supported. This is an asymptotic test under the moment, covariance, dimensional and regular-edge conditions in Li (2025). Very small tested ranks or dimensions may give poor finite-sample approximation.

Value

An object of class htest, with statistic, selected ridge in parameter, asymptotic upper-tail p.value, and standard test metadata. Additional fields include largest.root, center, scale, n1, n2, dimension, selection, tuning (candidate ridges, constants and scores), spectrum, and selected calibration. Fixed selection has an undefined (NA) tuning score.

References

Li, H. (2025). Ridge-Regularized Largest Root Test For High-Dimensional General Linear Hypotheses. doi:10.48550/arXiv.2504.15510. Accepted by the Annals of Statistics.

See Also

ridge_root

Examples

set.seed(8)
Y <- matrix(rnorm(20 * 80), 20)
X <- rbind(1, matrix(rnorm(10 * 80), 10))
C <- diag(11)[, -1, drop = FALSE]
ridge_glht(Y, X, C, selection = "minimax",
           control = list(size = 40, grid_size = 60))

Ridge-Regularized Largest Root Test for Two Covariance Matrices

Description

Tests using the largest eigenvalue of W1 times the inverse of W2 plus a positive ridge, with asymptotic real Tracy-Widom calibration.

Usage

ridge_root(W1, W2, n1, n2, lambda = NULL,
  selection = c("fixed", "bayes_identity", "bayes_covariance", "minimax"),
  lambda_grid = NULL, spectrum = NULL, control = list())

Arguments

Details

Bayes criteria are the identity or covariance signal estimates divided by the scaling constant. The minimax criterion is the smaller of the identity signal estimate and the trace-normalized covariance signal estimate, divided by that same scaling constant. This solves the minimization over nonnegative mixtures exactly at an endpoint. All adaptive choices use a finite candidate set. Their asymptotic validity requires the separation and consistency conditions in Section 4 of the article; choosing a ridge from the observed data does not provide a finite-sample error guarantee. The returned p-value is for the selected test and does not adjust for reporting multiple tests or searching multiple hypotheses. Inputs must be real; complex-valued Wishart models are not supported.

Value

An object of class htest, with statistic, selected ridge in parameter, asymptotic upper-tail p.value, and standard test metadata. Additional fields include largest.root, center, scale, n1, n2, dimension, selection, tuning (candidate ridges, constants and scores), spectrum, and selected calibration. Fixed selection has an undefined (NA) tuning score.

References

Li, H. (2025). Ridge-Regularized Largest Root Test For High-Dimensional General Linear Hypotheses. doi:10.48550/arXiv.2504.15510. Accepted by the Annals of Statistics.

See Also

ridge_glht, estimate_spectrum

Examples

set.seed(17)
Z1 <- matrix(rnorm(20 * 30), 20)
Z2 <- matrix(rnorm(20 * 60), 20)
ridge_root(tcrossprod(Z1) / 30, tcrossprod(Z2) / 60,
           n1 = 30, n2 = 60, control = list(size = 40, grid_size = 60))