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\title{RidgeLRGLH: Testing General Linear Hypotheses}
\author{Haoran Li}
\date{}
\begin{document}
\maketitle
\section{Model and hypothesis}
The package tests $BC=0$ in the multivariate model $Y=BX+E$.
Responses form a $p\times n$ matrix, the design is $m\times n$, and
contrasts form an $m\times q$ matrix. The design and contrast matrices must
have full row and column rank, respectively. Include an intercept as a row
of the design when appropriate. Residual and tested degrees of freedom
are $n_2=n-m$ and $n_1=q$.

The test uses the largest eigenvalue of
$W_1(W_2+\lambda I_p)^{-1}$, where both Wishart matrices are normalized by
their respective degrees of freedom. Under the conditions in Li (2025),
$p^{2/3}(\ell_{\max}-\Theta_1)/\Theta_2$ has an asymptotic real Tracy--Widom
distribution. This approximation may be inaccurate for small dimensions,
small tested rank, irregular spectral edges or violations of the model.

\section{An example}
Here the first row of the design is an intercept and the hypothesis jointly
tests the remaining ten predictors. Simulated data are generated under
the null.
<<example>>=
library(RidgeLRGLH)
set.seed(8)
Y <- matrix(rnorm(20 * 80), 20)
X <- rbind(1, matrix(rnorm(10 * 80), 10))
C <- diag(11)[, -1, drop = FALSE]
fit <- ridge_glht(Y, X, C, selection = "minimax",
                 control = list(size = 60, grid_size = 80))
fit
head(fit$tuning)
@
For production use, the default spectral grid has 500 masses and 500
transform points. The smaller grids above keep this example fast.

\section{Regularization}
A fixed ridge can be supplied as \texttt{lambda}. Its default is the mean
residual eigenvalue. Adaptive choices use a finite candidate grid:
\texttt{bayes\_identity}, \texttt{bayes\_covariance}, or \texttt{minimax}.
The default candidates range from
$\min(0.1p/n_2,1)\operatorname{tr}(W_2)/p$ to
$5\operatorname{tr}(W_2)/p$ at ten equally spaced points.
Minimax optimizes the worst signal-to-noise ratio over nonnegative
mixtures of identity and covariance priors normalized to equal total
signal strength. Adaptive calibration requires the consistency and
separation conditions of Section 4 of the article. It does not give an
exact finite-sample guarantee or adjust for testing multiple hypotheses.

\section{Estimation and diagnostics}
Algorithm 1 estimates a population spectral distribution by linear
programming, matching the real and imaginary relative discrepancies of
$Q_1$ and $Q_2$. Default grid selection and weight truncation follow
Supplement S.2. The scalar parameterization of Section 3 yields the edge
and derivatives by bracketed root finding and implicit differentiation,
without requiring a dense ODE trajectory. Only \texttt{lpSolve} and
\texttt{RMTstat} are required beyond standard R packages.
<<diagnostics>>=
fit$spectrum$loss
sum(fit$spectrum$weights)
fit$calibration
@
Inspect sensitivity to spectral grid size, fitting ridge and constraint
order. The loss is an approximation diagnostic before weight truncation;
it has no universal significance threshold. A user-supplied population
spectrum can be used for oracle studies:
<<oracle>>=
ridge_glht(Y, X, C, lambda = 1,
           spectrum = list(sigma = 1, weights = 1))
@
\section{Precomputed matrices}
Use \texttt{ridge\_root(W1, W2, n1, n2)} for a common-covariance double
Wishart problem. Both matrices must be positive semidefinite and normalized
by their respective degrees of freedom. The denominator may be singular,
but must have positive trace. This interface does not turn arbitrary
sample covariance pairs into an equality-of-covariance test.

\section{Reference}
Li, H. (2025). Ridge-Regularized Largest Root Test For High-Dimensional
General Linear Hypotheses. Accepted by the Annals of Statistics.
\url{https://arxiv.org/abs/2504.15510}.
\end{document}