% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/get_B.R
\name{.get_B_gee_gaussian}
\alias{.get_B_gee_gaussian}
\title{Path 1a: Gaussian Identity + GEE (Closed-Form Full-System Solve)}
\usage{
.get_B_gee_gaussian(
  X_block,
  X_tilde,
  y_tilde,
  VhalfInv_perm,
  Lambda_block,
  A,
  K,
  p_expansions,
  constraint_value_vectors,
  family,
  return_G_getB,
  quadprog,
  qp_Amat,
  qp_bvec,
  qp_meq,
  qp_score_function,
  order_list,
  observation_weights,
  ...
)
}
\arguments{
\item{X_block}{Full \eqn{N \times P} unwhitened block-diagonal design.}

\item{X_tilde}{Whitened design \eqn{\mathbf{V}^{-1/2}\mathbf{X}}.}

\item{y_tilde}{Whitened response \eqn{\mathbf{V}^{-1/2}\mathbf{y}}.}

\item{VhalfInv_perm}{\eqn{\mathbf{V}^{-1/2}} permuted to partition
ordering.}

\item{Lambda_block}{Full \eqn{P \times P} block-diagonal penalty.}

\item{A}{Constraint matrix (\eqn{P \times R}).}

\item{K, p_expansions}{Integer dimensions.}

\item{constraint_value_vectors}{Constraint RHS list encoding
\eqn{\mathbf{A}^{\top}\boldsymbol{\beta} = \mathbf{c}}.}

\item{family}{GLM family object.}

\item{return_G_getB}{Logical; return covariance components.}

\item{quadprog}{Logical; apply QP refinement.}

\item{qp_Amat, qp_bvec, qp_meq}{QP constraint specification.}

\item{qp_score_function}{Score function for QP step.}

\item{order_list, observation_weights}{Standard partition arguments.}

\item{...}{Passed to score function.}
}
\value{
If \code{return_G_getB = TRUE}: list with \code{B},
  \code{G_list}, and \code{qp_info}. Otherwise: list of \code{B} and
  \code{qp_info}.
}
\description{
Computes the constrained penalized GLS estimate for Gaussian response
with identity link when a correlation structure is present. Because
\eqn{\mathbf{V}^{-1/2}} couples all partitions, fitting must operate on
the full \eqn{P}-dimensional whitened system rather than partition-wise.
}
\details{
The unconstrained GLS estimate is:
\deqn{\hat{\boldsymbol{\beta}} = \mathbf{G}(\mathbf{X}^{*\top}
  \mathbf{y}^{*}), \quad
  \mathbf{G} = (\mathbf{X}^{*\top}\mathbf{X}^{*} +
  \boldsymbol{\Lambda})^{-1}}
where \eqn{\mathbf{X}^{*} = \mathbf{V}^{-1/2}\mathbf{X}} and
\eqn{\mathbf{y}^{*} = \mathbf{V}^{-1/2}\mathbf{y}}. The constrained
estimate is then obtained via the \eqn{\mathbf{G}^{1/2}\mathbf{r}^*}
trick in the full \eqn{P}-dimensional space.
}
\keyword{internal}
