computeGamma1 {WALS}

Internal function: Compute model-averaged estimator of focus regressors in walsNB

Description

Exploits the SVD of the design matrix of the focus regressors \bar{Z}_1, the model-averaged estimator for the auxiliary regressors \hat{\gamma}_{2} and the Sherman-Morrison-Woodbury formula for computing the model-averaged estimator of the focus regressors in walsNB.

Usage

computeGamma1(
  gamma2,
  Z2start,
  Z2,
  U,
  V,
  singularVals,
  ellStart,
  gStart,
  epsilonStart,
  qStart,
  y0Start,
  tStart,
  psiStart
)

Arguments

gamma2

Model-averaged estimate for auxiliary regressors from computePosterior.

Z2start

Transformed design matrix of auxiliary regressors \bar{Z}_2. See details.

Z2

Another transformed design matrix of auxiliary regressors Z_2. See details.

U

Left singular vectors of \bar{Z}_1 from svd.

V

Right singular vectors of \bar{Z}_1 from svd.

singularVals

Singular values of \bar{Z}_1 from svd.

ellStart

Vector \bar{\ell} see details.

gStart

Derivative of dispersion parameter \rho of NB2 with respect to \alpha = \log(\rho) evaluated at starting values of one-step ML. gStart is a scalar. See section "ML estimation" of Huynh (2024a).

epsilonStart

Scalar \bar{\epsilon}, see section "One-step ML estimator" of Huynh (2024a) for definition.

qStart

Vector \bar{q}, see section "One-step ML estimator" of Huynh (2024a) for definition.

y0Start

Vector \bar{y}_0, see section "One-step ML estimator" of Huynh (2024a) for definition.

tStart

Scalar \bar{t}, see section "One-step ML estimator" of Huynh (2024a) for definition.

psiStart

Diagonal matrix \bar{\Psi}, see section "One-step ML estimator" of Huynh (2024a) for definition.

Details

See section "Simplification for computing \hat{\gamma}_{1}" in the appendix of Huynh (2024b) for details of the implementation and for the definitions of argument ellStart.

All parameters that contain "start" feature the starting values for the one-step ML estimation of submodels. See section "One-step ML estimator" of Huynh (2024a) for details.

The argument Z2start is defined as (Huynh 2024a)

\bar{Z}_{2} := \bar{X}_{2} \bar{\Delta}_{2} \bar{\Xi}^{-1/2},

and Z2 is defined as

Z_{2} := X_{2} \bar{\Delta}_{2} \bar{\Xi}^{-1/2}.

Uses svdLSplus under-the-hood.

References

Huynh K (2024a). “Weighted-Average Least Squares for Negative Binomial Regression.” arXiv 2404.11324, arXiv.org E-Print Archive. doi:10.48550/arXiv.2404.11324.

Huynh K (2024b). “WALS: Weighted-Average Least Squares Model Averaging in R.” University of Basel. Mimeo.