Compare other methods for obtaining approximate p-values

[haziqj]

The Rao-Scott adjustment seems natural to use in the context of complex sampling. The problem is it requires a Choleski decomposition of $\boldsymbol\Omega_2$, whose estimate can be not full rank. Could we explore stable/modified Choleski decompositions and see how it affects the $p$-value calculations?

install.packages("Matrix")
library(Matrix)

# Create a non-positive definite matrix
A <- matrix(c(4, 2, 2, 2, 4, 2, 2, 2, 4), nrow = 3, ncol = 3)

# Find the nearest positive definite matrix
nearest_pd <- nearPD(A, corr = FALSE, do2eigen = TRUE, ensureSymmetry = TRUE)

# Perform Cholesky decomposition on the nearest positive definite matrix
L <- chol(nearest_pd$mat)

print(L)

[haziqj]

Alternative approximate chi-square distributions can be explored as well. Here's a good starting point:

Bodenham, D. A., & Adams, N. M. (2016). A comparison of efficient approximations for a weighted sum of chi-squared random variables. Statistics and Computing, 26(4), 917-928. https://link.springer.com/article/10.1007/s11222-015-9583-4