Compute the Selective Maximum Likelihood Estimate for beta1

This function numerically approximates the conditional maximum likelihood estimate (MLE) of a single regression coefficient using a Monte Carlo approximation to the selective likelihood, conditional on passing the overall F-test.

Usage

compute_MLE(
  X,
  y,
  beta_init,
  sigma_sq_init,
  alpha_ov,
  interval_beta = c(-10, 10),
  interval_sigma_sq = c(-10, 10),
  B = 1e+06
)

Arguments

X: A numeric matrix of predictor variables (n x p), with the first column corresponding to beta1.
y: A numeric response vector of length n.
alpha_ov: The significance level for the overall F-test (used in defining the selection region).
B: The number of Monte Carlo samples used to approximate the likelihood (default: 1e6).
interval: A numeric vector of length 2 giving the search interval for beta1 (default: c(-10, 10)).

Value

A list containing:

beta: The MLE of beta1 (a single coefficient) under the selective likelihood.
max_likelihood: The maximum log-likelihood value achieved at the optimal beta1.