 Solution

Given numerical values of $\mu$ and ${\sigma }^2$, maximize the joint log-likelihood over the response rates $0<p_{jk}<1$:This is a penalized binomial regression problem and can be solved by Newton method or another numerical optimizer. One may alternate this maximization with the closed-form updates for $\mu$ and ${\sigma }^2$ from part i until convergence. The selected solution should have a negative-definite Hessian in the fitted log-odds parameters.