(a) State the likelihood, score function, and Hessian matrix for the binomial distribution. (6) Program (in R) the Newton-Raphson algorithm to maximize the likelihood for a binomial random variable. Turn in your code. (c) For T = 3 based on n = 10, show results of the first 6 iterations for different starting values: 1, 2, 33. (d) Summarize the effects of the starting value on the speed of convergence. (e) Repeat part (c) for n = 50 and T = 1. What happens? You do not need to submit results; just describe in words.
(20 points) Fisher Scoring vs. Newton-Raphson Algorithms: (a) For n independent observations from a Poisson distribution, show that Fisher Scoring gives p(t+1) = y for all. By contrast, what happens with the Newton-Raphson algorithm? (b) For noncanonical links in 4 GLM, show that the observed information matrix depends on y and hence differs from the expected information. Relate this result to the Fisher Scoring vs. Newton-Raphson algorithms.
(25 points) Negative Binomial GLM: The negative binomial probability mass function is often written as:
Tl f(Y; μ, θ) = (θ+Y-1) / (θ+μ)^θ * (1+μ)^(-Y)
where Y ≥ 0, μ > 0, and θ > 0 are the dispersion parameters.