2. [14 marks] This question relates to the Iteratively reweighted least squares (IRLS) algorithm for GLM maximum likelihood estimation and provides an insight into the distribution of the parameter estimator 3. Let y; be independent random variables with mean ; such that g( ) = n = x, 8, where g is a link function, X is a design matrix with x; = (ra, . ..,Pip) and S a parameter vector. Let the variance of y be V( ), where V is a known function, and o a scale parameter. Define = = g'(m )(yi - m) +n and w; = {V(m.)g'(;)"}-1. a) Show that E(2,) = x, B. b) Show that the covariance matrix of z is We'd, where W is a diagonal matrix with Wii = wi. c) If 3 is estimated by minimization of [ wilz, - xTB)? show that the covariance matrix of the resulting estimates, 3, is o(XTW X)-1 and compute E(B). d) The multivariate version of the central limit theorem implies that as the dimension of z tends to infinity, X W z will tend to multivariate gaussian. What does this imply about the large sample distribution of 3