Exponential Distribution Gauss file(s) max_exp.g Matlab file(s) max_exp.m The aim of this exercise is to reproduce the convergence properties of the different algorithms in Table 3.1. Suppose that the following observations {3.5, 1.0, 1.5} are taken from the exponential distribution f(y; θ) = 1 θ exp h − y θ i , θ > 0 .

(a) Derive the log-likelihood function ln LT (θ) and also analytical expressions for the gradient, GT (θ), the Hessian, HT (θ), and the outer product of gradients matrix, JT (θ).

(b) Using θ(0) = 1 as the starting value, compute the first seven iterations of the Newton-Raphson, scoring and BHHH algorithms.

(c) Redo

(b) with GT (θ) and HT (θ) computed using numerical derivatives.

(d) Estimate var(θb) based on HT (θ), JT (θ) and I(θ).