Weibull Distribution Gauss file(s) max_weibull.g Matlab file(s) max_weibull.m (a) Simulate T = 20 observations with = { = 1, = 2} from the

Weibull Distribution Gauss file(s) max_weibull.g Matlab file(s) max_weibull.m

(a) Simulate T = 20 observations with θ = {α = 1, β = 2} from the Weibull distribution f (y; θ) = αβyβ−1 exp h −αyβ i .

(b) Derive ln LT (θ), GT (θ), HT (θ), JT (θ) and I(θ).

(c) Choose as starting values θ(0) = {α(0) = 0.5, β(0) = 1.5} and evaluate G(θ(0)), H(θ(0)) and J(θ(0)) for the data generated in part (a). Check the analytical results using numerical derivatives.

(d) Compute the update θ(1) using the Newton-Raphson and BHHH algorithms.

(e) Continue the iterations in part

(d) until convergence. Discuss the numerical performances of the two algorithms.

(f) Compute the covariance matrix, Ω, using the Hessian and also the b outer product of the gradients matrix. (g) Repeat parts

(d) and

(e) where the log-likelihood function is concentrated with respect to βb. Compare the parameter estimates of α and β with the estimates obtained using the full log-likelihood function. (h) Suppose that the Weibull distribution is re-expressed as f(y; θ) = β λ y λ β−1 exp  − y λ β  , where λ = α −1/β. Compute λb and se(λb) for T = 20 observations by the substitution method and also by the delta method using the maximum likelihood estimates obtained previously.