Suppose that x has theWeibull distribution f (x) = x1ex, x 0, , > 0.

Suppose that x has theWeibull distribution f (x) = αβxβ−1e−αxβ, x ≥ 0, α, β > 0. a. Obtain the log-likelihood function for a random sample of n observations.

b. Obtain the likelihood equations for maximum likelihood estimation of α and β.

Note that the first provides an explicit solution for α in terms of the data and β. But, after inserting this in the second, we obtain only an implicit solution for β. How would you obtain the maximum likelihood estimators?

c. Obtain the second derivatives matrix of the log-likelihood with respect to α and β. The exact expectations of the elements involving β involve the derivatives of the gamma function and are quite messy analytically. Of course, your exact result provides an empirical estimator. How would you estimate the asymptotic covariance matrix for your estimators in Part b?

d. Prove that αβ Cov [ln x, xβ] = 1.