Let 31, .. . . I, be a random sample from M(p, ). Derive the ML estimator f of a and prove its consistency. A random variable X is said to have a Pareto distribution with parameter A, denoted X ~ Pareto(A), if it is continuously distributed with density if z > 1, otherwise. A random sample 21, ..., 2, from the Pareto(A) population is available. (a) Derive the ML estimator A of A, prove its consistency and find its asymptotic distribution, (b) Derive the Wald, Likelihood Ratio and Lagrange Multiplier test statistics for testing the null hypothesis Ho : A = do against the alternative hypothesis H. : A # do. Do any of these statistics coincide? Berndt and Savin in 1977 showed that W 2 CR 2 CM for the case of a multivariate regression model with normal disturbances. Ullah and Zinde-Walsh in 1984 showed that this inequality is not robust to non-normality of the disturbances. In the spirit of the latter article, this problem considers simple examples from non-normal distributions and illustrates how this conflict among criteria is affected. 1. Consider a random sample 21, . .., In from a Poisson distribution with parameter A. Show that testing A = 3 versus A # 3 yields W 2 CM for a $ 3 and WS CM for 2 2 3. 2. Consider a random sample 1, ..., 2, from an exponential distribution with parameter 0. Show that testing 0 = 3 versus 0 * 3 yields W 2 CM for 0