Consider, sampling from a multivariate normal distribution with mean vector = (1, 2,...,M) and covariance matrix 2I. The log-likelihood function is ln L = nM 2 ln(2) nM 2 ln 2 1 22 n i=1 (yi ) (yi ). Show that the maximum likelihood estimates of the parameters are 2 ML = n i=1 M m=1 (yim ym)2 nM = 1 M M m=1 1 n n i=1 (yim ym) 2 = 1 M M m=1 2 m. Derive the second derivatives matrix and show that the asymptotic covariance matrix for the maximum likelihood estimators is E 2 ln L 1 = 2I/n 0 0 24/(nM) . Suppose that we wished to test the hypothesis that the means of the M distributions were all equal to a particular value 0. Show that the Wald statistic would be W = (y 0 i) 2 n I 1 (y 0 i), = ! n s2 " (y 0 i) (y 0 i), where y is the vector of sample means.

Week 10 Notes for FIN 305 10/24 Standard Deviation Equal to the amount of risk Probability vs Range 68% = +/ 1 STD 95% = +/ 2 STD 99% = +/ 3 STD *Usually easier to draw without the bell curve *The higher STD, the higher the risk CV = Coefficient Value Decision rule regarding this: Want...