Question: Consider, sampling from a multivariate normal distribution with mean vector = (1, 2, . . . , M) and covariance matrix 2I. The log-likelihood

Consider, sampling from a multivariate normal distribution with mean vector μ = (μ1, μ2, . . . , μM) and covariance matrix σ2I. The log-likelihood function is Show that the maximum likelihood estimates of the parameters are Derive the second derivatives matrix and show that the asymptotic covariance matrix for the maximum likelihood estimators is Suppose that we wished to test the hypothesis that the means of the Mdistributions were all equal to a particular value μ0. Show that the Wald statistic would be where ?y is the vector of sample means.br>

OML In L= = -nM N/3 2 In(27) - - nM 2

OML In L= = -nM N/3 2 In(27) - - nM 2 Ino -1 M-1 (Yim - Ym) m=1 nM = W = ( - nv (21) M M M1 20 In L {-E [2001]} = [0/ . 72 11 -( ). 1=1 n (x-i), = ( (yim - Ym) 20+ / M)] = M M M1 ) ( i)' (y i), 'm'

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