Exercise A We consider n observations y1, ...;y, of a variable and n vectors r; (t(r,) = ($1, ..., Fix)) where we have the observations of k variables. For ic {1, ..., n}, we assume that y; is an observation of Y, with Yi ~ N(t(x;).8, 0?) where B is a vector of dimension k (t(B) = (81, . . ., BE)). We assume that the variable Y, are independent. Let consider the following populations : - I1 = {1,..., nj} indices associated to the first population with n, elements 12 = {n + 1,..., nj + ny} indices associated to the second population with n2 elements .... - Ip = {n+ ...+ np_1+ 1,..., n} indices association to the population p with no elements We assume that if i e I;, o? = j.02. We want to estimate / and o' thanks to maximum likelihood. 1. What is the expression of fy (y:), where fy, denotes the density function of Y? 2. Show that 3 and o2 are solution of: ELil Lien,(y: - t(z.) 8)2 = no2 Vi = 1, .... k, Ell Zien,(yi - t(x.)B).x;; =0 3. Prove that the previous system is similar to: JIA(Y - XB)|12 = no2 ( X) A? ( Y - XB) = 0 where Y and X are the classical matrices involved in linear model, and A is a diagonal matrix whose elements are , for i such that i e It 4. We assume that t(X) A'X is invertible. Give the expression of , and &2. 5. Prove that no? = ||V| |2 where V is a centered gaussian vector. 6. Deduce that E(| |V| |2) is the sum of the diagonal elements of the covariance matrix of V. 7. Prove that no? is n k is an unbiased estimator of o