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TO do this one can take the partial derivative with respect to p to obtain and equating the partial derivative to zero, we get -j--=L=l xzi A P = -j-;="=,; - X'Y - X'X' where X is the column vector (zl, . . 2). The usual estimator for a2 in this instance would be ( l / n . ) S 2of , which we have no knowledge since the true P is unknown and, consequently, the 6,'s are unobserved. j] = E [ E ~ ] E=[ 0 E ~for ] i # j. Hence, an unbiased estimator of the error variance a2 is in this case S2 = S n / ( n- 1 ) .

Proof: For example, ifa is a column vector such that a'X' = 0, then a ' X I X = 0. Suppose now that afX'X = O;, then alX'Xa = 0, which implies that a'X' = 0. Hence, every vector orthogonal to X' is orthogonal to X ' X . Therefore, M ( X ' ) = M(XtX). 3 PROJECTIONS + Let U I and U2 be subspaces of vector space V. U1 U2 = {ul+u2: u1 E U 1 ,u2 E U2) is a subspace of V. If U1 n U2 = { 0 } ,denote CT1 U2 by W = U1 / U2. The space W is known as the direct sum of spaces U 1 and U z . 4. If v that v = u1 + u2.

L ~ ) ' ,write ~i Then Piyj= 0 for i # j. Also, Piyi= yi and PiZ = 0 because Z E ( L I $ . . $ Lk)' c L:. Therefore, From (i), That is, P projects on L1 @ . . @ Lk along the orthogonal complement. 12. Suppose that PI and P2 are orthogonal projection matrices on Li and La, respectively. Then the following are equivalent: (i) PI - P2is a projection matrix. (ii) PIP2 = P2P1= P2 if and only if L2 c L 1 . (iii) If PI- P2 is a projection matrix, it projects on L1 n L; along (L1 n L;)' L; @ L2.

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