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As the vectorized version of the estimated matrix (vecB ) can be considered as a function of the pr vectorized form of E, it follows from the asymptotic normality of vecE ~ that vecB is also asymptotically normally distributed. The expectation of pr ~ the limiting distribution of vecBpr is given by vecBp r and its covariance matrix can be obtained by applying the delta method. Therefore the derivatives of vecB pr with respect to (vecE)' are needed. 34) a(vecB ) pr () V Bpr;E - a(vecE)' aB 1 a(vecE), (kpxk 2 ) a(vecE), Obviously it is sufficient to obtain the derivative of one column of Bpr (the derivative of a characteristic vector of E in the Q-1-metric) with respect to (vecE)'.

P. 74). Setting this derivative equal to zero. 24) (kxp). The resulting system of equations seems a little obscure. as the parameter matrix "Bpr appears many times in the matrix equation. In order to get an idea of a possible solution. 23) is the minimum of a quadratic form in B and it is well-known that its solution is given by the set of characteristic vectors of E. corresponding with the p smalles characteristic values. 25) B' Q-IB - I ~ (I k - Q-I Bpr B'pr )EB pr - O. 25) is known to be solved by the characteristic vectors of E in the Q-l-metric.

17) with K the (p x 1) vector of Lagrange multipliers. 19) at dK 0 ~=X K=K and K equal to zero. -2QX + 2QX - 2BK = 0 0 ~=X K=K ~ -2B'X = o. 21) K o - -1 -(B'Q B) -1 B'X. Note that as Q-l is positive definite and B is of rank p. the matrix B'Q-IB is also positive definite and therefore non-singular (see for instance Muirhead (1982. p. 585». 15). 8). 22) min E(X'B(B'Q-I B)-I B,X). B Using the trace operator. 23) min tr(B(B'Q-IB)-I B'E). 23) with respect to B is known in the literature and can for instance be found in Balestr.!

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