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L', with equality only when XT = (1,0) and (0,1). 1 we wish to estimate 1 + 2 with minimum variance; that is we are interested in eTe with eT = (1 1). The geometric argument used by Elfving (1952) shows that the design measure which puts probability t at each of (1,0) and (0,1) is optimal. However practical intuition suggests that the measure rr* which puts probability 1 at t) is also optimal. This design measure has the singular information matrix e e

Let JtA be the subset of Jt consisting of matrices with this property; JtA contains all the non-singular matrices in Jt and also some singular ones. Again by analogy with results in least-squares theory, the variance matrix of the least-squares estimator of ATe arising from a design measure with information matrix M E JtA is proportional to ATM- A, where M- is any generalized inverse (g-inverse) of M, that is any matrix such that MM- M = M; see, for instance, Searle (1971, p. 184). Note that ATM - A does not depend on which g-inverse is used; also that A TM - A is a positive-definite s x s matrix.

Xx T ) < (j, xefI: at the stopping stage. 35 T ). OPTIMAL DESIGN Do we necessarily reach this stopping stage? Consider the Walgorithm with a sequence of step-lengths having the above properties. Again suppose that tjJ is a differentiable at all Mn and that maxXE~ F",(M n , xx T ) ~ b > 0 for all n. We have-see Appendix 3: tjJ(Mn) - tjJ(Mn- 1 ) = F",(M n- 1 ,Mn) + ollMn - Mn-111 = F", {Mn- 1, (1 - IXn)Mn-l + IXnX(n)XTn)} + o(lXn) + o(lXn) = IXn max F",(M n - 1 ,XXT) + r n , = IXnF",(M n- 1 ,x(n)xTn») XE~ where rn = o(lXn)· Since by assumption, for all n, maxXE~ F",(M n , xx T ) ~ b, we therefore have n tjJ(Mn) - tjJ(Mo) ~ b L i; 1 lXi + n Lr i; 1 i, and the right-hand side tends to infinity as n ~ 00, since L lXi diverges, rn = o(lXn), and consequently L rJL lXi ~ O.

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A short introduction to graphical algebra by H. S Hall

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