Wynn, H. P.: 1970, `The sequential generation of D-optimum experimental designs'. Ann. Math. Statist. 41, 1655-1664.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the minimisation here is restricted to only those k tuples x 1 ; x k that provide p with all non negative components. In the case of a single constraint on the total mass the described approach turns into a conventional method widely used in the optimal design literature, see, e.g. (Wynn, 1970). In this case Df( is minimised for having a single atom placed at a point of global minimum of d f ( Since this descent di ers from the steepest descent given algo sub.tex; 1 11 2000; 15:52; p.15 16 Molchanov and Zuyev in Corollary 4.2, an additional analysis is necessary to ensure ....

Wynn, H. P.: 1970, `The sequential generation of D-optimum experimental designs'. Ann. Math. Statist. 41, 1655-1664.

....D ( The steepest descent algorithm described in Section 3 emerges from our theoretical results on constrained optimisation in the space of measures presented in Section 1. In contrast to the classical sequential algorithms in the optimal design literature (Wu, 1978ab; Wu and Wynn, 1978; Wynn, 1970), we do not renormalise the obtained design measure on each step. Instead, the algorithm adds a signed measure chosen so to minimise the Fr echet derivative of the goal function among all measures satisfying the imposed constraints. This extends the ideas of Atwood (1973, 1976) Fedorov (1972) and ....

.... in space of measures and optimal design 7 The most basic method of the gradient descent type used in the optimal design suggests moving from n (the approximation on step n) to n 1 = 1 n ) n n n , where 0 n 1 and n minimises D ( over all probability measures (Wynn, 1970). It is easy to see that such n is concentrated at the points where the corresponding gradient function d (x; is minimised. Rearranging the terms, we obtain n 1 = n n ( n n ) 1.11) In this form the algorithm looks like a conventional descent algorithm that descends along the ....

Wynn, H. P. (1970). The sequential generation of D-optimum experimental designs. Ann. Math. Statist., 41:1655--1664.

....and ff k =k 0. The case ff k =k 1 is not considered, since H1 implies that for any ffl, 9K 1 such that kj( x) ff k ffl for any k K 1 and any x 2 X , see (9) Under H1 H3, the measure k therefore converges to a D optimal measure on X for the linear regression model (1) see Wynn (1970) for a proof of convergence when x k 1 = arg max x2X d k (x) 2.1 Compromise designs: ff k = kff When ff k = kff, the sequence generated by (11) makes a compromise between D optimal design and maximisation of j( x) as indicates the following theorem, the proof of which is given in the ....

Wynn, H.P. (1970) The sequential generation of D-optimum experimental designs. Annals of Math. Stat., 41, 1655--1664.

....due to Lindsay (1983) The mixing distribution b G maximizes the likelihood if and only if (i) D( b G; 0 for all ; ii) D( b G; 0 in the support of b G. This result is the basis of the gradient methods. We present a simple, but somewhat inefficient algorithm known as the VDM (Wynn 1970, Federov 1972) At the m th step let G (m) be the current estimator. At each step, find m to maximize D(G (m) then find ffl m to maximize the likelihood evaluated at (1 Gamma ffl)G (m) fflffi( m ) set G (m 1) 1 Gamma ffl m )G (m) fflffi( m ) Iterate until ....

Wynn H. P. (1970). The sequential generation of D-optimum experimental designs. The Annals of Mathematical Statistics, 41, 1655-1664.

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