By Hideo Kusuoka and Julien I.E. Hoffman
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Extra resources for Advice on Statistical Analysis for Circulation Research
Example text
The convex hull of the pOints (a11' g (au) and (ai2' g (ad) is a segment connecting these two points. The segment, by the convexity of the function g (xi)' is the upper boundary and is determined by the equation If r == 2, then, depending on the position of the random vector x E I, it is necessary to consider two cases, for each of which H3 is bounded above by two hyperplanes. A more precise upper estimate for the expected value of the function g(xi' x2, ... , x r) is obtained if fuller use is made of the convexity of g (x).
E. This proves the theorem. 7) Pl(Q)X 1 +PZX2 + ... +Pnxn
Let the vector x* be such that min EC (b (q), x)=EC (b (q), x*) x and let the vectors x*(q) be such that min C (b (q), x)=C (b (q), x* (q». x Then C (b (q), x*) ::> C (b (q), x*(q» for any q. Hence, EC (b (q), x*) > >- EC (b (q), x* (q», which yields on the basis of the notation introduced above min EC (b (q), x) >- E min C (b (q), x). 29) we consider for any q the dual of the problem min C (b (q), x). Let the functional of x the dual problem be b (q)T z. If z* (q) is the optimal solution of this 28 V.
Advice on Statistical Analysis for Circulation Research by Hideo Kusuoka and Julien I.E. Hoffman
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