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**O; j=1,2, ... ,n. The elements of the column P1 (q) are independent random variables a11 (q), a21 (q) •... , amI (q). 7) have a unique optimal plan in the mean x* (~ with positive component (~. Arbuzov proposes the change xi =~; Yz = ~; ... Yn =.!!!.... 7) transforms to the linear-fractional programming problem of variables y = q; (x); YI max c, -BYI + c,112 ~ + PIYc + ... + PnYn <;: - y __ + (nLl.!!... 8) = 1,2, ... , fl. 8) the column of free terms of the system of constraints is random. **

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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