# Ishiguro M., Sakamoto Y.'s A Bayesian approach to binary response curve estimation PDF By Ishiguro M., Sakamoto Y.

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7. e -Stability of a Solution in the Mean Unlike Tintner, in [1, 4] Arbuzov and Danilov consider one fixed point of the space Q, namely the point q whose coordinates are 31 STOCHASTIC PROGRAMMING the expectations of the random parameters. They deduce conditions under which the optimal basis of this problem are left optimal for all q E Q except a subset of given measure e > O. De fi nit ion. A solution of the SLP problem in the mean is said to be stochastically stable modulo e (or e-stable) if an optimal extreme point of the problem in the mean is left optimal under any realization q of the random parameters except a set of measure e.

X) x y + 'Ji (y) I g (x) + h (y) :;;.. b}. 8) The following theorem holds. 3. Let x(Eb) be a solution of the problem minl(Eb,x). The inequalities x E'( (b, x(Eb)) :;;.. min El (b, x) :;;.. E min"( (b, x) :;;.. min"( (Eb, x) x x x are valid, the truth of the last inequality requiring only the assumption that cp(x) and l/J (y) are convex continuous functions and the components of g(x) and h(y) are concave continuous functions of their arguments. Proof. The first inequality is obvious, since min E"( (b, x x) does not exceed the value of the function Ey (b, x) calculated at some point x = (Eb).

E. This proves the theorem. 7) Pl(Q)X 1 +PZX2 + ... +PnxnO; 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.