The condition of Proposition 1 does not hold if the conditional and marginal parameters ψ j and ψ ~ j are not distinct (i.e. if their joint parameter space is not the product of their separate parameter spaces), and in particular if the combined dimension of ψ j and ψ ~ j is greater than that of θ.
From the multivariate normal example above it is straightforward to show that the non-informative margins condition of Proposition 1 holds for imputing any W j. The conditional distribution of Y given W, p(Y ∣ W, ψ Y ), is the logistic regression model with covariates W[ 25].
Check optimality condition of Proposition 1 to ensure global optimization.
Thus, it may verify the existence of the space X satisfying the condition of Proposition 1.
For this income distribution, our Proposition 3.4 directly applies (while there is no need to check the necessary and sufficient condition of Proposition 3 in De Donder and Hindriks [3]).
Assume that is a complete metric space and that is a continuous surjective -times reasonable expansive self-mapping which satisfies the constraint (3.20) and the necessary condition of Proposition 3.9.
As before by Proposition 3.1 there exists (psi in G) such that begin{aligned}Omega subset B 0,R) subset psi (Omega )end{aligned}and let (omega) be a component of (psi ^{-1} (B(0,R)) subset Omega.) Then, (omega) satisfies all the condition of Proposition 3.4 and hence there exists a fixed point p of (psi) in (omega) and (O^-_{psi }(p cap omega) is finite.
Thus, although condition of Proposition 5 is less easy to check than the one corresponding to the existence of the GNE, from (66) it can be inferred that whenever the ratio between each direct and interference channel is small, matrix V is likely to satisfy the diagonal dominance property and thus to be a P-matrix, ensuring the uniqueness of the GNE.
To sum up, we have proved the following result: Under assumptions (H1) and (H2), there exists a unique solution (K, Z K), Λ,Z Λ ), Y,Z Y ),(χ,Z χ ) to the BSDE (20) satisfying the integrability condition of Proposition 3.1, and consequently we have an optimal control for the LQCMKV1 problem given by (21).
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