By Dorota Kurowicka; Roger Cooke
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Extra info for Uncertainty analysis : mathematical foundations and applications
First we sample u ∼ U (0, 1). If u ∈ (β, 1 − β), then the conditional distribution V |U = u is uniform on the interval (u − β, u + β). Sampling from this distribution, we obtain v. 8. piece-wise uniform: uniform on (0, β − u) and on (−u + β, u + β) with density on the first interval twice that on the second interval. The same holds mutatis mutandis when u is greater than 1 − β. For the diagonal band copula, we can find the conditional distribution and its inverse. 16 Let (U, V ) be distributed according to bα (u, v), −1 ≤ α ≤ 1.
18 Let U, V be uniform on [− 12 , 21 ]. 8. a. E(V |U = u) = ρU , b. Var(V |U = u) = 21 (1 − ρ 2 ) c. for ρu − 1 − ρ2 FV |U (v|u) = d. for − 21 ≤ t ≤ 1 2 1 4 + 1 4 − U2 , − u2 ≤ v ≤ ρu + 1 π arcsin √ 1 − ρ2 v−ρu 1−ρ 2 1 −u2 4 1 4 − u2 , 1 2 FV−1|U (t|u) = 1 − ρ2 1 4 − u2 sin(π t) + ρu. The elliptical copula inherits some attractive properties of the normal distribution. For example, it has linear regression (property (a) in the preceding text), conditional correlations are constant (see below) and are equal to partial correlations.
What is the nature of this dependence? How dependent are they? How can we measure the dependence? These questions must be addressed in building a dependence model. In this section, we present the most common measures of dependence based on linear or monotone relationship that are used later in this book. There are many other concepts of dependence proposed in the literature. For richer exposition and references to original papers, see for example, Doruet Mari and Kotz (2001); Joe (1997); Nelsen (1999).
Uncertainty analysis : mathematical foundations and applications by Dorota Kurowicka; Roger Cooke