By Rey Pastor, Julio; Pi Calleja, Pedro; Trejo, Cesar A.
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Extra info for Analisis matematico. Vol. III: Analisis funcional y aplicaciones. (Spanish) Mathematical analysis. Vol. III: Functional analysis and applications
3 Wave height parameters along a beach proﬁle . . . . . . . . . . 4 Independence test results for consecutive wave heights . . . . . . 5 Correlation parameter . . . . . . . . . . . . . . . . . . . . 6 Survival function for consecutive wave heights. Comparison between data and the model . . . . . . . . . . . . . . . . . . . . . 75, 1 . . . . . 2 Pickands dependence functions A (red) and A∗ (green) . . . . . . 3 Pickands dependence functions A∗ (green) and Aco (red) .
25) (iii) E( i (X, Y )) = ζi , i = 1, 2, . . , n3 for speciﬁed functions 1 , 2 , . . , n3 (iv) E(ϕi (X, Y )|φi (X, Y ) = λi ) = wi , for speciﬁed functions ϕi , φi and constants λi , i = 1, 2, . . , n4 All of these requirements can be stated in terms of linear equalities that the elements of P (the pij ’s) must satisfy. For example P ((X, Y ) ∈ Bi |(X, Y ) ∈ Ci ) = ηi is equivalent to pij − ηi (i,j)∈Bi ∩Ci pij = 0. 26) where A, a matrix of dimension M × (I × J)),and b are known. 26)). Techniques for solving such systems and for identifying all possible solutions are available in Castillo et al.
L2 ⎬ τj q2j (y) . 6) may then be shown to be itself an exponential family of densities. Speciﬁcally we have the following result of Arnold and Strauss (1991). Theorem 1. 6). 7) in which q (1) (x) = (q10 (x), q11 (x), q12 (x), . . , q1l1 (x)), q (2) (y) = (q20 (y), q21 (y), q22 (y), . . , (l1 + 1) × (l2 + 1)] subject to the requirement that f (x, y)dμ1 (x)dμ2 (y) = 1. S(X) S(Y ) For convenience we can partition the matrix M as follows: ⎞ ⎛ m00 | m01 · · · m0l2 ⎜ −− + −− −− −− ⎟ ⎟ ⎜ ⎟. 8) ˜ ≡ 0.