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Archimax copulas and invariance under transformations. (English) Zbl 1126.62040

From the introduction: M. Sklar’s theorem [Publ. Inst. Stat. Univ. Paris 8, 229–231 (1960; Zbl 0100.14202)] states that each random vector \((X,Y)\) is characterized by some copula \(C\) in the sense that for its joint distribution \(H_{XY}\) and for the corresponding marginal distributions \(F_X\) and \(F_Y\) we have \(H_{XY}(x,y)= C(F_X(x),F_Y(y))\). We investigate transformations of copulas by functions in one variable. Such transformations play a role in statistics: as an example, if \((X_1,Y_1),(X_2,Y_2),\dots,(X_n,Y_n)\) are i.i.d. random vectors (characterized by some copula \(C\)) then the random vector \((\max(X_1,X_2,\dots,X_n)\), \(\max(Y_1,Y_2,\dots, Y_n))\) is characterized by the \(\varphi_{1/n}\)-transform of \(C\) with \(\varphi_{1/n}(x)= x^{1/n}\). Recall that a (two-dimensional) copula is a function \(C:[0,1]^2\to [0,1]\) such that \(C(0,x)= C(x,0)=0\) and \(C(1,x)= C(x,1)=x\) for all \(x\in [0,1]\), and \(C\) is 2-increasing, i.e., for all \(x,x^*,y,y^*\in [0,1]\) with \(x\leq x^*\) and \(y\leq y^*\) for the volume \(\text{Vol}_C\) of the rectangle \([x,x^*]\times[y,y^*]\) we have
\[ \text{Vol}_C ([x,x^*]\times[y,y^*])= C(x,y)- C(x,y^*)+ C(x^*,y^*)- C(x^*,y)\geq 0. \]
Important examples of copulas are the Fréchet-Hoeffding bounds \(M\) and \(W\) given by \(M(x,y)= \min(x,y)\) and \(W(x,y)= \max(x+y-1,0)\), respectively, and the product \(\Pi\) given by \(\Pi(x,y)=x\cdot y\). Obviously, each copula \(C\) satisfies \(W\leq C\leq M\).

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62A01 Foundations and philosophical topics in statistics

Citations:

Zbl 0100.14202
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References:

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