## 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

Zbl 0100.14202
Full Text:

### References:

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