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is the least dual D -norm function, corresponding to the case of independent uni- variate margins, where ‖·‖$_{D}$ = ‖·‖$_{1}$ , and 

<img src='content_image/298880.jpg'>

is the largest dual D -norm function, corresponding to the perfect dependence case, where ‖·‖$_{D}$ = ‖·‖$_{∞}$ . Hence, we have for an arbitrary dual D -norm function the bounds 

For the next examples, the following abbreviation is useful. We define for x ∈ R d and a nonempty subset T ⊂ { 1 , . . . , d } 

<img src='content_image/298879.jpg'>

<img src='content_image/298878.jpg'>

Example 2.5 (Fr´ echet model) . It is well-known that a D -norm is given by the l$_{λ}$ -norm 

<img src='content_image/298877.jpg'>

usually referred to as the logistic model in the literature. Therefore, we obtain by 

<img src='content_image/298876.jpg'>

A generator Z = ( Z$_{1}$, . . . , Z$_{d}$ ) of ‖·‖$_{λ}$ can easily be found: Put Z$_{i}$ := ˜ Z$_{i}$/ Γ (1 − 1 /λ ), i = 1 , . . . , d , where ˜ Z$_{1}$, . . . , ˜ Z$_{d}$ are iid Fr´ echet distributed with parameter λ , and Γ denotes the gamma function. 

Example 2.6 (Weibull model) . We can define a generator Z = ( Z$_{1}$, . . . , Z$_{d}$ ) by taking independent Weibull distributed random variables ˜ Z$_{1}$, . . . , ˜ Z$_{d}$ , i. e. P ( ˜ Z$_{1}$ > t ) = exp( − t $^{α}$), t > 0, α > 0, and putting Z$_{i}$ := ˜ Z$_{i}$/ Γ(1 + 1 /α ). It is easy to show that the corresponding dual D -norm function is for x ∈ R $^{d}$, x$_{i}$ ̸ = 0, i = 1 , . . . , d , 

