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D -norm 

<img src='content_image/306768.jpg'>

where ‖·‖$_{D}$$_{t,s}$$_{1}$$_{,...,s}$$_{d}$ is the D -norm pertaining to ( η$_{t}$, η$_{s}$$_{1}$, . . . , η$_{s}$$_{d}$ ) . 

Proof. As Z = ( Z$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k is a generator of η , we have for x, y ≤ 0 

<img src='content_image/306769.jpg'>

Then the assertion follows from the fact that ˆ Z$_{t}$ ≥ 0 and E ( ˆ Z$_{t}$ ) = 1. 

We can now use the preceding Lemmas to compute the mean squared error. 

Proposition 2.5. The mean squared error of ˆ η$_{t}$ is given by 

<img src='content_image/306770.jpg'>

Proof. Due to Lemma 2.4, ( η$_{t}$, ˆ η$_{t}$ ) is standard max-stable. Therefore, Lemma 2.3 (i) and the fact that E ( η$_{t}$ ) = E (ˆ η$_{t}$ ) = − 1 and Var( η$_{t}$ ) = Var(ˆ η$_{t}$ ) = 1 yield 

Lemma 2.6. The mean squared error of ˆ η$_{t}$ satisfies 

Proof. We have 

<img src='content_image/306773.jpg'>

<img src='content_image/306771.jpg'>

<img src='content_image/306772.jpg'>

