8 MICHAEL FALK, MAXIMILIAN ZOTT 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 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 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 D -norm ‖ ( x, y ) ‖$_{D}$$_{t}$ = E ( max ( | x | Z$_{t}$, | y | ˆ Z$_{t}$ )) = ‖ ( x, g$_{1}$ ( t ) y, . . . , g$_{d}$ ( t ) y ) ‖$_{D}$$_{t,s}$$_{1}$$_{,...,s}$$_{d}$ , P ( η$_{t}$ ≤ x, ˆ η$_{t}$ ≤ y ) = P ( η$_{t}$ ≤ x, η$_{s}$$_{1}$ ≤ g$_{1}$ ( t ) y, . . . , η$_{s}$$_{d}$ ≤ g$_{d}$ ( t ) y ) = exp ( − E (max ( | x | Z$_{t}$, | y | max ( g$_{1}$ ( t ) Z$_{s}$$_{1}$, . . . , g$_{d}$ ( t ) Z$_{s}$$_{d}$ )))) = exp ( − E ( max ( | x | Z$_{t}$, | y | ˆ Z$_{t}$ ))) . MSE (ˆ η$_{t}$ ) := E ( ( η$_{t}$ − ˆ η$_{t}$ ) 2 ) = 2 ( 2 − ∫ ∞ 0 1 ‖ (1 , u ) ‖ 2 D$_{t}$ du ) , t ∈ [0 , 1] $^{k}$. MSE (ˆ η$_{t}$ ) = E ( η 2 t ) − 2 E ( η$_{t}$ ˆ η$_{t}$ ) + E ( ˆ η 2 t ) = 4 − 2 ∫ ∞ 0 1 ‖ (1 , u ) ‖ 2 D$_{t}$ du. MSE (ˆ η$_{t}$ ) ≤ 6 E (∣ ∣ Z$_{t}$ − ˆ Z$_{t}$ ∣ ) , t ∈ [0 , 1] $^{k}$. 2 − ∫ ∞ 0 1 ‖ (1 , u ) ‖ 2 D$_{t}$ du = ∫ ∞ 0 1 ‖ (1 , u ) ‖ 2 ∞ du − ∫ ∞ 0 1 ‖ (1 , u ) ‖ 2 D$_{t}$ du