content_image/318572.jpg
where Φ denotes the standard normal distribution function and the absolute value | t$_{1}$ − t$_{2}$ | is meant component wise, see Kabluchko (2009, Remark 24). 

This Brown-Resnick model could in particular be used for the generalized max- linear model in dimension k = 1 as in Example 2.1, since in this case the approxi- mation ˆ η of η only uses bivariate D -norms ‖·‖$_{t}$$_{1}$$_{,t}$$_{2}$ . 

## 3. A generalized max-linear model based on kernels 

The model. There is the need for the definition of d functions g$_{1}$, . . . , g$_{d}$ satisfying certain constraints in the ordinary generalized max-linear model with d = d ( n ) tending to infinity as the grid s$_{1}$, . . . , s$_{d}$ gets dense in the index set. For the kernel approach introduced in this section, this is reduced to the choice of just one kernel and a bandwidth. And in this case we can establish convergence to zero of MSE and IMSE as the grid gets dance, essentially without further conditions. This approach was briefly mentioned in Falk et al. (2015) and is evaluated here. 

The disadvantages are: The interpolation is not an exact one at the grid points, i.e., ˆ η$_{s}$$_{j}$ ̸ = η$_{s}$$_{j}$ . This is due to the fact that the generated functions do not satisfy the condition g$_{i}$ ( s$_{j}$ ) = δ$_{ij}$ exactly, but only in the limit as h tends to zero, see Lemma 3.1. The choice of an optimal bandwidth, which is statistical folklore in kernel density estimation, is still an open problem here. 

Again, throughout the whole section, let η = ( η$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k be an SMSP with generator Z = ( Z$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k and denote by ‖·‖$_{s}$$_{1}$$_{,...,s}$$_{d}$ the D -norm pertaining to ( η$_{s}$$_{1}$, . . . , η$_{s}$$_{d}$ ). 

Let K : [0 , ∞ ) → [0 , 1] be a continuous and strictly monotonically decreasing function (kernel) with the two properties 

<img src='content_image/318572.jpg'>

The exponential kernel K$_{e}$ ( x ) = exp( − x ), x ≥ 0, is a typical example. Choose an arbitrary norm ‖·‖ on R k and a grid of pairwise different points { s$_{1}$, . . . , s$_{d}$ } in 

