Upper tail dependence and smoothness of random fields
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The modeling of risk situations that occur in a space-time framework can be done using max-stable random fields. Although the summary coefficients of the spatial and temporal dependence do not characterize the finite-dimensional distributions of the random field, they have the advantage of being immediate to interpret and easier to estimate. As the joint tendency for extreme values of random variables is usually summarized in the literature by upper-tail dependence coefficients, the question arises: how to use these coefficients to summarize the degree of smoothness of a random field or a stochastic process? We invite the reader to follow us in a justified construction of a response to this question. The coefficients that we propose, give us information about the tendency of a random field for oscillations of its values in relation to real valued high levels. It is not the magnitude of the oscillations that is being evaluated, but rather the greater or lesser number of oscillations, that is, the tendency of the trajectories to oscillate. Therefore, we can observe surface trajectories more smooth for the random field according to higher smoothness coefficients values. The objective of this work is to quantify the smoothness of a random field through coefficients that are easy to estimate and use in applications. They take values in [0,1] and increase with the concordance of the variables of the random field.
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