On Numerical Computation of Pickands Constants

Pickands constant $H_{\alpha}$ appears in the classical result about tail probabilities of the extremes of Gaussian processes and there exist several different representations of Pickands constant. However, the exact value of $H_{\alpha}$ is unknown except for two special Gaussian processes. Significant effort has been made to find numerical approximations of $H_{\alpha}$. In this paper, we attempt to compute numerically $H_{\alpha}$ based on its representation derived by $H{\ddot{u}}sler$ (1999) and Albin and Choi (2010). Our estimates are compared with the often quoted conjecture $H_{\alpha}=1/{\Gamma}(1/{\alpha})$ for 0 < ${\alpha}$ ${\leq}$ 2. This conjecture does not seem compatible with our simulation result for 1 < ${\alpha}$ < 2, which is also recently observed by Dieker and Yakir (2014) who devised a reliable algorithm to estimate these constants along with a detailed error analysis.