Lipschitz-Killing curvatures of the Excursion Sets of Skew Student's t Random fields

In many real applications related with Geostatistics, medical imaging and material science, the real observations have asymmetric, and heavy-tailed multivariate distributions. These observations are spatially correlated and they could be modelled by the skew random fields. However, certain statistical analysis problems require giving analytical expectations of some integral geometric characteristics of these random fields, such as Lipschitz-Killing curvatures, specifically Euler-Poincaré characteristic. This paper considers a class of skew random fields, namely skew student’s t random fields. The goal is to give the analytical expressions of the LipschitzKilling curvatures of the skew student’s t excursion sets on a compact subset S of R. The motivation comes from the need to model the roughness of some engineering surfaces, involved in the total hip replacement application, which is characterized by the Euler-Poincaré characteristic function. The analytical and estimated Euler-Poincaré characteristics are fitted in order to test the skew student’s t random field with the surface roughness topography. Keywords-Skew student-t random field; Excursion sets; Lipschitz-Killing curvatures; Skewness; Surface roughness topography