Bivariate Covariance Functions of P\'olya Type

We provide sufficient conditions of P\'olya type which guarantee the positive definiteness of a $2\times 2$-matrix-valued function in $\mathbb{R}$ and $\mathbb{R}^3$. Several bivariate covariance models have been proposed in literature, where all components of the covariance matrix are of the same parametric family, such as the bivariate Mat\'{e}rn model. Based on the P\'olya type conditions, we introduce two novel bivariate parametric covariance models of this class, the powered exponential (or stable) covariance model and the generalized Cauchy covariance model. Both models allow for flexible smoothness, variance, scale, and cross-correlation parameters. The smoothness parameters are in $(0, 1]$. Additionally, the bivariate generalized Cauchy model allows for distinct long range parameters. We also show that the univariate spherical model can be generalized to the bivariate case within the above class only in a trivial way. In a data example on the content of copper and zinc in the top soil of Swiss Jura we compare the bivariate powered exponential model to the traditional linear model of coregionalization and the bivariate Mat\'{e}rn model.