3.14 Repeat Exercise 3.12 with the same test function ????(h), but this time using the de Wijsian generalized intrinsic covariance function

????GI(h)=−log h, h > 0, where ????GI(h) is defined up to an additive constant.

As before, define the regularized intrinsic covariance function ????I,????(h) by

(3.67).

(a) Show that for h > 0,

????I,????(h) = h2 log h +

3 2 − 1 2

(1 + h)

2 log(1 + h) − 1 2

(1 − h)

2 log |1 − h|.

(b) Show that lim|h|→0????I,????(h) = 3∕2 is finite. Hence, deduce that

????I,????(h) defines an ordinary intrinsic random field with semivariogram ????????(h) = ????I,????(0) − ????I,????(h) = 3∕2 − ????I,????(h). Show that

????????(h) = h2 log h + O(h2) as h → 0. That is, ????????(h) defines an ordinary intrinsic random field, whose semivariogram is nearly smooth enough to be twice-differentiable at the origin.