Explicit expressions for the variogram of first--order intrinsic autoregressions

TL;DR

This paper derives an exact, explicit formula for the variogram of first-order intrinsic autoregressions using Appell's hypergeometric function, providing a precise alternative to previous approximations.

Contribution

It introduces a novel exact expression for the variogram of first-order intrinsic autoregressions applicable across all parameters.

Findings

Provides explicit formula involving Appell's hypergeometric function

Derives particular cases of the variogram expression

Offers a precise alternative to asymptotic approximations

Abstract

Exact and explicit expressions for the variogram of first--order intrinsic autoregressions have not been known. Various asymptotic expansions and approximations have been used to compute the variogram. In this note, an exact and explicit expression applicable for all parameter values is derived. The expression involves Appell's hypergeometric function of the fourth kind. Various particular cases of the expression are also derived.