Polyvariograms and their Asymptotes

A definition of a polyvariogram (PV) γb(h)(h = 1, 2, ...) of order b (b≥ max(0, d− 1) is suggested for time series {Z(t})} satisfying {∇d(Z(t) = W(t) (where d is a non‐negative integer and {W(t)} is a second‐order stationary time series and is not over‐differenced). When b = 0, 1 and 2, this definition corresponds to Cressie's (J. Am. Stat. Assoc. 83 (1988), 1108–16; 85 (1990), 272) semivariogram linvariogram and quadvariogram respectively and is simpler. Under very general conditions about {W(t)}, we obtain the relationship between γb(h) and the autocovariance function of {W(t)} and show that the asymptote of γb(h) is a straight line having a positive slope when b = d− 1 and levelling out when b≥d. A definition of a sample polyvariogram (SPV) of order b is given and is shown to be an unbiased and consistent estimate of the PV; and further, some uniformly (in h) almost sure convergence rates are obtained. These properties provide theoretical support for using the SPV to replace the practically unknown PV and generalize the guidelines for identifying d given by Cressie, where {W(t)} was restricted to a white noise and b≤ 2. Some further asymptotic theorems and avenues for using them for statistically testing d and parameters of models for {W(t)} are briefly introduced.