Polyvariograms and their Asymptotes

A definition of a polyvariogram (PV) γ b ( 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 γ b ( h ) and the autocovariance function of { W ( t )} and show that the asymptote of γ b ( 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.