MLREML4: a program for the inference of the power variogram model by maximum likelihood and restricted maximum likelihood

The power variogram model γ(h)=α·|h|β, α>0, β∈]0, 2[, is an important theoretical model when only the intrinsic hypothesis is assumed for a random function and has been extensively used in practice, e.g. for variables such as piezometric level in groundwater hydrology and rainfall in surface hydrology. MLREML4 is an ANSI FORTRAN-77 program which provides maximum likelihood and restricted maximum likelihood estimates of the parameters α and β of the model, parameters of scale and shape, respectively. These parametric estimators have several advantages over other non-parametric estimators: the former are more efficient (as will be shown using the sampling distribution of the estimates), with only the parameters of interest being estimated (instead of estimating the variogram for different distances and fitting the model). Furthermore the uncertainty of the estimates is easily assessed by their standard errors, which means approximate confidence limits may be constructed. A good strategy is to use the non-parametric and the parametric approach complementarily. Firstly the non-parametric approach suggests which is the kind of variogram model that seems more adequate and secondly, the parameters are estimated by the parametric approach. Results from simulation and different sets of data are shown to illustrate the implementation of the program. MLREML4 is an upgrade of MLREML, i.e. it has all the capabilities of the latter plus the possibility of choosing the power variogram model, in addition to the three transition models, spherical, exponential and Gaussian, already included in MLREML.