On the Optimal Estimation of the Cumulative Distribution Function in Presence of Spatial Dependence

For spatial analysis of data, the optimal prediction of the cumulative distribution function (c.d.f.) is made difficult by the presence of spatial dependence, particularly when data are clustered. Several methods have been proposed in the literature to correct for the redundancy which is present when the observed values are correlated. Most of them are based on an unequal weighting of the values for the computation of the c.d.f. The methodology is extended by defining weights which depend both on the spatial correlation and on the threshold value. If a parametric multivariate distribution is assumed, optimal weights can be obtained conditionally to the choice of this distribution, and a nonparametric indicator estimate of the c.d.f. can be defined a posteriori. The method can be extended in order to provide kernel smoothed estimates of the c.d.f., by replacing the indicator functions by other functions taken here as complementary differentiable c.d.f., whose parameters are defined in order to preserve the unbiasedness and minimum variance property of the estimate, conditionally to the choice of the a priori distribution. A simulation study is conducted in order to emphasize the properties of the various methods. It is found that, even for bounded and positively skewed distributions, a simplified version of the method seems to perform quite well without significant loss of efficiency, and provides smooth estimates of the c.d.f. which can be differentiated in order to obtain density estimates. These estimates do not depend highly on the choice of the parameters for the a priori distribution, and yet they respect major characteristics of the a priori c.d.f., such as lower and upper bounds, which are mainly dictated by the nature of the variable under study. A practical case study is conducted on cobalt concentrations for various rock types in the Swiss Jura, illustrating the applicability of the methodology for real datasets.