It is common in geostatistics to use methods such as the variogram or the correlation coefficient to describe spatial dependence, and kriging to make interpolation and predictions, but these methods are sensitive to extreme values and are strongly influenced by marginal distribution of the random field. Hence they can lead to unreliable results. As an alternative to traditional models in geostatistics are considered the use of the copula functions. Copula is widely used in the finance and actuary fields and due to satisfactory results they started to be considered in other areas of application of statistical sciences. This work shows the effect of copulas as a tool that presents a geostatistical analysis under the range of quantiles and a dependence structure, considering models of spatial tendency, continuous and discrete marginal distributions and covariance functions. Three interpolation methods are shown: the first is the kriging indicator and disjunctive kriging, the second method is known as the simple kriging and the third method is a plug-in prediction and the generalization of the trans-Gaussian kriging, these methods are used based on the copula function due to the existing relationship between bivariate copulas and covariance indicators. Results are presented for a set of actual data in the city of Gomel that contains measurements of radioactive isotopes, consequence of the Chernobyl nuclear accident. Finally, discrete copulas are studied and applied to a set of simulated data, this allows an extension of the usual works of copulas in Geostatistics.

This article performs Bayesian estimation of Gaussian Markov random fields. In particular, it is proposed to perform a spatial dependency analysis by means of a graph that characterizes the observed intensities of an image with a model widely used in spatial statistics and geostatistics known as the conditional autoregressive model (CAR). This model is useful for obtaining multivariate joint distributions from a random vector based on univariate conditional specifications. These conditional specifications are based on the Markov properties, so that the conditional distribution of a component of the random vector depends only on a set of neighbors, defined by the graph. Conditional autoregressive models are particular cases of random Markov fields and are used as \textit{a priori} distributions, which, combined with the information contained in the sample data (likelihood function), induce a \textit{a posteriori} distribution on which the estimate is based. The CAR model has a particular case called IAR, in which the \textit{a priori} distribution is not proper, in this article both models are applied making a comparison between them. All model parameters are estimated in a completely Bayesian environment, using the Metropolis-Hastings algorithm. The complete estimation procedures are illustrated and compared using various artificial examples. For these experiments, the CAR model and the IAR model performed very favorably with homogeneous images.

Panjer’s algorithm used in the calculation actuarial basis taking class distributions (a,b), presents a recursive formula for calculating function distribution of sums of random variables in a model of collective risk. If the secondary distribution in this model is the ETNBD, Compound Poisson distribution is named PoissonPascal, this is a family of distributions very used in the mathematics of insurance and can generate models statistically appropriate. It illustrates the methodology application to a data set of a portfolio of policies cars, in addition the algorithm is implemented using the statistical software R.