The problem of estimation of a coregionalization of size q using cokriging will be discussed in this chapter. Cokriging—a multivariate extension of kriging—is the usual procedure applied to multivariate regionalized problems within the framework of geostatistics. Its fundament is a distribution-free, linear, unbiased estimator with minimum estimation variance, although the absence of constraints on the estimator is an implicit assumption that the multidimensional real space is the sample space of the variables under consideration. If a multivariate normal distribution can be assumed for the vector random function, then the simple kriging estimator is identical with the conditional expectation, given a sample of size N. See Journel (1977, pp. 576-577), Journel (1980, pp. 288-290), Cressie (1991, p. 110), and Diggle, Tawn, and Moyeed (1998, p. 300) for further details. This estimator is in general the best possible linear estimator, as it is unbiased and has minimum estimation variance, but it is not very robust in the face of strong departures from normality. Therefore, for the estimation of regionalized compositions other distributions must also be taken into consideration. Recall that compositions cannot follow a multivariate normal distribution by definition, their sample space being the simplex. Consequently, regionalized compositions in general cannot be modeled under explicit or implicit assumptions of multivariate Gaussian processes. Here only the multivariate lognormal and additive logistic normal distributions will be addressed. Besides the logarithmic and additive logratio transformations, others can be applied, such as the multivariate Box-Cox transformation, as stated by Andrews et al. (1971), Rayens and Srinivasan (1991), and Barcelo-Vidal (1996). Furthermore, distributions such as the multiplicative logistic normal distribution introduced by Aitchison (1986, p. 131) or the additive logistic skew-normal distribution defined by Azzalini and Dalla Valle (1996) can be investigated in a similar fashion. References to the literature for the fundamental principles of the theory discussed in this chapter were given in Chapter 2. Among those, special attention is drawn to the work of Myers (1982), where matrix formulation of cokriging was first presented and the properties included in the first section of this chapter were stated.
Different Random Distributions Research on Logistic-Based Sample Assumption
