In a design-based approach statistical inferences rest upon the randomization principle and conﬁdence intervals are interpreted under the hypothetical repetitions of samples. Even when models are used, such as in two-phase sampling schemes, the validity of an inference is insured by the scheme and not by the validity of the model in the classical statistical sense (i.e. with assumptions such as normally distributed, independent random variables). As mentioned for local density, Y (x) is random because point x is the realization of a random variable. This contrasts with a model-dependent approach, in which point x is ﬁxed and Y (x) is interpreted as the realization of a spatial stochastic process that depicts the actual forest under investigation. Inferences are then based on the hypothetical realization of all forests that can be generated by a given stochastic process. The actual forest is viewed as one out of an inﬁnite number of “similar” forests. Thus, in principle, one must build a highly complex process that determines the positioning of the trees as well as all variables Y (p)i associated with N trees, N also being a random variable under that model. Obviously, such models are complex mathematical objects (e.g. marked point processes) so that estimating the parameters is a highly non-trivial task. This approach could be viewed as the microscopic approach, analogous to thermodynamics, where one attempts to model pressure, temperature, entropy and other physical quantities at the molecular level. For practical problems encountered when conducting a forest inventory, it suﬃces to take a more pragmatic approach. This might be then considered macroscopic, in which we directly model the locally observed density Y (x) as a stochastic process or random function. There are of course hidden diﬃculties with this philosophy. First, microscopic models at the tree level and macroscopic models at the point level are generally incompatible in general, in the sense that the relationship
1 λ(F )
Yi = 1
can be violated (with EM1 and EM2 denoting expectations under their respective models). Second, we must deﬁne the stochastic integral, i.e. the integral of a random function.