## ABSTRACT

Here we consider a population P of N individuals and their associated response variables Y (m)i . A sampling scheme is a procedure that involves one or more random mechanisms to select a subset s ⊂ P of the population, i.e. the sample. The set of all possible samples s is denoted by S, which is a subset of the set of all subsets (the power set) of P. A well-known example might be a lottery machine that may choose 6 balls out of 45. In that case the set S consists of the

) potential outcomes, which are all equally possible with a

probability ( 45 6

)−1 . In a survey one usually needs a sampling frame, i.e. a

list of all individuals in the population, which are identiﬁed by a key in the data base (e.g. the social security number of Swiss residents). In this book the identifying key is an integer number called the label and is simply denoted by i = 1, 2 . . . N . Again, a forest inventory is peculiar in that no such list can exist, but this diﬃculty can be circumvented as we shall see. Using pseudorandom numbers (e.g. generated by a computer program and not by a physical mechanism of some kind) one can draw, in most instances sequentially, the individuals forming the sample. At this point it is not necessary to describe the practical implementation of such schemes; these will be discussed later.