A General Rho Theory on Survey Sampling

In the previous chapter we have seen how useful the sampling autocorrelation coefficient ρ _z is for deriving formulas for variance estimators in case of unequal probability sampling from a finite population. Especially in case of sampling with replacement, the formulas are straightforward because ρ _z = 0 or, equivalently, the sample observations are uncorrelated for such sampling designs. This applies to multistage sampling designs as well. From a geometric point of view this means that the random variables z _i are mutually orthogonal (i = 1,...,n). In this chapter we will give a more detailed geometric interpretation of ρ _z in case of unequal probability sampling without replacement. Before doing this we give a comprehensive and a somewhat more formal description of the alternative rho approach to unequal probability sampling. In the next section we first pay attention to the classical Horvitz-Thompson estimator (for short, the HT estimator).