Alternative Approach to Unequal Probability Sampling

In this chapter we examine an alternative approach to the problem of unequal probability sampling which is based on the so-called sampling autocorrelation coefficient, denoted by ρ _z (rho(z)) and also ρ _y . As a point of departure for our considerations we take the normal distribution and a discrete approximation of it. That is, in the simple situation of sampling with replacement we start without loss of generality with the assumption that the target variable in the population can be well described by the normal distribution. We also could have taken the more realistic lognormal distribution but, unfortunately, this distribution leads to more tedious algebra; note that most economic data such as income, investments, or returns are better described by a lognormal distribution than by a normal distribution. On the other hand, it is recalled from mathematical statistics that independent drawings from an arbitrary continuous or discrete distribution always lead to the same sampling formulas irrespective of the type of the underlying distribution or population.