5 EstimatorsIn this section the estimator used for computing averages or proportions, is described. The estimates may be proportions of the total survey population as well as of any sub-population, including those of the three main regions. As the issue of discussion is of a highly technical character, mathematical formulas have to be used extensively.
The variable of interest is denoted x. Given that x is a variable defined on the individual level (examples: level of education, age etc.), x(s,k,c,h,d,i) is the x-value of population unit (individual) (s,k,c,h,d,i). The sum of x-values for all (population) members of a household (s,k,c,h,d) is denoted x(s,k,c,h,d). We will use the latter notation also in cases where x is a variable defined on the basis of households.
We introduce a random variable Y(s,k,c,h,d,i) taking the value 1 if individual (s,k,c,h,d,i) is included in the sample, and 0 otherwise. For the sample of households the variable is denoted Y(s,k,c,h,d). The variable Y is a sample indicator in which all the information about the sample design is incorporated. In fact, Y is the only random element involved in the survey design, aiming at separating the sample units (having Y=1) from the non-sample units (Y=0). The probability of Y being 1 is of course the overall inclusion probability P. The sample indicator may be extended by attaching a subscript T indicating a subset of the population. Thus YT (s, k, c, h, d, i)=1 if unit (s,k,c,h,d,i) both is included in the sample and belongs to the sub-population T, and 0 otherwise (and similarly for the sample of households).T may also be the total survey population itself.
The survey observations may be expressed through the composite variable
defined on the basis of individuals, or in case x is defined on the basis of households.
If the variable is defined according to households, the "x-aggregate"
of the household is of course the observation itself:
The very size of sub-population T is estimated similarly by putting x(s,k,c,h,d,i)
or x(s,k,c,h,d) equal to 1 for all units of the population. For the sake
of clarity we will use the following notations for these estimates: