Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 14

1.3 The Predictive Approach: The Concept
of Superpopulations

The traditional approach to survey sampling has several limitations, which have
been discussed in the literature over the last 40 years. Among others, Godambe
( 1966 ) analyzed the singular effect of likelihood considerations on this approach to
survey sampling.
Design-based inference, as already noted, only considers s as the stochastic
element, and treats y as a constant. An alternative strategy assumes that the actual
finite population y i is a realization of a random vector. In particular y¼ ( y 1 y 2 ...
y N ) t represents a certain observation vector of an N -dimensional distribution that
defines a stochastic model ʾ . This is called the superpopulation model. For example,

we may be interested in estimating the population total, t ¼ X U y k . Using the
superpopulation approach, the population total represents the sum of only one
realization. If other realizations were generated, they would have different values

for t ¼ X U y k . Under this strategy, the sample values are also random variables.
The population total is a sum of random variables, so it is also a random variable. In
this case, the source of randomness in a sample is only derived from the stochastic
model,

. The superpopulation approach can be interpreted in terms of a long series
of realizations of the random process, for a fixed sample s . In reality, there is only
one finite population, so for the sake of simplicity, we will use y k to represent both
the random variable Y k and the observed value.
Now, consider a generic drawn sample s 2 S , and its complement s 2 U s . The
sample value y k is only observed for k 2 s . An estimator of t is defined as a function
of these observed values,

ʾ

t ¼ ^ t k

f ð Þ . Estimating t is equivalent to
predicting the population total using the available data. So, the predictor t of t is
said to be model unbiased if, given s

: k 2 s

E ʾ t t

½ ð Þ j ¼ 0 : ð 1 : 31 Þ

In this case, the model mean square error of t is

h i

2 j

MSE ʾ ðÞ ¼ E ʾ t t ð Þ : ð 1 : 32 Þ

The difference between the two definitions of MSE in Eqs. ( 1.18 ) and ( 1.32 )isonly
due to the source of randomness. In the design-based approach, the uncertainty is
ensured by p ( s ), while in the superpopulation approach, the randomness is provided
from the model

ʾ . For this reason, this approach is also called model-based survey

sampling.
Using the superpopulation approach, we estimate the population total as follows.
The population total, t, can be decomposed into