Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 10

If there is only one study variable, a parameter of the population can be
expressed

ʸ ¼ ʸ ð y 1 ; y 2 ; ...; y N Þ: ð 1 : 11 Þ

Generally, we are interested in the estimates of the population total of y:
t ¼ X U y k ; ð 1 : 12 Þ

the population mean of y,

N ¼ X U y k = y U ¼ t = N ; ð 1 : 13 Þ

or the population variance of y,

N 1 X

N

1

2

S y , U ¼

ð y i y U Þ : ð 1 :

14 Þ

i ¼1

Note that the expression X U y k means X k2U y k . Conversely, if we consider a

subset D of U (i.e., D U ), Eq. ( 1.12 ) can be written t ¼ X D y k . The estimator of ʸ

is denoted

ʸ ¼ ʸ

ðÞ: ð 1 : 15 Þ

that denotes an estimator ʸ

The combination pðÞ , ʸ

based on s , chosen according

to a design p ( s ), is defined as a sampling strategy.
To be useful, an estimator should have a number of properties. For example, an
estimator that varies little around the parameter to be estimated is intuitively better
than one with great variability. To make this consideration operational, we must
start from the concept of the sampling distribution of the estimator. To this end, it is
needed to specify, for each c , the following probability

¼ X
s2Ω c

ʸ ¼ c

Pr pðÞ; ð 1 : 16 Þ

Ω c is the set of samples s for which ʸ ¼ c . Therefore, a sampling distribution
is a distribution of the estimator over all possible samples. The main properties of
this estimator are summarized as follows. Note that these properties are defined
according to the expectation across all possible samples, which represents the only
random component of this approach (i.e., this is under the design-based hypothesis).

where