Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 9

A sampling scheme is a method for selecting a sample from a population. The
units are selected from the population one by one, using a pre-assigned set of
selection probabilities for each draw. Generally, we assume that the sample design
has strictly positive first-order inclusion probabilities,

π k . This conjecture ensures
that each element has a non-zero selection probability (some exceptions to this rule
are described in Chap. 8 ).
A statistic is a real-valued function that depends on the different outcomes of an
experiment. The sample membership indicator I k ( s ) is a statistic, and satisfies

EIðÞ ¼ π k ,
VIðÞ ¼ Var IðÞ ¼ π k 1 π k ð Þ ʔ kk , ð 1 : 9 Þ

CI k ; ð I l Þ ¼ Cov I k ; ð I l Þ ¼ π kl π k π l ¼ ʔ kl :

For the proofs of Eq. ( 1.9 ), see S¨rndal et al. ( 1992 , p. 36). The following R code
can be used to obtain the covariance matrix of the sample membership indicator
statistic (i.e., ʔ ).

> set.seed(160964)
> delta < - matrix(0,N,N)
> for (i in 1:(N-1))
{

for (j in (i+1):N)
{

delta[i,j] < - second_order[i,j]-first_order[i]*first_order[j]
}

}
> delta < - delta + t(delta)
> diag(delta) < - first_order*(1-first_order)

> delta

[,1] [,2] [,3] [,4] [,5]
[1,] 0.24155320 -0.06055668 -0.06479317 -0.05988446 -0.05631889
[2,] -0.06055668 0.23857135 -0.06043757 -0.05128016 -0.06629695
[3,] -0.06479317 -0.06043757 0.23856148 -0.06230817 -0.05102257
[4,] -0.05988446 -0.05128016 -0.06230817 0.24048494 -0.06701215
[5,] -0.05631889 -0.06629695 -0.05102257 -0.06701215 0.24065056

The sample size of s can be also denoted
n S ¼ X
U

I k ; ð 1 :

10 Þ

and is another simple example of a statistic. A statistic used to estimate a parameter
is referred to as an estimator. The numerical value of an estimator for a given
sample is called an estimate.