Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 12

corresponding response). In this way, the HT estimator uses probability to weight
the responses when estimating the total. This estimator can also be defined in terms
of I k

t HT ¼ X U I k y k

π k :

ð 1 : 23 Þ

The estimator in Eq. ( 1.23 ) can also take the form
t HT ¼ X U I k ^ k ¼ X s ^ k :

ð 1 : 24 Þ

The term ^ k ¼ y k =π k is defined as the

-expanded y -value of the k -th element. The
variance of the HT estimator for the population total is
V HT t H ðÞ ¼ Var HT t H ðÞ ¼ XX U ʔ kl ^ k ^ l :

π

ð 1 : 25 Þ

The variance in Eq. ( 1.25 ) can also be expressed in terms of the non-expanded
original values ( y k )as

y k y l :

V HT t H ðÞ ¼ Var HT t H ðÞ ¼ XX U

π kl
π k π l 1

ð 1 : 26 Þ

An unbiased estimator of V HT t H ðÞ is given by
V HT t H ðÞ ¼ XX s ʔ

^ kl ^ k ^ l ;

ð 1 : 27 Þ

^ kl ¼ ʔ kl =π kl

where ʔ represents the expanded ʔ value. Note that

^ kk ¼ 1 π k . Alternatively, in function of
non-expanded original values ( y k ) the estimator in Eq. ( 1.27 ) can be expressed as

^ kl ¼ 1 π k π l =π kl

ʔ ð Þ for k 6 ¼ l and ʔ

y k y l :

V HT t H ðÞ ¼ XX s

1
π kl

π kl
π k π l 1

ð 1 : 28 Þ

Yates and Grundy ( 1953 ), and Sen ( 1953 ) defined an alternative formula for the
variance of estimator t HT , obtained when p ( s ) is a fixed size sampling design,

2

2 XX U ʔ kl ^ k ^ l

1

V YGS t H ðÞ ¼ Var YGS t H ðÞ ¼

: ð 1 :

29 Þ

0 for all k , l 2 U , an unbiased estimator of V YGS t H ðÞ is Provided that π kl >