Geoscience Reference
In-Depth Information
Also, if the probability of selection is proportional to a variable
X
that has
the known value of
x
i
for the
i
th sampled unit, then Equation (2.35) becomes
n
n
∑∑
·
/
ˆ
ˆ
µ=
TN yp
=
(/)/ (1/)
p
,
(2.36)
y
y
i
i
i
i
=
1
i
=
1
and the estimated mean of the variable
X
itself becomes
n
∑
ˆ
/(1/ )
x
µ=
n
x
.
(2.37)
i
i
=
1
These estimators are called Horvitz-Thompson estimators (Horvitz and
Thompson, 1952). They provide unbiased estimates of the population param-
eters because of the weight given to different observations. For example,
suppose that there are a number of population units with
p
i
= 0.1. Then,
it is expected that only 1 in 10 of these units will appear in the sample of
observed units. Consequently, the observation for any of these units should
be weighted by 1/
p
i
= 10 to account for those units that are missed from
the sample. Variance equations for all three estimators were provided by
McDonald and Manly (1989), who suggested that replications of the sam-
pling procedure will be a more reliable way of determining variances. The
topic by Thompson (2012) gives a comprehensive guide to the many situa-
tions that occur for which unequal probability sampling is involved.
