Geoscience Reference
In-Depth Information
to estimate the equation
Y
=
a
+
bU
. The mean of
U
, μ
u
, is estimated either
from a much larger sample of units or for all units in the population, and the
regression estimator of the mean of
Y
is
reg u
where
y
and
u
are
the means for the smaller sample of
Y
and
U
values. This can be interpreted
as
y
from the small sample corrected by
b
(μ
u
−
u
) to allow for a low or high
mean of the
U
values in the small sample.
yyb u
=+µ−
(
)
2.12 Unequal Probability Sampling
Situations do arise for which the nature of the sampling mechanism makes
random or systematic sampling impossible because the availability of sam-
ple units is not under the control of the investigator. In particular, cases occur
for which the probability of a unit being sampled is a function of the char-
acteristics of that unit, which is called unequal probability sampling. For
example, large units might be more conspicuous than small ones, so that the
probability of a unit being selected depends on its size. If the probability of
selection is proportional to the size of units, then this special case is called
size-biased sampling.
It is possible to estimate population parameters allowing for unequal prob-
ability sampling. Thus, suppose that the population being sampled contains
N
units with values
y
1
,
y
2
, . . .
y
N
for a variable
Y
, and that sampling is carried
out so that the probability of including
y
i
in the sample is
p
i
. Assume that
estimation of the population size
N
, the population mean μ
y
, and the popula-
tion total
T
y
is of interest, and that the sampling process yields
n
observed
units. Then, the population size can be estimated by
n
∑
ˆ
N
=
(1/)
i
p
,
(2.33)
i
=
1
the total of
Y
can be estimated by
n
∑
·
T
(/)
y
p
,
(2.34)
=
y
i
i
i
=
1
and the mean of
Y
can be estimated by
n
n
∑∑
·
/
ˆ
µ=
ˆ
TN yp
=
(/)/ (1/)
p
.
(2.35)
y
y
i
i
i
i
=
1
i
=
1
