Geoscience Reference
In-Depth Information
This estimator is often called the Petersen estimate by fisheries biologists
and the Lincoln index by terrestrial wildlife biologists following its indepen-
dent use by Petersen (1896) and Lincoln (1930), respectively. However, the
principle involved was used even earlier by Laplace (1786) to estimate the
size of the human population of France and probably even before that. Here,
this is called the Petersen-Lincoln estimate.
The assumptions behind Equation (7.1) are the following:
a. The population is closed so that there are no losses and gains in the
time between the two samples.
b. The second sample is randomly chosen from the population.
c. No marks are lost before the second sample is taken, and all marked
animals are recognized as such in the second sample.
To meet assumption a, at least approximately, the study must generally be
carried out over a relatively short period of time. Actually, assumption a can
be relaxed slightly. If there are losses from the population but no gains, and
the losses are at the same rate for marked and unmarked animals, then
N
ˆ
estimates the population size at the time of the first sample.
A potential problem with using Equation (7.1) is that
m
2
can be zero, giv-
ing an infinite estimate for
N
. To overcome this, several modified estimators
have been proposed, but the estimator proposed by Chapman (1951) is pre-
ferred, which is
ˆ
*
Nn n m
=+
( )(
+
1)/(
+
1) 1
−
.
( 7. 2)
1
2
2
This estimator is unbiased (i.e., it would give the correct average value if a
study was repeated a large number of times) when
n
1
+
n
2
>
N
and is approx-
imately unbiased otherwise. Also, an estimate of the standard error of this
estimator is
·
{
}
ˆ
SE(
N
*
)
(
n
1)( )(
n
n mnmm m
)(
)/ ( )(
2
2)
=
+
+
−
−
+
+
.
( 7. 3)
1
2
1
2
2
2
2
2
One criticism of these estimators is that the equations for the standard
errors are derived under the assumption that the sample sizes
n
1
and
n
2
are
fixed before the study, so Sekar and Deming (1949) derived the estimate of
the standard error of the index without this assumption:
·
(
)
1
2
ˆ
*
3
SE(
N nn mnmm
)
=
(
−
)(
−
)/()
( 7. 4)
12 1
2
2
2
2
This estimator is not corrected for bias and is known to be valid only for
large sample sizes, but in practice it gives results that are almost the same
