Geoscience Reference
In-Depth Information
and
ˆ
ˆ
NnMm
*
=+
( )
*
/(
+
1)
(8.6)
j
j
j
j
respectively (Seber, 1982, p. 304), leaving the other estimation equations
unchanged, except that the new estimators are used for
M
j
and
N
j
.
Variance equations were provided by Jolly (1965) and Seber (1965). In using
these, it is necessary to distinguish between sampling errors (the difference
between estimated parameters and the parameters occurring in the real-
ized population) and stochastic errors (the difference between the realized
parameters and their mean values from hypothetical repeated generations
of the population). For the population size, it seems to be generally accepted
that only sampling errors are relevant, so that an estimate of the variance of
N
ˆ
j
given
N
j
is needed. This is approximately
·
ˆ
ˆ
[
ˆ
ˆ
1/ )/
ˆ
ˆ
ˆ
ˆ
*
Var(
NN NN nMmR r RMNMNm
)
≈
−
][(
−+
)(1/
−
+
(
−
)/(
)]
. (8.7)
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
On the other hand, it is usually considered that survival probability is being
estimated rather than the proportion surviving in the population. Therefore,
the variance of ϕ
j
is taken to include stochastic errors. It can then be approxi-
mated by
·
ˆ
ˆ
)/
ˆ
Var(
ˆ
ˆ
2
2
φ≈φ
)
[{(
MmMmRM r
−
)(
−+
}(1/
−
1/
R
)
(8.8)
j
j
j
+
1
j
+
1
j
+
1
j
+
1
j
+
1
j
+
1
j
+
1
j
+
1
ˆ
ˆ
)/
ˆ
ˆ
ˆ
2
+−
(
Mm MmRr
)/(
−+
)(1/
−
1/ )
R
+ φ−φ
(1
M
j
j
j
j
j
j
j
j
j
j
+
1
Variance equations for
M
ˆ
j
and
ˆ
j
can be found in the original works of Jolly
(1965) and Seber (1965) and in reviews such as those of Seber (1982, 1986,
1992), Pollock et al. (1990), and Schwarz and Seber (1999). Covariance equa-
tions are also available from the same sources.
Before the JS model was developed, there was a crucial article by Cormack
(1964). He derived one component of the likelihood used by Jolly and Seber
in their more general model. To recognize Cormack's contribution to the
development of these models, the term
Cormack-Jolly-Seber
(CJS) model is
often used when referring to the marked animal component of the likeli-
hood function, which allows the estimation of survival and capture prob-
abilities. Jolly (1965) and Seber (1965) used different likelihoods but came up
with the same estimators.
The JS likelihood can be viewed as three conditionally independent com-
ponents with the overall likelihood the product of these. The overall like-
lihood is then
L
1
×
L
2
×
L
3
, where
L
1
is a product binomial likelihood that
relates the unmarked population and sample sizes at each time to the cap-
ture probabilities,
L
2
only contains parameters for probabilities of being lost
