1. Detecting Extinction in Sighting Data
3
following property of the stationary Poisson process. Conditional on their number
n, the sightings are independently, uniformly distributed over the interval (0, T).
Thus, under H 0 , the time T n of the most recent sighting has the same distribution as
the maximum of n independent random variables uniformly distributed over the
interval (0, T). It follows from the distribution theory of the sample maximum
(David 1981) that the significance level of the observed value of T n (or p value) is
prob 0 (T n ≤ t n ) = (t n /T)
n
(1.3)
Alternatively, the critical value for testing at significance level α is
c(α) = α
1/n T
(1.4)
The same argument can be used to find the power of this test. Under the alternative
hypothesis, the n sightings are independently, uniformly distributed over the
interval (0, T E ) with T E < T. The power of the α-level test is given by
prob 1 [T n < c(α)] = 1
α (T / T E )
n
0 ≤ T E ≤ α
1/n T
α
1/n T < T E
(1.5)
with the subscript 1 denoting that the probability is calculated under the alternative hypothesis H 1 . For example, if n = 10 and α = 0.05, H 0 is sure to be rejected if
T E /T ≤ 0.74 (if extinction occurs earlier than 74% of the way through the observation period).
Testing for Extinction in a Declining Population
The test described in the previous section assumes that the pre-extinction sighting
rate is approximately constant. If the pre-extinction population size is declining,
then to the extent that sighting rate depends on population size, this test will tend
to give spuriously significant results. The reason for this is that, even under the
null hypothesis that extinction has not occurred, the sightings will tend to be
concentrated in the earlier part of the record. In particular, the time of the most
recent sighting will not represent the maximum of n independent random variables uniformly distributed over (0, T).
In this section, we describe a test for extinction that can be used when the preextinction sighting rate declines. Specifically, suppose that the sighting record
follows a nonstationary Poisson process with rate function
λ(t) = exp(β 0 + β 1 t)
0
0 ≤ t ≤ T E
t > T E
(1.6)
with β 1 ≤ 0. As before, the sighting rate parameters β 0 and β 1 and the extinction
time T E are unknown, and interest centers on testing the null hypothesis H 0 : T E = T
(or, equivalently, T E ≥ T) against the alternative hypothesis H 1 : T E < T.
The particular choice of the form of λ(t) in Equation (1.6) is to some extent
arbitrary. It is consistent, however, with a model in which population size follows
Brownian motion with drift parameter β 1 (Lande and Orzack 1988) and the
sighting rate is proportional to population size.
3
following property of the stationary Poisson process. Conditional on their number
n, the sightings are independently, uniformly distributed over the interval (0, T).
Thus, under H 0 , the time T n of the most recent sighting has the same distribution as
the maximum of n independent random variables uniformly distributed over the
interval (0, T). It follows from the distribution theory of the sample maximum
(David 1981) that the significance level of the observed value of T n (or p value) is
prob 0 (T n ≤ t n ) = (t n /T)
n
(1.3)
Alternatively, the critical value for testing at significance level α is
c(α) = α
1/n T
(1.4)
The same argument can be used to find the power of this test. Under the alternative
hypothesis, the n sightings are independently, uniformly distributed over the
interval (0, T E ) with T E < T. The power of the α-level test is given by
prob 1 [T n < c(α)] = 1
α (T / T E )
n
0 ≤ T E ≤ α
1/n T
α
1/n T < T E
(1.5)
with the subscript 1 denoting that the probability is calculated under the alternative hypothesis H 1 . For example, if n = 10 and α = 0.05, H 0 is sure to be rejected if
T E /T ≤ 0.74 (if extinction occurs earlier than 74% of the way through the observation period).
Testing for Extinction in a Declining Population
The test described in the previous section assumes that the pre-extinction sighting
rate is approximately constant. If the pre-extinction population size is declining,
then to the extent that sighting rate depends on population size, this test will tend
to give spuriously significant results. The reason for this is that, even under the
null hypothesis that extinction has not occurred, the sightings will tend to be
concentrated in the earlier part of the record. In particular, the time of the most
recent sighting will not represent the maximum of n independent random variables uniformly distributed over (0, T).
In this section, we describe a test for extinction that can be used when the preextinction sighting rate declines. Specifically, suppose that the sighting record
follows a nonstationary Poisson process with rate function
λ(t) = exp(β 0 + β 1 t)
0
0 ≤ t ≤ T E
t > T E
(1.6)
with β 1 ≤ 0. As before, the sighting rate parameters β 0 and β 1 and the extinction
time T E are unknown, and interest centers on testing the null hypothesis H 0 : T E = T
(or, equivalently, T E ≥ T) against the alternative hypothesis H 1 : T E < T.
The particular choice of the form of λ(t) in Equation (1.6) is to some extent
arbitrary. It is consistent, however, with a model in which population size follows
Brownian motion with drift parameter β 1 (Lande and Orzack 1988) and the
sighting rate is proportional to population size.