2
Andrew Solow and Thomas Helser
would be appropriate if there were no dependence of the sighting rate on population size. In the second case, population size is assumed to decline exponentially
prior to extinction. This model would be appropriate for a species experiencing a
negative net rate of population growth (e.g., due to overharvesting).
Testing for Extinction in a Stable Population
Suppose that during the period of observation (0, T) sightings occur at the set of
ordered times t 1 < t 2 . . . < t n . A word is in order about the definition of the
observation period. If the observation program has a clearly defined beginning
and a clearly defined end, then these will define the beginning and end of the
observation period. In many cases, it is difficult to define the beginning of the
observation program because historical data usually only record the presence of a
species and neglect to record absences. Indeed, it may be difficult to define the
observation program itself. In that case, the beginning of the observation period
should be taken to coincide with the initial sighting (in which case, the initial
sighting is dropped from the sighting record) and the end of the observation period
should be taken to coincide with the present time.
In statistical terminology, the sighting record is said to arise from a point
process (Cox and Lewis 1978). A point process is characterized by a rate function
(i.e., the interval between any pair of consecutive events follows a common
probability distribution), which gives the expected number of events (in this case,
sightings) in a unit time interval. In the stationary case, the rate function does not
depend on time. The first model that we consider is that the sighting record
follows a stationary Poisson process with rate function
λ(t) = λ
0
0 ≤ t ≤ T E
t > T E
(1.1)
where both the pre-extinction sighting rate λ and the extinction time T E are
unknown. Properties of the Poisson process are described in Taylor and Karlin
(1984). As noted, the stationary model is appropriate for cases in which preextinction population size is stable.
Under this formulation, interest centers on testing the null hypothesis that
extinction has not occurred—that is, H 0 : T E = T (or, equivalently, T E ≥ T)—
against the alternative hypothesis that it has—that is, H 1 : T E < T. Let the random
variable T i be time of the ith sighting (i.e., the random variable of which t i is a
realization). A natural statistic for testing H 0 against H 1 is the time of the most
recent sighting, T n , with H 0 being rejected if T n is small (or, equivalently, if the
time since the most recent sighting, T − T n , is large). Formally, H 0 is rejected at
significance level α if T n < c(α) where the critical value c(α) is chosen to satisfy
prob 0 [T n < c(α)] = α
(1.2)
with the subscript 0 denoting that the probability is calculated under H 0 .
Under H 0 , the sighting record is a realization of a stationary Poisson process on
the interval (0, T). To choose the critical value c(α), it is useful to exploit the
Andrew Solow and Thomas Helser
would be appropriate if there were no dependence of the sighting rate on population size. In the second case, population size is assumed to decline exponentially
prior to extinction. This model would be appropriate for a species experiencing a
negative net rate of population growth (e.g., due to overharvesting).
Testing for Extinction in a Stable Population
Suppose that during the period of observation (0, T) sightings occur at the set of
ordered times t 1 < t 2 . . . < t n . A word is in order about the definition of the
observation period. If the observation program has a clearly defined beginning
and a clearly defined end, then these will define the beginning and end of the
observation period. In many cases, it is difficult to define the beginning of the
observation program because historical data usually only record the presence of a
species and neglect to record absences. Indeed, it may be difficult to define the
observation program itself. In that case, the beginning of the observation period
should be taken to coincide with the initial sighting (in which case, the initial
sighting is dropped from the sighting record) and the end of the observation period
should be taken to coincide with the present time.
In statistical terminology, the sighting record is said to arise from a point
process (Cox and Lewis 1978). A point process is characterized by a rate function
(i.e., the interval between any pair of consecutive events follows a common
probability distribution), which gives the expected number of events (in this case,
sightings) in a unit time interval. In the stationary case, the rate function does not
depend on time. The first model that we consider is that the sighting record
follows a stationary Poisson process with rate function
λ(t) = λ
0
0 ≤ t ≤ T E
t > T E
(1.1)
where both the pre-extinction sighting rate λ and the extinction time T E are
unknown. Properties of the Poisson process are described in Taylor and Karlin
(1984). As noted, the stationary model is appropriate for cases in which preextinction population size is stable.
Under this formulation, interest centers on testing the null hypothesis that
extinction has not occurred—that is, H 0 : T E = T (or, equivalently, T E ≥ T)—
against the alternative hypothesis that it has—that is, H 1 : T E < T. Let the random
variable T i be time of the ith sighting (i.e., the random variable of which t i is a
realization). A natural statistic for testing H 0 against H 1 is the time of the most
recent sighting, T n , with H 0 being rejected if T n is small (or, equivalently, if the
time since the most recent sighting, T − T n , is large). Formally, H 0 is rejected at
significance level α if T n < c(α) where the critical value c(α) is chosen to satisfy
prob 0 [T n < c(α)] = α
(1.2)
with the subscript 0 denoting that the probability is calculated under H 0 .
Under H 0 , the sighting record is a realization of a stationary Poisson process on
the interval (0, T). To choose the critical value c(α), it is useful to exploit the