4
Andrew Solow and Thomas Helser
The following theoretical development is necessarily a little complicated, although the test itself is easy to apply. It can be shown that, conditional on n, the
sighting times represent an ordered sample from the exponential distribution with
mean β 1
−1 truncated on the right at T E (Cox and Lewis 1978). The probability
density function of this distribution, which does not depend on β 0 , is
f(t) = β 1 exp(−β 1 t)/[1 − exp(−β 1 T E )]
0 ≤ t ≤ T E
(1.7)
Let S = Σ T I with realized value s. The uniformly most powerful unbiased test of
H 0 against H 1 at significance level α is to reject H 0 if T n < c′(α) where the critical
value c′(α) is chosen to satisfy
prob 0 [T n < c′(α) | S = s] = α
(1.8)
(Beg 1982). The statistic S is sufficient for β 1 under H 0 . It follows that the
conditional distribution of T n given S = s is the same as the largest gap in n − 1
points independently uniformly distributed over the interval (0, s) such that the
largest gap does not exceed T. Finally, from the results of Fisher (1929) on the
distribution of the largest gap, it follows that
prob 0 (T n ≤ t n | S = s) = F s (t n )/F s (T)
(1.9)
where F s is defined by Solow (1993b). The significance level of the observed
value of T n is given by Equation (1.9), and H 0 is rejected at significance level α if
the p value is less than α.
The conditional power of this test can be found by noting that, under H 1 , the
conditional distribution of T n given S = s is the same as that of the largest gap in
n − 1 points independently uniformly distributed over the interval (0, s) such that
the largest gap does not exceed T E . Thus, the test has conditional power 1 if 0 ≤ T E
≤ c′(α) and αF s (T)/F s (T E ) if c′(α) < T E ≤ s. Solow (1993b) presented some
simulation results indicating the conditions under which the unconditional power
of this test is reasonably high.
Mark Burgman (personal communication) pointed out that, for t n fixed, the
significance level in Equation (1.9) does not fall to zero as T approaches ∞ but is
equal to F s (t n ) for T ≥ s. However, this is the conditional significance level given
S = s. Because the distribution of S under H 0 depends on T (specifically, S tends to
be larger for larger T), the unconditional probability that T n ≤ t n —which is the
unconditional significance level—does fall to zero as T approaches ∞.
Illustrative Examples
In this section, the tests outlined above are illustrated by using sighting data for
two northwest Atlantic fish species. These data were taken from the annual
autumn bottom trawl survey of Georges Bank conducted by the Northeast Fisheries Science Center (Clark 1979). These examples are for illustrative purposes
only. It is important to note that, because of the limited nature of the survey, in
these applications extinction refers to local species loss (strictly, extirpation during the autumn season).
Andrew Solow and Thomas Helser
The following theoretical development is necessarily a little complicated, although the test itself is easy to apply. It can be shown that, conditional on n, the
sighting times represent an ordered sample from the exponential distribution with
mean β 1
−1 truncated on the right at T E (Cox and Lewis 1978). The probability
density function of this distribution, which does not depend on β 0 , is
f(t) = β 1 exp(−β 1 t)/[1 − exp(−β 1 T E )]
0 ≤ t ≤ T E
(1.7)
Let S = Σ T I with realized value s. The uniformly most powerful unbiased test of
H 0 against H 1 at significance level α is to reject H 0 if T n < c′(α) where the critical
value c′(α) is chosen to satisfy
prob 0 [T n < c′(α) | S = s] = α
(1.8)
(Beg 1982). The statistic S is sufficient for β 1 under H 0 . It follows that the
conditional distribution of T n given S = s is the same as the largest gap in n − 1
points independently uniformly distributed over the interval (0, s) such that the
largest gap does not exceed T. Finally, from the results of Fisher (1929) on the
distribution of the largest gap, it follows that
prob 0 (T n ≤ t n | S = s) = F s (t n )/F s (T)
(1.9)
where F s is defined by Solow (1993b). The significance level of the observed
value of T n is given by Equation (1.9), and H 0 is rejected at significance level α if
the p value is less than α.
The conditional power of this test can be found by noting that, under H 1 , the
conditional distribution of T n given S = s is the same as that of the largest gap in
n − 1 points independently uniformly distributed over the interval (0, s) such that
the largest gap does not exceed T E . Thus, the test has conditional power 1 if 0 ≤ T E
≤ c′(α) and αF s (T)/F s (T E ) if c′(α) < T E ≤ s. Solow (1993b) presented some
simulation results indicating the conditions under which the unconditional power
of this test is reasonably high.
Mark Burgman (personal communication) pointed out that, for t n fixed, the
significance level in Equation (1.9) does not fall to zero as T approaches ∞ but is
equal to F s (t n ) for T ≥ s. However, this is the conditional significance level given
S = s. Because the distribution of S under H 0 depends on T (specifically, S tends to
be larger for larger T), the unconditional probability that T n ≤ t n —which is the
unconditional significance level—does fall to zero as T approaches ∞.
Illustrative Examples
In this section, the tests outlined above are illustrated by using sighting data for
two northwest Atlantic fish species. These data were taken from the annual
autumn bottom trawl survey of Georges Bank conducted by the Northeast Fisheries Science Center (Clark 1979). These examples are for illustrative purposes
only. It is important to note that, because of the limited nature of the survey, in
these applications extinction refers to local species loss (strictly, extirpation during the autumn season).