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Fishery Bulletin 93(2). 1995 



nally obtained from Equation 7. Finally, to complete 

 the bootstrap replication procedures we made two 

 final calculations; the mean of the lower quartile ob- 

 tained from the n -fitted pseudo- values of the m boot- 

 strap replicates, and the mean-fitted survey abun- 

 dance index for each year. These are given as 





and 



i " 

 — 1 v-i -* 



(16) 



(17) 



(=i 



This resampling process is conditioned on the set of 

 residuals from a fitted model to the observed data, 

 and no distributional assumption is made concern- 

 ing the structure of the error. The sample size of re- 

 siduals for the example given here (25 to 30) should 

 be adequate to characterize the tails of the underly- 

 ing error distribution, and bootstrap estimates of the 

 mean and lower quartile should have converged suf- 

 ficiently as the number of resampled replicates m 

 was performed 1,000 times. As such, the mean and 

 variances generated by these new realizations of the 

 time series for both the fitted index and the refer- 

 ence point, as well as the shape of their parent dis- 

 tributions, provide the necessary information for 

 making inferences about the population. Further, this 

 approach to generating variances and confidence in- 

 tervals is particularly useful because explicit solu- 

 tions to the "normal equations" cannot be derived 

 because of the nonlinear nature of the equations rep- 

 resenting the underlying population process 

 (Rawlings, 1988). 



Making inferences 



Decisions made by fishery managers are often based 

 on the state of the resource relative to a chosen man- 

 agement target or reference point. One question that 

 forms the basis of management action is the follow- 

 ing: Given the uncertainty in both the index of abun- 

 dance and the value of the chosen reference point, 

 what is the probability that the index in year t is 

 less than or equal to the reference point? Statements 

 of probability and inferences regarding the state of 

 the resource can be formulated by using the boot- 

 strap generated data in the following manner: Let 

 the fitted index for any particular year ( y t ) and the 

 lower quartile ( q ) generated from m bootstrap rep- 

 lications be random variates Yj and Y 2 , respectively. 

 In addition, assume that Y, and Y 2 are jointly con- 

 tinuous random variates with the density function 



f(yj,y 2 )- In practice, we want to determine the 

 P(Yj<Y 2 ), that is, the probability that Y r which takes 

 of the value ofy ; , is less than Y 9 , taking on the value 

 of y. ? . This probability is computed as 



P{Y 1 <Y 2 ) = ^ \ y y{y 1 ,y 2 )dy l 



dy 2 



(18) 



Further, we may wish to determine the probability 

 of Yj being less than Y 9 for each possible value of y 2 

 in its domain and sum the resulting probabilities 

 giving the cumulative probability distribution func- 

 tion. Thus, for any possible value of the reference 

 point, q , we can determine the probability that the 

 fitted index, y t , lies below the value of the lower 

 quartile for all possible values. By analogy, this can 

 be extended to consider the joint probability that two 

 consecutive years of the fitted survey abundance in- 

 dices lie below the reference point. This approach 

 takes into account the uncertainty in both the value 

 of the fitted survey index for any given year (or two 

 consecutive years) and the reference point to which 

 the population level is compared. 



An example 



The Atlantic wolffish, Anarhichas lupus, is a cold 

 water, bottom-dwelling species distributed from the 

 Newfoundland banks to Nantucket (Bigelow and 

 Schroeder, 1953). Little is known about the biology 

 of wolffish in the western Gulf of Maine and Georges 

 Bank region. Catches of wolffish in research surveys 

 are low owing to its rather sedentary behavior and 

 small, localized populations. 



In U.S. waters, wolffish are taken primarily as 

 bycatch in a mixed groundfish fishery and in other 

 large mesh otter trawl fisheries. Commercial land- 

 ings of wolffish in the Gulf of Maine-Georges Bank 

 region averaged only about 220 metric tons (t) be- 

 fore 1970, after which they increased by nearly a fac- 

 tor of six; from 200 t in 1970 to 1,300 t in 1984 (Fig. 

 3). After peaking in 1984, commercial landings have 

 steadily declined by about 100 to 200 t per year and 

 reached 500 t in 1990, the lowest in nearly a decade. 

 NEFSC spring survey abundance indices have shown 

 a consistent downward trend, particularly since the 

 early 1980's (Fig. 3). 



Although no formal definition of overfishing pres- 

 ently exists for wolffish, this resource is presently 

 considered overexploited and depleted on the basis 

 of declining trends in commercial landings and sur- 

 vey indices (NEFSC, 1993). While this may intu- 

 itively be the correct conclusion regarding the sta- 

 tus of the resource, there is little guidance in terms 

 of its present level relative to its long-term abundance 

 or to an appropriate reference point for fisheries 



