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Fishery Bulletin 90(2). 1992 



et al. 1971, Doubleday 1975, Ulltang 1977, Pope and 

 Shepherd 1982). Sullivan et al. (1990) suggest expand- 

 ing the number of terms in the sum of squares to in- 

 clude effort and abundance indices if meaningful 

 weights for these auxiliary data can be found (guidance 

 for finding such weights is not provided). The statistical 

 characteristics of the estimator are not yet described. 



The lack of Monte Carlo tests of the performance of 

 these estimators is a particular concern because, 

 without knowledge of their statistical behavior, little 

 certainty can be placed on the resulting estimates. 

 Some of the estimators were developed by the likeli- 

 hood method, but the justification for assuming the 

 chosen sampling distributions often seems weak or 

 lacking. The usefulness of those estimators that require 

 total fishing effort seem limited, since that statistic is 

 often estimated from catch and effort samples rather 

 than enumerated. Most of the methods estimate the 

 parameters of individual growth as part of the solution 

 vector. This seems questionable in view of findings in 

 a study of the separation of central moments of indi- 

 vidual distributions from distribution mixtures (Has- 

 selbald 1966), studies of the magnitude of correlation 

 between estimates of growth-equation parameters 

 (Gallucci and Quinn 1979), and of the performance of 

 methods that estimate growth parameters from size 

 distributions (Hampton and Majkowski 1987, Rosen- 

 berg and Beddington 1987, Basson et al. 1988). Also, 

 most of the methods are based on elaborate population 

 models, a characteristic that leads to two problems. 

 First, such models often include deterministic stock- 

 recruitment functions, and such functions are regarded 

 as unrealistic representations of the dynamics of fish 

 stocks. Second, since the population model is extensive, 

 it includes a large number of parameters that must be 

 estimated. It is well known that an exact representa- 

 tion of a real- world system is not possible; hence, a 

 suitably parsimonious model that is a useful approx- 

 imation with an informative structure is superior (Box 

 1979). The most germane variables are the current size- 

 specific abundances since they will determine stock pro- 

 duction in the immediate future. 



The object of this study was to develop an abundance 

 estimator that would be appropriate in almost all cases, 

 whether or not the population is composed of cohorts, 

 or whether or not age data is available. Effort was 

 taken to write the estimation model as parsimonious 

 as possible, to base estimation on data commonly 

 collected from most fisheries, and to insure that the 

 correct sampling distribution was used in the likeli- 

 hood procedure. The bulk of the study was directed at 

 describing the statistical characteristics of the esti- 

 mator over a broad range of conditions from Monte 

 Carlo simulations. 



Methods 



Abundance estimator 



An abimdance estimator was developed that uses a 

 model of individual growth, size-specific catches and 

 catch dates, and size-specific abundance observations 

 (sighting data, research cruise catch-per-tow, etc.). The 

 estimator makes three assumptions: 



(1) Unobserved phenomena that change stock abun- 

 dance (immigration, emigration, unrecorded catch, 

 predation, and disease) are a (continuous) Poisson 

 process with combined rate z, 



(2) the size of an individual on a date is a known deter- 

 ministic function of size on another date, and 



(3) the sample average of relative abundance obser- 

 vations is a normally-distributed random variable 

 with an expectation equal to a portion of absolute 

 abundance. 



The estimator uses a growth model to relate sizes and 

 dates and an abundance model to project abundance 

 from observed catches scaled to relative abundance 

 observations. 



Consider T time-periods, not necessarily of equal 

 duration, so that 0<t<T. Within period t, relative 

 abundance was observed on date yt , then a catch oc- 

 curred on date q . The number of fish caught on date 

 Ct was Ct. Abimdance on the date of the relative abun- 

 dance observation (date yt) is of interest; let this abun- 

 dance (numbers of fish) be Nj. From assumption (1), 



Nt+i = [Nt e-^'tlc.-y.) _ Q] e-^(i'fi-<H). 



Abundance on the date of the final abundance sample 

 (i.e., Nt) is of most interest because stock production 

 in the immediate future depends on it. Writing the 

 above equation in terms of Nj as a time-series gives 

 a simple forward projection of abundance on each 

 relative abundance sampling date: 



XT XT ^''kb'k^i-yk) -r- ^ Zk(Ck-yk)+^2i(yi»i-yi) 

 Nt = N^e'" -^ 2. Cue 



k = t 



If the unobserved change rate is assumed temporally 

 invariant, this simplifies to 



T-l 



Nt = Nt e'^'^T-yt) -i- 2! Ck e^<'^k-y,). 



k = t 



Each catch is subtracted separately; catching is not 

 assumed to occur continuously at a constant rate. 

 Abundance changes due to unobserved events are, 

 however, assumed to occur continuously at a constant 

 rate. 



