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Fishery Bulletin 90(2|. 1992 



and 



if b = a: I Cb „ dw = 



; 



V2(b'-a')fo+(a'-a)(^a+i-fe) + «b 



or 



; 



if b < a: Ck w dw = 



V2(a+l-a')(<a + (a'-a)(ca+i-«a) + ^a+i)+ • • • 

 . . .V2(b'-b)(2ft + (b'-b)Ub+i-<b)), 



where rjs = 2(Nt,s-Nt,s+i + Nt,s+2-Nt,s+3+ • • • ±Nt,s. 

 and Cs = 2(Ck,s-Ck,s+i + Ck.s+2-Ck.s+3+ • • • ±Ck.s- 



On a sampling date, r measures are recorded and 

 sample mean calculated for each size class: 



Yt,s = I 



Yt,s,k 



According to assumption (3), the expectation of relative 

 abundance is 



E[Yt.s] =/[/3|Y,C] = qsNt.s, 



where p contains the sampling-gear efficiency coeffi- 

 cients (the Qs), the unobserved change rate (z), and the 

 abundance of each size-class on date yx (the Nt,s)- 

 Nt,s is as defined by (2), Y indicates a matrix of 

 relative abundance observations and C catches. Since 

 it is a mean, clearly 



Yt.s~N /[/}|Y,C], 



o2[Yt,3]\ 



(assumption 3). This implies the likelihood. 



L(/5) = n (2")"''^ o[Yt,s]-' e-v^(Y,,-/[PIY,ci)'^+<,^[Y,,i_ 

 t,s 



where n is the product of the number of size-classes 



and sampling dates. Maximizing its logarithm (constant 

 terms ignored), 



lI(Y,s-/[/?|Y,C])2^oii[Yt,3] 



(4) 



with respect to p yields maximum-likelihood estimates 

 of the Qs, the Nx.s, and z. Maximization was achieved 

 by minimizing the negative of (4) by the "Marquardt" 

 method (Morrison 1960, Marquardt 1963, Conway et 

 al. 1970, Gallant 1975, Press et al. 1986). 



This estimator is equivalent to common least-squares 

 if size and date variances are equal, but that restric- 

 tion seems unlikely. Since 



o'-[Y,J = Nt,3- Var[qJ, 



(5) 



abundance is the dominant term. Abundance is depen- 

 dent on reproductive success and a mortality history. 

 Both are time-variant, so an assumption of equal 

 variances is inappropriate. 



This abundance estimator possesses few restrictions. 

 Relative-abundance measures and catches can occur on 

 any date. Any number of catches, or none at all, can 

 occur between relative abundance samples or visa 

 versa. The period of data collection may be short; the 

 time-series may be brief. Individual growth can follow 

 any form. Most important, recruitment to the exploited 

 stock can occur continuously so that breeding (spawn- 

 ing) and birth (hatching) need not happen just once dur- 

 ing each period. Reproduction may be continuous so 

 age-specific cohorts need not exist. This estimator is 

 not a cohort analysis, but it uses similar data. 



Monte Carlo tests 



Each test was designed to collect a history of estimator 

 performance over many applications of the method in 

 similar circumstances. Each test was composed of 

 several trials. On each trial, a new exploited popula- 

 tion was simulated, followed by relative abundance 

 sampling, growth rate estimation, and catch estima- 

 tion. Next, p was estimated by (4) from the data col- 

 lected in the second step. Last, estimation error for 

 each element of p was calculated. The familiar mea- 

 sure of error, e = (ji-P), where /? is the vector of 

 population parameters estimated by p, was not ap- 

 propriate because p changed from one simulation to the 

 next. Error was measured by the sufficient statistic 

 E. = p^p. The bias of each element of p was estimated 

 as the average € over the n trials (Monte Carlo sam- 

 ples). If a particular estimate was unbiased, then 

 /i[€] = 1 for that parameter. The estimated error vari- 

 ance of each parameter, s2[E], was also calculated. 



