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Fishery Bulletin 103(2) 



A more convenient form for computation arises after 

 changing the integration variable from the asymptotic 

 length x to time since recruitment, t-a+a , 



'-*Sf3 



(8) 



The expression (Eq. 7) then becomes 



f t il\p)^j°°p(x(T))exp(-f' ^z(y)dy)r(t-T) ^- (9) 



In the special case of constant recruitment, i.e., r(t)=l, 

 and constant mortality, z(.t)=z, f,U\p) becomes indepen- 

 dent of time as first obtained by Powell (1979). 



Maximum likelihood estimation 



Let p,,(/3) be the expected proportion of individuals in 

 the i th length class (/,_j, /) on the j ,h occasion, where 

 j=l, 2, • •• , N; and let n :] be the corresponding observed 

 numbers. The value of Pj.ifi) can be obtained from the 

 density function f t (l;P) given by Equation 2. Thus 



P,/Py- 





(10) 



in which fAl;P) is the (unnormalized) density function on 

 the y th occasion. Under a multinomial model, estimation 

 of the parameter vector fi relies on the procedure 



maximize £ n lt log Py(.fi) with respect to ft. (11) 



The sum is the log-likelihood function up to a constant 

 independent of the parameters. The probability, p , can 

 be approximated as fj(.l i+ i/2^ifj^i+V2^> wnic h is the nor- 

 malized value of the density function for thej th occasion 

 at the midpoint of the ;' th length class. 



If sampling effort is known and expected catch is as- 

 sumed to be a known function of effort and population 

 abundance, the log-likelihood function in Equation 11 

 can be easily modified to incorporate effort informa- 

 tion. For example, if the total number of individuals on 

 each occasion, n i =I. i =n ll , is assumed to follow a Poisson 

 model with overdispersion parameter v, the log-likeli- 

 hood function becomes 



sampling effort, <p is the total abundance index over all 

 occasions; andp is the expected proportion of individu- 

 als on the./" 1 occasion (i.e., the relative abundance), so 

 that (pp is the expected catch per unit of effort. In this 

 case we can obtain the maximum likelihood estimate of 

 <p as 2yi-/2.-ej7.-. The probability, p } , can be approximated 

 as EjjWj+i^V^ ,/J(/, + i/ 2 >- Here v is introduced to allow for 

 overdispersion in the Poisson model. It plays a weighting 

 role for the two terms in Equation 12, and the second 

 summation can be regarded as auxiliary information. 

 If?;, is assumed to follow a Poisson distribution exactly, 

 we have v=l. 



In our simulation and tiger prawn studies we specify 

 a case of fixed, known recruitment length, / , and fM;P) 

 is obtained from Equation 7 or 9. For definiteness we 

 set the constant of proportionality implicit in these 

 equations to one. 



The integrals in Equations 7 and 9 present some 

 subtleties for their evaluation, so that some details 

 of the numerical implementation might be of inter- 

 est. For the simulation study we used Equation 7. 

 The integral was performed on an /-dependent grid 

 of 41 and 81 quantiles of the L x distribution p(x) and 

 then improved upon by using the Richardson extrap- 

 olation. Note that there is an apparent singularity 

 at x = l. However, by decomposing the mortality into 

 a mean and deviation term, z(y)=z +z(y)— z , we find 

 that the factor involving mortality is proportional to 

 (x-lY lk . Hence the integrand is proportional to (x-lY lk , 

 and, because zlk— 1>— 1, the singularity is integrable 

 (i.e., the integral is finite). We used a quadrature scheme 

 designed for integrands of the form (x— lrf(x)£>— 1, to 

 perform the integral in the neighborhood of x = l. 



For the tiger-prawn study we used Equation 9. The 

 integral was performed on uniform grids of 41 and 81 

 points over the interval tE(0,1.5) years and, as before, 

 was improved by using the Richardson extrapolation. 

 We used our knowledge that tiger prawns live for about 

 18 months to determine the upper limit of integration. 

 Note that despite appearances, this integral contains no 

 singularity because 3c(t)-><* as t-»0), and therefore the 

 factor p(x(T))/(l-e _, ' T )^0. The effort integral within the 

 integrand was computed by linear interpolation between 

 cumulative totals of the weekly effort. 



The prototype implementation of our maximum like- 

 lihood method was written in S-plus software (Lucent 

 Technologies) by using the optimizer "nlminb." However, 

 to improve the performance for a large number of simu- 

 lations, the program was recoded in C by using Powell's 

 optimization routine with numerical derivatives (Press 

 et al., 1992). The C code and some relevant reports are 

 available on request. 



5X-]ogpj,(j8) + vX{n,logA,(0)-A//»}, (12) 



where A(/3) is the expected total number in the sample 

 on the j-th occasion and depends on effort. One way to 

 model this dependence is A / (/3) = 0p / (/3)e / , where e f is the 



Results 



Simulation studies 



We simulated length-frequency data based on the 

 recruitment pattern of tiger prawns P. esculentus in 



