274 



Fishery Bulletin 90(2). 1992 



f({n'i}|n',(qi}) = 



V I 



n 



n'i ! I . 



n qi" . 



where Z qi = 1. Estimators for {qi} and the elements 



i 



of the variance-covariance matrix of {qi} are 



n'i 



qi = — - 



To obtain a distribution for {njj}, it is convenient to 

 condition on the fixed subsamples nj. As in the case 

 of the first phase of samphng, it is assumed that n-, is 

 much less than the number in the population of that 

 length category, so that a product of multinomial 

 distributions is obtained for the second phase of 

 sampling, 



f({nij} I (n,}, {q',j}) = fl 



rin,! 



n q'ij"' 



where Z q'jj = 1 for all i. Estimators for (q'ij} and the 



j 

 elements of the variance-covariance matrix of {q'jj} 

 are 



Var(q,) = ^^X^^^M, 



While they were able to obtain estimators which cor- 

 rectly modeled the sampling procedures, they also 

 found that their exercise in theoretical rigor did not 

 result in any appreciable difference in practice. 



Estimation of mean length-at-age 



An unbiased estimate of Ij is given by 



^j = Z li qij. 



where qjj is the probability of length i given age j. An 

 expression for qjj is obtained using Bayes theorem, 



qu 



qiqi 



Z qi q'i 



A variance approximation 

 for mean length-at-age 



Because the above expression for mean length-at-age 

 is nonlinear in the observations {n'j} and (n|j}, a delta- 

 method approximation is derived. Delta-method esti- 

 mators can be algebraically complex but all have the 

 same simple structure. For mean length-at-age, a delta- 

 method approximation is given by 



Var(Tj) = djT Vdj. 



where dj is the vector of partial derivatives of Ij with 

 respect to {qj} and {q'ij}, and V is the variance-covari- 

 ance matrix of (qj} and {q'ij}. Defining 



and 



Aj = Z li qi q'ij. 

 Pj = Z qi q'ij. 



C6v(q'ij, q'hk) = 



q ij q hk 



Hi 



for i = h and zero otherwise. 



A troublesome inconsistency with this approach is 

 that the n; are assumed to be predetermined quan- 

 tities. In fact, n; is necessarily less than or equal to n'i, 

 the number in the ith length category from the first- 

 phase sampling, and n'j is a random variable that can 

 take values between zero and the min(n',Ni) where 

 Ni is the number of fish of length category i in the 

 total catch. Singh and Singh (1965) address this issue 

 while developing variance estimators for what in this 

 fisheries application would correspond to mean age. 



the elements of the vector of partial derivatives are 

 given by 



and 



3 1j ^ q'ij (liPj - Aj) 

 9 qi Pj' 



dlj ^ qj (liPj - Aj) 

 a q'ij Pj^ 



Combining these expressions with the estimators for 

 the variance-covariance matrix of qi and q'ij given 

 earlier, 



