NOTE Lai: Age-length key to estimate age composition of fish population 



383 



Var(pj) = variance of p, ; 



Cov(p J? p k ) = covariance between Pj and p k . 



A caret denotes the estimate of each variable. 



The unbiased estimate of p (Tanaka 1953) from an 

 ALKis 



A i. A A , , , 



The variance of Pj has been derived by Tanaka (1953) 

 and Kimura (1977); however, the approximate form of 

 variance is more frequently used (Kutkuhn 1963, 

 Southward 1963, Doubleday & Rivard 1983, Lai 1987): 



Vbr(p,)= X 



7o A , , A . r ) . A A ,.-> 



N 



(2) 



The terms in the right-hand side of Eq. 2 represent 

 the portion of the total variance due to variation within 

 length-strata and that due to variation between strata, 

 respectively. 



The covariance of p and p\ is derived using the 

 method of Kimura ( 1977): 



A , . A . A A A 9 A A , A A A A A 



AA L-l (I- l)q n , l -l-q q , l I q q p p 



p,p.)=£ — i — ' V| "'* + X-—^- + X - H '->;'* -qpi 



Cov(p /Pk 



=i n 



y, a a y A A A A 



l -I- a a , l I q q pp. 



<=i rc <=i N N 



The approximate form of covariance omits the first 

 term because this term is small compared with the 

 sum of the other two terms. A quadratic loss function 

 (Jinn et al. 1987) is used to infer the precision of the 

 estimated age composition p'=(pi,p 2 ,...,p A ): 



Ap,p)= (p-p) 'W(fi-p) 



= X w Var (p)+I«) Cov ( p , p, 



r a A 

 = E I w (p-p 



+ E 



JW.ipr-pp^-p,) 



.(4) 



The loss function presented in Eq. 4 is identical to 

 Kimura's Vartot provided that W is an identity ma- 

 trix, i.e., w n =1 and w lk -0 for j*k. 



Substituting Eq. 2 and the approximate form of Eq. 

 3 into Eq. 4 and collecting terms, we obtain: 



i / a,-u,\ b+v-m 

 i=i \ n. I N 



(5) 



LA A A A 



v = 51 S^ jk l.qij q ik - 



i=l j *k 



A A . 



m ^u^PjPk, 



and where a, u„ b, m, and v are all positive. 



A linear cost function is used for the optimal sam- 

 pling design: 



C - c,N + Y.c 2l n it 



(6) 



where C is total cost, q is per-unit cost of collecting a 

 random length sample, and c 2 , is per-unit cost for age- 

 ing a fish in the ith length-stratum. 



Survey designs generally are based on two 

 constraints: (i) a fixed total cost, i.e., minimize the 

 loss function in Eq. 4 at a fixed cost; or (ii) a desired 

 precision of the estimators, i.e., minimize the total cost 

 at a given level of the loss function. Therefore, the 

 problem for optimal allocation becomes one of deter- 

 mining the optimal set of N' and n,"'s which minimizes 

 L at a given total cost or which minimizes total cost at 

 a desired precision level of L (N* and n,* are the opti- 

 mal sample sizes of length and age samples, respec- 

 tively). Kendall & Stuart (1977, Sect. 39.20) and 

 Cochran (1977, Sect. 5.5) show that choosing the opti- 

 mal set of N* and n,"s to minimize L for a fixed C or to 

 minimize C for a fixed L are both equivalent to mini- 

 mizing the product of L and C: 



1C 



l a,-u, b + v-m 

 I— - + 



,=1 n, N 



C;N + X c 2 ,n, 



(7) 



Applying the Cauchy-Schwarz inequality to Eq. 7, the 

 product LC is 



J1C 



1 1 | a.-u, V ' I b+v-m \* 



Al—U.. N 



L 



5 



(Vc,A0 2 +X(Vc 2 a) 2 



i=l 



XVc 2 ,-(a,-u,) + ^lb+v- 



»i 



Kendall & Stuart (1977) showed that the minimum 

 value of the product LC occurs when 



Vc 21 ra! 



a,-«, 



^c 2l n, 



a-u, 



Vc 2L n L Vc~^V~ 



a,-u, 



b+v-m 



N 



= constant>0 

 (8) 



where a, = Xm^ Pq,j (l-q\j), 

 A A > A A 



L A 



b = XS^l/q.j-p,,) 2 , 

 1=1 j =i 



Use the terms of the equality between the ith and the 

 (L+l)th terms and rearrange the variables to obtain 

 r^n^/N*, the optimal subsampling ratio between age 

 subsamples and length samples in the ith length stra- 

 tum. The solution of r<" is: 



