Gaughan et al.: Distribution of Sardinops sagax off southwestern Australia 



627 



Table 3 



Parameter estimates for two models (III and IV, see text for details) of Sardinops sagax spawning biomass, including tests for 

 zero coefficients, at each of four regions in southwestern Australia. 



West coast 



SE of estimate: 15,280 



B sp 

 Intercept 5316.10 



PPS 63,192.69 



West coast 



SE of estimate: 15,330 



SEofBgp ((32) 



4826.58 1.10 



23,161.29 2.73 



P-level 

 0.278929 

 0.010251 



PPS 



B SP 



84,615.52 



12,615.76 



r(33l 

 6.71 



P-level 

 1.22E-07 



zero and one, and A was also close to 1, model II was 

 able to be recast in a more tractable form by using the 

 Taylor series expansion of the RHS of model II about 

 PPS = 1, leading to the relationship 



iB .sp.pps»/= <x PPS, + S+e t , 



(3) 



where the expected value of 8 is approximately 

 -0.25a„(A-l). Details of the derivation are provided in 

 Appendix 1. 



Fitting the regression model III to the DEPM-based 

 estimates of B sp gave the estimated coefficients shown 

 in the left hand column of Table 3. Residual diagnostics 

 showed that model III was satisfactory. Because none 

 of the intercept terms were significantly different from 

 zero, the parsimonious model 



{B SP-pps ) t = a (PPS) t + e t 



(4l 



was fitted, giving the results in the right hand column 

 of Table 3. Residual diagnostics were also satisfactory 

 for these models. 



Optimal estimation of spawning biomass 



We now have available two unbiased estimates of B sp : 

 estimator 1 (i.e., B sp _ DEPM ) with associated error e'. 



which has an expected value of and variance Var(e') 

 = rjj 2 ; and estimator 2 (i.e., B sppps ) which model IV 

 of the previous section fitted to the values of B sp _ DEPM 

 with error e, which had an expected value of and 

 variance Var(e) = o~ 2 . Thus estimator 2 can be seen to 

 be unbiased and with full error term (e+e). In order to 

 obtain an optimal predictor, i.e., with minimum vari- 

 ance, of spawning biomass *>B sp „„„,„/), we considered 

 the weighted average of the two estimators above: 



Bsp-Optimai = (( "'' estimator 1 + (1-wO estimator 2), 

 with weight w: 0<«'<1. 



We must choose the weight w of estimator 1 in order 

 to minimize the variance ( Var(B sp „,„„„/)) of the esti- 

 mator B. S p. 0p ,„„ o/ . 



Var(S sp.o P ,„„ n /» = Var(B»e+(l-w)(e+e')) 

 = Varie +{l-w)e I 

 = Var(e) + ( l-u;) 2 Var(e') + 2( 1-w i 

 covariance(e.e') 



= a 2 +il-w)' 2 a l 2 +2(l-wtc><7 1 p 



where p = correlation between e and e'. For Var (B sp 

 Optimal t° be a minimum, the w derivative must be zero, 

 yielding 



