296 



Fishery Bulletin 93(2), 1995 



Pennington ( 1985, 1986) noted similar problems and 

 concluded that the relative shortness of the time se- 

 ries of trawl data makes reliable estimation of the 

 moving average parameter difficult. Although 

 Pennington suggested using a value ranging from 

 0.3 to 0.4 appropriate for many stocks, we used the 

 value of 0.50 because an analysis of the residuals 

 from the model indicated that they were normally 

 distributed (P>0. 10). 



The fitted estimates of abundance over the entire 

 time series produce an index whose variance is con- 

 siderably less than the variance of the observed se- 

 ries (Pennington, 1986). A comparison of a fitted in- 

 dex to the reference level (i.e. the lower quartile or 

 25 th percentile) derived from the fitted indices should 

 provide a reasonable evaluation of the stock's status 

 because these reflect "true" trends in population 

 abundance from only process variability (the effect 

 of survey variance has been reduced). The fitted in- 

 dices indicate that wolffish abundance has declined 

 since the early 1980's and that by 1990 the index 

 had fallen below the reference point (lower quartile= 

 -1 .90 ) to -2.2 ( Fig. 4 ). In a hypothetical sense, if man- 

 agers of this resource considered the lower quartile 

 of the fitted indices a reasonable reference point, the 

 1990 index might have triggered some action, al- 

 though it could be argued that the downward trend 



itself might be cause for concern. However, given the 

 relative closeness of the fitted value of the 1990 in- 

 dex (-2.2) to the reference point (-1.9), as well as the 

 uncertainty in both values, a logical question to ask 

 is: What is the probability that the fitted 1990 index 

 lies below the reference point? 



Probability statements addressing this question 

 can be derived from the parent distributions of both 

 the fitted indices and reference points from the 1,000 

 bootstrap replications (Fig. 5). For the wolffish ex- 

 ample, each of these distributions appear log-nor- 

 mally distributed, the 1990 fitted index exhibiting 

 slightly more dispersion about its mean (as would 

 be expected) compared with the reference point (Fig. 

 5). Both the fitted index and lower quartile means 

 are nearly identical to the expected values (computed 

 from the initial integrated moving average model fit), 

 indicating little or no bias and uncorrelated model 

 residual errors. Thus, for the time series on wolffish, 

 an a priori integrated moving average model specifi- 

 cation appears appropriate to describe the underly- 

 ing population process. 



For any value of the lower quartile we can state 

 the probability that the fitted 1990 index lies below 

 that value of the reference point using a discrete 

 approximation of Equation 18. This simply repre- 

 sents the area integrated under the joint density 



250 



200 



L? 150 



c 



100 



50 



Wolffish spring survey 



 1990 Index 



 Lower quartile 



ll 



-2.8 



-2.6 



-22 



-2.0 



Indices (log e [mean number per tow]) 



Figure 5 



Comparison of empirical distributions of the 1990 spring survey index predicted by an 

 integrated moving average (IMA) model fitted to 1,000 bootstrapped generated realiza- 

 tions of the survey time series 1968-92 and the lower quartile (25th percentile) of those 

 predicted indices. The 1,000 realizations of the time series were generated by sampling 

 with replacement the IMA model residual errors and randomly adding these to the 

 predicted survey indices. 



