Helser and Hayes: Quantitative management advice based on stock abundance indices 



293 



4and(l-0) 2 =4- 

 at at 



(8) 



Therefore, fitting the model (Eq. 7) to the observed 

 survey abundance indices provides an estimate of 8 

 and from Equation 8, an estimate of o 2 Ja 2 c . Pen- 

 nington (1986) notes that this approach has several 

 advantages over using the raw indices in that: 1 ) the 

 resulting model variance is more precisely estimated, 

 as survey variance is affected by varying catchability 

 from year to year; and 2) relevant information con- 

 tained in the other years of the survey is used in 

 estimates for a particular year. Further, the fitted 

 survey series is considered to be more precise than 

 the original series (Pennington, 1985, 1986; Fogarty 

 et al., 1986). At this point the fitted index, with suf- 

 ficient length of time, may be used to characterize 

 trends in abundance relative to a chosen reference 

 point. An estimate of the forecast variance of the fit- 

 ted time series can be calculated (Box and Jenkins, 

 1976), although the reference point (i.e. interquartile) 

 is deterministic. Further, if the time series is short, 

 a correct specification and estimation of the model 

 parameters will be difficult and parametric estima- 

 tion of the variance uncertain. To overcome these 

 constraints, nonparametric methods with boot- 

 strapping techniques (Efron, 1982) were used to es- 

 timate the variances of the fitted index and the ref- 

 erence point as well as to determine the shape of their 

 parent distributions. This approach is particularly 

 useful in making inferences between the observed 

 population level as estimated by the fitted index and 

 the reference point, within a probablistic framework. 



Bootstrap procedure 



Once a maximum likelihood estimate of the inte- 

 grated moving average parameter, 8, has been ob- 

 tained from Equation 7, "fitted" estimates of the sur- 

 vey population abundance, y t , with known residual 

 errors are available at equally spaced points in time, 

 such that 



a-B)\ny t =(1-8)0, = a t . 



(9) 



where Var(a t ) = Var[(l-8)c t ] = (l-8) 2 a 2 = a a 2 (see Eq. 

 8). The variance of y t is given by 



transformed survey abundance estimates. For the ith 

 bootstrap replicate, n values of the model residuals 

 were randomly selected with replacement (redefined 

 as c' t ) and added to the predicted abundance esti- 

 mates (y t ) to obtain n new pseudo- values y*. Thus, a 

 particular realization with the same underlying pro- 

 cess was generated, where 



Vt = Vt + 



fief 



(11) 



Assuming that Equation 7 provides an adequate rep- 

 resentation of the actual population levels of the time 

 series, it can be shown from Equations 8—11 that the 

 bootstrap generated realizations take on the random 

 component because of measurement error, since (from 

 Eq. 8) 



v[Wt 



8a 2 = a 2 . 



(12) 



Conceptually, this random resampling of the residu- 

 als mimicked a hypothetical resampling of the en- 

 tire time series of abundance estimates with random 

 variation generated from measurement error super- 

 imposed on the underlying process variation (i.e. 

 variation in population levels). Random sampling of 

 residuals and generation of n new time-series pseudo- 

 values were repeated m times (i.e. m bootstrap rep- 

 licates were performed). For each i th bootstrap repli- 

 cate, the prespecified integrated moving average 

 model in Equation 7 was again fitted to the n new 

 pseudo-values of the time series by using the same 

 moving average parameter estimate, 8. The n new 

 pseudo-values of the times series and the new fitted 

 values for the m bootstrap replicates are given as 



and 



hV^.yv-,^ 





(13) 



(14) 



respectively. The lower quartile corresponding to each 

 of the n new pseudo- values of the m bootstrap repli- 

 cated time series as 



9,>9,,9, 



(15) 



V(.y t ) = a< 



2\ 



al 



(10) 



We applied bootstrapping procedures (Efron, 1982) 

 to the vector of residual errors generated by the in- 

 tegrated moving average model (Eq. 7) applied to log- 



Rather than obtaining new estimates of 6 for the 

 model fitting to each bootstrap replicate, we followed 

 Pennington's ( 1985) suggestion that, given the large 

 variability inherent in marine trawl surveys, a pre- 

 liminary estimate of 8 between 0.3 and 0.4 appears 

 to be an appropriate value for estimating an abun- 

 dance index, and we set the value of 8 to that origi- 



