532 



Fishery Bulletin 100(3) 



o 



CO 



o 



O 

 O 



d 



Mean 



[- 



Gamma 

 Bootstrap 



" Normal 



— Lognormal 



Q10% 



0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 



Relative biomass (BtJK) 



Figure 10 



Mean, median, mode, Qi,,,;. Qy,,.;. and Qj,^,, ten year stock-rebuilding 

 time isopleths for Georges Bank yellowtail flounder, simulated with 

 model type 3 (no uncertainty about F^j^y and uncorrelated r^ values). 

 Stochastic r^, values were from a gamma distribution (same as Figs. 

 6 and 8), normal distribution, lognormal distribution, or bootstrap of 

 observed r values. All distributions had the same mean and variance. 



be modified to include negative or extreme values not 

 evident in short observed time series, and most can give 

 the same mean, variance and autocorrelation levels as 

 estimated from available data. However, as in the case of 

 Georges Bank yellowtail flounder and cowcod, the shape 

 of theoretical and observed distributions may not match 

 closely (Figs. 2-3). 



In comparing theoretical and observed distributions 

 for model parameters (e.g. Figs. 2-3), it is important to 

 remember that most observed distributions are based on 

 relatively few observations (Table 2). Furthermore, ob- 

 served values may be autocorrelated (p=0.33 for Georges 

 Bank yellowtail flounder and p=0.94 for cowcod rockfish. 

 Tables 2-3). High levels of autocorrelation reduce the 



"effective" number of observations dramatically so that 

 observed values may provide a poor estimate of the shape 

 of their distribution (Bartlett, 1946; Bayley and Ham- 

 mersley, 1946). For example. Equation 16 in Bayley and 

 Hammersley, with estimated autocorrelations (lags of 

 1-13 years) for Georges Bank yellowtail flounder, gives an 

 effective sample size n*=Yl (compared to ;!=25 r^, values). 

 For cowcod rockfish (with autocorrelations for lags 1-17), 

 n*=l\ (compared to n=47 r^, values). Thus, autocorrela- 

 tion in r^, values may reduce the effective sample size and 

 information used to estimated the shape, mean and vari- 

 ance of statistical distributions for r^, values by about 32% 

 for Georges Bank yellowtail flounder and 77% for cowcod 

 rockfish. 



