536 



Fishery Bulletin 100(3) 



The mode for empirical r^, values in our study was always 

 larger than -M so that a^ was larger than one. 



Maximum likelihood estimates of the parameters a and 

 ji for each stock (conditional on the assumption y=-M) 

 were obtained iteratively in a spreadsheet by maximum 

 likelihood with observed r^ values as data. In simulations, 

 we used the simpler method of moments to calculate gam- 

 ma distribution parameters. Given y= -M and estimates 

 of the mean (jj) and variance (a~) for r^ values from real 

 data, the method of moments solves the two equations 

 ^=(Ba+y} and &-= a/3- for the two unknowns a and fi. In 

 particular {i=a'^l iii-y) and a=&^/ P'^. The maximum likeli- 

 hood approach and method of moments gave similar pa- 

 rameter estimates for real data sets (Figs. 2-3) suggesting 

 that the method of moments was acceptable for simula- 

 tions. In what comes later, maximum likelihood parameter 

 estimates from data (a, j3, y= -M) are distinguished from 

 simulated values calculated by the method of moments 

 (dj, P^,y=-M). 



In each simulation run, the assumed "true" value of 

 ^MSY^ was used to calculate T\ = 2F^^,jy . In simulation 

 runs with no process error the simple calculation r^^= r^ 

 was used. In runs with process error, r^, was the mean of the 

 distribution of stochastic r,^, values (see below). 



For runs with independent process errors (no autocor- 

 relation I, annual logistic parameter values r^ ^, were drawn 

 from a gamma distribution with parameters (a, fi, y) cal- 

 culated by the method of moments from the mean r^ and 

 variance cr^ estimated from empirical data. 



average parameter ( G) was 



2p 



(2) 



with 6 in the range (0,1). The random numbers d^^ in 

 algorithm 1 were drawn from a gamma distribution with 

 parameters: 



- (1-i-r) 



a, =a, 5" 



A. = A-*'"'' 



(i+r) 



(3) 



(4) 



and y=-M. The adjusted parameter values (d,, P^, y) make 

 the mean and variance of autocorrelated r^ ^, values from 

 algorithm 1 the same as for an independent series drawn 

 from a gamma distribution with parameters a^,p^,y (see 

 Appendix 1). 



Algorithm 2 was for autocorrelations (p) in the range 

 (0.5,1): 



Z^»o-i-. 



J=l 



(5) 



Appendix 2— Autocorrelated process errors 



This appendix describes two algorithms for generating 

 autocorrelated production process errors from gamma 

 distributions. The algorithms are based on first-order 

 autoregressive and moving average error structures used 

 in time-series analysis (Nelson, 1973). The shapes of origi- 

 nal uncorrelated gamma distributions and new, correlated 

 probability distributions for both algorithms appeared 

 identical in plots. 



Algorithm 1 was for autocorrelations (p) in the range 

 zero to 0.5: 



's.v ~ "s,V +"'^s,y-l' 



(1) 



where r^^= was the logistic population growth parameter 

 for yeary in simulation run s, and the moving 



where the random numbers d^ ,, were drawn from gamma 

 distributions with parameters d,^ = a.^JL,L an integer >3, 

 ?>sr- A^r^' ^■^'^ y- '^he adjusted parameter values make the 

 mean and variance of the autocorrelated and independent 

 r^ ^, values the same. 



The autocorrelation in algorithm 2 is p = (L-l)IL. For 

 simulations, we chose the smallest value of L that gave 

 an autocorrelation that was at least as large as the value 

 desired. Extremely high autocorrelations (e.g. p>0.9) can 

 give dj,, = O-^Jl^ ^ 1 when a^^ is near L. This was a minor 

 problem in some cases because gamma distributions for 

 rfj.^., and autocorrelated values of r^^,, have no mode when 

 dj^ < 1. We avoided this, where necessary, by setting the 

 maximum value of L to 10 (p=0. 90) and constraining dj,^,< 1 

 in simulations. 



