Prager: A nonequilibrium surplus-production model 



379 



gear for finding or catching the fish is introduced. In 

 such cases, the formulation represented by Equations 

 11 and 12 can be used to estimate different 

 catchability coefficients for segments of a single time 

 series. In formulating such a model, the time seg- 

 ments would be treated as separate fisheries, each 

 having nonzero catch and effort data only during its 

 respective time period. Each additional time segment 

 would add one additional parameter to the model. 



A common concern is determining whether the 

 improvement in fit obtained from a more complex 

 model is statistically significant. A production model 

 with added catchability parameters can be tested 

 against the simpler model (with one estimated q) with 

 a standard F-ratio test. (Here F refers to the F dis- 

 tribution of statistics, not to fishing mortality rate.) 

 The test statistic F* is 



F* 



(SSE s -SSE c )/ t ; 1 

 SSE c /t; 2 



(13) 



where SSE s and SSE c are the error sums of squares 

 of the simple and complex models, respectively; v l is 

 the difference in number of estimated parameters 

 between the two models; and v 2 is the number of data 

 points less the total number of estimated parameters. 

 The significance probability of F* can be obtained 

 from tables of the F-distribution with v x and v 2 de- 

 grees of freedom. As pointed out by a referee, this is 

 equivalent to to a likelihood-ratio test assuming log- 

 normal error structure, which is implicit in using the 

 SSE from log-transformed data. Because of the pos- 

 sibility of specification error, any such significance 

 test must be considered approximate. 



A nonparametric test of the null hypothesis q x = 

 q 2 can also be conducted by examining a bias-cor- 

 rected confidence interval on the ratio of the two 

 catchability coefficients. (Construction of bias-cor- 

 rected confidence intervals is described later.) As an 

 example, assume that the alternative hypothesis is 

 q x * q 2 . The null hypothesis would be rejected at 

 P<0.05 if a 95% confidence interval on the ratio q x I 

 q 2 did not include the value 1.0. Like the F-test, this 

 test is approximate because of the possibility of speci- 

 fication error. 



In other cases, catchability is thought to vary in 

 more subtle ways than the step function just sug- 

 gested (Paloheimo and Dickie, 1964; Gulland, 1975; 

 MacCall, 1976; Peterman and Steer, 1981; Winters 

 and Wheeler, 1985), and one could incorporate any 

 number of catchability models into the estimation 

 framework. It would be straightforward to model a 

 linear trend (increase or decrease) in catchability 

 with time. This could be parameterized by estimat- 

 ing the first and last years' values of q and generat- 

 ing intermediate years' values by linear interpola- 



tion, so that only one additional parameter would be 

 estimated. One could also add some form of density- 

 dependent catchability model with a minimal cost 

 in terms of number of parameters estimated; the 

 foundation of such an approach was presented by Fox 

 (1975). However, it might prove difficult to distin- 

 guish varying catchability from trends in biomass 

 itself. If so, the use of external estimates or indices 

 of biomass, as explained above, might be especially 

 valuable. 



Bootstrap estimates of bias and variability 



The bootstrap (Efron, 1982; Stine, 1990) is a sample 

 reuse technique often used to estimate sampling vari- 

 ances, confidence intervals, bias, and similar prop- 

 erties of statistics, including parameter estimates. 

 Major advantages of the bootstrap, compared to al- 

 ternative methods (such as those based on the infor- 

 mation matrix), are its flexibility and relative free- 

 dom from distributional assumptions. A minor draw- 

 back is that it demands a great deal of computer time. 



Bootstrapping is often performed by resampling 

 the original observations. However, in fitting non- 

 equilibrium production models, the order of the catch- 

 effort pairs is as significant as the data themselves. 

 For time-series models (in the broad sense), Efron 

 and Tibshirani (1986) describe a method, used here, 

 that preserves the original time structure of the data. 

 For each bootstrap trial (of which there may be 250 

 to several thousand), a set of synthetic observations 

 is constructed by combining the ordered predictions 

 from the original fit with residuals chosen at ran- 

 dom (with replacement) from the set of residuals from 

 the original fit. The model is then refit to this set of 

 synthetic observations. 



The bootstrap can be used to estimate bias in pa- 

 rameter estimates. The median estimation bias B g 

 in a parameter 6 is estimated by 



Be -Q m -0, 



(13a) 



where is the conventional estimator of 6, and 8 m 

 is the median value of 6 obtained from the bootstrap 

 trials (Efron,1982; Efron and Gong, 1983). A bias- 

 corrected estimator 6 BC of a parameter 6 can there- 

 fore be given by 







BC 



e-B, 



'6  



(13b) 



It appears that the median bias correction, rather 

 than a mean correction, has been adopted in the 

 bootstrapping literature because a mean correction 

 (which would be expected to produce an "unbiased" 

 estimate in the usual sense) can have extremely high 

 variance (Hinkley, 1978). The resulting problems are 



