FOX: RANDOM VARIABILITY AND PARAMETER ESTIMATION 



duces a complex statistical model on integrating 

 equation (3). Therefore, his statistical model 

 was given no further consideration. 



Pella and Tomlinson (1969) also mentioned 

 that the generalized production model is not 

 deterministic. They pointed out several sources 

 of error in Schaefer's finite difference approxi- 

 mation of population change and estimation pro- 

 cedure, and advanced a "least-squares" searching 

 procedure as an alternative. In doing so, how- 

 ever, apparently no consideration was given to 

 statistical implications of their technique. The 

 Pella-Tomlinson procedure integrates equation 

 (2) over the time period during which the fish- 

 ing effort is assumed constant, A(, to give 



X e 



_, I 



-IK (*) g/) (1-m) t 1-"" 



(4) 



where Po is the population size at the beginning 

 of the time period, and the upper signs applying 

 when TO < 1 and the lower when to > 1. Start- 

 ing with initial guesses of the parameter values, 

 an estimated catch history, { C, } where i = 

 1 . . . n time periods, is calculated from the known 

 fishing effort history, \ fi} , by the formula 



N . ^ ^ 

 Ci = qfi- 2 ^(Pui + Pi.i + i) • ^t,/N (5) 

 3 = 1 



where Pi„- are found from equation (4) over 

 j = 1 . . .N subintervals of each time interval /. 

 The fitting criterion, S, is computed from the 

 known catch history, \Ci], of /( time periods as 



n 

 S = 2 (C. 

 i = 1 



i = 1 ' 



(6) 



where the £; are residuals. The initial parameter 

 guesses are then modified in a searching routine 

 with their computer program GENPROD until 

 those parameter values which minimize S are 

 located. 



The statistic S is a "least-squares" criterion. 

 For the parameters of a nonlinear model which 

 minimize S to be the best least-squares estimates, 

 the residuals, ti, must: 1) be independent, 2) 



have an expected value (or mean) of zero, and 

 3) have^ constant variance (i.e., not correlated 

 with t, C„ or /,).' Consequently, the proper sta- 

 tistical model for the Pella-Tomlinson fitting 

 technique must both fulfill the three assumptions 

 and be biologically rational. It is also important 

 that the statistical model be simple, i.e., one 

 which requires no additional parameters to be 

 estimated. 



Ignoring for the moment that equation (5) 

 is an approximation, the choice of equation (6) 

 as the least-squares estimate criterion tacitly 

 assumes 



givmg 



Ci = Ci 



P, = P, + (l/qf,) . ei 



(7) 

 (8) 



where P = 



n 



P dt for ease of notation. Equa- 



tion (7), referred to hereafter as statistical 

 Model 0, is biologically tantamount to assuming 

 random variation in population size approaches 

 being infinitely great in an unexploited popula- 

 tion. This denies the concept of an environ- 

 mentally limited maximum population size or 

 "carrying capacity" which is usually a founda- 

 tion of the production model. Therefore, Model 

 assumed by Pella and Tomlinson is intrinsically 

 unattractive even though it may be a reasonable 

 approximation at intermediate exploited pop- 

 ulation levels. 



There are three simple statistical models 

 (among many) which are commonly assumed, 

 biologically reasonable, and involve calculating 

 S as a weighted sum of squares or from trans- 

 formed data. 



Model 1. Additive Error 



so 



Pi = Pi + eii (9) 



Ci = Ci + (qf,)  e,i (10) 



' Additionally, if the g, are normally distributed then 

 it can be shown that the least-squares estimates are 

 also the maximum likelihood estimates which have min- 

 imum variance as the number of data grows large — 

 hence are global best estimates (e.g., see Draper and 

 Smith, 1966). 



571 



