FISHERY BULLETIN: VOL. 74, NO. 3 



APPENDIX 



Estimators II and III are based on probabilities 

 of not sighting daily units of fishing effort (i.e., 

 vessel days which were on grounds or on grounds 

 and fishing). These estimates require the calcula- 

 tion of these probabilities during some time period 

 when reported days on grounds and days fished are 

 available which can be correlated with sighted 

 days on grounds, sighted days fished, and fleet size 

 as determined from surveillance overflight data. 

 These estimators were fashioned by considering 

 the possible daily events, constructing the 

 probability space in units of vessel days from 

 reported and observed effort, and then deriving 

 needed probabilities from the constructed space. 

 Let the possible events be symbolized: 



G = the daily event of a vessel on the grounds; 

 F = the daily event of a vessel on the grounds 



and fishing; 

 E = the daily event of a vessel not on the 



grounds, i.e., elsewhere; 

 S = the daily event of a unit of daily effort 



observed on overflights; and 

 N = the daily event of a unit of daily effort not 



observed on overflights. 



Further, by defining the fleet size during some 

 time period Af , where t is in days, as the number of 

 vessels that were present at some time during that 

 period, the total event space (which is the sum of 

 all possible events) is easily calculated: 



n = Vlt 



where n = the total number of all possible 

 daily events, 



It = the time period in days, and 

 V = the fleet size. 



Even though certain cells in the event space (of 

 little consequence to us) cannot be observed di- 

 rectly, the possible events defined in units of vessel 

 days may be broken down: 



From effort reported to ICNAF and from that 

 observed on overflights, the number of daily 

 events in each cell is easily defined. The number of 

 events of vessel days fished (/) are reported to 

 ICNAF and are either observed on overflights (/') 

 or are not seen on overflights (/"). The number of 

 events of vessel days on grounds (g) may be 

 reported to ICNAF and are either observed on 

 overflights {g') or not observed on overflights {g"). 

 The event of a vessel day on grounds spent not 

 fishing (o) is either observed (o') or not observed 

 (o"). It is important to note that if a vessel day on 

 grounds was fished but was observed on an over- 

 flight as not in the fishing mode, which may 

 possibly occur if fishing operations were not 

 initiated until late in the day after the flight 

 occurred, that event would be incorrectly categor- 

 ized as o' rather than as /'. The effect of this 

 occurrence on estimator III will be discussed; it has 

 no effect on estimator II. If a possible daily event 

 were not on the grounds, then it was elsewhere (e) 

 and was not observed {e"). It was not possible for a 

 vessel day elsewhere to be observed; overflights 

 were directed within the fishing grounds so that e' 

 is zero in all cases. The numbers of daily events, 

 then, are categorized as on the grounds (g) and 

 fishing (/) or not fishing (o), or as elsewhere (e). 

 The numbers of daily events in each category are 

 symbolized by a single prime ( ') if observed on 

 overflights, as a double prime (") if not observed, 

 and without a symbol if the value is a total number 

 of events. 



Estimator II is best explained by considering 

 the probability of a day on grounds: 



''PiG) = PiG,S) + PiG^ 



= P{G,S) + P{G/N)PiN). 



From the above event space, the probabilities of 

 on grounds and observed [Pr{G,S)], of on grounds 

 if not observed [Pr(G/AO], and of not observed 

 [P{N)], are easily defined: 



PiG,S) = g'^n, 

 P{G/N) = g" ^ {n - g'), and 

 P{N) = {n - g') H- n. 



Item Observed (S) 



On the grounds (G) g' 



Fishing (F) /' 



Not fishing <>' = (/' -f" 



Elsewhere (E) f" = zero 



Total o' 



Then by substitution 

 P{G) = {g'^n) + [{g - g') ^ {n - g')\{n - g') ^ n 



®For an explanation of probabilities and the theorems used in 

 the development of these expressions, see Hoel (1962:4-17). 



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