Andrew and Chen: Estimating size structure and mean size of Haliotis rubra 
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with the above density function, E(x ) - a + a and A is 
a constant defined by a, a, and T. For a discrete vari- 
able X with the interval width d, constant A can be 
approximately estimated from 
N _ (x,~a) 
a J V = h 
where 
A T T -a 
N = + 1. 
d 
Solving this equation, we have 
1 
A = 
N _ ( x, -a ) 
i=i 
The lower bound (parameter a) is the size limit of 
115 mm and is the same for all diver-days. Para- 
meter T ranges from 117 to 150 mm among diver- 
days. For a diver-day i, parameter T. was determined 
by randomly drawing an integer between 117 and 
150 mm on the basis of the uniform distribution. For 
diver-day i, the mean size s- was randomly selected 
from the frequency distribution of mean size esti- 
mated from data gathered in 1993-94 (Fig. 20. The 
error a for diver-day i is estimated as cr = s. — a. 
Thus, the probability of abalone in length interval j 
being caught on diver-day i, P* i ., was calculated. The 
number of abalone in length interval j caught in 
diver-day i, C ; ., was then calculated as 
Evaluation of the simulation 
The size-frequency distribution and mean size of 
abalone at harvest calculated from the total catch of 
all diver-days were used as the “true” population of 
landed catch. Different sample sizes were used for 
each sampling scenario and compared to this known 
population. For each sampling scenario, 100 simula- 
tion runs were conducted. An error index, after mea- 
suring the difference between length-frequency dis- 
tributions calculated from the catch of all diver-days 
in a year and from the sampled catch, was calcu- 
lated as 
100 j~N 
Error index = h ~ l k ~ x * 
100 
where T k is the frequency of abalone in length inter- 
val k calculated from the total catch of all diver-days. 
This population was fixed among runs h calculated 
from the total catch of all diver-days, and O h k is the 
frequency of abalone in length interval k in simulation 
run h calculated from the sampled catch. Thus the er- 
ror index provides an index of the summed deviations 
from the true population across all size classes. 
The difference between mean sizes of abalone es- 
timated from the catch of all diver-days and from 
the sampled catch was evaluated by using an index 
defined as 
100 
X K-v| 
Average absolute difference in mean = h=l ’ 
where M h is the estimated mean of the h th simula- 
tion run for the sample catch and M is the mean of 
the total catch (i.e. the true mean size). The distri- 
bution of the calculated difference in means for 100 
simulation runs was used to evaluate the variation 
in estimated average difference in mean size. 
Results 
We concentrated on results of simulations appropri- 
ate to estimating the size structure and mean size of 
abalone for the smaller spatial scale — that of zones 
or groups of zones. Results will be presented across 
a range of sample sizes for zones or groups of zones 
with up to 600 diver-days per year. We briefly dis- 
cuss results for the whole fishery by sampling 100 
diver-days in a fishery of 4,000 diver-days under sce- 
nario 3. Probabilities of detecting changes in mean 
size of abalone under this scheme are provided. 
Under scenario 1, in which all abalone caught in a 
randomly selected diver-day are measured, there was 
a sharp decline in the error index of size composition 
as the number of diver-days sampled increased from 
1 to 10 (Fig. 4). After 10 diver-days, the rate of de- 
cline in the error index slowed markedly. As expected, 
as the number of diver-days sampled approached the 
number of diver-days in the fishery, the error ap- 
proached zero (Fig. 4). There was little variation in 
the error index among fisheries with 400 to 600 diver- 
days per year ( Fig. 4 ). A similar pattern was observed 
in comparing differences in the mean size between 
the sampled catch and the total population for all 
sizes of the fishery (Fig. 4). For example, irrespec- 
tive of the number of diver-days per year, when 10 
diver-days were sampled, the average difference in 
mean size was 5 mm over 100 simulation runs. 
Under scenario 2, diver-days were ignored as a 
source of variation and all abalone caught during the 
year had an equal probability of being sampled. This 
