152 
Fishery Bulletin 95 ( 1 ), 1997 
Table 2 
West Atlantic bluefin tuna, Thunnus thynnus, catch at age data (in numbers) used for the sequential population analysis (from 
ICC AT, 1995). 
Age 
Year 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 + 
1970 
64,886 
105,064 
127,518 
21,455 
3,677 
914 
176 
172 
535 
3,726 
1971 
62,998 
153,364 
38,360 
46,074 
672 
1,673 
2,109 
1,350 
1,133 
5,957 
1972 
45,402 
98,578 
33,762 
3,730 
3,857 
118 
569 
576 
261 
5,519 
1973 
5,105 
74,311 
30,482 
7,161 
2,132 
1,451 
953 
1,544 
555 
4,444 
1974 
55,958 
20,056 
21,094 
6,506 
3,170 
683 
916 
913 
1,081 
12,508 
1975 
43,556 
148,027 
8,328 
11,963 
821 
547 
317 
671 
1,651 
9,472 
1976 
5,412 
19,781 
72,393 
2,910 
2,899 
344 
206 
1,168 
558 
14,033 
1977 
1,274 
22,419 
9,717 
32,139 
4,946 
3,633 
957 
513 
1,109 
13,532 
1978 
5,133 
10,863 
20,015 
6,315 
10,530 
4,061 
655 
472 
341 
11,982 
1979 
2,745 
10,552 
16,288 
14,916 
3,448 
3,494 
2,612 
599 
557 
12,283 
1980 
3,160 
16,183 
11,068 
8,881 
2,866 
2,982 
5,533 
3,454 
1,061 
12,213 
1981 
6,087 
9,616 
16,541 
5,244 
6,023 
3,721 
2,884 
3,211 
2,764 
10,621 
1982 
3,528 
3,729 
1,654 
498 
342 
751 
477 
519 
896 
3,077 
1983 
4,173 
2,438 
3,268 
894 
866 
911 
1,402 
1,353 
1,039 
5,628 
1984 
868 
7,504 
1,848 
2,072 
2,077 
1,671 
594 
759 
1,091 
4,574 
1985 
568 
5,523 
12,310 
2,814 
4,329 
4,019 
1,024 
612 
698 
5,603 
1986 
563 
5,939 
7,135 
3,442 
1,128 
1,726 
931 
520 
345 
5,335 
1987 
1,513 
13,340 
9,137 
5,491 
4,385 
2,318 
1,566 
1,251 
1,014 
3,856 
1988 
4,850 
9,149 
11,745 
3,933 
4,144 
4,220 
2,258 
1,631 
1,600 
4,555 
1989 
787 
12,877 
1,679 
3,815 
1,713 
2,082 
2,677 
1,864 
1,461 
5,356 
1990 
2,368 
4,238 
17,958 
1,947 
2,747 
1,825 
1,629 
2,388 
1,522 
4,253 
1991 
3,327 
14,533 
10,761 
2,924 
1,650 
2,166 
2,347 
1,946 
1,915 
4,485 
1992 
420 
5,985 
1,997 
711 
1,425 
737 
1,916 
1,870 
1,323 
4,383 
1993 
329 
1,130 
5,215 
3,689 
2,089 
1,883 
1,598 
2,456 
1,479 
2,922 
where the notation j:n i indicates that the squared 
residuals are sorted in ascending order from j= 1 to 
n-; note that nJ2 +1 is actually an integer value equal 
to nj 2 when n i is even and equal to nJ2 +1 when n i is 
odd. Equations 2 and 3 are two different minimiza- 
tion objectives that differ in the way they treat mul- 
tiple series of relative abundance data. In Equation 
2, the trimmed sums of squared residuals are com- 
puted separately for each index and then added to 
the objective function being minimized. Thus, the 
individual indices are de facto given equal weight- 
ing. In Equation 3, the trimming is done over all avail- 
able data points, regardless of which relative abundance 
series they belong to. Thus, the LTSj formulation forces 
each available series to contribute to the objective func- 
tion, whereas the LTS 2 formulation could plausibly 
eliminate indices that fit very poorly in comparison with 
the others. An analogous distinction can be made for 
the LS fit by giving either equal weight to all available 
data series (as in Eq. 1) or by assigning weights to each 
series in proportion to their mean squared errors. The 
latter has often been accomplished by means of itera- 
tive reweighting (Powers and Restrepo, 1992) or maxi- 
mum likelihood (Punt, 1994). 
Algorithms for high-breakdown robust regression 
are notoriously computation-intensive, even in the 
simplest univariate linear regression case (Rousseeuw, 
1984; Rousseeuw and Leroy, 1987; Steele and Steiger, 
1986) . A typical algorithm for a linear robust regres- 
sion with p parameters goes like this: For a large 
number of times, s, select p data points, do a least- 
squares regression (LS) and compute the correspond- 
ing robust objective function (e.g. sum of trimmed 
squares) for the complete data set. The LTS solution 
is given by the parameter estimates and results in 
the lowest robust objective function value. In the lin- 
ear case, the value ofs is chosen such that, for a given 
fraction of data contamination and a given p, at least 
one of the s subsamples is not contaminated 
(Rousseeuw and Leroy, 1987). The choice of s in the 
nonlinear case is not clearcut. However, in the lin- 
ear case the values of s grow very rapidly with p and 
percent contamination; therefore many available al- 
gorithms set s = 3,000 for p > 9 (Rousseeuw and Leroy, 
1987) . Similar values were used here for the nonlin- 
ear case. 
Algorithms for nonlinear robust regression are 
rare, owing partly to the increased computational 
