LO: MORTALITY RATES OF NORTHERN ANCHOVY 



l « n p^ = 



i = 2 



k 



n (SKAc-i) - £W 



i = 2 



S(«i) - 5(20) 



k 



I TV, 



i-2 



and 



ln(L)= I [JVi ln(P,)] = I [JV,- ln(P ? )] + I [N t ln(P t )] 



i = 2 



i = 2 



i = c + l 



(10) 



where A/", = m(y i _ 1 - y t ) = ra-r^ and c is max(i) for ^ < 3 d (u^. Substituting Equation 

 (9) for P, in Equation (10) yields 



ln(L) = 1 ty ln(e-" f .-i - e~ at ) + S N l {-au 1 + p In Mj) + ty ln(^ - ^-P) 



1 NAn 



i = 2 



g-<"l _ g-«™, 



'20W 



Mi 



(11) 



Solving simultaneous equations = and for a and /? gives their 



MLEs. 



da 



8< 



The asymptotic variance-co variance (ASVAR-COV) of MLEs of a and ft was computed 

 according to Kulldorff (1961, p. 86-87): 



As var(a) 



As cov(a,(i) As var(/J) 



= (UN) 



3,P 



a 



li 



3-21 a 22 



= (UN) A s3 



s,P 



«,p 



(12) 



For detailed derivation of the MLEs, see the Appendix. 



Conceptually, abundance declines monotonically with increasing age, but this may not 

 occur in the sample. Although its absence does not complicate regression analysis, cor- 

 rections are required when the MLEs are used. The MLEs are functions of sample totals 

 fy = (Hi-i - y~i)m, Ni > 0, and can also be expressed as function of sample proportions 

 NJN (Equations (Al) and (A2)), which are equal to the ratios of differences of sam- 

 ple mean daily productions (^_j - y^K^i - Vk)- n ^ n ( see Appendix). The quantity n^ = 

 NJm is the sample mean daily death between two adjacent groups. The MLEs require 

 N { > 0. Due to sampling error, it is possible to observe more individuals in the older 

 group than the adjacent younger group, i.e., y i _ x < y { . If so, some adjacent groups 



401 



