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Fishery Bulletin 100(2) 



Pollock, K. H., C. M. Jones, and T. L. Brown. 



1994. Angler survey methods and their applications in fish- 

 eries management. Am. Fish. Soc. Special Publication 2.5, 

 Bethesda, MD, 371 p. 



Sober, G. A. F. 



1982. The estimation of animal abundance and related 

 parameters, second ed. Griffin, London, 6.54 p. 

 Sokal, R. R.,andF.J. Rohlf 



1995. Biometry, third ed. Freeman and Company, New 

 York, NY, 887 p. 



Thompson. S. K. 



1992. Sampling. John Wiley and Sons, New York, NY', .343 p. 



Appendix 1 : The covariance of two estimators 

 from sample means 



First we consider the covariance of two total estimates. 



Let X, and Yj be simple random samples (i=l n) from 



a population of size N with mean //^ and ;/^, and A' and Y 

 be two sample means. 



Cochran (1977. p .2.5) derived the covariance of two- 

 sample mean, that is 



Cov( X. y ) = ^^^!— ^ - Cov( X, y ) 



N n 



N-n 



N 



tAY^X -^ )(Y -u ). 



This is estimated by 



Cov(x, y ) = ^^-^ - Cov( X, y 



N n 

 > (A, -X){Y, -Y 



Nnin- 



W, = C,tv, + w, (C,-C,) + C,iw,-IU,). 



From Taylor's series (mentioned above), the approximate 

 covariance is obtained. 



Cov( w' " , w;.' ' ) = £( w; ■" - w; -^ ' ) ( w'. ' -w:?') 



-£[^,(Cl'^'-Cr') + C-'([Z7,. 



A^ 





Here Q''|' and iZij , are independent, and both Wk and u'a; are 

 estimated from different samples. Therefore Cov (QVilij. ) = 

 Cov(w,,Cj:^ = Covi w,,w,. ) = 0, then we get the covariance 

 as only the first term. 



Appendix 3: Approximate variance of ^ 



Taylor's series of R with respect to C and M is obtained 



by" 



^ = — -F— (C-C)--^(M-M). 

 MM M- 



Then the approximate variance is obtained by 



V(i?) = £;(^-i?)-"- J-V(C) + -£-V'(M) 



M' M' 



-'iS-CoviCM). 



The covariance between two population total estimators 

 is defined by 



Cov(X,y ) = CmtNX.NY) = EiNX - N^, HNY - N/u^. ) 

 ., -^ NiN-n) — 



= Af-Cov(X,y)= cov(X.y). 



n 

 This is estimated by 



Appendix 4: Covariance between C'lf' and C, 

 By expanding C,.'' and C",''' we get 



CI'' ' = Dp,R, + DR, ( p, - p, ) + Dp, ( i?, - «, ). 

 Q'^' = Dp,R, + DR,Ap,. -p,) + Dp.AR,. - R, 



(d) 

 k 



c7viX,Y) = ^^^^^^^^±iX,-XnY,-Y). 



For the monthly total catches, we get 



c^(cr',c-' ) = ^'^";'' X ic,, - ^, )(c„. - 1 



nin - 1) ■'— ' 



Appendix 2: Approximate covariance between 



Wl"and Wl" 



Taylor's series of W^ with respect to the random variables 

 is obtained by 



then the approximate covariance is given by 



Cov(c;,'",c;'" ) = £(c;.'" - Dp,,/?,, kc;;" - Dp,yR,, 



= D- 



R,M,Covi p, , p, I + /?,p, Cov( p, .R,) 

 +p,M,XoviR,.p,, ) + p,p, Cov(i?,.i?,. I 



If the first three covariance components are ecjual to 

 because of independent sampling, then we have 



Cov(c;;",c;;" i = £»-p,,p,,Cov( /?,.,/?, ). 



Here we can write R/. and Rf. as the ratio of two random 

 variables from Equation 1 by 



