Bottom Trawl Surveys - Proceedings of a Workshop Held at Ottawa ...

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163
APPENDIX 1: FIND AN UNBIASED VARIANCE ESTIMATOR
FOR VAR(W3).
1 n
Var (113) Var { n+ 1 }; xi }
_L
n+l
2
i=l
n
Var ( L xi)
n
i=1
1 2 }; Var(xi)
ii+I i=l
2
n.cr , assuming
Xi ••• , Xn are i.i.d. random
variables
Therefore, the variance estimator, in order
to be unbiased, must have an expected value of
[ntr] 2
n
• n.cr2. Note that the estimator
2 = }; (xi-x)2 has an expected value equal to
cr i=l ---
n-1
cr2 (Mood et al. 1974). Therefore, an unbiased
estimator of Var(W3) would be
Var(W3) ·= n
n
(n+1)2
l:
i=l