Page 416 - Contributed Paper Session (CPS) - Volume 6

CPS2007 Jai-Hua Yen et al.
which is formed by singleton species at the same time. Accordingly, we have
the expected sampled doubleton richness:
( ) ≈ { [(1 − ) + × (1 − )] × (− × )}
2
2
1 1
+ { × × × (1 − ) × × (− × )}.
1

,
where denotes the number of sampling unit. By solving the two equations
above, we have the singleton and doubleton richness adjustment:
1
1 = (1−̅ ̂ ×̂)(−̅ ̂ ×̂)′ (4)
and
1
2 − 1 ×̅ ̂ ×̂×(1− )× 1 ×(−̅ ̂ ×̂)

,
2 = (1−̅ ̂ ×̂)×(−̅ ̂ ×̂) . (5)
However, the estimation of traditional Chao2 estimator will be inaccurate
even though and are asymptoticly unbiased. It causes the value of
2
1
2
1 overestimated. Hence, we choose first-order Jackknife and Chao2 richness
2 2
estimator as the theoretical foundation of deriving the adjusted richness
estimator. We propose an adjusted richness estimator by Taylor series
2
expansion of ( 1 ) by the mean and . Then we get the difference
1
2
2 2
between [( 2 )] 2 and ( 1 2 ) to have the adjust term:
(2 2 ) 2 2
2
1 2 [( )] 2 ̂( ) ( )̂( , ) [( )] ̂( )
1
2
1
1
1
2
1
( ) ≈ + − + ,
2 2 (2 ) 2( ) [( )] 2 2[( )] 3
2
2
2
2

where ̂( , ) = − 1 2 , ̂( ) = (1 − ). Therefore, we have the

1

2
̂
̂
adjusted richness estimator:
2
1
1
1
̂ = , + −1 {( 2 2 − 2 2 − 2 2 2 ) , 0}. (6)

2
When 0 ≤ 2 ≤ 1, by simulation studies, the adjusted richness estimator will
be:
̂ = , + −1 . (7)
1

3. Result
a. Simulation Results
To test the performance of the adjusted richness estimator, we presented
the simulation results by several species detection models and different
settings of number of sampling units. We fixed = 40 and = 100. 500
simulation data sets were generated and 200 bootstrapping trials were
conducted by each simulation data. The bootstrapping method is
regenerating , , 1′ and by binomial distribution independently in
2
order to increase the estimated standard error while the traditional
bootstrapping method usually underestimates the standard error in this case.
In true method, the estimation of species richness used the traditional Chao2
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