Estimating the number of shared species by a jackknife procedure

Abstract

A sequence of jackknife estimators is developed to estimate the number of shared species in two communities. The estimators have simple and explicit formulae. A sequential testing criterion is also developed to determine a proper order for these jackknife estimators. The performance of the estimators is evaluated using empirical data on two forests from Malaysia, where 209 shared species present in both forests, and using simulated data. Results for the empirical data and simulated scenarios (for sampling fraction ranging from 0.5 to 20 %) show that the jackknife estimator, compared with other existing estimators, has a smaller bias and provides more reliable interval estimation in most cases. Additionally, two avian datasets from Taiwan and Hong Kong are used to demonstrate the proposed method. To extend the proposed method to three communities, we also list the first six orders of the jackknife estimators explicitly.

Appendix: A general result of the jackknife estimators \(\hat{S}_k \)

Define a 2-dimensional array of coefficients $d_{t,u}$ as:

$$\begin{aligned} \left\{ {{\begin{array}{ll} {d_{1,1} =1} &{} \\ {d_{t,t} =-td_{t-1,t-1}} &{} {\forall t\ge 2;} \\ {d_{t,1} =2^{t}-1} &{} {\forall t\ge 2;} \\ {d_{t,u} =d_{t-1,u} +u\left( {d_{t-1,u} -d_{t-1,u-1}} \right) } &{} {\forall t\ge 2\hbox { and }2\le u<t} \\ {d_{t,u} =0} &{} {\hbox {otherwise}.} \\ \end{array}}} \right. \end{aligned}$$

These coefficients are used to simplify the expressions of jackknife estimators. The formulae can be summarized with the following Theorem.

Theorem 1

For each nonnegative integer $v$, we have:

$$\begin{aligned} \hat{S}_{2\nu ,X}= & {} D+\sum _{t=1}^{\nu +1} {d_{\nu +1,t} \frac{n_1 -t}{n_1}f_{t+} +} \sum _{u=1}^\nu {d_{\nu ,u} \frac{n_2 -u}{n_2}f_{+u}} \nonumber \\&+\,\sum _{t=1}^{\nu +1} {\sum _{u=1}^\nu {d_{\nu +1,t} d_{\nu ,u} \frac{(n_1 -t)(n_2 -u)}{n_1 n_2}f_{tu}}} \end{aligned}$$

(7)

and

$$\begin{aligned} \hat{S}_{2\nu ,Y}= & {} D+\sum _{t=1}^\nu {d_{\nu ,t} \frac{n_1 -t}{n_1}f_{t+}} + \sum _{u=1}^{\nu +1} {d_{\nu +1,u} \frac{n_2 -u}{n_2}f_{+u}} \nonumber \\&+\,\sum _{t=1}^\nu {\sum _{u=1}^{\nu +1} {d_{\nu ,t} d_{\nu +1,u} \frac{(n_1 -t)(n_2 -u)}{n_1 n_2}f_{tu}}} . \end{aligned}$$

(8)

Therefore, $\hat{S}_{2\nu +1} =(n_1 \hat{S}_{2\nu ,X} +n_2 \hat{S}_{2\nu ,Y})/(n_1 +n_2)$ is a linear combination of the frequencies $f_{tu}$. Furthermore, the $(2\nu +2)$-th order jackknife estimator is:

$$\begin{aligned} \hat{S}_{2\nu +2}= & {} D+\sum _{t=1}^{\nu +1} {d_{\nu +1,t} \frac{n_1 -t}{n_1}f_{t+}} + \sum _{u=1}^{\nu +1} {d_{\nu +1,u} \frac{n_2 -u}{n_2}f_{+u}} \nonumber \\&+\,\sum _{t=1}^{\nu +1} {\sum _{u=1}^{\nu +1} {d_{\nu +1,t} d_{\nu +1,u} \frac{(n_1 -t)(n_2 -u)}{n_1 n_2}f_{tu}}} . \end{aligned}$$

(9)

The proof is established by mathematical induction and is shown in the Supplementary Materials due to lengthy algebra. We can further simplify the formulae in the next Corollary.

Corollary 1

When the sample sizes $n_1$ and $n_2$ are sufficiently large, define $\lambda _j =(n_j -h)/(n_1 +n_2)$ for any finite number $h$ and $j=1,2$. Asymptotically, the explicit forms of the jackknife estimators $\hat{S}_k$ for $k=1,\ldots ,6$, are as follows:

$$\begin{aligned} \hat{S}_1= & {} D+\lambda _1 f_{1+} +\lambda _2 f_{+1} ; \\ \hat{S}_2= & {} D+f_{1+} +f_{+1} +f_{11} ; \\ \hat{S}_3= & {} D+(1+2\lambda _1)f_{1+} -2\lambda _1 f_{2+} +(1+2\lambda _2)f_{+1} -2\lambda _2 f_{+2} \\&+\,3f_{11} -2\lambda _1 f_{12} -2\lambda _1 \lambda _2 f_{21} ; \\ \hat{S}_4= & {} D+3f_{1+} -2f_{2+} +3f_{+1} -2f_{+2} +9f_{11} -6f_{12} -6f_{21} +4f_{22} ; \\ \hat{S}_5= & {} D+(3+4\lambda _1)f_{1+} -2(1+5\lambda _1)f_{2+} +6\lambda _1 f_{3+} \\&+\,(3+4\lambda _2)f_{+1} -2(1+5\lambda _2)f_{+2} +6\lambda _2 f_{+3} \\&+\,21f_{11} +(22\lambda _1 -36)f_{12} -(22\lambda _2 -36)f_{21} +24f_{22} \\&+\,18\lambda _1 f_{31} +18\lambda _2 f_{13} -12\lambda _1 f_{32} -12\lambda _2 f_{23} ; \\ \hat{S}_6= & {} D+7f_{1+} -12f_{2+} +6f_{3+} +7f_{+1} -12f_{+2} +6f_{+3} \\&+\,49f_{11} -84f_{12} +42f_{13} -84f_{21} +144f_{22} -72f_{23} \\&+\,42f_{31} -72f_{32} +36f_{33} . \end{aligned}$$