## The jackknife estimator |
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Jackknife estimators are used in ecology in two situations:- mark-recapture estimation of number of animals in a closed population;
- species richness estimation for a defined assemblage.
Sobs) is often too low, as some animals/species are missed. The
raw number is thus a biased estimator. The jackknife aims to
produce unbiased estimates.
The data consist of detected/not-detected observations on a series of occasions, usually recorded as a matrix of ones and zeros, with a column for each occasion and a row for each animal or species. The box on the right shows data for 26 tigers and 10 capture occasions from a study in Kanha Tiger Reserve (Karanth et al, 2004). ## Estimating biasIf we had data from an infinite number of occasions, we would
know the true number of animals/species ( If we have data for From these, we can calculate a set of \[S_i^*=nS_{obs}-(n-1)S_{-i} \] Partial estimates and pseudo-values for the tiger data are shown in the table below:
If the bias is proportional to 1/ You may have noticed that the species or animals which drop out are those which were detected on only one occasion, called "singletons". Burnham and Overton (1979) showed that the jackknife estimator can be calculated from the number of singletons, \( f_1 \): \[ S^*=S_{obs}+\frac{n-1}{n} f_1 \] The tiger data has 10 animals caught only once, \(f_1 = 10\), so \(S^* = 35 \). The ‘drop-one-out’ method is a first-order jackknife; we can drop out more than one occasion. ## Higher-order jackknives
If we drop out \[ S_2^*=S_{obs}+\frac{2n-3}{n} f_1 - \frac{(n-2)^2}{n(n-1)} f_2 \] where \(f_2\) is the number of animals or species recorded twice; \(f_2\) = 6 for the tiger data, so we have \[S_2^*=26+\frac{2×10-3}{10} 10 - \frac{(10-2)^2}{10(10-1)} 6=26+17-4.2667=38.733\] Since a whole series of jackknife estimates are available, which should we use? Higher-order jackknives have smaller bias, but they also have larger standard errors, so there is a trade-off between bias and precision. Burnham and Overton (1979) provide a stopping rule which compares the difference between successive jackknives with the standard error, which is implemented in Program CAPTURE and the R code in the wiqid package. CAPTURE and wiqid add a further refinement: they allow for interpolation between the two best jackknife estimates. So their estimate of the number of tigers in the Kanha Reserve is 33.32, between the values from the first-order jackknife and "zero-order" jackknife, which is just \(S_{obs} \).
## ReferencesBurnham, K. P. & Overton, W. S. (1979) Robust estimation of
population size when capture probabilities vary among animals.
Karanth, K. U., Nichols, J. D., Kumar, N. S., Link, W. A., &
Hines, J. E. (2004) Tigers and their prey: Predicting carnivore
densities from prey abundance. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Updated 23 December 2013 by Mike Meredith |