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HjPk PjPk HjHk where p.j = E(Pj) denotes the average capture probability for the /th sample. Researchers in fishery sciences have suggested that correlation bias due to heterogeneity could be reduced if two different sampling schemes were used (for example, trapping and then resighting, or netting and then angling). This was justified by Seber (reference [10], p. 86); it also could be seen from formula (6b) because there is almost no covariance between the distributions for two distinct samplings.

The CCV for more than two samples can be similarly defined and interpreted. For example, the CCV for samples ki,k2,...,k„ in a random-effect model is defined as

E[(Pkl - nki )(Ph - fik ) ■ ■ • (Pkm - Hkm )]

The CCV for the general cases with two types of dependencies has been developed [53], but it will not be addressed here. We only remark that all CCVs in the general cases measure the overall effect of the two types of dependencies.

For two lists, the usual independence assumption is equivalent to setting the two-sample CCV at 0 (y12 = 0). It is not possible to model dependence between two lists as we discussed in Section 3.2. For the three-list cases, there are seven observable categories (as in the HAV data in Table II) and eight parameters: N\ n2\712', V13; 723; 7i23- One constraint is still needed, yet it is possible to model dependence. Consequently, at least three samples are required to reasonably estimate any dependence parameters.

The concept of sample coverage was originally proposed by Turing and Good [55]. This concept has played an important role in the classical species estimation for heterogeneous communities [56] and has been modified for multiple-sample cases [53], in which the sample coverage is used as a measure of overlap fraction. The basic idea is that the sample coverage can be well estimated even in the presence of two types of dependencies. Thus an estimate of population size can be derived via the relationship between the population size and the sample coverage. Chao and Tsay [53] dealt mainly with the three-sample case. Extension to cases with more than three samples was provided by Tsay and Chao [54]. Below we will separately summarize the estimation procedures for the three-list case and the general case.

If an additional case were selected from the third list, then a proper overlapping measure would be the conditional probability of finding this case that had already been identified in the combined list of the other two sources (that is, finding a case i for which Xlt + Xi2 > 0). The overlap fraction can be quantified as ]Tfli P^liX^ + Xi2 Pn,- Considering that this additional individual could be selected from any of the three lists, we define the sample coverage as the average of the three possible overlap fractions as follows:

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