Abstract
We consider two sample survey situations. In one, we have a list of ‘selection units’ (su's) out of which one may suitably draw a sample. But the interest is to estimate the total value of a variable defined on an unknown number of individuals called ‘observational units’ (ou's) which are not directly identifiable but may be contacted through the above su's to “one or more” of which each ‘ou’ is ‘linked’ in a ‘well-defined’ manner. Each collection of ou's so linked is called a ‘network’. Each network is ‘disjoint’ from every other and together they exhaust all the ou's of interest. Through a sample of su's one may observe the variate-values for the ou's in the networks linked to the sampled su's. This approach of reaching samples of ou's through such networks is called ‘network’ sampling. Thompson (1990, 1992) and Thompson and Seber (1996) have given theories of estimation of total or mean and variance estimation when the su's are selected by simple random sampling (SRS) without replacement (WOR) or by stratified SRSWOR methods. Here we extend to general sampling schemes with unequal probabilities.
In the second kind of surveys an initial sample is chosen from a list of identifiable units for many of which the value of a variable is ‘zero’ and only for a few ‘unknown and unidentified’ units the values are positive. The purpose is to estimate the population total with a high coverage of the positive-valued units in the sample. A concept of a ‘neighbourhood’ is then well defined such that given a unit with a positive value some of those in its neighbourhood may also hopefully be positive-valued. So, from an initial sample one may go on ‘adaptively’ extending it by adding successively to each positive-valued unit in the sample each unit in its neighbourhood and stopping only on encountering a ‘neighbouring’ unit that is zero-valued. The relevant theory is developed in the above locations cited. But this covers only SRS with and without replacement (WR and WOR) for the initial sample. We extend here to more general sampling schemes.
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