A general purpose unequal probability sampling plan

Smallest enclosing ball for probabilistic data

“…In order to translate our algorithm into a single-pass algorithm with a space bound even independent of n (though exponential in d), note that by Lemma 10 in both of the cases in which our algorithm operates, we only need a constant size sample of the elements in order to get a good approximation. In the first case we need to sample s = Θ( 1 ε 2 log 1 εδ ) of the locations qij = ⊥ proportional to their probabilities pij with repetition which can be done by running s independent copies of the weighted sampling algorithm by Chao [7] which is a straightforward generalization of the well-known reservoir sampling approach [36] to the weighted case; see also [10]. At the same time we also sample everything we need for the second case.…”

Atlantic bluefin tuna in the Gulf of Maine, II: precision of sampling designs in estimating seasonal abundance accounting for tuna behaviour

“…Strategic sampling of bluefin tuna will likely continue to provide insights on individual and school movements, and the application of survey designs comparable to spotter-search surveys in providing crucial information on the statistical behaviour of sighting distance as a function of school size and observer-school encounter rate as well as information required to estimate and partition spatial variance arising from aggregation and environmental heterogeneity. To this end, an adaptive survey design could be tested in the future based on unequal detection probabilities by forming a database of gridded values of environmental conditions across the GOM region (Chao 1982;Little 1983;Apostolaki et al 2003) and could be combined with a spatially explicit model of tuna population dynamics similar to ones currently developed (Bestley and Hobday 2002;Newlands 2002). Reliable and efficient monitoring strategies, capable of yielding precise estimates of abundance, will likely see increasing future application, especially in regions such as marine protected areas (Godö 1998;Buckland et al 2000;Mangel 2000;Patterson et al 2001;Hastings 2004).…”

Efficient Estimators with Simple Variance in Unequal Probability Sampling

“…For example, whether the cited schemes of Chao (1982), Sampford (1967), and Sunter (1977) have exceptional merit from the low variance perspective is largely unknown. Analytical comparisons of variances are difficult because of 1299 the complexity of the 7rkl; comparison by simulations for different populations would have to be used.…”

“…Considerable ingenuity is often evidenced in the construction. For example, Chao (1982) proposed a general scheme satisfying a and b as well as 7l"kl -7l"k7l"1 < 0 for all k =1= l, which guarantees nonnegativity of the Yates-Grundyvariance estimator. Chao started out with declaring the first n of the N listed units to be an initial sample, then admitted the possibility of adding a unit appearing later in the list at the expense of one that is already in the sample, so that requirement b is satisfied in every step; in the end, a is also satisfied.…”