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An Exact Formula for the Probability That Two Specified Sampling Units Will Occur in a Sample Drawn with Unequal Probabilities and without Replacement

Connor, W. S.

Journal of the American Statistical Association, 1966-06, Vol.61 (314), p.384-390 [Periódico revisado por pares]

Taylor & Francis Group

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  • Título:
    An Exact Formula for the Probability That Two Specified Sampling Units Will Occur in a Sample Drawn with Unequal Probabilities and without Replacement
  • Autor: Connor, W. S.
  • Assuntos: Approximation ; Probabilities
  • É parte de: Journal of the American Statistical Association, 1966-06, Vol.61 (314), p.384-390
  • Descrição: Given a population of N units, it is required to draw a sample of n distinct units in such a way that the probability π i , (i = 1, ..., N) for the ith unit to be in the sample is proportional to its 'size' x i . One way to achieve this is as follows: The N units in the population are listed in a random order and their x i are cumulated and a systematic selection of n elements from a "random start" is then made on the cumulation. The mathematical theory associated with this procedure has been presented in [1], where with the help of asymptotic theory, compact expressions for the variance of the estimate of the population total are derived. These expressions contain probabilities P ii ′ (i ≠ i′ i, i′ = 1, ..., N) that units i and i′ both are in the sample. For n = 2, N = 3 and n = 2, N = 4, exact formulae are derived in [1], but for other n and N only approximate formulae are obtained. It is the purpose of this paper to derive an exact formula for P ii ′ for any values of n and N. The argument makes use of the solution given in [1] for the case n = 2 and N = 4. The new formula for n = 2 is derived in Section 1 and illustrated in Section 2 by a numerical example for N = 10. An exact formula for general n is derived in Section 3 and examplified in Section 4.
  • Editor: Taylor & Francis Group
  • Idioma: Inglês
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  • Realização: ABCD-USP USP   Logos de Redes Sociais: Freepik

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