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Sampling with probabilities proportional to the variable of interest

Author

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  • J. J. A. Moors
  • R. Smeets
  • F. W. M. Boekema

Abstract

To estimate the mean sojourn time, a sample of Tilburg fair visitors was asked for the duration of their stay on the fair grounds. The longer a visitor's sojourn, the larger his/her probability of being interviewed will be; therefore, longer sojourn times will be overrepresented in the sample. As a consequence, the arithmetic sample mean is not a good estimator. The paper places this problem against a theoretical background. Sampling with unequal probabilities is considered in a general context. The special case that the sampling probabilities are a function of the variable under investigation, is discussed in detail. As a better estimator the harmonic mean of the observations is presented. Most properties of this estimator are difficult to derive analytically, but a suitable variance estimator is derived. The behavior of estimator and variance estimator is studied in a number of quite different examples.

Suggested Citation

  • J. J. A. Moors & R. Smeets & F. W. M. Boekema, 1998. "Sampling with probabilities proportional to the variable of interest," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 52(2), pages 129-140, June.
    • Handle: RePEc:bla:stanee:v:52:y:1998:i:2:p:129-140
    DOI: 10.1111/1467-9574.00073
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