A note on exponential dispersion models which are invariant under length-biased sampling

Shaul K. Bar-Lev, Frank A. Van Der Duyn Schouten

Research output: Contribution to journalArticlepeer-review


Length-biased sampling (LBS) situations may occur in clinical trials, reliability, queueing models, survival analysis and population studies where a proper sampling frame is absent. In such situations items are sampled at rate proportional to their "length" so that larger values of the quantity being measured are sampled with higher probabilities. More specifically, if f(x) is a p.d.f. presenting a parent population composed of non-negative valued items then the sample is practically drawn from a distribution with p.d.f. g(x) = xf(x)/E(X) describing the length-biased population. In this case the distribution associated with g is termed a length-biased distribution. In this note, we present a unified approach for characterizing exponential dispersion models which are invariant, up to translations, under various types of LBS. The approach is rather simple as it reduces such invariance problems into differential equations in terms of the derivatives of the associated variance functions.

Original languageEnglish
Pages (from-to)275-284
Number of pages10
JournalStatistics and Probability Letters
Issue number4
StatePublished - 15 Dec 2004


  • Exponential dispersion model
  • Length-biased sampling
  • Variance function

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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