Boundedly complete families which are not complete

S. K. Bar-Lev, D. Plachky

Research output: Contribution to journalArticlepeer-review


Completeness of a family of probability distributions implies its bounded completeness but not conversely. An example of a family which is boundedly complete but not complete was presented by Lehmann and Scheffe [5]. This appears to be the only such example quoted in the statistical literature. The purpose of this note is to provide further examples of this type. It is shown that any given family of power series distributions can be used to construct a class containing infinitely many boundedly complete, but not complete, families. Furthermore, it is shown that the family of continuous distributions(Formula presented.), is boundedly complete, but not complete, where U denotes the uniform distribution on [a, b] and {Pϑ, ϑ ∈ IR}, is a translation family generated by a distribution P0 with mean value zero, which is continuous with respect to the Lebesgue measure.

Original languageEnglish
Pages (from-to)331-336
Number of pages6
Issue number1
StatePublished - Dec 1989


  • Boundedly complete family
  • complete family

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


Dive into the research topics of 'Boundedly complete families which are not complete'. Together they form a unique fingerprint.

Cite this