Truncation and Variance in Scale Mixtures

  • William C. Bauldry
  • Jaimie L. Hebert


Let X be a continuous nonnegative random variable with density f, mean µ, and finite variance σ 2. Mullooly (1988), hereafter simply Mullooly, has shown that if f is positive on the interior of its support, \(\mathop {\lim }\limits_{x \to 0} f(x) >0\frac{\sigma }{\mu } > 1\), and \(\frac{\sigma }{\mu } >1\), then σ 2 may be increased by truncation. Denote by σ 2(t), the variance of the truncated random variable X t ≡ I(t, ∞)(X), where I A is an indicator on the set A. Specifically, Mullooly demonstrates that for densities satisfying these conditions, there exists a real number T > 0 such that σ 2(t) > σ 2 for all t∈ (0,T). We shall call T the variance inflation boundary for X t . When σ 2(t) > σ 2 for all t ∈ (0, ∞), we say that T = ∞.


Hazard Rate Random Environment Lifetime Distribution Graph Covering Scale Mixture 
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Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • William C. Bauldry
    • 1
  • Jaimie L. Hebert
    • 1
  1. 1.Dept. of Mathematical SciencesAppalachian State UniversityBooneUSA

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