Truncation and Variance in Scale Mixtures

  • William C. Bauldry
  • Jaimie L. Hebert

Abstract

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 = ∞.

Keywords

Doyle 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Barlow, R E. and. F. Proschan (1975), Statistical Theory of Reliability and Life Testing: Probability Models (Holt, Rinehart, and Winston, New York).Google Scholar
  2. Harris, C. M. and Singpurwalla, N. (1968), Life Distributions Derived from Stochastic Hazard Functions, IEEE Trans. on Reliability, R-17, 70–79.CrossRefGoogle Scholar
  3. Hebert, J.L. and Seaman, J.W. Jr. (1994), The variance of a truncated mixed exponential process, Journal of Applied Probability, 31, No. 1 (In Press).MathSciNetGoogle Scholar
  4. Jewell, N. P. (1982), Mixtures of Exponential Distributions, The Annals of Statistics 10, 479–484.MathSciNetMATHCrossRefGoogle Scholar
  5. McNolty, F. (1964), Reliability Density Functions when the Failure Rate is Randomly Distributed, Sankhyā, A-26, 287–292.Google Scholar
  6. McNolty, F., Doyle, J. and Hansen, E. (1980), Properties of the Mixed Exponential Failure Process, Technometrics, 22, 555–565.MathSciNetMATHCrossRefGoogle Scholar
  7. Mullooly, J. P. (1988), The Variance of Left-Truncated Continuous Nonnegative Distributions, The American Statistician 42, 208–210.MathSciNetGoogle Scholar

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

Personalised recommendations