The Exponential Power-Law Process for the Nonhomogeneous Poisson Process and Bathtub Data

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

Research output: Contribution to journalArticlepeer-review


Recently, Bar-Lev, Bshouty and Van der Duyn Schouten [Math. Methods Stat. 25 (2016) 79-980] developed a systematic method, called operator-based intensity function, for constructing huge classes of nonmonotonic intensity functions (convex or concave) for the nonhomogeneous Poisson process, all of which are suitable for modeling bathtub data. Each class is parametrized by several parameters (as scale and shape parameters) in addition to the operator index parameter n . For the sake of demonstration only, we focus in this paper on a special subclass called the exponential power law process (EXPLP(n)) whose base function is the intensity function of the power-law process. We describe various properties of such a subclass and use one of its special case, namely EXPLP(1) intensity function, to analyze failure data which lack monotonicity. Maximum likelihood estimation of the parameters involved and relevant functions thereof is discussed with respect various aspects as existence, uniqueness, asymptotic behavior and statistical inference facets. Using two real datasets from the literature we provide evidence that the EXPLP(1) intensity function is well suited to analyze data which exhibit a bathtub behavior.

Original languageEnglish
Article number2050011
JournalInternational Journal of Reliability, Quality and Safety Engineering
Issue number4
StatePublished - 1 Aug 2020

Bibliographical note

Publisher Copyright:
© 2020 World Scientific Publishing Company.


  • Bathtub data
  • concave or convex intensity function
  • exponential power-law process
  • nonhomogeneous poisson process

ASJC Scopus subject areas

  • General Computer Science
  • Nuclear Energy and Engineering
  • Safety, Risk, Reliability and Quality
  • Aerospace Engineering
  • Energy Engineering and Power Technology
  • Industrial and Manufacturing Engineering
  • Electrical and Electronic Engineering


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