The Shapes of a Probability Density Function and Its Hazard Function

  • Masaaki Sibuya


The shape of a probability density function restricts that of its hazard function, and vice versa. We are concerned with the six specific ‘shapes’: increasing, decreasing, unimodal, anti-unimodal, increasing-decreasing-increasing, and decreasing-increasing-decreasing.


Probability Density Function Hazard Function Distribution Range Finite Interval Discontinuous Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Efron, B. and Johnstone, I.M. (1990) “Fisher’s information in terms of the hazard rate”, Annals of Statistics, 18, 38–62.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Lawless, J.F. (1982) Statistical Models and Methods for Lifetime Data, New York, NY: Wiley.zbMATHGoogle Scholar
  3. Sibuya, M. (1984) “ Unimodal hazard functions,” Japanese Journal of Applied Statistics, 13, 115–124. (in Japanese)CrossRefGoogle Scholar
  4. Sibuya, M. (1994) “ Unimodal and anti-unimodal hazard function,” (in preparation).Google Scholar
  5. Zelterman, D. (1992) “ A statistical distribution with an unbounded hazard function and its application to a theory from demography,” Biometrics, 48, 807–818.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Masaaki Sibuya
    • 1
  1. 1.Department of MathematicsKeio UniversityYokohamaJapan 223

Personalised recommendations