Properties of the elasticity of a continuous random variable. A special look at its behavior and speed of change

ABSTRACT

Belzunce et al. (1995) define the elasticity for non negative random variables as the reversed proportional failure rate (RPFR). Veres-Ferrer and Pavía (2012, 2014b) interpret it in economic terms, extending its definition to variables that can also take negative values, and briefly present the role of elasticity in characterizing probability distributions. This paper highlights a set of properties demonstrated by elasticity, which shows many similar properties to the reverse hazard function. This paper pays particular attention to studying the increase/decrease and the speed of change of the elasticity function. These are important properties because of the characterizing role of elasticity, which makes it possible to introduce our hypotheses and knowledge about the random process in a more meaningful and intuitive way. As a general rule, it is observed the need for distinguishing between positive and negative areas of the support.

KEYWORDS:

• Elasticity function
• Hazard function
• Reversed hazard function
• Model comparison
• Statistical characterization
• Fisher information
• Shannon entropy.

Acknowledgments

The authors wish to thank two anonymous referees for their valuable comments and suggestions and Marie Hodkinson for translating into English the text of the paper. This research has been supported by the Spanish Ministry of Economics and Competitiveness through the project CSO2013-43054-R.

Notes

1 The r(x) function has received different names in the literature. It has been called reverse hazard rate (Chechile, 2011), quotient function (Veres-Ferrer and Pavía, 2012) and reverse failure rate (Oliveira and Torrado, 2015).

2 A proof of this can be found in Theorem 12 in Chechile (2011).