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Quantile-Based Reliability Analysis pp 235–279Cite as

Birkhäuser

Nonmonotone Hazard Quantile Functions

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Abstract

The existence of nonmonotonic hazard rates was recognized from the study of human mortality three centuries ago. Among such hazard rates, ones with bathtub or upside-down bathtub shape have received considerable attention during the last five decades. Several models have been suggested to represent lifetimes possessing bathtub-shaped hazard rates. In this chapter, we review the existing results and also discuss some new models based on quantile functions. We discuss separately bathtub-shaped distributions with two parameters, three parameters, and then more flexible families. Among the two-parameter models, the Topp-Leone distribution, exponential power, lognormal, inverse Gaussian, Birnbaum and Saunders distributions, Dhillon’s model, beta, Haupt-Schäbe models, loglogistic, Avinadev and Raz model, inverse Weibull, Chen’s model and a flexible Weibull extension are presented along with their quantile functions. The quadratic failure rate distribution, truncated normal, cubic exponential family, Hjorth model, generalized Weibull model of Mudholkar and Kollia, exponentiated Weibull, Marshall-Olkin family, generalized exponential, modified Weibull extension, modified Weibull, generalized power Weibull, logistic exponential, generalized linear failure rate distribution, generalized exponential power, upper truncated Weibull, geometric-exponential, Weibull-Poisson and transformed model are some of the distributions considered under three-parameter versions. Distributions with more than three parameters introduced by Murthy et al., Jiang et al., Xie and Lai, Phani, Agarwal and Kalla, Kalla, Gupta and Lvin, and Carrasco et al. are presented as more flexible families. We also introduce general methods that enable the construction of distributions with nonmonotone hazard functions. In the case of many of the models so far specified, the hazard quantile functions and their analysis are also presented to facilitate a quantile-based study. Finally, the properties of total time on test transforms and Parzen’s score function are utilized to develop some new methods of deriving quantile functions that have bathtub hazard quantile functions.

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Authors and Affiliations

  1. Department of Statistics, Cochin University of Science and Technology, Cochin, Kerala, India

    N. Unnikrishnan Nair & P. G. Sankaran

  2. Department of Mathematics and Statistics, McMasters University, Hamilton, ON, Canada

    N. Balakrishnan

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  1. N. Unnikrishnan Nair
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Nair, N.U., Sankaran, P.G., Balakrishnan, N. (2013). Nonmonotone Hazard Quantile Functions. In: Quantile-Based Reliability Analysis. Statistics for Industry and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-8361-0_7

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