Integrated beta model for bathtub-shaped hazard rate data

Abstract

In this paper, we introduce a new parametric distribution for modelling lifetime data with bathtub-shaped hazard rate, derived by representing the cumulative hazard as proportional to a generalized beta density. The distribution has a finite range, useful when there is a maximum possible lifetime, a common scenario when there are additional failure modes leading to failure after a certain time. Reliability and other distributional properties of the distribution are discussed. Parameter issues are also studied. The application of the model is illustrated by fitting the model to several data sets from the literature.

Keywords:

• Generalized beta
• lifetime data
• probability distribution
• reliability

Acknowledgements

The authors are grateful to the two reviewers for their helpful comments.

Disclosure statement

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of this article.

Notes on contributors

Chin-Diew Lai is a Professor of Statistics at Massey University, Palmerston North, New Zealand. He received his PhD in Statistics in 1975 from the Victoria University of Wellington. He spent a year as an Associate Professor at National Chiao Tung University, Hsinchu, Taiwan. Professor Lai’s research is mainly in reliability modelling and bivariate distributions. He has over 100 journal publications and authored several books, including ‘Stochastic Ageing and Dependence for Reliability’ (with M. Xie), ’Continuous Bivariate Distributions’ (with N. Balakrishnan) and ‘Generalized Weibull Distributions’ published by Springer in 2006, 2009 and 2014, respectively.

Geoff Jones, obtained his PhD in Statistics in 1996 from the University of California at Davis, and is currently an Associate Professor of Statistics at Massey University in New Zealand. He is a Fellow of the Royal Statistical Society and member of the American Statistical Association and the New Zealand Statistical Association. He has over 100 publications in a range of areas of statistics including calibration, survival analysis, experimental design and small-area estimation. He has worked on several consulting projects with local industries and government agencies, and since 2003 he has been working on poverty estimation projects overseas with the UN World Food Programme and the World Bank in a range of Third World countries.

Min Xie, is currently a Chair Professor of Industrial Engineering and serves as Associate Dean (Internationalization) at College of Science and Engineering, City University of Hong Kong. He received his MSc from Royal Inst of Technology in Sweden in 1984 and his PhD from Linkoping University in 1987. Dr Xie joined the National University of Singapore in 1991 as one of the first recipients of the prestigious LKY fellowship. Prof Xie has published numerous papers and eight books, including ‘Software Reliability Modelling’ by World Scientific in 1991, ‘Weibull Models’ by John Wiley in 2002, ‘Computing Systems Reliability’ by Springer in 2003. He is an editor, associate editor and on the editorial board of a number of international journals, and has organized many international conferences. Prof Xie is an elected fellow of IEEE and an elected member of ISI.