# A Transformation for the Analysis of Unimodal Hazard Rate Lifetimes Data

## Abstract

The family of distributions introduced by [34] is the best known, best understood, most extensively investigated, and commonly employed model used for lifetimes data analysis. A variety of software packages are available to simplify its use. Yet, as is well known, the model is appropriate only when the hazard rate is monotone. However, as suggested in an overview by [23], the software packages may be usefully employed by transforming data when exploratory tools such as TTT transform or nonparametric estimates indicate unimodal, bathtub or J-shaped hazard rates, which are also commonly encountered in practice. Mudholkar et al. [22] discussed the details of one such transformation relevant for the bathtub case. In this paper, specifics of another transformation which is appropriate when data exploration indicates a unimodal hazard rate is discussed. The details of parameter estimation and hypothesis testing are considered in conjunction with earlier alternatives and illustrated using examples from the fields of biological extremes and finance.

## Keywords

Weibull distribution Exponentiated Weibull Generalized Weibull Lifetimes data Maximum likelihood Transformation## MSC:

62F03 62F10 62P20 62P20## References

- 1.Aarset, M.V. 1987. How to identify bathtub hazard rate.
*IEEE Transactions on Reliability*36: 106–108.CrossRefMATHGoogle Scholar - 2.Balakrishnan, N., and C.R. Rao. 2004. Advances in survival analysis.
*Handbook of statistics*, vol. 23. Amsterdam: North-Holland.Google Scholar - 3.Bartlett, M.S. 1947. The use of transformations.
*Biometrics*3: 39–52.MathSciNetCrossRefGoogle Scholar - 4.Box, G.E.P., and D.R. Cox. 1964. An analysis of transformations.
*Journal of the Royal Statistical Society, Series B*26: 211–243.MathSciNetMATHGoogle Scholar - 5.Cox, D.R., and D. Oakes. 1984.
*Analysis of survival data*. London: Chapman & Hall.Google Scholar - 6.Cox, D.R., and D.V. Hinkley. 1974.
*Theoretical statistics*. London: Chapman & Hall.CrossRefMATHGoogle Scholar - 7.Diciccio, T.J., and J.P. Romano. 1988. A review of bootstrap confidence intervals.
*Journal of Royal Statistical Society B*50: 338–354.MathSciNetMATHGoogle Scholar - 8.Efron, B. 1987. Better bootstrap confidence intervals.
*Journal of the American Statistical Association*82: 171–185.MathSciNetCrossRefMATHGoogle Scholar - 9.Efron, B. 1988. Logistic regression, survival analysis and the Kaplan-Meier curve.
*Journal of the American Statistical Association*83: 414–425.MathSciNetCrossRefMATHGoogle Scholar - 10.Fisher, R.A. 1915. Frequency distribution of the values of the correlation coefficient in samples of an indefinitely large population.
*Biometrika*10: 507–521.Google Scholar - 11.Harman, Y.S., and T.W. Zuehlke. 2007. Nonlinear duration dependence in stock market cycles.
*Review of Financial Economics*16: 350–362.CrossRefGoogle Scholar - 12.Heckman, J.J., and S. Burton. 1984. Econometric duration analysis.
*Journal of Econometrics*24: 63–132.MathSciNetCrossRefMATHGoogle Scholar - 13.Kalbfleisch, J.D., and R.L. Prentice. 1980.
*The statistical analysis of failure data*. New York: Wiley.MATHGoogle Scholar - 14.Klein, J.P., and M.L. Moeschberger. 2003.
*Survival analysis, techniques for censored and truncated data*, 2nd ed. New York: Springer.MATHGoogle Scholar - 16.Lawless, J.F. 1982.
*Statistical methods and model for lifetime data*. New York: Wiley.MATHGoogle Scholar - 17.Le Cam, L. 1986.
*Asymptotic methods in statistical decision theory*. New York: Springer.CrossRefMATHGoogle Scholar - 18.Le Cam, L. 1990. Maximum likelihood: An introduction.
*International Statistical Institute (ISI)*58: 153–171.MATHGoogle Scholar - 19.Le Cam, L., and G. Lo Yang. 2000.
*Asymptotics in statistics: Some basic concepts*, 2nd ed. New York: Springer.Google Scholar - 20.Lehmann, E.L., and G. Casella. 1998.
*Theory of point estimation*, 2nd ed. New York: Springer.MATHGoogle Scholar - 21.McQueen, G., and S. Thorley. 1994. Bubbles, stock returns, and duration dependence.
*Journal of Financial and Quantitative Analysis*29: 379–401.CrossRefGoogle Scholar - 22.Meeker, W.Q., and L.A. Escobar. 1995. Assessing influence in regression analysis with censored data.
*Biometrics*48: 50728.Google Scholar - 23.Mudholkar, G.S., K.O. Asubonteng, and D.A. Hutson. 2009. Transformation for Weibull model analysis of bathtub failure rate data in reliability analysis.
*Statistical Methodology*6: 622–633.MathSciNetCrossRefMATHGoogle Scholar - 24.Mudholkar, G.S., and K.O. Asubonteng. 2010. Data-transformation approach to lifetimes data analysis: An overview.
*Journal of Statistical Planning and Inference*140: 2904–2917.MathSciNetCrossRefMATHGoogle Scholar - 25.Mudholkar, G.S., and D.K. Srivastava. 1993. Exponentiated Weibull family for analyzing bathtub failure rate data.
*IEEE Transactions on Reliability*42: 299–302.CrossRefMATHGoogle Scholar - 26.Mudholkar, G.S., and G.D. Kollia. 1994. Generalized Weibull family: A structural analysis.
*Communication Statistics-Theory and Methods*23: 1149–1171.MathSciNetCrossRefMATHGoogle Scholar - 27.Mudholkar, G.S., D.K. Srivastava, and M. Freimer. 1995. Exponentiated Weibull family: A reanalysis of the bus motor failure data.
*Technometrics*37: 436–445.CrossRefMATHGoogle Scholar - 28.Mudholkar, G.S., D.K. Srivastava, and G. Kollia. 1996. A generalization of Weibull distribution with application to the analysis of survival data.
*Journal of the American Statistical Association*91: 1575–1583.MathSciNetCrossRefMATHGoogle Scholar - 29.Mudholkar, G.S., and D.A. Hutson. 1996. The exponentiated Weibull family: Some properties and a flood data application.
*Communication Statistics—Theory and Methods*25: 3059–3083.MathSciNetCrossRefMATHGoogle Scholar - 30.Newcomb, S. 1881. Note on the frequency of use of the different digits in natural numbers.
*American Journal of Mathematics*4: 39–40.MathSciNetCrossRefMATHGoogle Scholar - 31.Pawitan, Y. 2001.
*In all likelihood: Statistical modeling and inference using likelihood method*. New York: Oxford University Press.MATHGoogle Scholar - 33.Rao, C.R. 1973.
*Linear statistical inference and its application*. New York: Wiley.CrossRefGoogle Scholar - 35.Terrell, G.R. 2003. The Wilson-Hilfety transformation is locally saddlepoint.
*Biometrika*445–453.Google Scholar - 36.United States Water Resources Council. 1977. Guidelines For determining flood flow frequency. Hydrology Committee. Washington, D.C.: Water Resources Council.Google Scholar
- 37.Weibull, W. 1939. Statistical theory of the strength of materials.
*Ingenioor Vetenskps Akademiens Handlingar*151: 1–45.Google Scholar