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Measures of ageing tendency

Part of: Distribution theory Survival analysis and censored data Distribution theory - Probability

Published online by Cambridge University Press:  30 July 2019

Magdalena Szymkowiak
Magdalena Szymkowiak*
Affiliation:
Poznan University of Technology
*
*Postal address: Institute of Automation and Robotics, Poznan University of Technology, Pl. M. Skłodowskiej-Curie 5, 60-965 Poznań, Poland.
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Abstract

A family of generalized ageing intensity functions of univariate absolutely continuous lifetime random variables is introduced and studied. They allow the analysis and measurement of the ageing tendency from various points of view. Some of these generalized ageing intensities characterize families of distributions dependent on a single parameter, while others determine distributions uniquely. In particular, it is shown that the elasticity functions of various transformations of distributions that appear in lifetime analysis and reliability theory uniquely characterize the parent distribution. Moreover, the recognition of the shape of a properly chosen generalized ageing intensity estimate admits a simple identification of the data lifetime distribution.

Keywords

Generalized failure rategeneralized ageing intensityelasticity functioncharacterization

MSC classification

Primary: 60E05: Distributions
Secondary: 62E10: Characterization and structure theory 62N05: Reliability and life testing
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 2 , June 2019 , pp. 358 - 383
Copyright
© Applied Probability Trust 2019 

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References

Barlow, R.E. and Van Zwet, W. R. (1969a). Asymptotic properties of isotonic estimators for the generalized failure rate function. Part I. Strong consistency. In Nonparametric Techniques in Statistical Inference (Proc. Symp. Indiana University, Bloomington, IN, 1969). Cambridge University Press, pp. 159–176.Google Scholar
Barlow, R.E. and Van Zwet, W. R. (1969b). Asymptotic properties of isotonic estimators for the generalized failure rate function. Part II. Asymptotic distributions. Operations Research Center Report ORC, University of California, Berkeley, pp. 69–110.Google Scholar
Barlow, R.E. and Van Zwet, W. R. (1971). Comparison of several nonparametric estimators of the failure rate function. Operations Research Center Report ORC, Gordon Breach, New York, pp. 375–399.Google Scholar
Bhattacharjee, S., Nanda, A. K. and Misra, S. Kr. (2013). Reliability analysis using ageing intensity function. Stat. Prob. Lett. 83, 1364–1371.CrossRefGoogle Scholar
Bowman, A.W. and Azzalini, A. (1997). Applied Smoothing Techniques for Data Analysis. Oxford University Press.Google Scholar
Case, K.E. and Fair, R. C. (2007). Principles of Economics, 8th edn. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Chang, M. N. (1996). On the asymptotic distribution of an isotonic window estimator for the generalized failure rate function. Commun. Statist. Theory Meth. 25, 2239–2249.CrossRefGoogle Scholar
Cheng, K. F. (1982). Contributions to nonparametric generalized failure rate function estimation. Metrika 29, 215–225.CrossRefGoogle Scholar
Chiang, A. C. and Wainwright, K. (2005). Fundamental Methods of Mathematical Economics, 4th edn. McGraw-Hill, New York.Google Scholar
Denneberg, D. (1990). Premium calculation: Why standard deviation should be replaced by absolute deviation. ASTIN Bull. 20, 181–190.CrossRefGoogle Scholar
Deshpande, J. V., Kochar, S.C. and Singh, H. (1986). Aspects of positive ageing. J. Appl. Prob. 23, 748–758.CrossRefGoogle Scholar
Jiang, R., Ji, P. and Xiao, X. (2003). Aging property of unimodal failure rate models. Reliab. Eng. Syst. Safety 79, 113–116.CrossRefGoogle Scholar
Johnson, N. L, Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions. John Wiley, New York.Google Scholar
Lai, C.D., Xie, M. and Murthy, D. N. P. (2003). A modified Weibull distribution. IEEE Trans. Reliab. 52, 33–37.CrossRefGoogle Scholar
Lee, E. and Wang, J. (2003). Statistical Methods for Survival Data Analysis, 3rd edn. John Wiley, New York.CrossRefGoogle Scholar
Marshall, A. (1890). Principles of Economics. Macmillan, London.Google Scholar
Marshall, A. W. and Olkin, I. (2007). Life Distributions: Structure of Nonparametric, Semiparametric and Parametric Models. Springer Series in Statistics. Springer, New York.Google Scholar
Navarro, J., Del Aguila, Y., Sordo, M.A. and Suárez-Llorens, A. (2013). Stochastic ordering properties for systems with dependent identically distributed components. Appl. Stoch. Models Business Industry 29, 264–278.CrossRefGoogle Scholar
Navarro, J., Del Aguila, Y., Sordo, M.A. and Suárez-Llorens, A. (2014). Preservation of reliability classes under the formation of coherent systems. Appl. Stoch. Models Business Industry 30, 444–454.CrossRefGoogle Scholar
Phani, K. K. (1987). A new modified Weibull distribution function. Commun. Amer. Ceramic Soc. 70, 182–184.Google Scholar
Pickands, J. (1975). Statistical inference using extreme order statistics. Ann. Statist. 3, 119–131.Google Scholar
Rady, E. A., Hassanein, W. A. and Elhaddad, T. A. (2016). The power Lomax distribution with an application to bladder cancer data. SpringerPlus 5, 1838.CrossRefGoogle Scholar
Shaked, M. (1979). An estimator for the generalized hazard rate function. Commun. Statist. Theory Meth. A8, 17–33.CrossRefGoogle Scholar
Sordo, M. A., Suárez-Llorens, A. and Bello, A. (2015). Comparison of conditional distributions in portfolios of dependent risks. Insurance Math. Econom. 61, 62–69.CrossRefGoogle Scholar
Sydsæter, K. and Hammond, P. (2012). Essential Mathematics for Economic Analysis. Pearson Education, London.Google Scholar
Szymkowiak, M. (2018a). Characterizations of distributions through aging intensity. IEEE Trans. Reliab. 67, 446–458.CrossRefGoogle Scholar
Szymkowiak, M. (2018b). Generalized aging intensity functions. Reliab. Eng. Syst. Safety 178, 198–208.CrossRefGoogle Scholar
Veres-Ferrer, E. J. and Pavía, J. M. (2014). On the relationship between the reversed hazard rate and elasticity. Statist. Papers 55, 275–284.CrossRefGoogle Scholar
Veres-Ferrer, E. J. and Pavía, J. M. (2017). Properties of the elasticity of a continuous random variable. A special look to its behaviour and speed of change. Commun. Statist. Theory Meth. 46, 3054–3069.CrossRefGoogle Scholar
Weibull, W. (1951). A statistical distribution function of wide applicability. J. Appl. Mech. 18, 293–296.Google Scholar