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HOW TO OBTAIN BATHTUB-SHAPED FAILURE RATE MODELS FROM NORMAL MIXTURES

Published online by Cambridge University Press:  01 October 2004

Jorge Navarro  and
Pedro J. Hernandez
Jorge Navarro
Affiliation:
Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain, E-mail: jorgenav@um.es
Pedro J. Hernandez
Affiliation:
Universidad de Politécnica de Cartagena, Cartagena, Murcia, Spain, E-mail: pedraj.hernandez@upct.es
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Abstract

We obtain some techniques to study the shape of reliability functions (failure rate, mean residual life, etc.) by using the s-equilibrium distribution of a renewal process defined by Fagiuoli and Pellerey (Naval Res. Logist., 1993). We apply these techniques to study how to obtain distributions with bathtub shaped failure rate (BFR) from mixtures of two positive truncated normal distributions.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 18 , Issue 4 , October 2004 , pp. 511 - 531
Copyright
© 2004 Cambridge University Press

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References

REFERENCES

Al-Hussaini, E.K. & Sultan, K.S. (2001). Reliability and hazard models based on finite mixture models. Handbook of Statistics 20, Advances in Reliability 139–183.CrossRef
Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing. New York: Holt, Rinehart and Winston.
Block, H.W. & Joe, H. (1997). Tail behavior of the failure rate functions of mixtures. Lifetime Data Analysis 3: 269–288.Google Scholar
Block, H.W., Jong, Y., & Savits, T.H. (1999). Bathtub functions and burn-in. Probability in the Engineering and Informational Sciences 13: 497–507.Google Scholar
Block, H.W., Savits, T.H., & Singh, H. (2002). A criterion for burn-in that balances mean residual life and residual variance. Operations Research 50: 290–296.Google Scholar
Fagiuoli, E. & Pellerey, F. (1993). New partial orderings and applications. Naval Research Logistics 40: 829–842.Google Scholar
Finkelstein, M.S. (2002). On the shape of the mean residual lifetime function. Applied Stochastic Models in Business and Industry 18: 135–146.Google Scholar
Glaser, R.E. (1980). Bathtub and related failure rate characterizations. Journal of the American Statistical Association 75: 667–672.Google Scholar
Gupta, R.C. & Warren, R. (2001). Determination of change points of non-monotonic failure rates. Communications in Statistical Theory and Methods 30: 1903–1920.Google Scholar
Jiang, R. & Murthy, D.N.P. (1998). Mixture of Weibull distributions—Parametric characterization of failure rate function. Applied Stochastic Models and Data Analysis 14: 47–65.Google Scholar
Klar, B. & Müller, A. (2003). Characterizations of classes of lifetime distributions generalizing the NBUE class. Journal of Applied Probability 40: 20–32.Google Scholar
Kotz, S. & Shanbhag, D.N. (1980). Some new approaches to probability distributions. Advances in Applied Probability 12: 903–921.Google Scholar
Krohn, C.A. (1969). Hazard versus renewal rate of electronic items. IEEE Transactions on Reliability 18(2): 64–73.Google Scholar
Lai, C.D., Xie, M., & Murthy, D.N.P. (2001). Bathtub-shaped failure rate life distributions. Advances in Reliability 69–104.CrossRef
Linch, J.D. (1999). On conditions for mixtures of increasing failure rate distributions to have an increasing failure rate. Probability in the Engineering and Informational Sciences 13: 33–36.Google Scholar
Mi, J. (1995). Bathtub failure rate and upside-down bathtub mean residual life. IEEE Transactions on Reliability 44(3): 388–391.Google Scholar
Rajarshi, S. & Rajarshi, M.B. (1988). Bathtub distributions: A review. Communications in Statistical Theory and Methods 17: 2597–2621.Google Scholar
Rojo, J. (1996). Relationships between pure tail orderings of lifetime distributions and some concepts of residual life. Annals of the Institute of Statistical Mathematics 48: 247–255.Google Scholar
Ruiz, J.M. & Navarro, J. (1994). Characterization of distributions by relationships between failure rate and mean residual life. IEEE Transactions on Reliability 43(4): 640–644.Google Scholar
Shaked, M. & Shanthikumar, J.G. (1994). Stochastic orders and their applications. San Diego, CA: Academic Press.
Shaked, M. & Spizzichino, F. (2001). Mixtures and monotonicity of failure rate functions. Advances in Reliability 185–198.CrossRef
Vaupel, J.W. & Yashin, A.I. (1985). Some surprising effects of selection on population dynamics. American Statistics 39: 176–184.Google Scholar
Wondmagegnehu, E.T. (2002). Mixtures of distributions with increasing failure rates. Ph.D. thesis. University of Pittsburgh, Pittsburgh, PA.