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Journal of Statistical Theory and Practice

, Volume 11, Issue 4, pp 670–692 | Cite as

Analysis of lifetime model with discrete mass at zero and one

  • K. Muralidharan
  • Bavagosai Pratima
Article

Abstract

Inliers (instantaneous or early failures) are natural occurrences of a life test, where some of the items fail immediately or within a short time of the life test due to mechanical failure, inferior quality, or faulty construction of items and components. A similar situation is observed in mortality studies, where the life pattern of newborn babies includes stillbirths (no life), neonatal births (life less than 28 days may be recorded as 1), and babies surviving a month and longer. The inconsistency of such life data is modeled using a nonstandard mixture of distributions, with degeneracy occurring at 0 and 1, and a probability distribution for positive observations. Keeping the underlying distribution as exponential distribution, we model the inliers situation and propose various estimators and characteristics of the model. The model is implemented on mortality data obtained through the NFHS-3 studies.

Keywords

Asymptotic distribution early failures failure time distribution infant mortality rate inliers instantaneous failures 

AMS Subject Classification

62F10 62P10 62P25 62P30 

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References

  1. Aitchison, J. 1955. On the distribution of a positive random variable having a discrete probability mass at the origin. Journal of the American Statistical Association 50:901–8.MathSciNetzbMATHGoogle Scholar
  2. Charalambides, C. H. 1974. Minimum variance unbiased estimation for a class of left truncated distributions. Sankhya A 36:392–418.MathSciNetzbMATHGoogle Scholar
  3. Gupta, R. C. 1977. Minimum variance unbiased estimation in modified power series distribution and some of its applications. Communication in Statistics 6:977–91. doi:10.1080/03610927708827546.MathSciNetCrossRefGoogle Scholar
  4. Gupta, R. C., and D. Kundu. 2001. Generalized exponential distribution: Different method of estimation. Journal of Statistical Computation and Simulation 69:315–37. doi:10.1080/00949650108812098.MathSciNetCrossRefGoogle Scholar
  5. Hanna, H. A.-Z. 2014. Six method of estimations for the shape parameter of Exponentiated Gompertz distribution. Applied Mathematical Sciences 8:4349–59. doi:10.12988/ams.2014.46503.CrossRefGoogle Scholar
  6. Jani, P. N. 1977. Minimum variance unbiased estimation for some left-truncated modified power series distributions. Sankhya 39:258–78.MathSciNetzbMATHGoogle Scholar
  7. Jani, P. N. 1993. A characterization of one-parameter exponential family of distributions. Calcutta Statistical Association Bulletin 43 (3–4):253–56. doi:10.1177/0008068319930310.MathSciNetCrossRefGoogle Scholar
  8. Jani, P. N., and H. P. Dave. 1990. Minimum variance unbiased estimation in a class of exponential family of distributions and some of its applications. Metron 48:493–507.MathSciNetzbMATHGoogle Scholar
  9. Jani, P. N., and A. K. Singh. 1995. Minimum variance unbiased estimation in multi-parameter exponential family of distributions. Metron 53:93–106.MathSciNetzbMATHGoogle Scholar
  10. Jayade, V. P., and M. S. Prasad. 1990. Estimation of parameters of mixed failure time distribution. Communications in Statistics: Theory and Methods 19 (12):4667–77. doi:10.1080/03610929008830466.MathSciNetCrossRefGoogle Scholar
  11. Johnson, N. L., S. Kotz, and N. Balakrishnan. 1995. Continuous Univariate Distribution, vol. 2, 2nd ed. New York, NY: John Wiley.zbMATHGoogle Scholar
  12. Joshi, S. W., and C. J. Park. 1974. Minimum variance unbiased estimation for truncated power series distributions. Sankhya A 36:305–14.MathSciNetzbMATHGoogle Scholar
  13. Kale, B. K., and K. Muralidharan. 2000. Optimal estimating equations in mixture distributions accommodating instantaneous or early failures. Journal of the Indian Statistical Association 38:317–29.MathSciNetGoogle Scholar
  14. Kale, B. K., and K. Muralidharan. 2008. Maximum likelihood estimation in presence of inliers. Journal of the Indian Society for Probability and Statistics 10:65–80.Google Scholar
  15. Kao, J. H. K. 1958. Computer methods for estimating Weibull parameters in reliability studies. Transactions of IRE-Reliability and Quality Control 13:15–22. doi:10.1109/IRE-PGRQC.1958.5007164.CrossRefGoogle Scholar
  16. Kao, J. H. K. 1959. A graphical estimation of mixed Weibull parameters in life testing electron tube. Technometrics 1:389–407. doi:10.1080/00401706.1959.10489870.CrossRefGoogle Scholar
  17. Khatri, C. G. 1959. On certain properties of power series distributions. Biometrica 46:486–90. doi:10.1093/biomet/46.3-4.486.MathSciNetCrossRefGoogle Scholar
  18. Murthy, D. N. P., M. Xie, and R. Jiang. 2004. Weibull models. Hoboken, NJ: John Wiley.zbMATHGoogle Scholar
  19. Patel, S. R. 1978. Minimum variance unbiased estimation of multivariate modified power series distribution. Metrika 25:155–61. doi:10.1007/BF02204360.MathSciNetCrossRefGoogle Scholar
  20. Patil, G. P. 1963a. Minimum variance unbiased estimation and certain problem of additive number theory. Annals of Mathematics and Statistics 34:1050–56. doi:10.1214/aoms/1177704029.MathSciNetCrossRefGoogle Scholar
  21. Patil, G. P. 1965c. On multivariate generalized power series distribution and its application to the multinomial and negative multinomial. In Classical and contagious distributions, ed. G. P. Patel, 183–94. Calcutta, India: Statistical Publishing Society.Google Scholar
  22. Roy, J., and S. K. Mitra. 1957. Unbiased minimum variance estimation in a class of discrete distributions. Sankhya 18:371–78.MathSciNetzbMATHGoogle Scholar
  23. Swain, J., S. Venkatraman, and J. Wilson. 1988. Least squares estimation of distribution function in Johnson’s translation system. Journal of Statistical Computation and Simulation 29:271–97. doi:10.1080/00949658808811068.CrossRefGoogle Scholar
  24. International Institute for Population Sciences (IIPS) and Macro International. 2007. National Family Health Survey (NFHS-3), India, 2005–06, Vol II. Gujarat, Mumbai: IIPS. https://doi.org/www.measuredhs.com Google Scholar

Copyright information

© Grace Scientific Publishing, 20 Middlefield Ct, Greensboro, NC 27455 2017

Authors and Affiliations

  1. 1.Department of Statistics, Faculty of ScienceMaharaja Sayajirao University of BarodaVadodaraIndia

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