A New Weibull Model with Inliers at Zero and One Based on Type-II Censored Samples

Research Article
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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. The inconsistency of such life data is modeled using a nonstandard mixture of distributions; with degeneracy occurring at zero and one, and a probability distribution for positive observations. In this paper, the estimation of parameters based on type-II censored sample from a Weibull distribution with discrete mass at zero and one is studied. The maximum likelihood estimators (MLE) are developed for estimating the unknown parameters. The Fisher information matrix, as well as the asymptotic variance–covariance matrix of the MLEs are derived. Uniformly minimum variance unbiased estimate (UMVUE) of model parameters as well as UMVUE of density function, reliability function and some parametric function is obtained along with UMVUE of the different estimators. The model is implemented on a real data of tumor size in invasive ductal breast carcinoma of female patients.

Keywords

Early failures Failure time distribution Inliers Instantaneous failures Type-II censored sample 

Notes

Acknowledgements

Both the author thanks the referee and Editor for their valuable comments and suggestions. The authors are also grateful to Prof. M Sreehari (Retired) of Department of Statistics, The Maharaja Sayajirao University of Baroda for his valuable inputs for revising the paper.

References

  1. Aitchison J (1955) On the distribution of a positive random variable having a discrete probability mass at the origin. J Am Stat Assoc 50:901–908MathSciNetMATHGoogle Scholar
  2. Charalambides CH (1974) Minimum variance unbiased estimation for a class of left truncated distributions. Sankhya Indian J Stat Ser A 36:392–418MathSciNetMATHGoogle Scholar
  3. Gupta RC (1977) Minimum variance unbiased estimation in modified power series distribution and some of its applications. Commun Stat Theory Methods 6(10):977–991MathSciNetCrossRefMATHGoogle Scholar
  4. Jani PN (1977) Minimum variance unbiased estimation for some left-truncated modified power series distributions. Sankhya Indian J Stat Ser B 39:258–278MathSciNetMATHGoogle Scholar
  5. Jani PN (1993) A characterization of one-parameter exponential family of distributions. Calcutta Stat Assoc Bull 43(3–4):253–256MathSciNetCrossRefMATHGoogle Scholar
  6. Jani PN, Dave HP (1990) Minimum variance unbiased estimation in a class of exponential family of distributions and some of its applications. Metron 48:493–507MathSciNetMATHGoogle Scholar
  7. Jani PN, Singh AK (1995) Minimum variance unbiased estimation in multi-parameter exponential family of distributions. Metron 53:93–106MathSciNetMATHGoogle Scholar
  8. Jayade VP (1993) Statistical inference for AR-process with mixed error. Unpublished Ph. D. thesis, Shivaji University, Kolhapur, MaharashtraGoogle Scholar
  9. Jayade VP, Prasad MS (1990) Estimation of parameters of mixed failure time distribution. Commun Stat Theory Methods 19(12):4667–4677MathSciNetCrossRefMATHGoogle Scholar
  10. Joshi SW, Park CJ (1974) Minimum variance unbiased estimation for truncated power series distributions. Sankhya Indian J Stat Ser A 36:305–314MathSciNetMATHGoogle Scholar
  11. Kale BK, Muralidharan K (2000) Optimal estimating equations in mixture distributions accommodating instantaneous or early failures. J Indian Stat Assoc 38(2):317–329MathSciNetGoogle Scholar
  12. Kale BK, Muralidharan K (2006) Maximum likelihood estimation in presence of inliers. J Indian Soc Prob Stat 10:65–80Google Scholar
  13. Khatri CG (1959) On certain properties of power series distributions. Biometrica 46:486–490MathSciNetCrossRefMATHGoogle Scholar
  14. Muralidharan K (2010) Inlier prone models: a review. ProbStat Forum 3:38–51MathSciNetMATHGoogle Scholar
  15. Muralidharan K, Bavagosai P (2017) Analysis of lifetime model with discrete mass at zero and one. J Stat Theory Pract 11(4):670–692MathSciNetCrossRefGoogle Scholar
  16. Muralidharan K, Lathika P (2006) Analysis of instantaneous and early failures in Weibull distribution. Metrika 64(3):305–316MathSciNetCrossRefMATHGoogle Scholar
  17. Patel SR (1978) Minimum variance unbiased estimation of multivariate modified power series distribution. Metrika 25:155–161MathSciNetCrossRefMATHGoogle Scholar
  18. Patil GP (1963) Minimum variance unbiased estimation and certain problem of additive number theory. Ann Math Stat 34:1050–1056MathSciNetCrossRefMATHGoogle Scholar
  19. Roy J, Mitra SK (1957) Unbiased minimum variance estimation in a class of discrete distributions. Sankhya Indian J Stat 18:371–378MATHGoogle Scholar

Copyright information

© The Indian Society for Probability and Statistics (ISPS) 2018

Authors and Affiliations

  1. 1.Department of Statistics, Faculty of ScienceThe Maharaja Sayajirao University of BarodaVadodaraIndia

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