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On maximum likelihood estimates of a proportional hazard rate model parameters based on record values

The proportional hazard rate models have wide acceptance in modeling complex data from different engineering, reliability and survival applications. The hazard rate function of such models is able to model data with a monotonic or a bathtubshape hazard rate. In this paper, we propose a subclass of proportional hazard rate models with three-parameters for which we have proved the existence and uniqueness of the maximum likelihood estimators based on upper kth record values. The proposed inference is found especially useful in some real illustrations within mechanical and medical analysis. Finally, several remarks are presented in the final stage of this paper.
PAPER REVISED: 2022-10-11
PAPER ACCEPTED: 2022-10-15
  1. K. N. Chandler. The distribution and frequency of record values. Journal of the Royal Statistical Society Series B (Methodological), 14, 1952.
  2. B. C. Arnold, N. Balakrishnan, and H. N. Nagaraja. Records. Wiley, 1998.
  3. W. Dziubdziela and B. Kopoci´nski. Limiting properties of the k-th record values. Applicationes Mathematicae, 2(15):187-190, 1976.
  4. V. B. Nevzorov. Records: Mathematical Theory. Translations of Mathematical Monographs. AMS, 2000.
  5. C. D. Lai, X. Min, and D. N. P. Murthy. A modified Weibull distribution. IEEE Transactions on reliability, 52(1):33-37, 2003.
  6. X. Peng and Z. Yan. Estimation and application for a new extended Weibull distribution. Reliability Engineering & System Safety, 121, 2014.
  7. H. Pham and C. D. Lai. On recent generalizations of the Weibull distribution. IEEE Transactions on Reliability, 56(3):454-458, 2007.
  8. S. J. Almalki and S. Nadarajah. Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124:32-55, 2014.
  9. S. Jiang and D. Kececioglu. Graphical representation of two mixed-Weibull distributions. IEEE Transactions on Reliability, 41(2):241-247, 1992.
  10. H. J. Ma and Z. Z. Yan. Discrete Weibull-Rayleigh distribution properties and parameter estimations. Thermal Science, 26(3 Part B):2627-2636, 2022.
  11. T. M¨akel¨ainen, K. Schmidt, and G. P. H. Styan. On the existence and uniqueness of the maximum likelihood estimate of a vector-valued parameter in fixed-size samples. The Annals of Statistics, pages 758-767, 1981.
  12. V. D. Barnett. Evaluation of the maximum-likelihood estimator where the likelihood equation has multiple roots. Biometrika, 53(1/2):151-165, 1966.
  13. N. Glick. Breaking records and breaking boards. American Mathematical Monthly, pages 2-26, 1978.
  14. J. Ahmadi and N. R. Arghami. Comparing the Fisher information in record values and iid observations. Statistics, 37(5):435-441, 2003.
  15. Lai C. D. Bebbington, M. and R. Zitikis. Useful periods for lifetime distributions with bathtub shaped hazard rate functions. IEEE Transactions on Reliability, 55(2): 245-251, 2006.
  16. H. Jiang, M. Xie, and L. C. Tang. On MLEs of the parameters of a modified Weibull distribution for progressively type-2 censored samples. Journal of Applied Statistics, 37(4):617-627, 2010.
  17. H. K. T. Ng. Parameter estimation for a modified Weibull distribution, for progressively type-II censored samples. IEEE Transactions on Reliability, 54(3):374-380, 2005.
  18. Z. Vidovi´c. On MLEs of the parameters of a modified Weibull distribution based on record values. Journal of Applied Statistics, 46(4):715-724, 2018.
  19. N. Balakrishnan and E. Cramer. The art of progressive censoring. Springer, 2014.
  20. A. Pak and S. Dey. Statistical inference for the power lindley model based on record values and inter-record times. Journal of Computational and Applied Mathematics, 347:156-172, 2019.
  21. R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2015.
  22. J. F. Lawless. Statistical models and methods for lifetime data, volume 362. John Wiley & Sons, 2011.
  23. M. Crowder. Tests for a family of survival models based on extremes. In Recent Advances in Reliability Theory, pages 307-321. Springer, 2000.
  24. E. T. Lee and J. Wang. Statistical methods for survival data analysis, volume 476. John Wiley & Sons, 2003.
  25. A. Alzaatreh, C. Lee, and F. Famoye. A new method for generating families of continuous distributions. Metron, 71(1):63-79, 2013.
  26. M. C¸ . Korkmaz. A new family of the continuous distributions: The extended Weibull- G family. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1):248-270, 2018.