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Maximum Likelihood Estimation Using Probability Density Functions of Order Statistics

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Computational Probability Applications

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 247))

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Abstract

A variation of maximum likelihood estimation (MLE) of parameters that uses PDFs of order statistic is presented. Results of this method are compared with traditional maximum likelihood estimation for complete and right-censored samples in a life test. Further, while the concept can be applied to most types of censored data sets, results are presented in the case of order statistic interval censoring, in which even a few order statistics estimate well, compared to estimates from complete and right-censored samples. Population distributions investigated include the exponential, Rayleigh, and normal distributions. Computation methods using APPL are simpler than existing methods using various numerical method algorithms.

Originally published in Computers and Industrial Engineering, Volume 58, Issue 4, in 2010, this paper relied extensively on the APPL environment to explore and analyze censored data techniques. At the heart of the research was the need to find likelihood functions for various censoring schemes. These functions needed the PDFs of order statistics, and the APPL OrderStat procedure produced them. The simulated results reported in the last table all came from APPL-based simulations.

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References

  1. David, H. A., & Nagaraja, H. N. (2003). Order statistics (3rd ed.). Hoboken, NJ: Wiley.

    Book  Google Scholar 

  2. Deshpande, J. V., & Purohit, S. G. (2005). Life time data: Statistical models and methods. London: World Scientific.

    Google Scholar 

  3. Drew, J.H., Evans, D., Glen, A., & Leemis, L. (2008). Computational probability: Algorithms and applications in the mathematical sciences. New York: Springer.

    Google Scholar 

  4. Kalbfleisch, J. D., & Prentice, R. L. (2002). The statistical analysis of failure time data (2nd ed.). Hoboken, NJ: Wiley.

    Book  Google Scholar 

  5. Kendall, M. G., & Stuart, A. (1963). The advanced theory of statistics: Volume II inference and relationship. New York: Hafner.

    Google Scholar 

  6. Klein, J., & Moeschberger, M. (1997). Survival analysis: Techniques for censored and truncated data. New York: Springer.

    Book  Google Scholar 

  7. Larsen, R. J., & Marx, M. L. (2001). An introduction to mathematical statistics and its applications (3rd ed.). Upper Saddle River: Prentice–Hall.

    Google Scholar 

  8. Lawless, J. F. (2003). Statistical models and methods for lifetime data (2nd ed.). New York: Wiley.

    Google Scholar 

  9. Lee, E. T. (1992). Statistical methods for survival data analysis. New York: Wiley.

    Google Scholar 

  10. Leemis, L. (1995). Reliability: Probabilistic models and statistical methods. Upper Saddle River: Prentice–Hall.

    Google Scholar 

  11. Leemis, L., & Shih, L. (1989). Exponential parameter estimation for data sets containing left- and right-censored observations. Communications in Statistics – Simulation, 18(3), 1077–1085.

    Article  Google Scholar 

  12. Nelson, W. (1982). Applied life data analysis. New York: Wiley.

    Book  Google Scholar 

  13. Odell, P., Anderson, K., & D’Agostino, R. (1992). Maximum likelihood estimation for interval-censored data using a Weibull–based accelerated failure time model. Biometrics, 48, 951–959.

    Article  Google Scholar 

  14. Oller, R., Gomez, G., & Luz Calle, M. (2004). Interval censoring: Model characterizations for the validity of the simplified likelihood. The Canadian Journal of Statistics, 32(3), 315–326.

    Article  Google Scholar 

  15. Sun, J. (2004). Statistical analysis of doubly interval-censored failure time data. Handbook of statistics (Vol. 3). Amsterdam: Elsevier.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, The Colorado College, Colorado Springs, CO, USA

    Andrew G. Glen

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  1. Andrew G. Glen
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Correspondence to Andrew G. Glen .

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Editors and Affiliations

  1. Department of Mathematics and Computer Science, The Colorado College, Colorado Springs, Colorado, USA

    Andrew G. Glen

  2. Department of Mathematics, The College of William and Mary, Williamsburg, Virginia, USA

    Lawrence M. Leemis

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Glen, A.G. (2017). Maximum Likelihood Estimation Using Probability Density Functions of Order Statistics. In: Glen, A., Leemis, L. (eds) Computational Probability Applications. International Series in Operations Research & Management Science, vol 247. Springer, Cham. https://doi.org/10.1007/978-3-319-43317-2_7

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