Statistical Methods and Applications

, Volume 11, Issue 2, pp 187–200 | Cite as

Computing maximum likelihood estimates from type II doubly censored exponential data

  • Arturo J. fernández
  • José I. Bravo
  • Íñigo De Fuentes
Statistical Methods

Abstract

It is well-known that, under Type II double censoring, the maximum likelihood (ML) estimators of the location and scale parameters, θ and δ, of a twoparameter exponential distribution are linear functions of the order statistics. In contrast, when θ is known, theML estimator of δ does not admit a closed form expression. It is shown, however, that theML estimator of the scale parameter exists and is unique. Moreover, it has good large-sample properties. In addition, sharp lower and upper bounds for this estimator are provided, which can serve as starting points for iterative interpolation methods such as regula falsi. Explicit expressions for the expected Fisher information and Cramér-Rao lower bound are also derived. In the Bayesian context, assuming an inverted gamma prior on δ, the uniqueness, boundedness and asymptotics of the highest posterior density estimator of δ can be deduced in a similar way. Finally, an illustrative example is included.

Key words

Maximum likelihood estimation Type II double censoring exponential distributions order statistics Bayes estimators 

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References

  1. Bain LJ (1978) Statistical Analysis of Reliability and Life-Testing Models. Marcel Dekker, New YorkMATHGoogle Scholar
  2. Balakrishnan N, Chan PS (1995) Maximum likelihood estimation for the log-gamma distribution under Type II censored samples and associated inference. In: Balakrishnan N (ed) Recent Advances in Life-Testing and Reliability. CRC Press, Inc., pp. 409–421Google Scholar
  3. Bhattacharyya GK (1985) On asymptotics of maximum likelihood and related estimators based on Type II censored data. J. Amer. Statist. Assoc. 80:398–404MATHMathSciNetCrossRefGoogle Scholar
  4. David HA (1970) Order Statistics. John Wiley, New YorkMATHGoogle Scholar
  5. Dempster AP, Laird NM, Rubin DB (1977) Maximun likelihood from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc. B 39:1–38MATHMathSciNetGoogle Scholar
  6. Elfessi A (1997) Estimation of a linear function of the parameters of an exponential distribution from doubly censored data. Statist. and probab. Letters 36:251–259MATHMathSciNetCrossRefGoogle Scholar
  7. Fernández AJ (2000) Bayesian inference from type II doubly censored Rayleigh data. Statist. and Probab. Letters 48:393–399MATHCrossRefGoogle Scholar
  8. Harter HL, Moore AH (1968) Maximum likelihood estimation, from doubly censored samples, of the parameters of the first asymptotic distribution of extreme values. J. Amer. Statist. Assoc. 63:889–901MathSciNetCrossRefGoogle Scholar
  9. Jeffrey H. (1961) Theory of Probability. Clarendom Press, OxfordGoogle Scholar
  10. Kambo NS (1978) Maximum likelihood estimators of the location and scale parameters of the exponential distribution from a censored sample. Commun. Statist. A 12:1129–1132MathSciNetGoogle Scholar
  11. Lalitha S, Mishra A (1996) Modified maximum likelihood estimation for Rayleight distribution. Commun. Statist. A 25:389–401MATHMathSciNetCrossRefGoogle Scholar
  12. Lawless JF (1982) Statistical Models and Methods for Lifetime Data. John Wiley, New YorkMATHGoogle Scholar
  13. Leemis LM, Shih LH (1989) Exponential parameter estimation for data sets containing left and right censored observations. Commun. Statist. B 18:1077–1085MATHMathSciNetCrossRefGoogle Scholar
  14. Nelson W (1982) Applied Life Data Analysis. John Wiley, New YorkMATHCrossRefGoogle Scholar
  15. Rao CR (1973) Linear Statistical Inference and Its Applications. John Wiley, New YorkMATHGoogle Scholar
  16. Raqab MZ (1995) On the maximum likelihood prediction of the exponential distribution based on doubly Type II censored samples. Pak. J. Statist. 11:1–10MATHMathSciNetGoogle Scholar
  17. Sarhan AE (1955) Estimation of the mean and standard deviation by order statistics, Part. III. Ann. Math. Statist. 26:576–592MATHMathSciNetGoogle Scholar
  18. Stoer J, Bulirsch R (1983) Introduction to Numerical analysis. Springer, Berlin Heidelberg New YorkGoogle Scholar
  19. Tiku ML (1967) A note on estimating the location and scale parameters of the exponential distribution from a censored sample. Aust. J. Statist. 9:48–54MathSciNetCrossRefGoogle Scholar
  20. Tiku ML, Tan WY, Balakrishnan N (1986) Robust Inference. Marcel Dekker. New YorkMATHGoogle Scholar

Copyright information

© Springer-Verlag 2002

Authors and Affiliations

  • Arturo J. fernández
    • 1
  • José I. Bravo
    • 1
  • Íñigo De Fuentes
    • 1
  1. 1.Departamento de Estadística, I. O. y Computación, Facultad de MatemáticasUniversidad de La LagunaLa LagunaSpain

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