A Class of Truncated Binomial Lifetime Distributions

Author(s)
Said Hofan Alkarni

ABSTRACT

In this paper, a new lifetime class with decreasing failure rate is introduced by compounding truncated binomial distribution with any proper continuous lifetime distribution. The properties of the proposed class are discussed, including a formal proof of its probability density function, distribution function and explicit algebraic formulae for its reliability and failure rate functions. A simple EM-type algorithm for iteratively computing maximum likelihood estimates is presented. The Fisher information matrix is derived in order to obtain the asymptotic covariance matrix. This new class of distributions generalizes several distributions which have been introduced and studied in the literature.

Cite this paper

S. Alkarni, "A Class of Truncated Binomial Lifetime Distributions,"*Open Journal of Statistics*, Vol. 3 No. 5, 2013, pp. 305-311. doi: 10.4236/ojs.2013.35036.

S. Alkarni, "A Class of Truncated Binomial Lifetime Distributions,"

References

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[2] C. Kus, “A New Lifetime Distribution,” Computational Statistics and Data Analysis, Vol. 51, 2007, pp. 44974509.

[3] R. Tahmasbi and S. Rezaei, “A Two-Parameter Lifetime Distribution with Decreasing Failure Rate,” Computational Statistics and Data Analysis, Vol. 52, 2008, pp. 38893901.

[4] M. Chahkandi and M. Ganjali, “On Some Lifetime Distributions with Decreasing Failure Rate,” Computational Statistics and Data Analysis, Vol. 53, 2009, pp. 44334440.

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[7] S. Saunders and J. Myhre, “Maximum Likelihood Estimation for Two-Parameter Decreasing Hazard Rate Distributions Using Censored Data,” Journals of American Statistical Association, Vol. 78, 1983, pp. 664-673.

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[12] F. Hemmati, E. Khorram and S. Rezakhah, “A New ThreeParameter Distribution,” Journal of Statistical Planning and Inference, Vol. 141, 2011, pp. 2255-2275.

[13] R. Silva, W. Barreto-Souza and G. Cordeiro, “A New Distribution with Decreasing, Increasing and Upside-Down Bathtub Failure Rate,” Computational Statistics & Data Analysis, Vol. 54, 2010, pp. 935-944.

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[1] K. Adamidis and S. Loukas, “A Lifetime Distribution with Decreasing Failure Rate,” Statistics and Probability Letters, Vol. 39, No. 1, 1998, pp. 35-42.

[2] C. Kus, “A New Lifetime Distribution,” Computational Statistics and Data Analysis, Vol. 51, 2007, pp. 44974509.

[3] R. Tahmasbi and S. Rezaei, “A Two-Parameter Lifetime Distribution with Decreasing Failure Rate,” Computational Statistics and Data Analysis, Vol. 52, 2008, pp. 38893901.

[4] M. Chahkandi and M. Ganjali, “On Some Lifetime Distributions with Decreasing Failure Rate,” Computational Statistics and Data Analysis, Vol. 53, 2009, pp. 44334440.

[5] K. Lomax, “Business Failures: Another Example of the Analysis of Failure Data,” Journals of American Statistical Association, Vol. 49, 1954, pp. 847-852.

[6] F. Proschan, “Theoretical Explanation of Observed Decreasing Failure Rate,” Journal of Technometrics, Vol. 5, 1963, pp. 375-383.

[7] S. Saunders and J. Myhre, “Maximum Likelihood Estimation for Two-Parameter Decreasing Hazard Rate Distributions Using Censored Data,” Journals of American Statistical Association, Vol. 78, 1983, pp. 664-673.

[8] F. McNolty, J. Doyle and E. Hansen, “Properties of the Mixed Exponential Failure Process,” Technometrics, Vol. 22, 1980, pp. 555-565.

[9] L. Gleser, “The Gamma Distribution as a Mixture of Exponential Distributions,” Journals of American Statistical Association, Vol. 43, 1989, pp. 115-117.

[10] J. Gurland and J. Sethuraman, “Reversal of Increasing Failure Rates When Pooling Failure Data,” Technometrics, Vol. 36, 1994, pp. 416-418.

[11] K. Adamidis, T. Dimitrakopoulou and S. Loukas, “On an Extension of the Exponential-Geometric Distribution,” Statistics & Probability Letters, Vol. 73, 2005, pp. 259269.

[12] F. Hemmati, E. Khorram and S. Rezakhah, “A New ThreeParameter Distribution,” Journal of Statistical Planning and Inference, Vol. 141, 2011, pp. 2255-2275.

[13] R. Silva, W. Barreto-Souza and G. Cordeiro, “A New Distribution with Decreasing, Increasing and Upside-Down Bathtub Failure Rate,” Computational Statistics & Data Analysis, Vol. 54, 2010, pp. 935-944.

[14] A. Morais and W. Barreto-Souza, “A Compound Class of Weibulland Power Series Distributions,” Computational Statistics and Data Analysis, Vol. 55, 2011, pp. 14101425.

[15] A. Morais, “A Class of Generalized Beta Distributions, Pareto Power Series and Weibull Power Series,” M.s. Thesis, Universidade Federal de Pernambuco, Recife-PE, 2009.

[16] S. Alkarni and A. Oraby, “A Compound Class of Poisson and Lifetime Distributions,” Journal of Statistics Applications & Probability, Vol. 1, 2012, pp. 45-51.

[17] S. Alkarni, “New Family of Logarithmic Lifetime Distributions,” Journal of Mathematics and Statistics, Vol. 8, No. 4, 2012, pp. 435-440.

[18] G. Mudholkar and D. Srivastava, “Exponentiated Weibull Family for Analyzing Bathtub Failure-Rate Data,” IEEE Transaction on Reliability, Vol. 42, 1993, pp. 299-302.

[19] G. Mudholkar, D. Srivastava and M. Freimer, “The Exponentiated Weibull Family,” Journal of Technometrics, Vol. 37, 1995, pp. 436-445.

[20] G. Mudholkar and A. Hutson, “The Exponentiated Weibull Family: Some Properties and a Flood Data Application,” Communications in Statistics, Theory and Methods, Vol. 25, 1996, pp. 3059-3083.

[21] M. Nassar and F. Eissa, “On the Exponentiated Weibull Distribution,” Communications in Statistics, Theory and Methods, Vol. 32, 2003, pp. 1317-1336.

[22] S. Nadarajah and S. Kotz, “The Beta Exponential Distribution,” Reliability Engineering and System Safety, Vol. 91, 2006, pp. 689-697.

[23] W. Barreto-Souza and F. Cribari-Neto, “A Generalization of the Exponential-Poisson Distribution,” Statistics & Probability Letters, Vol. 79, 2009, pp. 2493-2500.

[24] H. Bakouch, M. Ristic, A. Asgharzadah and L. Esmaily, “An Exponentiated Exponential Binomial Distribution with Application,” Statistics and Probability Letters, Vol. 82, 2012, pp. 1067-1081.

[25] R. Barlow, A. Marshall and F. Proschan, “Properties of Probability Distributions with Monotone Hazard Rate,” The Annals of Mathematical Statistics, Vol. 34, 1963, pp. 375-389.

[26] D. Cox and D. Hinkley, “Theoretical Statistics,” Chapman and Hall, London, 1974.

[27] A. Dempster, N. Laird and D. Rubin, “Maximum Likelihood from Incomplete Data via the EM Algorithm,” Journal of The Royal Statistical Society Series B, Vol. 39, 1977, pp. 1-38.

[28] G. McLachlan and T. Krishnan, “The EM Algorithm and Extension,” Wiley, New York, 1997.

[29] R. Little and D. Rubin, “Incomplete Data,” In: S. Kotz and N. L. Johnson, Eds., Encyclopedia of Statistical Sciences, Vol. 4, Wiley, New York, 1983.

[30] K. Adamidis, “An EM Algorithm for Estimating Negative Binomial Parameters,” Austral, New Zealand Statistics, Vol. 41, No. 2, 1999, pp. 213-221.

[31] H. Ng, P. Chan and N. Balakrishnan, “Estimation of Parameters from Progressively Censored Data Using EM Algorithm,” Computational Statistics & Data Analysis, Vol. 39, 2002, pp. 371-386.

[32] D. Karlis, “An EM Algorithm for Multivariate Poisson Distribution and Related Models,” Journal of Applied Statistics, Vol. 301, 2003, pp. 63-77.

[33] C. Shannon, “A Mathematical Theory of Communication,” Bell System Technical Journal, Vol. 27, 1948, pp. 379432.