A New Five Parameter Lifetime Distribution: Properties and Application
DOI:
https://doi.org/10.24297/jam.v13i3.6045Keywords:
T-X family, Moment generating function, Quantile function, Maximum likelihood estimationAbstract
This paper deals with a new generalization of the Weibull distribution. This distribution is called exponentiated exponentiated exponential-Weibull (EEE-W) distribution. Various structural properties of the new probabilistic model are considered, such as hazard rate function, moments, moment generating function, quantile function, skewness, kurtosis, Shannon entropy and Rényi entropy. The maximum likelihood estimates of its unknown parameters are obtained. Finally, areal data set is analyzed and it observed that the present distribution can provide a better fit than some other known distributions.
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References
[1] Jhonson, N. L., Kotz, S., and Balakrishnan, N. (1994). Continuous univariate
distributions. I. Wiley, Newyork.
* A
* Distribution W
W 1.018880 5.20603
Ga 0.402684 2.63673
MW 0.516374 2.62049
TW 0.336408 1.73131
EEEW 0.26702 0.49363
- 11 -
[2] Jhonson, N. L., Kotz, S., and Balakrishnan, N. (1995). Continuous univariate distributions. II. Wiley, Newyork.
[3] Hansen, B. E., (1994). Auto regressive conditional density estimation. Intern-ational Economic Review 35, 705-730.
[4] Azzalini, A., and Capitanio, A. (2003). Distributions generated by perturbation of summetry with emphasis an a multivariate skew t-distribution. Journal of the Royal Statistical Society. Series B 65, 397-389.
[5] Eugene, N., Lee, C., and Famoye, F. (2002). Beta-normal distribution and its applications. Communication in Statistics-Theory and Methods 31, 497-512.
[6] Aas, K., and Haff, I. H. (2006). The generalized hyperbolic skew student’s t-distribution. Journal of Financial Econometrics 4, 275-309.
[7] Gurvich, M. R., Dibenedetto, A. T. and Ranade, S. V. (1997). A new statistical distribution for characterizing the random strength of brittle materials. Journal of Materials Science 32, 2559-2564.
[8] Bebbington, M. S., Lai, C. D. and Zitikis, R. (2007). A flexible Weibull extension. Reliability Engineering and System Safety 92(6), 719-726.
[9] Zhang, T. and Xie, M. (2011). On the upper truncated Weibull distribution and its reliability implications. Reliability Engineering and System Safety 96(1), 194-200.
[10] Xie, M., Tang, Y. and Goh, T. N. (1995). A modified Weibull extension with bathtub-shaped failure rate function. Reliability Engineering and System Safety 52, 87-93.
[11] Carrasco, M., Ortega, E. M. and Cordeiro, G. M. (2008). A generalized modified Weibull distribution for lifetime modeling. Computational Statistics and Data Analysis 53(2), 450-462.
[12] Sarhan, A. M. and Zaindin, M. (2009). Modified Weibull distribution. Applied Sciences 11, 123-136.
[13] Almalki, S. J. and Yuan, J. (2013). A new modified Weibull distribution. Reliability Engineering and System Safety 111, 164-170.
[14] Xie, M. and Lai, C. D. (1995). Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliability Engineering and System Safety 52, 87-93.
- 18 -
[15] Cordeiro, G. M., and de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical.81(7),883-898.
[16] Nadarajah, S. and Kotz, S. (2004). The beta Gumbel distribution. Mathematical Problems in Engineering 10, 323-332.
[17] Nadarajah, S. and Gupta, A. K. (2004). The beta Fréchet distribution. Far East Journal of Theoretical Statistics 15, 15-24.
[18] Nadarajah, S. and Kotz, S. (2005). The beta exponential distribution. Reliability Engineering and System Safety 91, 689-697.
[19] Alzaatreh, A., Lee, C., and Famoye, F. (2013). A new method for generating families of continuous distributions. METRON 71(1), 63-79.
[20] Alzaatreh, A., Famoye, F., and Lee, C. (2013). Weibull-Pareto distribution and its applications. Communications in Statistics-Theory and Methods 42(9), 1673-1691.
[21] Alzaatreh, A., Famoye, F., and Lee, C. (2014). The gamma-normal distribution: properties and applications. Computational Statistics and Data Analysis 69, 67-80.
[22] Alzaghal, A., Famoye, F., and Lee, C. (2013). Exponentiated T-X family of distributions with some applications. International Journal of Statistics and Probability 2(3), 31-49.
[23] Galton, F. (1883) Enquiries on to Human Faculty and its Developments. Macmillan & Company. London.
[24] Moors, J. J. (1998). A quantile alternative for kurtosis. Journal of The Royal Statistical Society. Series D(the statistician) 37, 25-32.
[25] Shannon, C. E. (1948). A Mathematical Theory of Communication .Bell System Technical Journal 27, 379-432.
[26] Rényi, A. (1961). On measures of entropy and information. In Proceedings of the 4th Berkeley symposium on Mathematical Statistics and Probability I, 547-561. Berkeley: Univ. California Press. MR0132570.
[27] Nichols, M. D. and Padgett, W. J. (2006). A boot strap Control for Weibull Percentiles. Quality and Reliability Engineering International 22, 141-151.
distributions. I. Wiley, Newyork.
* A
* Distribution W
W 1.018880 5.20603
Ga 0.402684 2.63673
MW 0.516374 2.62049
TW 0.336408 1.73131
EEEW 0.26702 0.49363
- 11 -
[2] Jhonson, N. L., Kotz, S., and Balakrishnan, N. (1995). Continuous univariate distributions. II. Wiley, Newyork.
[3] Hansen, B. E., (1994). Auto regressive conditional density estimation. Intern-ational Economic Review 35, 705-730.
[4] Azzalini, A., and Capitanio, A. (2003). Distributions generated by perturbation of summetry with emphasis an a multivariate skew t-distribution. Journal of the Royal Statistical Society. Series B 65, 397-389.
[5] Eugene, N., Lee, C., and Famoye, F. (2002). Beta-normal distribution and its applications. Communication in Statistics-Theory and Methods 31, 497-512.
[6] Aas, K., and Haff, I. H. (2006). The generalized hyperbolic skew student’s t-distribution. Journal of Financial Econometrics 4, 275-309.
[7] Gurvich, M. R., Dibenedetto, A. T. and Ranade, S. V. (1997). A new statistical distribution for characterizing the random strength of brittle materials. Journal of Materials Science 32, 2559-2564.
[8] Bebbington, M. S., Lai, C. D. and Zitikis, R. (2007). A flexible Weibull extension. Reliability Engineering and System Safety 92(6), 719-726.
[9] Zhang, T. and Xie, M. (2011). On the upper truncated Weibull distribution and its reliability implications. Reliability Engineering and System Safety 96(1), 194-200.
[10] Xie, M., Tang, Y. and Goh, T. N. (1995). A modified Weibull extension with bathtub-shaped failure rate function. Reliability Engineering and System Safety 52, 87-93.
[11] Carrasco, M., Ortega, E. M. and Cordeiro, G. M. (2008). A generalized modified Weibull distribution for lifetime modeling. Computational Statistics and Data Analysis 53(2), 450-462.
[12] Sarhan, A. M. and Zaindin, M. (2009). Modified Weibull distribution. Applied Sciences 11, 123-136.
[13] Almalki, S. J. and Yuan, J. (2013). A new modified Weibull distribution. Reliability Engineering and System Safety 111, 164-170.
[14] Xie, M. and Lai, C. D. (1995). Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliability Engineering and System Safety 52, 87-93.
- 18 -
[15] Cordeiro, G. M., and de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical.81(7),883-898.
[16] Nadarajah, S. and Kotz, S. (2004). The beta Gumbel distribution. Mathematical Problems in Engineering 10, 323-332.
[17] Nadarajah, S. and Gupta, A. K. (2004). The beta Fréchet distribution. Far East Journal of Theoretical Statistics 15, 15-24.
[18] Nadarajah, S. and Kotz, S. (2005). The beta exponential distribution. Reliability Engineering and System Safety 91, 689-697.
[19] Alzaatreh, A., Lee, C., and Famoye, F. (2013). A new method for generating families of continuous distributions. METRON 71(1), 63-79.
[20] Alzaatreh, A., Famoye, F., and Lee, C. (2013). Weibull-Pareto distribution and its applications. Communications in Statistics-Theory and Methods 42(9), 1673-1691.
[21] Alzaatreh, A., Famoye, F., and Lee, C. (2014). The gamma-normal distribution: properties and applications. Computational Statistics and Data Analysis 69, 67-80.
[22] Alzaghal, A., Famoye, F., and Lee, C. (2013). Exponentiated T-X family of distributions with some applications. International Journal of Statistics and Probability 2(3), 31-49.
[23] Galton, F. (1883) Enquiries on to Human Faculty and its Developments. Macmillan & Company. London.
[24] Moors, J. J. (1998). A quantile alternative for kurtosis. Journal of The Royal Statistical Society. Series D(the statistician) 37, 25-32.
[25] Shannon, C. E. (1948). A Mathematical Theory of Communication .Bell System Technical Journal 27, 379-432.
[26] Rényi, A. (1961). On measures of entropy and information. In Proceedings of the 4th Berkeley symposium on Mathematical Statistics and Probability I, 547-561. Berkeley: Univ. California Press. MR0132570.
[27] Nichols, M. D. and Padgett, W. J. (2006). A boot strap Control for Weibull Percentiles. Quality and Reliability Engineering International 22, 141-151.
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Published
2017-04-30
How to Cite
Dessoky, S. A., & Abd El-Bar, A. M. T. (2017). A New Five Parameter Lifetime Distribution: Properties and Application. JOURNAL OF ADVANCES IN MATHEMATICS, 13(3), 7205–7218. https://doi.org/10.24297/jam.v13i3.6045
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