Abstract
In this paper, we start with a distribution of a random sum of N independent exponential random variables when the random number N has a discrete uniform distribution with positive integer parameter k and show that the probability density function (pdf) and cumulative distribution function (cdf) of this new distribution remain pdf and cdf for any positive real number k. This distribution is proportional to the upper tail probability of the gamma distribution. The hazard rate function of the new model can be constant, decreasing, and increasing depending on its parameters. The statistical properties of the distribution are studied. Estimation of the unknown parameters are discussed by the moments and maximum likelihood methods. The usefulness of the distribution is illustrated by two real datasets.
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References
Adamidis K, Loukas S (1998) A lifetime distribution with decreasing failure rate. Stat Prob Lett 39:35–42
Bakouch HS, Aghababaei Jazi M, Nadarajah S, Dolati A, Roozegar R (2014) A lifetime model with increasing failure rate. Appl Math Model 38:5392–5406
Cancho VG, Louzada-Neto F, Barriga GDC (2011) The poisson-exponential lifetime distribution. Comput Stat Data Anal 55(1):677–686
Casella G, Berger RL (2002) Statistical inference, 2nd edn. Duxbury, Pacific Grove
Carlton MA, Devore JL (2014) Probability with applications in engineering, Science, and technology. Springer, New York
Finkelstein M (2008) Failure rate modeling for reliability and risk. Springer, London
Gleser LJ (1989) The gamma distribution as a mixture of exponential distributions. Am Stat 43:115–117
Gupta RD, Kundu D (1999) Generalized exponential distributions. Aust NZ J Stat 41:173–188
Gupta RD, Kundu D (2001) Exponentiated exponential family: an alternative to gamma and Weibull distributions. Biom J 33:117–130
Klugman SA, Panjer H, Willmot G (2004) Loss models: from data to Decisions. Wiley, New York
Kus C (2007) A new lifetime distribution. Comput Stat Data Anal 51:4497–4509
McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management: concepts, techniques and tools. Princeton University Press, New Jersey
Marshall AW, Olkin I (2007) Life distributions. Springer, New York
Nadarajah S, Haghighi F (2011) An extension of the exponential distribution. Statistics 45(6):543–558
Proschan F (1963) Theoretical explanation of observed decreasing failure rate. Technometrics 5:375–383
Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer Verlag, New York
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We would like to thank the two referees for their valuable comments and suggestions which greatly improved the paper.
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Habibi, M., Asgharzadeh, A. Exponential–Uniform Distribution. Iran J Sci Technol Trans Sci 42, 1439–1450 (2018). https://doi.org/10.1007/s40995-017-0222-0
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DOI: https://doi.org/10.1007/s40995-017-0222-0
Keywords
- Collective risk model
- Exponential distribution
- Maximum likelihood estimator
- Standby system
- Uniform distribution