Generalized Exponential Distribution
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Probability density function ![]() | |||
Parameters |
(shape ) (rate) (or inverse scale) | ||
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Support | |||
CDF | |||
Quantile | |||
Mean | |||
Variance | |||
Skewness | |||
MGF |
In the theory of probability and statistics, the Generalized Exponential (GE) distribution is a two-parameter family of continuous probability distribution. This distribution was introduced by Gupta and Kundu[1] in a 1999 paper and arises as a special case of the three-parameter exponentiated Weibull distribution that was originally introduced by Mudholkar and Srivastava (1994).[2]
Definition[edit]
A continuous random variable is said to follow the Generalized Exponential (GE) distribution with parameters and if its CDF is given as[1]
Alternative Parametrization[edit]
The GE distribution offers an alternative parameterization utilizing the scale parameter , which is defined as the inverse of the rate parameter .
The CDF and PDF of the GE distribution, expressed with respect to the scale parameter, are given as
Property[edit]
Hazard Function[edit]
The hazard function of the GE distribution is given by[1]
The GE distribution has varying hazard rate depending on the value of the shape parameter. The hazard function is decreasing for , constant for , and increasing for .
Moment Generating Function and Moments[edit]
The moment generating function (MGF) of the GE distribution is given by[1]
and differentiating the log of the MGF with respect to t repeatedly and then setting t = 0, we can get the expectation, variance, and skewness of GE distribution as[1]
where is the polygamma function of order ; for , it denotes the digamma function.
Information Matrix[edit]
If random variable, then the log-likelihood function with respect to the data is given as
Use in Literature[edit]
With its shape and rate parameters, GE Distribution felicitates more rigorous skewness attributes than many other distributions. This is why it is considered to be a better choice as a potential flexible model to incorporate high positive skewness in the data. Additionally, depending on the shape parameter, the varying nature of the hazard function provides GE distribution a lot more compatibility in fitting into complex data structures. Numerous researchers have modeled the experimental data using GE distribution across several disciplines. For instance, Iwona et al. (2015)[3] studies the usefulness of GE distribution in Flood frequency analysis (FFA) in fitting flood extremes data. Madi and Raqab (2007)[4] uses GE distribution for the Bayesian prediction of rainfall data. Kannan, Kundu, and Nair (2010)[5] considers the cure rate model based on the GE distribution. Maximov, Rivas-Davalos, and Cota-Felix (2009)[6] uses GE distribution to evaluate the mean life and standard deviation of power system equipment with limited end-of-life failure data. Al-Nasser and Gogah (2017)[7] uses GE distribution for developing reliability test plans with median ranked set sampling theory. Biondi et al.[8] studies a new stochastic model for episode peak and duration in the context of eco-hydro-climatic studies with the assumption that the conditional distribution of the peak value of an episode follows a GE distribution.
References[edit]
- ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Gupta, Rameshwar D.; Kundu, Debasis (June 1999). "Generalized exponential distributions". Australian New Zealand Journal of Statistics. 41 (2): 173–188. doi:10.1111/1467-842X.00072. ISSN 1369-1473. Unknown parameter
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ignored (help) - ↑ "Exponentiated Weibull family for analyzing bathtub failure-rate data". Microelectronics Reliability. 34 (12): 1959. December 1994. doi:10.1016/0026-2714(94)90359-x. ISSN 0026-2714.
- ↑ Markiewicz, Iwona; Strupczewski, Witold G.; Bogdanowicz, Ewa; Kochanek, Krzysztof (2015-12-10). "Generalized Exponential Distribution in Flood Frequency Analysis for Polish Rivers". PLOS ONE. 10 (12): e0143965. Bibcode:2015PLoSO..1043965M. doi:10.1371/journal.pone.0143965. ISSN 1932-6203. PMC 4684336. PMID 26657239.
- ↑ Madi, Mohamed T.; Raqab, Mohammad Z. (2007). "Bayesian prediction of rainfall records using the generalized exponential distribution". Environmetrics. 18 (5): 541–549. Bibcode:2007Envir..18..541M. doi:10.1002/env.826. Unknown parameter
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ignored (help) - ↑ Kannan, Nandini; Kundu, Debasis; Nair, P.; Tripathi, R. C. (2010-09-21). "The generalized exponential cure rate model with covariates". Journal of Applied Statistics. 37 (10): 1625–1636. Bibcode:2010JApSt..37.1625K. doi:10.1080/02664760903117739. ISSN 0266-4763. Unknown parameter
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ignored (help) - ↑ Cota-Felix, J. E.; Rivas-Davalos, F.; Maximov, S. (2009). "An alternative method for estimating mean life of power system equipment with limited end-of-life failure data". 2009 IEEE Bucharest PowerTech. IEEE. pp. 1–4. doi:10.1109/ptc.2009.5281863. ISBN 978-1-4244-2234-0. Unknown parameter
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ignored (help) Search this book on - ↑ Al-Nasser, Amjad D.; Gogah, Fatima S. (2017-12-01). "On Using The Median Ranked Set Sampling for Developing Reliability Test Plans Under Generalized Exponential Distribution". Pakistan Journal of Statistics and Operation Research. 13 (4): 757. doi:10.18187/pjsor.v13i4.1721. ISSN 2220-5810.
- ↑ Biondi, Franco; Kozubowski, Tomasz J.; Panorska, Anna K.; Saito, Laurel (2008). "A new stochastic model of episode peak and duration for eco-hydro-climatic applications". Ecological Modelling. 211 (3–4): 383–395. Bibcode:2008EcMod.211..383B. doi:10.1016/j.ecolmodel.2007.09.019. ISSN 0304-3800.
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