In section 4 we showed the applications by analyzing a set of real time data and compared the results with similar three-parameter distributions. In section 3 we described the parameter estimation procedure using maximum likelihood method and method of moments. The rest of the article is organized as follows: Section 2 introduces the generalized exponential distribution and some its important properties. A real time numerical example is analyzed and compared with four other similar distributions. A sufficient condition for the existence of unique solution for the parameters estimated by the method of moments is derived.
In this article we have considered the estimation of parameters of the three-parameter generalized exponential distribution introduced by Hossain and Ahsanullah by using the maximum likelihood estimation and the method of moments. This distribution approaches a two parameter exponential distribution when the shape parameter approaches zero, whereas the distribution in Gupta and Kundu approaches two-parameter exponential if the shape parameter approaches one. Hossain and Ahsanullah introduced a new three-parameter generalized exponential distribution that represents a different type of generalization than Gupta and Kundu. They reanalyzed one set of data and the results have shown that the bivariate generalized exponential distribution provides a better fit than the bivariate exponential distribution. Recently, Kundu and Gupta introduced the bivariate generalized exponential distribution so that the marginal have generalized exponential distributions. One of its drawbacks being it is less flexible than the other two families for graduating tail thickness. Since its distribution function has closed form the inference based on the censored data can be handled more easily than gamma family. This distribution is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava, and Freimer. Ī three-parameter generalized exponential distribution was suggested by Gupta and Kundu. For more on exponentiated Weibull, beta Gumbel, and beta exponentiated distribution see Nadarajah, Nadarajah and Kotz, Nadarajah and Kotz, Nassar and Eissa, and Raqab and Ahsanullah. Gupta proposed special cases of the exponentiated Weibull and exponentiated exponential models and compared their performances with the two-parameter gamma family and two-parameter Weibull family, mainly through data analysis and computer simulations. The usefulness and flexibility of the family is illustrated by reanalyzing five classical data sets on bus-motor failures. It is a right skewed unimodal density function. This new family is suitable for modeling data that indicate non-monotone hazard rates and can be adopted for testing goodness of fit of Weibull as a submodel.
Mudholkar considered a three-parameter exponentiated Weibull distribution.
For more information on these distributions see Alexander and Jackson. These distributions have many desirable statistical properties. The three-parameter gamma and Weibull distributions are commonly used in life time data analysis. Let $$X_1, \ldots, X_n \overset$ is unbiased.The estimation of parameters and drawing conclusions based on the estimated parameters is one of the important aspects of inferential statistics.
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