Exponentiated Gumbel Family Of Distributions: Properties And Applications

ABSTRACT

In this dissertation, a new family of distribution called the Exponentiated Gumbel-G (EGu-G) family of distributions was developed using the T – X approach proposed by Alzaatreh et al., (2013a). The probability density function (PDF) of exponentiated Gumbel distribution was used as the generator while the logit of the Cumulative Distribution Function (CDF) of any continuous random variable is the transformation function. The PDF, CDF, Survival Function (SF) and Hazard Rate Function (HRF)) of the new family was explicitly defined. Various properties of the proposed family were investigated. The PDF of the new family was expressed as an infinite linear combination of exponentiated – G distribution of the baseline distribution. Its bivariate extension of the proposed family was derived while estimation of the parameters of the family were discussed based on Maximum Likelihood Estimation (MLE) method. Taking the baseline distribution as Exponential, Power, Lomax and Weibull distributions, we obtained Exponentiated Gumbel Exponential (EGuE), Exponentiated Gumbel Power (EGuP),  Exponentiated Gumbel Lomax (EGuL) and Exponentiated Gumbel Weibull (EGuW) distributions respectively as members of the EGu – G family. Shapes of HRF obtainable from these members of the family include increasing, decreasing, bathtub and inverted bathtub shaped. The properties of EGuL and EGuW were studied. The effect of the shape parameters on the shape of the studied members was investigated using quantile based measures of coefficient of skewness and kurtosis. A simulation study was carried out on MLEs of parameters of the EGuL and EGuW distribution to ascertain their stability. The potentiality of the EGu-G family was illustrated using EGuL and EGuW, through the applications to four different datasets.