ABSTRACT

In this study, a new family of distributions called the Gumbel Marshall Olkin-G (GMO-G) family of distributions was developed using Transformed – Transformer ( T-X ) method of generating family of distributions. The probability density function (pdf) of Gumbel distribution was used as the generator while the log- logit of the Marshall Olkin family of distributions was the transformation function. The pdf, cdf, Survival Function (sf) and Hazard Rate Function (hrf)) of the new family of distributions were defined. The properties of the proposed family were derived and studied. The pdf of the new family was expressed as an infinite linear combination of exponentiated – G distribution of the baseline distribution. The bivariate extension of the proposed family was derived while estimation of the parameters of the family was discussed based on Maximum Likelihood Estimation (MLE) method. The sub-models of the new family of distributions, namely Gumbel Marshall Olkin Exponential distribution (GMO-E), Gumbel Marshall Olkin Normal distribution (GMO-N), and Gumbel Marshall Olkin Weibull (GMO-W) were derived. The plots of pdf and hrf of GMO-E and   GMO-N were illustrated, and GMO-W properties studied. Shapes of hrf obtainable from these members of the family include increasing, decreasing, constant, right-skewed, left-skewed, bathtub shaped, reversed bathtub shaped, and reversed J-shaped. More so, the shapes of the pdf include increasing, decreasing, constant, unimodal, bimodal, symmetric, right-skewed, and left-skewed shaped. A simulation study was carried out on MLEs of parameters of the GMO-W distribution to ascertain its stability. The potentiality of the GMO-G family was illustrated by applying three different data sets each on GMO-W, the results from the goodness of fit statistics showcased that GMO-W provided a better fit amongst the competing baseline distributions.