A Six Parameters Beta Distribution with Application for Modeling Waiting Time of Muslim Early Morning Prayer

Abstract

Beta distribution is a well-known and widely used distribution for modeling and analyzing lifetime data, due to its interesting characteristics. In this paper, a six parameters beta distribution is introduced as a generalization of the two (standard) and the four parameters beta distributions. This distribution is closed under scaling and exponentiation, and has reflection symmetry property, has some well-known distributions as special cases, such as, the two and four parameters beta, generalized modification of the Kumaraswamy, generalized beta of the first kind, the power function, Kumaraswamy power function, Minimax, exponentiated Pareto, and the generalized uniform distributions. Its moments about the origin, moment generating function, incomplete moments, mean deviations, are derived. The maximum likelihood estimation method is used for estimating its parameters and applied to estimate the parameters of the six different simulated data sets of this distribution, in order to check the performance of the estimation method through the estimated parameters mean squares errors computed from the different simulated sample sizes. Finally, two real life data sets, represent the waiting period of Muslim worshipers from the time of entering the mosque till the actual time of starting Alfajir pray in two different mosques, were used to illustrate the usefulness and the flexibility of this distribution, as well as, presents better fitting than the other gamma, exponential, the four parameters beta, and the generalized beta of the first kind distributions