A new family of univariate probability distributions called the T − R {Y} power series family of probability distributions is introduced in this paper by compounding the T − R {Y} family of distributions and the power series family of discrete distributions. A treatment of the general mathematical properties of the new family is carried out and some sub-families of the new family are specified to depict the broadness of the new family. The maximum likelihood method of parameter estimation is suggested for the estimation of the parameters of the new family of distributions. A special member of the new family called the Gumbel–Weibull–{logistic}–Poisson (GUWELOP) distribution is defined and found to exhibit both unimodal and bimodal shapes. The GUWELOG distribution is further applied to a real multi-modal data set to buttress its applicability.
Six methods for estimating the Weibull shape and scale parameters are considered and compared in this paper. These methods are: the least squares method, weighted least squares method, method of moments, energy pattern factor method, method of L-moments and the maximum likelihood method. A simulation study as well as application to a real data set (wind speeds sample) is used to test the performance of the different methods using the smallest mean square error criterion. Results from the simulation study indicate that the maximum likelihood method is the most efficient method when dealing with large sample sizes, while the weighted least squares method, method of moments and the method of L-moments are quite efficient for small and moderate sample sizes. The maximum likelihood method produced the best method when all six methods were applied to a wind speeds sample by possessing the smallest mean square error. A very useful result obtained from the study is that the weighted least squares method, performed considerably well in estimating the Weibull parameters. This is a rare incidence in many studies.
The so-called Kumaraswamy distribution is a special probability distribution developed to model doubled bounded random processes for which the mode do not necessarily have to be within the bounds. In this article, a generalization of the Kumaraswamy distribution called the T-Kumaraswamy family is defined using the T-R {Y} family of distributions framework. The resulting T-Kumaraswamy family is obtained using the quantile functions of some standardized distributions. Some general mathematical properties of the new family are studied. Five new generalized Kumaraswamy distributions are proposed using the T-Kumaraswamy method. Real data sets are further used to test the applicability of the new family.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.