Abstract
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.
Highlights
Within the last two centuries, various methods for generating continuous univariate distributions have been put forward in the literature. These methods include the method based on differential equations (Pearson [1]; Burr [2]), method based on transformation (Johnson [3]), method based on quantiles (Tukey [4]; Aldeni et al [5]), method for generating skewed distributions (Azzalini [6]), method of addition of parameter(s) and generalization (Mudholkar and Srivastava [7]; Marshall and Olkin [8]; Shaw and Buckley [9]), method of compounding the continuous univariate distributions and the discrete univariate distributions (Adamidis and Loukas [10]), method based on generators (Eugene et al [11]; Jones [12]; Cordeiro and de Castro [13]), method based on the composition of densities (Cooray and Ananda [14]) and the Transformed–Transformer method (Alzaatreh et al [15]; Alzaatreh et al [16])
Summary and conclusion A new family of probability distributions called the T–R {Y}—power series family of distributions has been introduced in this paper
The new family was realized by compounding the T–R {Y} family of distribution and the power series family
Summary
Within the last two centuries, various methods for generating continuous univariate distributions have been put forward in the literature. : the cdf of the T–R {Y}–power series (T–R {Y}–PS) family of distributions is given by FT ðQY ð CðθÞ The survival and hazard functions of the T–R {Y}–PS family of distributions are given respectively by 1⁄2θð1− FT ðQY ð FRðxÞÞÞÞ f X C1⁄2θð1− FT ðQY ð FRðxÞÞÞÞ
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