The Exponentiated Kumaraswamy-G Class: General Properties and Application

Abstract

We propose a new family of distributions called the exponentiated Kumaraswamy-G class with three extra positive parameters, which generalizes the Cordeiro and de Castro's family. Some special distributions in the new class are discussed. We derive some mathematical properties of the proposed class including explicit expressions for the quantile function, ordinary and incomplete moments, generating function, mean deviations, reliability, Rényi entropy and Shannon entropy. The method of maximum likelihood is used to fit the distributions in the proposed class. Simulations are performed in order to assess the asymptotic behavior of the maximum likelihood estimates. We illustrate its potentiality with applications to two real data sets which show that the extended Weibull model in the new class provides a better fit than other generalized Weibull distributions.

Normal-G Class of Probability Distributions: Properties and Applications

“…Setting N = 2n + 1 and a k = 1 for all k ≥ 0, we get to the expression A2 in Equation (15) and we can use the result in Equation (17) to write as follows:…”

“…Since g k−j (x) is the pdf of a random variable of the exponentiated family, as described in [15,16], one can say that (20) is the Normal-G pdf (5) expressed as a linear combination of pdfs of exponentiated distributions. Such useful property is typically found and detailed in works on new classes of distributions; see for instance: [17][18][19][20].…”

“…The second application concerns to a dataset representing waiting times (in seconds) between 65 successive eruptions of water through a hole in the cliff at the coastal town of Kiama (New South Wales, Australia), known as the Blowhole. These data can be obtained in [17,28]. Here are they: 83, 51, 87, 60, 28,95,8,27,15,10,18,16,29,54,91,8,17,55,10,35,47,77,36,17,21,36,18,40,10,7,34,27,28,56,8,25,68,146,89,18,73,69,9,37,10,82,29,8,60,61,61,18,169,25,8,26,…”

“…Nonetheless, in order to pick a more parsimonious model, one should prefer the NLL, since it has fewer parameters than GoLL. It is worth mentioning that [17] proposed the new class Exponentiated Kumaraswamy-G and fitted one of its submodels (with Weibull as baseline) to the same eruption dataset. It presented A * = 0.7594 and W * = 0.1037, whereas NLL presented lower values of these statistics as one can check in Table 8.…”

On the Topp Leone Exponentiated-G Family of Distributions: Properties and Applications

“…The techniques for modifying the classical distributions are usually referred to as generators in literature and are capable of improving the goodness-of-fit of the modified distributions. Some well-known generators are Marshal-Olkin generated family (MO-G) by Marshal and Olkin [1], the Beta-G by Eugene et al [2] and Jones [3], Generalized Beta-generated distributions by Alexander et al [4], Gamma-G (type 1) by Zografos and Balakrishanan [5], Gamma-G (type 2) by Ristic and Balakrishanan [6], Log-gamma-G by Amini et al [7], Exponentiated generalized-G by Cordeiro et al [8], Transformed-Transformer (T-X) by Alzaatreh et al [9], exponentiated (T-X) by Alzaghal et al [10], Weibull-G by Bourguignon et al [11], Exponentiated half logistic generated family by Cordeiro et al [12], Lomax-G by Cordeiro et al [13], Kumaraswamy Odd log-logistic-G by Alizadeh et al [14], Kumaraswamy Marshall-Olkin by Alizadeh et al [15], Beta Marshall-Olkin by Alizadeh et al [16], Kummer-beta generalized distributions by Pescimet al [17], A new family of Marshall-Olkin extended distributions by Alshangiti et al [18], A new family of distributions: Libby-Novick beta by Cordeiro et al [19], Type 1 Half-Logistic family of distributions by Cordeiro et al [20], The generalized transmuted-G family by Nofal et al [21], Generalized transmuted family by Alizadeh et al [22], Another generalized transmuted family by Merovci et al [23], Transmuted exponentiated generalized-G family by Yousof et al [24], Transmuted geometric G family by Afify et al [25], Beta transmuted-H family by Afify et al [26], Kumaraswamy transmuted-G family by Afify et al [27], Topp-Leone Family of Distributions by Al-Shomrani et al [28], The transmuted transmuted-G family by Mansour et al [29], The Exponentiated Kumaraswamy-G Class by Silva et al [30], The extended Weibull-G family of distributions by Korkmaz…”