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Exponentiated Kumaraswamy-Dagum Distribution with Applications to Income and Lifetime Data Shujiao Huang and Broderick O Oluyede*

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Exponentiated Kumaraswamy-Dagum Distribution with Applications to Income and Lifetime Data Shujiao Huang and Broderick O Oluyede* Huang and Oluyede Journal of Statistical Distributions and Applications 2014, 1:8 http://www.jsdajournal.com/content/1/1/8 RESEARCH Open Access Exponentiated Kumaraswamy-Dagum distribution with applications to income and lifetime data Shujiao Huang and Broderick O Oluyede* *Correspondence: [email protected] Abstract Department of Mathematical A new family of distributions called exponentiated Kumaraswamy-Dagum (EKD) Sciences, Georgia Southern University, Statesboro, GA 30458, distribution is proposed and studied. This family includes several well known USA sub-models, such as Dagum (D), Burr III (BIII), Fisk or Log-logistic (F or LLog), and new sub-models, namely, Kumaraswamy-Dagum (KD), Kumaraswamy-Burr III (KBIII), Kumaraswamy-Fisk or Kumaraswamy-Log-logistic (KF or KLLog), exponentiated Kumaraswamy-Burr III (EKBIII), and exponentiated Kumaraswamy-Fisk or exponentiated Kumaraswamy-Log-logistic (EKF or EKLLog) distributions. Statistical properties including series representation of the probability density function, hazard and reverse hazard functions, moments, mean and median deviations, reliability, Bonferroni and Lorenz curves, as well as entropy measures for this class of distributions and the sub-models are presented. Maximum likelihood estimates of the model parameters are obtained. Simulation studies are conducted. Examples and applications as well as comparisons of the EKD and its sub-distributions with other distributions are given. Mathematics Subject Classification (2000): 62E10; 62F30 Keywords: Dagum distribution; Exponentiated Kumaraswamy-Dagum distribution; Maximum likelihood estimation 1 Introduction Camilo Dagum proposed the distribution which is referred to as Dagum distribution in 1977. This proposal enable the development of statistical distributions used to fit empir- ical income and wealth data, that could accommodate heavy tails in income and wealth distributions. Dagum’s proposed distribution has both Type-I and Type-II specification, where Type-I is the three parameter specification and Type-II deals with four parameter specification. This distribution is a special case of generalized beta distribution of the sec- ond kind (GB2), McDonald (1984), McDonald and Xu (1995), when the parameter q = 1, where the probability density function (pdf) of the GB2 distribution is given by: ayap−1 f (y; a, b, p, q) = ,fory > 0. GB2 ap + y a p+q b B(p, q)[1 b ] See Kleiber and Kotz (2003) for details. Note that a > 0, p > 0, q > 0, are the = (p)(q) shape parameters and b is the scale parameter and B(p, q) (p+q) is the beta func- tion. Kleiber (2008) traced the genesis of Dagum distribution and summarized several © 2014 Huang and Oluyede; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. Huang and Oluyede Journal of Statistical Distributions and Applications 2014, 1:8 Page 2 of 20 http://www.jsdajournal.com/content/1/1/8 statistical properties of this distribution. Domma et al. (2011) obtained the maximum like- lihood estimates of the parameters of Dagum distribution for censored data. Domma and Condino (2013) presented the beta-Dagum distribution. Cordeiro et al. (2013) proposed the beta exponentiated Weibull distribution. Cordeiro et al. (2010) introduced and stud- ied some mathematical properties of the Kumaraswamy Weibull distribution. Oluyede and Rajasooriya (2013) developed the Mc-Dagum distribution and presented its statistical properties. See references therein for additional results. The pdf and cumulative distribution function (cdf) of Dagum distribution are given by: − − −δ−1 −δ β 1 gD(x; λ, β, δ) = βλδx 1 + λx (1) and −δ −β GD(x; λ, β, δ) = 1 + λx ,(2) for x > 0, where λ is a scale parameter, δ and β are shape parameters. Dagum (1977) refers to his model as the generalized logistic-Burr distribution. The kth raw or non central moments are given by k k k E Xk = βλδ B β + ,1− , δ δ for k <δ,andλ, β>0, where B(·, ·) is the beta function. The qth percentile is − 1 1 − 1 δ xq = λ δ q β − 1 . In this paper, we present generalizations of the Dagum distribution via Kumaraswamy distribution and its exponentiated version. This leads to the exponentiated Kumaraswamy Dagum distribution. The motivation for the development of this distribution is the modeling of size dis- tribution of personal income and lifetime data with a diverse model that takes into consideration not only shape, and scale but also skewness, kurtosis and tail variation. Also, the EKD distribution and its sub-models has desirable features of exhibiting a non-monotone failure rate, thereby accommodating different shapes for the hazard rate function and should be an attractive choice for survival and reliability data analysis. This paper is organized as follows. In section 3, we present the exponentiated Kumaraswamy-Dagum distribution and its sub models, as well as series expansion, haz- ard and reverse hazard functions. Moments, moment generating function, Lorenz and Bonferroni curves, mean and median deviations, and reliability are obtained in section 4. Section 5 contains results on the distribution of the order statistics and Renyi entropy. Estimation of model parameters via the method of maximum likelihood is presented in section 6. In section 7, various simulations are conducted for different sample sizes. Section 8 contains examples and applications of the EKD distribution and its sub-models, followed by concluding remarks. 2 Methods, results and discussions Methods, results and discussions for the class of EKD distributions are presented in sections 3 to 8. These sections include the sub-models, series expansion of the pdf, closed form expressions for the hazard and reverse hazard functions, moments, moment gen- erating function, Bonferroni and Lorenz curves, reliability, mean and median deviations, Huang and Oluyede Journal of Statistical Distributions and Applications 2014, 1:8 Page 3 of 20 http://www.jsdajournal.com/content/1/1/8 distribution of order statistics and entropy, as well as estimation of model parameters and applications. 3 The exponentiated Kumaraswamy-Dagum distribution In this section, we present the proposed distribution and its sub-models. Series expansion, hazard and reverse hazard functions are also studied in this section. 3.1 Kumaraswamy-Dagum distribution Kumaraswamy (1980) introduced a two-parameter distribution on (0, 1).Itscdfisgiven by φ G(x) = 1 − 1 − xψ , x ∈ (0, 1), for ψ>0andφ>0. = dF(x) For an arbitrary cdf F(x) with pdf f (x) dx , the family of Kumaraswamy-G distributions with cdf Gk(x) is given by ψ φ GK (x) = 1 − 1 − F (x) , for ψ>0andφ>0. By letting F(x) = GD(x), we obtain the Kumaraswamy-Dagum (KD) distribution, with cdf φ = − − ψ GKD(x) 1 1 GD (x) . 3.2 The EKD distribution θ In general, the EKD distribution is GEKD(x) = [FKD(x)] ,whereFKD(x) is a base- line (Kum-Dagum) cdf, θ>0, with the corresponding pdf given by gEKD(x) = θ−1 θ[FKD(x)] fKD(x). For large values of x,andforθ>1(< 1), the multiplicative factor θ−1 θ[FKD(x)] > 1(< 1), respectively. The reverse statement holds for smaller values of x. Consequently, this implies that the ordinary moments of gEKD(x) are larger (smaller) than those of fKD(x) when θ>1(< 1). Replacing the dependent parameter βψ by α, the cdf and pdf of the EKD distribution are given by − φ θ = − − + −δ α GEKD (x; α, λ, δ, φ, θ) 1 1 1 λx ,(3) and − − − φ−1 = −δ−1 + −δ α 1 − + −δ α gEKD (x; α, λ, δ, φ, θ) αλδφθx 1 λx 1 1 λx θ−1 − −α φ × 1 − 1 − 1 + λx δ ,(4) for α, λ, δ, φ, θ>0, and x > 0, respectively. The quantile function of the EKD distribution is in closed form, ⎧ ⎫− 1 − 1 δ ⎨ 1 α ⎬ − 1 1 φ G 1 (q) = x = λ δ 1 − 1 − q θ − 1 .(5) EKD q ⎩ ⎭ Plots of the pdf for some combinations of values of the model parameters are given in Figure 1. The plots indicate that the EKD pdf can be decreasing or right skewed. The EKD distribution has a positive asymmetry. Huang and Oluyede Journal of Statistical Distributions and Applications 2014, 1:8 Page 4 of 20 http://www.jsdajournal.com/content/1/1/8 Figure 1 Graph of pdfs. 3.3 Some sub-models Sub-models of EKD distribution for selected values of the parameters are presented in this section. 1. When θ = 1, we obtain Kumaraswamy-Dagum distribution with cdf: − −α φ G(x; α, λ, δ, φ) = 1 − 1 − 1 + λx δ , for α, λ, δ, φ>0 and x > 0. 2. When φ = θ = 1, we obtain Dagum distribution with cdf: − −α G(x; α, λ, δ) = 1 + λx δ , for α, λ, δ>0 and x > 0. 3. When λ = 1, we obtain exponentiated Kumaraswamy-Burr III distribution with cdf: θ − −α φ G(x; α, δ, φ, θ) = 1 − 1 − 1 + x δ , for α, δ, φ, θ>0 and x > 0. 4. When λ = θ = 1, we obtain Kumaraswamy-Burr III distribution with cdf: − −α φ G(x; α, δ, φ) = 1 − 1 − 1 + x δ , for α, δ, φ>0 and x > 0. 5. When λ = φ = θ = 1, we obtain Burr III distribution with cdf: − −α G(x; α, δ) = 1 + x δ , for α, δ>0 and x > 0. 6. When α = 1, we obtain exponentiated Kumaraswamy-Fisk or Kumaraswamy-Log-logistic distribution with cdf: θ − −1 φ G(x; λ, δ, φ, θ) = 1 − 1 − 1 + λx δ , for λ, δ, φ, θ>0 and x > 0. Huang and Oluyede Journal of Statistical Distributions and Applications 2014, 1:8 Page 5 of 20 http://www.jsdajournal.com/content/1/1/8 7.
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