- Research
- Open Access
Exponentiated Kumaraswamy-Dagum distribution with applications to income and lifetime data
- Shujiao Huang^{1} and
- Broderick O Oluyede^{1}Email author
https://doi.org/10.1186/2195-5832-1-8
© Huang and Oluyede; licensee Springer. 2014
- Received: 22 November 2013
- Accepted: 11 March 2014
- Published: 16 June 2014
Abstract
A new family of distributions called exponentiated Kumaraswamy-Dagum (EKD) distribution is proposed and studied. This family includes several well known 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
See Kleiber and Kotz (2003) for details. Note that a > 0,p > 0,q > 0, are the shape parameters and b is the scale parameter and $B(p,q)=\frac{\Gamma \left(p\right)\Gamma \left(q\right)}{\Gamma (p+q)}$ is the beta function. Kleiber (2008) traced the genesis of Dagum distribution and summarized several statistical properties of this distribution. Domma et al. (2011) obtained the maximum likelihood 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 studied 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.
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 distribution 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, hazard 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 generating function, Bonferroni and Lorenz curves, reliability, mean and median deviations, 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
for ψ > 0 and ϕ > 0.
3.2 The EKD distribution
In general, the EKD distribution is G_{ EKD }(x) = [ F_{ KD }(x)]^{ θ }, where F_{ KD }(x) is a baseline (Kum-Dagum) cdf, θ>0, with the corresponding pdf given by g_{ EKD }(x) = θ [ F_{ KD }(x)]^{θ-1}f_{ KD }(x). For large values of x, and for θ > 1(< 1), the multiplicative factor θ [ F_{ KD }(x)]^{θ-1}> 1(< 1), respectively. The reverse statement holds for smaller values of x. Consequently, this implies that the ordinary moments of g_{ EKD }(x) are larger (smaller) than those of f_{ KD }(x) when θ > 1(< 1).
3.3 Some sub-models
- 1.When θ = 1, we obtain Kumaraswamy-Dagum distribution with cdf:$G(x;\alpha ,\lambda ,\delta ,\varphi )=1-{\left[\phantom{\rule{0.3em}{0ex}}1-{\left(1+\lambda {x}^{-\delta}\right)}^{-\alpha}\right]}^{\varphi},$
- 2.When ϕ = θ = 1, we obtain Dagum distribution with cdf:$G(x;\alpha ,\lambda ,\delta )={\left(1+\lambda {x}^{-\delta}\right)}^{-\alpha},$
- 3.When λ = 1, we obtain exponentiated Kumaraswamy-Burr III distribution with cdf:$G(x;\alpha ,\delta ,\varphi ,\theta )={\left\{1-{\left[\phantom{\rule{0.3em}{0ex}}1-{\left(1+{x}^{-\delta}\right)}^{-\alpha}\right]}^{\varphi}\right\}}^{\theta},$
- 4.When λ = θ = 1, we obtain Kumaraswamy-Burr III distribution with cdf:$G(x;\alpha ,\delta ,\varphi )=1-{\left[1-{\left(1+{x}^{-\delta}\right)}^{-\alpha}\right]}^{\varphi},$
- 5.When λ = ϕ = θ = 1, we obtain Burr III distribution with cdf:$G(x;\alpha ,\delta )={\left(1+{x}^{-\delta}\right)}^{-\alpha},$
- 6.When α = 1, we obtain exponentiated Kumaraswamy-Fisk or Kumaraswamy-Log-logistic distribution with cdf:$G(x;\lambda ,\delta ,\varphi ,\theta )={\left\{1-{\left[1-{\left(1+\lambda {x}^{-\delta}\right)}^{-1}\right]}^{\varphi}\right\}}^{\theta},$
- 7.When α = θ = 1, we obtain Kumaraswamy-Fisk or Kumaraswamy-Log-logistic distribution with cdf:$G(x;\lambda ,\delta ,\varphi )=1-{\left[1-{\left(1+\lambda {x}^{-\delta}\right)}^{-1}\right]}^{\varphi},$
- 8.When α = ϕ = θ = 1, we obtain Fisk or Log-logistic distribution with cdf:$G(x;\lambda ,\delta )={\left(1+\lambda {x}^{-\delta}\right)}^{-1},$
for λ,δ > 0 and x > 0.
3.4 Series expansion of EKD distribution
for b > 0 and |z| < 1, to obtain the series expansion of the EKD distribution.
where $\omega (i,j)=\alpha \lambda \delta \varphi \theta \frac{{(-1)}^{i+j}\Gamma (\theta )\Gamma (\varphi i+\varphi )}{\Gamma (\theta -i)\Gamma (\varphi i+\varphi -j)i!j!}$.
Note that in the Dagum (α,δ,λ) distribution, α and δ are shape parameters, and λ is a scale parameter. In the Exponentiated-Kumaraswamy (ψ,ϕ,θ) distribution, ψ is a skewness parameter, ϕ is a tail variation parameter, and the parameter θ characterizes the skewness, kurtosis, and tail of the distribution.
Consequently, for the EKD (α,λ,δ,ϕ,θ) distribution, α is shape and skewness parameter, δ is shape parameter, λ is a scale parameter, ϕ is a tail variation parameter, and the parameter θ characterizes the skewness, kurtosis, and tail of the distribution.
3.5 Hazard and reverse hazard function
4 Moments, moment generating function, Bonferroni and Lorenz curves, mean and median deviations, and reliability
In this section, we present the moments, moment generating function, Bonferroni and Lorenz curves, mean and median deviations as well as the reliability of the EKD distribution. The moments of the sub-models can be readily obtained from the general results.
4.1 Moments and moment generating function
where $\omega (i,j,s)=\alpha {\lambda}^{\frac{s}{\delta}}\varphi \theta \frac{{(-1)}^{i+j}\Gamma (\theta )\Gamma (\varphi i+\varphi )}{\Gamma (\theta -i)\Gamma (\varphi i+\varphi -j)i!j!}$, and s < δ.
for r < δ.
4.2 Bonferroni and Lorenz curves
for δ > 1, where t (a) = (1 + λ a^{-δ})^{-1}, and ${B}_{G\left(x\right)}(c,d)=\underset{0}{\overset{G\left(x\right)}{\int}}{t}^{c-1}{(1-t)}^{d-1}\mathit{\text{dt}}$ for |G (x)| < 1 is incomplete Beta function.
4.3 Mean and median deviations
respectively. The mean μ is obtained from equation (10) with s=1, and the median M is given by equation (5) when $q=\frac{1}{2}$.
4.4 Reliability
where $\zeta (i,j,k,l)={\alpha}_{1}{\lambda}_{1}{\delta}_{1}{\varphi}_{1}{\theta}_{1}\frac{{(-1)}^{i+j+k+l}\Gamma \left({\theta}_{1}\right)\Gamma ({\varphi}_{1}i+{\varphi}_{1})\Gamma ({\theta}_{2}+1)\Gamma ({\varphi}_{2}k+1)}{\Gamma ({\theta}_{1}-i)\Gamma ({\varphi}_{1}i+{\varphi}_{1}-j)\Gamma ({\theta}_{2}+1-k)\Gamma ({\varphi}_{2}k+1-l)i!j!k!l!}$.
5 Order statistics and entropy
In this section, the distribution of the k^{ th }order statistic and Renyi entropy (Renyi 1960) for the EKD distribution are presented. The entropy of a random variable is a measure of variation of the uncertainty.
5.1 Order statistics
where $K(i,j,p,k)=\frac{{(-1)}^{i+j+p}\Gamma (n-k+1)\Gamma (\theta k+\theta i)\Gamma (\varphi j+\varphi )}{\Gamma (n-k+1-i)\Gamma (\theta k+\theta i-j)\Gamma (\varphi j+\varphi -p)i!j!p!}k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)\alpha \lambda \delta \varphi \theta $.
5.2 Entropy
for $\alpha \tau +\mathrm{\alpha j}+\frac{1-\tau}{\delta}>0$ and $\tau +\frac{\tau -1}{\delta}>0$. Renyi entropy for the sub-models can be readily obtained.
6 Estimation of model parameters
In this section, we present estimates of the parameters of the EKD distribution via method of maximum likelihood estimation. The elements of the score function are presented. There are no closed form solutions to the nonlinear equations obtained by setting the elements of the score function to zero. Thus, the estimates of the model parameters must be obtained via numerical methods.
6.1 Maximum likelihood estimation
respectively. The MLE of the parameters α,λ,δ,ϕ, and θ, say $\widehat{\alpha},\widehat{\lambda},\widehat{\delta},\widehat{\varphi},$ and $\widehat{\theta}$, must be obtained by numerical methods.
6.2 Asymptotic confidence intervals
In this section, we present the asymptotic confidence intervals for the parameters of the EKD distribution. The expectations in the Fisher Information Matrix (FIM) can be obtained numerically. Let $\widehat{\Theta}=(\widehat{\alpha},\widehat{\lambda},\widehat{\delta},\widehat{\varphi},\widehat{\theta})$ be the maximum likelihood estimate of Θ = (α,λ,δ,ϕ,θ). Under the usual regularity conditions and that the parameters are in the interior of the parameter space, but not on the boundary, we have: $\sqrt{n}(\widehat{\Theta}-\Theta )\stackrel{d}{\to}{N}_{5}(\underline{0},{I}^{-1}(\Theta \left)\right)$, where I (Θ) is the expected Fisher information matrix. The asymptotic behavior is still valid if I (Θ) is replaced by the observed information matrix evaluated at $\widehat{\theta}$, that is $J\left(\widehat{\Theta}\right)$. The multivariate normal distribution ${N}_{5}\left(\underline{0},J{\left(\widehat{\Theta}\right)}^{-1}\right)$, where the mean vector $\underline{0}={(0,0,0,0,0)}^{T}$, can be used to construct confidence intervals and confidence regions for the individual model parameters and for the survival and hazard rate functions.
respectively, where ${Z}_{\frac{\eta}{2}}$ is the upper ${\frac{\eta}{2}}^{\mathit{\text{th}}}$ percentile of a standard normal distribution.
where $\widehat{\alpha},\phantom{\rule{0.3em}{0ex}}\widehat{\lambda},\phantom{\rule{0.3em}{0ex}}\widehat{\delta},\phantom{\rule{0.3em}{0ex}}\widehat{\varphi}$ and $\widehat{\theta}$ are the unrestricted estimates, and $\stackrel{~}{\alpha},\phantom{\rule{0.3em}{0ex}}\stackrel{~}{\lambda},$$\stackrel{~}{\delta}$ and $\stackrel{~}{\varphi}$ are the restricted estimates. The LR test rejects the null hypothesis if $\omega >{\chi}_{{d}_{}}^{2},$ where ${\chi}_{{d}_{}}^{2}$ denote the upper 100d % point of the χ^{2} distribution with 1 degrees of freedom.
7 Simulation study
Monte Carlo simulation results: mean estimates and RMSEs
I | II | |||||
---|---|---|---|---|---|---|
n | Parameter | Mean | RMSE | Mean | RMSE | |
200 | α | 4.41621 | 3.979304324 | 1.7899006 | 1.992043574 | |
λ | 1.3580866 | 2.642335804 | 1.4287071 | 1.528578784 | ||
δ | 3.1167852 | 2.601663026 | 1.0337146 | 0.5898521 | ||
ϕ | 5.7270324 | 6.535452517 | 2.4702434 | 3.712081559 | ||
θ | 4.5560563 | 4.306946865 | 2.8884959 | 3.689669972 | ||
400 | α | 3.5972873 | 3.071770841 | 1.5456974 | 1.513782811 | |
λ | 1.1196079 | 0.900800533 | 1.1382897 | 0.732002869 | ||
δ | 2.9333424 | 1.821450521 | 1.0064105 | 0.377302664 | ||
ϕ | 4.6989703 | 5.277876069 | 1.5488732 | 1.872088246 | ||
θ | 4.1188983 | 3.616692978 | 2.4213684 | 2.969367761 | ||
800 | α | 3.1040595 | 2.417025941 | 1.4359333 | 1.278449373 | |
λ | 1.0626388 | 0.609066006 | 1.0432761 | 0.346996974 | ||
δ | 2.8960167 | 1.36814261 | 1.0017278 | 0.250650155 | ||
ϕ | 3.7437056 | 3.919777583 | 1.176675 | 0.766203302 | ||
θ | 3.4890255 | 2.748229594 | 1.9733522 | 2.197844717 | ||
1200 | α | 2.8399564 | 2.058703427 | 1.3884174 | 1.169251427 | |
λ | 1.0429655 | 0.501712467 | 1.021836 | 0.258884917 | ||
δ | 2.9152476 | 1.133666485 | 1.0014919 | 0.193825437 | ||
ϕ | 3.1751818 | 3.043071803 | 1.083574 | 0.392293513 | ||
θ | 3.164176 | 2.346236284 | 1.731924 | 1.788360814 |
8 Application: EKD and sub-distributions
respectively.
Air conditioning system data
194 | 413 | 90 | 74 | 55 | 23 | 97 | 50 | 359 | 50 | 130 | 487 | 57 |
102 | 15 | 14 | 10 | 57 | 320 | 261 | 51 | 44 | 9 | 254 | 493 | 33 |
18 | 209 | 41 | 58 | 60 | 48 | 56 | 87 | 11 | 102 | 12 | 5 | 14 |
14 | 29 | 37 | 186 | 29 | 104 | 7 | 4 | 72 | 270 | 283 | 7 | 61 |
100 | 61 | 502 | 220 | 120 | 141 | 22 | 603 | 35 | 98 | 54 | 100 | 11 |
181 | 65 | 49 | 12 | 239 | 14 | 18 | 39 | 3 | 12 | 5 | 32 | 9 |
438 | 43 | 134 | 184 | 20 | 386 | 182 | 71 | 80 | 188 | 230 | 152 | 5 |
36 | 79 | 59 | 33 | 246 | 1 | 79 | 3 | 27 | 201 | 84 | 27 | 156 |
21 | 16 | 88 | 130 | 14 | 118 | 44 | 15 | 42 | 106 | 46 | 230 | 26 |
59 | 153 | 104 | 20 | 206 | 5 | 66 | 34 | 29 | 26 | 35 | 5 | 82 |
31 | 118 | 326 | 12 | 54 | 36 | 34 | 18 | 25 | 120 | 31 | 22 | 18 |
216 | 139 | 67 | 310 | 3 | 46 | 210 | 57 | 76 | 14 | 111 | 97 | 62 |
39 | 30 | 7 | 44 | 11 | 63 | 23 | 22 | 23 | 14 | 18 | 13 | 34 |
16 | 18 | 130 | 90 | 163 | 208 | 1 | 24 | 70 | 16 | 101 | 52 | 208 |
95 | 62 | 11 | 191 | 14 | 71 |
Descriptive statistics
Data | Mean | Median | Mode | SD | Variance | Skewness | Kurtosis | Min. | Max. |
---|---|---|---|---|---|---|---|---|---|
I | 92.07 | 54.00 | 14.00 | 107.92 | 11646 | 2.16 | 5.19 | 1.0 | 603.0 |
II | 3.26 | 1.15 | 0.40 | 4.36 | 19.05 | 2.10 | 5.13 | 0.4 | 33.0 |
III | 17.71 | 16.80 | 9.30 | 8.80 | 77.38 | 0.80 | 0.73 | 2.7 | 53.6 |
Estimation of models for air conditioning system data
Estimates | Statistics | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Distribution | α | λ | δ | ϕ | θ | -2 log likelihood | AIC | AICC | BIC | SS | |
EKD | 20.6164 | 4.7323 | 0.6192 | 18.1616 | 0.1657 | 2065.0 | 2075.0 | 2075.3 | 2091.2 | 0.0309 | |
(1.2347) | (0.4174) | (0.0459) | (5.8028) | (0.0089) | |||||||
KD | 5.0354 | 4.3846 | 0.3762 | 21.7047 | 1 | 2066.9 | 2074.9 | 2075.2 | 2087.9 | 0.0368 | |
(2.1177) | (3.0727) | (0.1253) | (27.9167) | - | |||||||
D | 1.2390 | 94.1526 | 1.2626 | 1 | 1 | 2078.4 | 2084.4 | 2084.5 | 2094.1 | 0.1344 | |
(0.1749) | (33.7549) | (0.0663) | - | - | |||||||
a | b | c | λ | θ | |||||||
EKW | 3.7234 | 0.1219 | 1.0595 | 0.0495 | 0.3784 | 2063.7 | 2073.7 | 2074.0 | 2089.8 | 0.0254 | |
(0.8783) | (0.0183) | (0.1448) | (0.0224) | (0.1136) | |||||||
a | b | α | β | c | λ | ||||||
BKW | 1.4342 | 0.0830 | 2.0054 | 1.9100 | 0.7412 | 0.1809 | 2064.6 | 2076.6 | 2077.1 | 2096.1 | 0.0338 |
(1.2507) | (0.0875) | (1.6573) | (1.9807) | (0.0343) | (0.0388) |
Estimation of models for baseball player salary data
Estimates | Statistics | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Distribution | α | λ | δ | ϕ | θ | -2 log likelihood | AIC | AICC | BIC | SS | |
EKD | 69.1586 | 0.000043 | 7.6321 | 0.0591 | 0.4075 | 2864.1 | 2874.1 | 2874.2 | 2897.7 | 7.8153 | |
(0.000036) | (0.0000058) | (0.0557) | (0.0044) | (0.0327) | |||||||
KD | 69.0839 | 0.000011 | 7.2375 | 0.0996 | 1 | 2957.2 | 2965.2 | 2965.2 | 2984.0 | 7.7095 | |
(0.000061) | (0.00000133) | (0.037) | (0.0036) | - | |||||||
D | 70.0780 | 0.0116 | 1.0312 | 1 | 1 | 3225.6 | 3231.6 | 3231.6 | 3245.7 | 6.4568 | |
(34.4988) | (0.0058) | (0.0301) | - | - | |||||||
a | b | c | λ | θ | |||||||
EKW | 15.0514 | 0.1368 | 0.6376 | 8.8903 | 0.5419 | 3209.8 | 3219.8 | 3219.9 | 3243.3 | 5.3289 | |
(2.0692) | (0.0266) | (0.0756) | (4.9198) | (0.2098) | |||||||
a | b | α | β | c | λ | ||||||
BKW | 24.0047 | 0.03783 | 14.4799 | 4.6029 | 0.5168 | 32.1184 | 3088.4 | 3100.4 | 3100.5 | 3128.7 | 18.0516 |
(0.6879) | (0.0039) | (0.2069) | (0.4549) | (0.006) | (2.4559) |
Estimation of models for poverty rate data
Estimates | Statistics | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Distribution | α | λ | δ | ϕ | θ | -2 log likelihood | AIC | AICC | BIC | SS | |
EKD | 75.5803 | 0.851500 | 0.8183 | 60.9069 | 0.3091 | 3750.7 | 3760.7 | 3760.8 | 3782.1 | 0.1305 | |
(11.1276) | (0.32) | (0.0714) | (29.1324) | (0.02229) | |||||||
KD | 60.8898 | 0.304000 | 0.4666 | 90.2889 | 1 | 3758.9 | 3766.9 | 3767.0 | 3784.0 | 0.2604 | |
(17.5714) | (0.0963) | (0.0555) | (54.8283) | - | |||||||
D | 1.7954 | 350.0100 | 2.4175 | 1 | 1 | 3831.8 | 3837.8 | 3837.9 | 3850.7 | 0.9210 | |
(0.2034) | (105.94) | (0.0784) | - | - | |||||||
a | b | c | λ | θ | |||||||
EKW | 0.1013 | 2.2289 | 2.741 | 0.02545 | 20.0336 | 3752.8 | 3762.8 | 3762.9 | 3784.2 | 0.1071 | |
(0.0944) | (1.8026) | (2.2276) | (0.0199) | (30.0233) | |||||||
a | b | α | β | c | λ | ||||||
BKW | 0.9985 | 1.0006 | 1.9999 | 0.03989 | 2.0006 | 0.1141 | 4727.5 | 4739.5 | 4739.7 | 4765.2 | 80.9942 |
(0.0069) | (0.0431) | (0.0584) | (0.0017) | (0.2564) | (0.0075) |
was calculated for each plot. The plot with the smallest SS corresponds to the model with points that are closer to the diagonal line. Plots of the empirical and estimated survival functions for the models are also presented in Figure 6.
For the air conditioning system data, initial values α = 1,λ = 2,δ = 0.6,ϕ = 3,θ = 1 are used in SAS code for EKD model. The LR statistics for the test of the hypothesis H_{0} : KD against H_{ a }: EKD and H_{0} : D against H_{ a }: EKD are 1.9 (p-value = 0.17) and 13.4 (p-value = 0.0012). Consequently, KD distribution is the best distribution based on the LR statistic. The KD distribution gives smaller SS value than Dagum distribution and slightly bigger than EKD. For the non nested models, the values of AIC and AICC for KD and EKW models are very close, however the BIC value for KD distribution is slightly smaller than the corresponding value for the EKW distribution. We conclude that KD model compares favorably with the EKW distribution and thus provides a good fit for the air conditioning system data.
For the baseball player salary data set, initial values for EKD model in SAS code are α = 70,λ = 0.01,δ = 1.026,ϕ = 0.1,θ = 1. The EKD distribution is a better fit than KD and Dagum distributions for this data, as well as the other distributions. The values of the statistics AIC, AICC and BIC for KD distribution are smaller compared to the non nested distributions. The LR statistics for the test of the hypotheses H_{0} : KD against H_{ a }: EKD and H_{0} : D against H_{ a }: EKD are 93.1 (p-value < 0.0001) and 361.5 (p-value <0.0001). Consequently, we reject the null hypothesis in favor of the EKD distribution and conclude that the EKD distribution is significantly better than the KD and Dagum distributions based on the LR statistic. The value of AIC, AICC and BIC statistics are lower for the EKD distribution when compared to those for the EKW and BKW distributions.
For poverty rate data, initial values for EKD model are α = 73,λ = 0.1,δ = 0.15,ϕ = 60,θ = 0.33. The LR statistic for the test of the hypotheses H_{0} : KD against H_{ a }: EKD and H_{0} : D against H_{ a }: EKD are 8.2 (p-value = 0.0042) and 81.1 (p-value < 0.0001), respectively. The values of AIC, AICC and BIC statistics shows EKD distributions is a better model and the SS value of EKD model is comparatively smaller than the corresponding values for the KD and D distributions. Consequently, we conclude that EKD distribution is the best fit for the poverty rate data.
9 Conclusions
Authors’ information
Shujiao Huang is a graduate student at Georgia Southern University and Broderick O. Oluyede is Professor of Mathematics and Statistics at Georgia Southern University.
Declarations
Authors’ Affiliations
References
- Chambers J, Cleveland W, Kleiner B, Tukey P: Graphical Methods for Data Analysis. Chapman and Hall, London;Google Scholar
- Cordeiro GM, Ortega EMM, Nadarajah S: The Kumaraswamy Weibull distribution with application to failure data. J. Franklin Inst 347: 1399–1429. (2010)MathSciNetView ArticleGoogle Scholar
- Cordeiro GM, Gomes AE, de-Silva CQ, Ortega EMM: The Beta Exponentiated Weibull Distribution. J. Stat. Comput. Simulat 83(1):114–138. (2013)MathSciNetView ArticleGoogle Scholar
- Dagum CA: New model of personal income distribution: specification and estimation. Economie Applique’e 30: 413–437. (1977)Google Scholar
- Domma F, Condino F: The Beta-Dagum distribution: definition and properties. Communications in Statistics-Theory and Methods 44(22):4070–4090. (2013)MathSciNetView ArticleGoogle Scholar
- Domma F, Giordano S, Zenga M: Maximum likelihood estimation in Dagum distribution with censored samples. J. Appl. Stat 38(21):2971–2985. (2011)MathSciNetView ArticleGoogle Scholar
- Kleiber C: A Guide to the Dagum Distributions. In Modeling Income Distributions and Lorenz Curve Series: Economic Studies in Inequality, Social Exclusion and Well-Being 5. Edited by: Duangkamon C. Springer, New York; (2008)Google Scholar
- Kleiber C, Kotz S: Statistical size distributions in economics and actuarial sciences. Wiley, New York; (2003)View ArticleGoogle Scholar
- Kumaraswamy P: Generalized probability density function for double-bounded random process. J. Hydrol 46: 79–88. (1980)View ArticleGoogle Scholar
- McDonald B: Some generalized functions for the size distribution of income. Econometrica 52(3):647–663.(1995)View ArticleGoogle Scholar
- McDonald B, Xu J: A generalization of the beta distribution with application. J. Econometrics 69(2):133–152. (1995)MathSciNetView ArticleGoogle Scholar
- Oluyede BO, Rajasooriya S: The Mc-Dagum Distribution and Its Statistical Properties with Applications. Asian J. Mathematics and Applications 2013(85): , 1–16 http://scienceasia.asia/index.php/ama/article/view/85/44 , 1–16Google Scholar
- Proschan F: Theoretical explanation of observed decreasing failure rate. Technometrics 5: 375–383.(1963)View ArticleGoogle Scholar
- Renyi A: On measures of entropy and information. Berkeley Symp. Math. Stat. Probability 1(1):547–561.(1960)MathSciNetGoogle Scholar
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