Exponentiated Kumaraswamy-Dagum distribution with applications to income and lifetime data

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-1f _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).

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.

3.3 Some sub-models

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

Plots of the hazard function are presented in Figure 2. The plots show various shapes including monotonically decreasing, unimodal, and bathtub followed by upside down bathtub shapes with five combinations of the values of the parameters. This attractive flexibility makes the EKD hazard rate function useful and suitable for non-monotone empirical hazard behaviors which are more likely to be encountered or observed in real life situations. Unfortunately, the analytical analysis of the shape of both the density (except for zero modal when α δ ≤ 1, and unimodal if α δ > 1, both for ϕ = θ = 1,) and hazard rate function seems to be very complicated. We could not determine any specific rules for the shapes of the hazard rate function.

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

Bonferroni and Lorenz curves are widely used tool for analyzing and visualizing income inequality. Lorenz curve, L(p) can be regarded as the proportion of total income volume accumulated by those units with income lower than or equal to the volume a, and Bonferroni curve, B(p) is the scaled conditional mean curve, that is, ratio of group mean income of the population. Plots of Bonferroni and Lorenz curves are given in Figure 3.

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 $\zeta (i,j,k,l)={\alpha }_1{\lambda }_1{\delta }_1{\varphi }_1{\theta }_1\frac{{(-1)}^{i+j+k+l}\Gamma \left({\theta }_1\right)\Gamma ({\varphi }_{1}i+{\varphi }_1)\Gamma ({\theta }_2+1)\Gamma ({\varphi }_{2}k+1)}{\Gamma ({\theta }_1-i)\Gamma ({\varphi }_{1}i+{\varphi }_1-j)\Gamma ({\theta }_2+1-k)\Gamma ({\varphi }_{2}k+1-l)i!j!k!l!}$.

5.1 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.

6.1 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 $\widehat{\alpha },\widehat{\lambda },\widehat{\delta },\widehat{\varphi }$ and $\widehat{\theta }$ are the unrestricted estimates, and $\tilde{\alpha },\tilde{\lambda },$$\tilde{\delta }$ and $\tilde{\varphi }$ 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 χ² distribution with 1 degrees of freedom.