In this section, we present the proposed distribution and its submodels. Series expansion, hazard and reverse hazard functions are also studied in this section.
3.1 KumaraswamyDagum distribution
Kumaraswamy (1980) introduced a twoparameter distribution on (0,1). Its cdf is given by
G\left(x\right)=1{\left(1{x}^{\psi}\right)}^{\varphi},x\in (0,1),
for ψ > 0 and ϕ > 0.
For an arbitrary cdf F (x) with pdf f\left(x\right)=\frac{\mathit{\text{dF}}\left(x\right)}{\mathit{\text{dx}}}, the family of KumaraswamyG distributions with cdf G_{
k
}(x) is given by
{G}_{K}\left(x\right)=1{\left(1{F}^{\psi}\left(x\right)\right)}^{\varphi},
for ψ > 0 and ϕ > 0. By letting F (x) = G_{
D
}(x), we obtain the KumaraswamyDagum (KD) distribution, with cdf
{G}_{\mathit{\text{KD}}}\left(x\right)=1{\left(1{G}_{D}^{\psi}\left(x\right)\right)}^{\varphi}.
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 (KumDagum) cdf, θ>0, with the corresponding pdf given by g_{
EKD
}(x) = θ [ F_{
KD
}(x)]^{θ1}f_{
KD
}(x). For large values of x, and for θ > 1(< 1), the multiplicative factor θ [ F_{
KD
}(x)]^{θ1}> 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).
Replacing the dependent parameter β ψ by α, the cdf and pdf of the EKD distribution are given by
\begin{array}{lcr}{G}_{{\mathit{\text{EKD}}}_{}}(x;\alpha ,\lambda ,\delta ,\varphi ,\theta )& =& {\left\{1{\left[1{\left(1+\lambda {x}^{\delta}\right)}^{\alpha}\right]}^{\varphi}\right\}}^{\theta},\end{array}
(3)
and
\begin{array}{ll}{g}_{{\mathit{\text{EKD}}}_{}}(x;\alpha ,\lambda ,\delta ,\varphi ,\theta )=& \phantom{\rule{2.77626pt}{0ex}}\alpha \lambda \delta \varphi \theta {x}^{\delta 1}{\left(1+\lambda {x}^{\delta}\right)}^{\alpha 1}{\left[1{\left(1+\lambda {x}^{\delta}\right)}^{\alpha}\right]}^{\varphi 1}\phantom{\rule{2em}{0ex}}\\ \times {\left\{1{\left[1{\left(1+\lambda {x}^{\delta}\right)}^{\alpha}\right]}^{\varphi}\right\}}^{\theta 1},\phantom{\rule{2em}{0ex}}\end{array}
(4)
for α,λ,δ,ϕ,θ >0, and x > 0, respectively. The quantile function of the EKD distribution is in closed form,
{G}_{{\mathit{\text{EKD}}}_{}}^{1}\left(q\right)={x}_{q}={\lambda}^{\frac{1}{\delta}}{\left\{{\left[1{\left(1{q}^{\frac{1}{\theta}}\right)}^{\frac{1}{\varphi}}\right]}^{\frac{1}{\alpha}}1\right\}}^{\frac{1}{\delta}}.
(5)
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 submodels
Submodels of EKD distribution for selected values of the parameters are presented in this section.

1.
When θ = 1, we obtain KumaraswamyDagum 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},
for α,λ,δ,ϕ > 0 and x>0.

2.
When ϕ = θ = 1, we obtain Dagum distribution with cdf:
G(x;\alpha ,\lambda ,\delta )={\left(1+\lambda {x}^{\delta}\right)}^{\alpha},
for α,λ,δ > 0 and x > 0.

3.
When λ = 1, we obtain exponentiated KumaraswamyBurr 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},
for α,δ,ϕ,θ > 0 and x > 0.

4.
When λ = θ = 1, we obtain KumaraswamyBurr III distribution with cdf:
G(x;\alpha ,\delta ,\varphi )=1{\left[1{\left(1+{x}^{\delta}\right)}^{\alpha}\right]}^{\varphi},
for α,δ,ϕ > 0 and x > 0.

5.
When λ = ϕ = θ = 1, we obtain Burr III distribution with cdf:
G(x;\alpha ,\delta )={\left(1+{x}^{\delta}\right)}^{\alpha},
for α,δ > 0 and x > 0.

6.
When α = 1, we obtain exponentiated KumaraswamyFisk or KumaraswamyLoglogistic distribution with cdf:
G(x;\lambda ,\delta ,\varphi ,\theta )={\left\{1{\left[1{\left(1+\lambda {x}^{\delta}\right)}^{1}\right]}^{\varphi}\right\}}^{\theta},
for λ,δ,ϕ,θ > 0 and x > 0.

7.
When α = θ = 1, we obtain KumaraswamyFisk or KumaraswamyLoglogistic distribution with cdf:
G(x;\lambda ,\delta ,\varphi )=1{\left[1{\left(1+\lambda {x}^{\delta}\right)}^{1}\right]}^{\varphi},
for λ,δ,ϕ > 0 and x > 0.

8.
When α = ϕ = θ = 1, we obtain Fisk or Loglogistic 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
We apply the series expansion
{(1z)}^{b1}=\sum _{j=0}^{\infty}\frac{{(1)}^{j}\Gamma \left(b\right)}{\Gamma (bj)j!}{z}^{j},
(6)
for b > 0 and z < 1, to obtain the series expansion of the EKD distribution.
By using equation (6),
{g}_{{}_{\mathit{\text{EKD}}}}\left(x\right)=\sum _{i=0}^{\infty}\sum _{j=0}^{\infty}\omega (i,j){x}^{\delta 1}{\left(1+\lambda {x}^{\delta}\right)}^{\alpha (j+1)1},
(7)
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 ExponentiatedKumaraswamy (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
The hazard function of the EKD distribution is
\begin{array}{ll}{h}_{{\mathit{\text{EKD}}}_{}}\left(x\right)& =\frac{{g}_{{\mathit{\text{EKD}}}_{}}\left(x\right)}{1{G}_{\mathit{\text{EKD}}}\left(x\right)}\phantom{\rule{2em}{0ex}}\\ =\mathrm{\alpha \lambda \delta \varphi \theta}{x}^{\delta 1}{\left(1+\lambda {x}^{\delta}\right)}^{\alpha 1}{\left[1{\left(1+\lambda {x}^{\delta}\right)}^{\alpha}\right]}^{\varphi 1}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{0.5em}{0ex}}\times {\{1{\left[\phantom{\rule{0.3em}{0ex}}1{\left(1+\lambda {x}^{\delta}\right)}^{\alpha}\right]}^{\varphi}\}}^{\theta 1}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{0.5em}{0ex}}\times {\left(1{\left\{1{\left[1{\left(1+\lambda {x}^{\delta}\right)}^{\alpha}\right]}^{\varphi}\right\}}^{\theta}\right)}^{1}.\phantom{\rule{2em}{0ex}}\end{array}
(8)
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 nonmonotone 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.
The reverse hazard function of the EKD distribution is
\begin{array}{ll}{\tau}_{{\mathit{\text{EKD}}}_{}}\left(x\right)& =\frac{{g}_{{\mathit{\text{EKD}}}_{}}\left(x\right)}{{G}_{{\mathit{\text{EKD}}}_{}}\left(x\right)}\phantom{\rule{2em}{0ex}}\\ =\phantom{\rule{2.77626pt}{0ex}}\mathrm{\alpha \lambda \delta \varphi \theta}{x}^{\delta 1}{\left(1+\lambda {x}^{\delta}\right)}^{\alpha 1}{\left[\phantom{\rule{0.3em}{0ex}}1{\left(1+\lambda {x}^{\delta}\right)}^{\alpha}\right]}^{\varphi 1}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{0.5em}{0ex}}\times {\left\{1{\left[1{\left(1+\lambda {x}^{\delta}\right)}^{\alpha}\right]}^{\varphi}\right\}}^{1}.\phantom{\rule{2em}{0ex}}\end{array}
(9)