Modified Kumaraswamy distribution

Modified Kumaraswamy
Probability density function
Cumulative distribution function
Parameters	${\displaystyle \alpha >0\,}$ (real) ${\displaystyle \beta >0\,}$ (real)
Support	${\displaystyle x\in (0,1)\,}$
PDF	${\displaystyle {\frac {\alpha \beta \mathrm {e} ^{\alpha -\alpha /x}(1-\mathrm {e} ^{\alpha -\alpha /x})^{\beta -1}}{x^{2}}}}$
CDF	${\displaystyle 1-(1-\mathrm {e} ^{\alpha -\alpha /x})^{\beta }}$
Quantile	${\displaystyle {\frac {\alpha }{\alpha -\log(1-(1-u)^{1/\beta })}}}$
Mean	${\displaystyle \alpha \beta \mathrm {e} ^{\alpha }\sum _{i=0}^{\infty }(-1)^{i}{\begin{pmatrix}\beta -1\\i\end{pmatrix}}\mathrm {e} ^{\alpha i}\Gamma \left[0,\left(i+1\right)\alpha \right]}$
Variance	${\displaystyle \alpha ^{2}\beta e^{\alpha }\sum _{i=0}^{\infty }(-1)^{i}{\begin{pmatrix}\beta -1\\i\end{pmatrix}}\mathrm {e} ^{\alpha i}(i+1)\Gamma \left[-1,\left(i+1\right)\alpha \right]-\mu ^{2}}$
MGF	${\displaystyle \alpha \beta e^{\alpha }\sum _{i=0}^{\infty }(-1)^{i}{\begin{pmatrix}\beta -1\\i\end{pmatrix}}\mathrm {e} ^{\alpha i}(\alpha +\alpha i)^{h-1}\Gamma \left[1-h,\left(i+1\right)\alpha \right]}$

In probability theory, the Modified Kumaraswamy (MK) distribution is a two-parameter continuous probability distribution defined on the interval (0,1). It serves as an alternative to the beta and Kumaraswamy distributions for modeling double-bounded random variables. The MK distribution was originally proposed by Sagrillo, Guerra, and Bayer ^[1] through a transformation of the Kumaraswamy distribution. Its density exhibits an increasing-decreasing-increasing shape, which is not characteristic of the beta or Kumaraswamy distributions. The motivation for this proposal stemmed from applications in hydro-environmental problems.

The probability density function of the Modified Kumaraswamy distribution is

${\displaystyle f_{X}\left(x;{\boldsymbol {\theta }}\right)={\frac {\alpha \beta x^{\alpha -\alpha /x}(1-\mathrm {e} ^{\alpha -\alpha /x})^{\beta -1}}{x^{2}}}}$

where ${\displaystyle {\boldsymbol {\theta }}=(\alpha ,\beta )^{\top }}$ , ${\displaystyle \alpha >0}$ and ${\displaystyle \beta >0}$ are shape parameters.

The cumulative distribution function of Modified Kumaraswamy is given by

${\displaystyle F_{X}\left(x;{\boldsymbol {\theta }}\right)=1-(1-\mathrm {e} ^{\alpha -\alpha /x})^{\beta }}$

where ${\displaystyle {\boldsymbol {\theta }}=(\alpha ,\beta )^{\top }}$ , ${\displaystyle \alpha >0}$ and ${\displaystyle \beta >0}$ are shape parameters.

The inverse cumulative distribution function (quantile function) is

${\displaystyle Q_{X}\left(u;{\boldsymbol {\theta }}\right)={\frac {\alpha }{\alpha -\log(1-(1-u)^{1/\beta })}}}$

The hth statistical moment of X is given by:

${\displaystyle {\textrm {E}}\left(X^{h}\right)=\alpha \beta \mathrm {e} ^{\alpha }\sum _{i=0}^{\infty }(-1)^{i}{\begin{pmatrix}\beta -1\\i\end{pmatrix}}\mathrm {e} ^{\alpha i}(\alpha +\alpha i)^{h-1}\Gamma \left[1-h,\left(i+1\right)\alpha \right]}$

Measure of central tendency, the mean ${\displaystyle (\mu )}$ of X is:

${\displaystyle \mu ={\text{E}}(X)=\alpha \beta \mathrm {e} ^{\alpha }\sum _{i=0}^{\infty }(-1)^{i}{\begin{pmatrix}\beta -1\\i\end{pmatrix}}\mathrm {e} ^{\alpha i}\Gamma \left[0,\left(i+1\right)\alpha \right]}$

And its variance ${\displaystyle (\sigma ^{2})}$:

${\displaystyle \sigma ^{2}={\text{E}}(X^{2})=\alpha ^{2}\beta \mathrm {e} ^{\alpha }\sum _{i=0}^{\infty }(-1)^{i}{\begin{pmatrix}\beta -1\\i\end{pmatrix}}\mathrm {e} ^{\alpha i}(i+1)\Gamma \left[-1,\left(i+1\right)\alpha \right]-\mu ^{2}}$

Sagrillo, Guerra, and Bayer^[1] suggested using the maximum likelihood method for parameter estimation of the MK distribution. The log-likelihood function for the MK distribution, given a sample ${\displaystyle x_{1},\ldots ,x_{n}}$, is:

${\displaystyle {\begin{aligned}\ell ({\boldsymbol {\theta }})=&\,n\alpha +n\log \left(\alpha \right)+n\log \left(\beta \right)-\alpha \sum _{i=1}^{n}{\frac {1}{x_{i}}}-2\sum _{i=1}^{n}\log(x_{i})\\&+(\beta -1)\sum _{i=1}^{n}\log(1-\mathrm {e} ^{\alpha -\alpha /x_{i}}).\end{aligned}}}$

The components of the score vector ${\displaystyle U\left({\boldsymbol {\theta }}\right)=\left[{\frac {\partial \ell ({\boldsymbol {\theta }})}{\partial \alpha }},{\frac {\partial \ell ({\boldsymbol {\theta }})}{\partial \beta }}\right]}$ are

${\displaystyle {\begin{aligned}{\frac {\partial \ell ({\boldsymbol {\theta }})}{\partial \alpha }}=n+{\frac {n}{\alpha }}+(\beta -1)\mathrm {e} ^{\alpha }\sum _{i=1}^{n}{\frac {x_{i}-1}{x_{i}(\mathrm {e} ^{\alpha }-\mathrm {e} ^{\alpha /x_{i}})}}-\sum _{i=1}^{n}{\frac {1}{x_{i}}}\end{aligned}}}$

and

${\displaystyle {\begin{aligned}{\frac {\partial \ell ({\boldsymbol {\theta }})}{\partial \beta }}={\frac {n}{\beta }}+\sum _{i=1}^{n}\log(1-\mathrm {e} ^{\alpha -\alpha /x_{i}})\end{aligned}}}$

The MLEs of ${\displaystyle {\boldsymbol {\theta }}}$, denoted by ${\displaystyle {\hat {\boldsymbol {\theta }}}=\left({\hat {\alpha }},{\hat {\beta }}\right)^{\top }}$, are obtained as the simultaneous solution of ${\displaystyle {\boldsymbol {U}}({\boldsymbol {\theta }})={\boldsymbol {0}}}$, where ${\displaystyle {\boldsymbol {0}}}$ is a two-dimensional null vector.

• If ${\displaystyle X\sim {\textrm {MK}}(\alpha ,\beta )}$, then ${\displaystyle \left\{1-{\frac {1}{X}}\right\}\sim {\textrm {K}}(\alpha ,\beta )}$ (Kumaraswamy distribution)
• If ${\displaystyle X\sim {\textrm {MK}}(\alpha ,\beta )}$, then ${\displaystyle {\frac {1}{X}}-1\sim }$ Exponentiated exponential (EE) distribution^[2]
• If ${\displaystyle X\sim {\textrm {MK}}(1,\beta )}$, then ${\displaystyle \exp \left\{1-{\frac {1}{X}}\right\}\sim {\textrm {Beta}}(1,\beta )}$. (Beta distribution)
• If ${\displaystyle X\sim {\textrm {MK}}(\alpha ,1)}$, then ${\displaystyle \exp \left\{1-{\frac {1}{X}}\right\}\sim {\textrm {Beta}}(\alpha ,1)}$.
• If ${\displaystyle X\sim {\textrm {MK}}(\alpha ,\beta )}$, then ${\displaystyle {\frac {1}{X}}-1\sim {\textrm {Exp}}(\alpha )}$ (Exponential distribution).

The Modified Kumaraswamy distribution was introduced for modeling hydro-environmental data. It has been shown to outperform the Beta and Kumaraswamy distributions for the useful volume of water reservoirs in Brazil.^[1] It was also used in the statistical estimation of the stress-strength reliability of systems.^[3]

• Kumaraswamy distribution

• Github with the distribution functions in R