Modified Kumaraswamy distribution

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 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.

Modified Kumaraswamy

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

Definitions

Probability density function

The probability density function of the Modified Kumaraswamy distribution is

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

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

Cumulative distribution function

The cumulative distribution function of Modified Kumaraswamy is given by

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

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

Quantile function

The inverse cumulative distribution function (quantile function) is

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

Properties

Moments

The hth statistical moment of X is given by:

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

Mean and Variance

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

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

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

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

Parameter estimation

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

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

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

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

and

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

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

Related distributions

• If $X\sim {\textrm {MK}}(\alpha ,\beta )$, then $\left\{1-{\frac {1}{X}}\right\}\sim {\textrm {K}}(\alpha ,\beta )$ (Kumaraswamy distribution)

• If $X\sim {\textrm {MK}}(\alpha ,\beta )$, then ${\frac {1}{X}}-1\sim$ Exponentiated exponential (EE) distribution

• If $X\sim {\textrm {MK}}(1,\beta )$, then $\exp \left\{1-{\frac {1}{X}}\right\}\sim {\textrm {Beta}}(1,\beta )$. (Beta distribution)

• If $X\sim {\textrm {MK}}(\alpha ,1)$, then $\exp \left\{1-{\frac {1}{X}}\right\}\sim {\textrm {Beta}}(\alpha ,1)$.

• If $X\sim {\textrm {MK}}(\alpha ,\beta )$, then ${\frac {1}{X}}-1\sim {\textrm {Exp}}(\alpha )$ (Exponential distribution).

Applications

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.

See also

• Kumaraswamy distribution