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 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.

Definitions

Properties

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 ${\displaystyle x_{1},\ldots ,x_{n}}$, 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.

Related distributions

• 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
• 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).

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. It was also used in the statistical estimation of the stress-strength reliability of systems.

See also

• Kumaraswamy distribution

References

External links

• Github with the distribution functions in R