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 ^[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.

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

Definitions

Probability density function

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.

Cumulative distribution function

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.

Quantile function

The inverse cumulative distribution function (quantile function) is

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

Properties

Moments

The hth statistical moment of X is given by:

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

Mean and Variance

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 ${\displaystyle (\sigma ^{2})}$ :

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

Parameter estimation

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.

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^[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).

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.^[1] It was also used in the statistical estimation of the stress-strength reliability of systems.^[3]

See also

• Kumaraswamy distribution

References

1. Sagrillo, M.; Guerra, R. R.; Bayer, F. M. (2021). "Modified Kumaraswamy distributions for double bounded hydro-environmental data". Journal of Hydrology. 603. Bibcode:2021JHyd..60327021S. doi:10.1016/j.jhydrol.2021.127021.
2. Gupta, R.D.; Kundu, D (1999). "Theory & Methods: Generalized exponential distributions". Australian & New Zealand Journal of Statistics. 41 (2): 173–188. doi:10.1111/1467-842X.00072.
3. Kohansal, Akram; Pérez-González, Carlos J; Fernández, Arturo J (2023). "Inference on the stress-strength reliability of multi-component systems based on progressive first failure censored samples". Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability. 238 (5): 1053–1073. doi:10.1177/1748006X231188075.

External links

• Github with the distribution functions in R