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
Probability density function The probability density function of the Modified Kumaraswamy distribution is
f X ( x ; θ ) = α β x α − α / x ( 1 − e α − α / x ) β − 1 x 2 {\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 θ = ( α , β ) ⊤ {\displaystyle {\boldsymbol {\theta }}=(\alpha ,\beta )^{\top }} , α > 0 {\displaystyle \alpha >0} and β > 0 {\displaystyle \beta >0} are shape parameters.
Cumulative distribution function The cumulative distribution function of Modified Kumaraswamy is given by
F X ( x ; θ ) = 1 − ( 1 − e α − α / x ) β {\displaystyle F_{X}\left(x;{\boldsymbol {\theta }}\right)=1-(1-\mathrm {e} ^{\alpha -\alpha /x})^{\beta }}
where θ = ( α , β ) ⊤ {\displaystyle {\boldsymbol {\theta }}=(\alpha ,\beta )^{\top }} , α > 0 {\displaystyle \alpha >0} and β > 0 {\displaystyle \beta >0} are shape parameters.
Quantile function The inverse cumulative distribution function (quantile function) is
Q X ( u ; θ ) = α α − log ( 1 − ( 1 − u ) 1 / β ) {\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:
E ( X h ) = α β e α ∑ i = 0 ∞ ( − 1 ) i ( β − 1 i ) e α i ( α + α i ) h − 1 Γ [ 1 − h , ( i + 1 ) α ] {\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 ( μ ) {\displaystyle (\mu )} of X is:
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