Continuous probability distribution
Modified Kumaraswamy Probability density function
Cumulative distribution function
Parameters α > 0 {\displaystyle \alpha >0\,} (real) β > 0 {\displaystyle \beta >0\,} (real) Support x ∈ ( 0 , 1 ) {\displaystyle x\in (0,1)\,} PDF α β e α − α / x ( 1 − e α − α / x ) β − 1 x 2 {\displaystyle {\frac {\alpha \beta \mathrm {e} ^{\alpha -\alpha /x}(1-\mathrm {e} ^{\alpha -\alpha /x})^{\beta -1}}{x^{2}}}} CDF 1 − ( 1 − e α − α / x ) β {\displaystyle 1-(1-\mathrm {e} ^{\alpha -\alpha /x})^{\beta }} Quantile α α − log ( 1 − ( 1 − u ) 1 / β ) {\displaystyle {\frac {\alpha }{\alpha -\log(1-(1-u)^{1/\beta })}}} Mean α β e α ∑ i = 0 ∞ ( − 1 ) i ( β − 1 i ) e α i Γ [ 0 , ( i + 1 ) α ] {\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 α 2 β e α ∑ i = 0 ∞ ( − 1 ) i ( β − 1 i ) e α i ( i + 1 ) Γ [ − 1 , ( i + 1 ) α ] − μ 2 {\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 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 \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] 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.
Probability density function [ edit ] 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} β > 0 {\displaystyle \beta >0}
Cumulative distribution function [ edit ] 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} β > 0 {\displaystyle \beta >0}
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 })}}} 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]} Measure of central tendency , the mean ( μ ) {\displaystyle (\mu )}
μ = E ( X ) = α β e α ∑ i = 0 ∞ ( − 1 ) i ( β − 1 i ) e α i Γ [ 0 , ( i + 1 ) α ] {\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 ( σ 2 ) {\displaystyle (\sigma ^{2})}
σ 2 = E ( X 2 ) = α 2 β e α ∑ i = 0 ∞ ( − 1 ) i ( β − 1 i ) e α i ( i + 1 ) Γ [ − 1 , ( i + 1 ) α ] − μ 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 [ edit ] Sagrillo, Guerra, and Bayer[ 1] x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}}
ℓ ( θ ) = n α + n log ( α ) + n log ( β ) − α ∑ i = 1 n 1 x i − 2 ∑ i = 1 n log ( x i ) + ( β − 1 ) ∑ i = 1 n log ( 1 − e α − α / x i ) . {\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 U ( θ ) = [ ∂ ℓ ( θ ) ∂ α , ∂ ℓ ( θ ) ∂ β ] {\displaystyle U\left({\boldsymbol {\theta }}\right)=\left[{\frac {\partial \ell ({\boldsymbol {\theta }})}{\partial \alpha }},{\frac {\partial \ell ({\boldsymbol {\theta }})}{\partial \beta }}\right]}
∂ ℓ ( θ ) ∂ α = n + n α + ( β − 1 ) e α ∑ i = 1 n x i − 1 x i ( e α − e α / x i ) − ∑ i = 1 n 1 x i {\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
∂ ℓ ( θ ) ∂ β = n β + ∑ i = 1 n log ( 1 − e α − α / x i ) {\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 θ {\displaystyle {\boldsymbol {\theta }}} θ ^ = ( α ^ , β ^ ) ⊤ {\displaystyle {\hat {\boldsymbol {\theta }}}=\left({\hat {\alpha }},{\hat {\beta }}\right)^{\top }} U ( θ ) = 0 {\displaystyle {\boldsymbol {U}}({\boldsymbol {\theta }})={\boldsymbol {0}}} 0 {\displaystyle {\boldsymbol {0}}}
If X ∼ MK ( α , β ) {\displaystyle X\sim {\textrm {MK}}(\alpha ,\beta )} { 1 − 1 X } ∼ K ( α , β ) {\displaystyle \left\{1-{\frac {1}{X}}\right\}\sim {\textrm {K}}(\alpha ,\beta )} Kumaraswamy distribution ) If X ∼ MK ( α , β ) {\displaystyle X\sim {\textrm {MK}}(\alpha ,\beta )} 1 X − 1 ∼ {\displaystyle {\frac {1}{X}}-1\sim } [ 2] If X ∼ MK ( 1 , β ) {\displaystyle X\sim {\textrm {MK}}(1,\beta )} exp { 1 − 1 X } ∼ Beta ( 1 , β ) {\displaystyle \exp \left\{1-{\frac {1}{X}}\right\}\sim {\textrm {Beta}}(1,\beta )} Beta distribution ) If X ∼ MK ( α , 1 ) {\displaystyle X\sim {\textrm {MK}}(\alpha ,1)} exp { 1 − 1 X } ∼ Beta ( α , 1 ) {\displaystyle \exp \left\{1-{\frac {1}{X}}\right\}\sim {\textrm {Beta}}(\alpha ,1)} If X ∼ MK ( α , β ) {\displaystyle X\sim {\textrm {MK}}(\alpha ,\beta )} 1 X − 1 ∼ Exp ( α ) {\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] [ 3]
^ a b c 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 . ^ 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 . ^ 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 . 
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![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/171dbb7889ae741f09303efe30ab536495de3fc0)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b133d8af46e2f92d69435bf00a9b654e87ce7861)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf5798ffa1289b40fa707075d068b8215e5153fd)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b93579597d8e80e5b6259cb74d9880b220595a7)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91105bdb721a4cbabdc9bed1c519568683051bf1)
![{\displaystyle U\left({\boldsymbol {\theta }}\right)=\left[{\frac {\partial \ell ({\boldsymbol {\theta }})}{\partial \alpha }},{\frac {\partial \ell ({\boldsymbol {\theta }})}{\partial \beta }}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/851f42dd82477be63eab6bcd7a2ffb8e26edd32f)
