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Modified Kumaraswamy distribution
Continuous probability distribution From Wikipedia, the free encyclopedia
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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.
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Definitions
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Probability density function
The probability density function of the Modified Kumaraswamy distribution is
where , and are shape parameters.
Cumulative distribution function
The cumulative distribution function of Modified Kumaraswamy is given by
where , and are shape parameters.
Quantile function
The inverse cumulative distribution function (quantile function) is
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Properties
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Moments
The hth statistical moment of X is given by:
Mean and Variance
Measure of central tendency, the mean of X is:
And its variance :
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Parameter estimation
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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 , is:
The components of the score vector are
and
The MLEs of , denoted by , are obtained as the simultaneous solution of , where is a two-dimensional null vector.
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Related distributions
- If , then (Kumaraswamy distribution)
- If , then Exponentiated exponential (EE) distribution[2]
- If , then . (Beta distribution)
- If , then .
- If , then (Exponential distribution).
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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]
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