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Modified Kumaraswamy distribution

Continuous probability distribution From Wikipedia, the free encyclopedia

Modified Kumaraswamy distribution
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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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  • 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]

See also

References

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