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
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
Properties
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 :
Parameter estimation Sagrillo, Guerra, and Bayer 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.
Related distributions If , then (Kumaraswamy distribution) If , then Exponentiated exponential (EE) distribution If , then . (Beta distribution) If , then . If , then (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. It was also used in the statistical estimation of the stress-strength reliability of systems.
See also
Kumaraswamy distribution
References
External links Github with the distribution functions in R