Modified Kumaraswamy distribution


In probability theory, the Modified Kumaraswamy distribution is a two-parameter continuous probability distribution defined on the interval. 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 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

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.