Kumaraswamy Distribution

Where do you meet this distribution?

• Hydrology

  Fletcher, S.G., and Ponnambalam, K. (1996). "Estimation of reservoir yield and storage distribution using moments analysis". Journal of Hydrology 182: 259-275.

Shape of Distribution

Basic Properties

• Two parameters $a, b$ are required.
• Continuous distribution defined on bounded range $0\leq x \leq 1$
• This distribution can be symmetric or asymmetric.

Probability

• Cumulative distribution function

  $$F(x)=1-\left(1-x^a\right)^b$$

• Probability density function

  $$f(x)=abx^{a-1}\left(1-x^a\right)^{b-1}$$

• How to compute these on Excel.

	A	B
1	Data	Description
2	0.5	Value for which you want the distribution
3	8	Value of parameter A
4	2	Value of parameter B
5	Formula	Description (Result)
6	=1-(1-A2^A3)^A4	Cumulative distribution function for the terms above
7	=A3*A4*A2^(A2-1)*(1-A2^A3)^(A4-1)	Probability density function for the terms above

Quantile

• Inverse function of cumulative distribution function

  $$F^{-1}(P)=\left[1-(1-P)^{1/b}\right]^{1/a}$$

• How to compute this on Excel.

	A	B
1	Data	Description
2	0.5	Probability associated with the beta distribution
3	1.7	Value of parameter A
4	0.9	Value of parameter B
5	Formula	Description (Result)
6	=POWER(1-(1-A2)^(1/A4),1/A3)	Inverse of the cumulative distribution function for the terms above

Characteristics

Mean -- Where is the "center" of the distribution? (Definition)

• Mean

  $$bB\left(1+\frac{1}{a},b\right)$$

• How to compute this on Excel

	A	B
1	Data	Description
2	8	Value of parameter A
3	2	Value of parameter B
4	Formula	Description (Result)
5	=A3*EXP(GAMMALN(1+1/A2)+GAMMALN(A3)-GAMMALN(1+1/A2+A3))	Mean of the distribution for the terms above

Random Numbers

• Random number x is generated by inverse function method, which is for uniform random U,

  $$x=\left[1-\left(1-U\right)^{1/b}\right]^{1/a}$$

• How to generate random numbers on Excel.

	A	B
1	Data	Description
2	0.5	Value of parameter A
3	0.5	Value of parameter B
4	Formula	Description (Result)
5	=POWER(1-(1-NTRAND(100))^(1/A3),1/A2)	100 Kumaraswamy deviates based on Mersenne-Twister algorithm for which the parameters above

Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A5:A104 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

NtRand Functions

Not supported yet

Reference

• Wikipedia -- Kumaraswamy distribution
• ModelAssist for ModelRisk