The Kumaraswamy GEV distribution - University of Manchester

The Kumaraswamy GEV distribution 

with applications to the rainfall data 

Sumaya Eljabri 

University of Manchester 

MRSc, Sep 28,2012 

S.Eljabri (University of Manchester) Kum-GEV Distribution MRSc, Sep 28,2012 1 / 39

1 Generalized Extreme Value distribution 

GEV distribution 

Applications of GEV distribution 

2 DB-PDF 

DB-PDF 

Applications of DB-PDF 

3 Kum-GEV distribution 

Sub-models of the Kum-GEV distribution 

PDF of the Kum-GEV distribution 

HRF of the Kum-GEV distribution 

Simulation Study 

Applications 


Why Study Extremes? 


Why Study Extremes? 


Extreme Events 

Q: What does it mean to be an ”Extreme Event”? 

A: It depends whom you ask: 

Financial analysts: ”Shocks” to a time series (Market Crash) 

Insurance companies: Costly Phenomena (Hurricanes) 

Meteorologists: Rare Weather Phenomena. 

Applied Mathematicians: Extremes of Gaussian/other processes, Large Deviations 

Statisticians (like me!): Extreme Value Theory. 

Many possible answers! 


Generalized Extreme Value distribution 

The Generalized extreme value (GEV) distribution was proposed by Jenkinson(1955). Since 

then, it become one of the most widely applied models for univariate extreme values. The cdf 

and pdf of the GEV distribution are defined as 

Gξ,µ,σ(x) = exp(−u), 

Where u = {1 + ξ(x − µ)/σ} −1/ξ > 0, µ, ξ ∈ R and σ > 0. 

Three Types of EVD 

gξ,µ,σ(x) = σ −1 u 1+ξ exp(−u). (1) 

The parameter ξ determines tail behavior and is difficult to estimate in practice. 

ξ < 0 : Negative Weibull Distribution 

ξ = 0 Gumbel Distribution 

ξ > 0 Frechét Distribution 


Applications of GEV distribution 

GEV distribution cover most fields of researches 

science and engineering 

analysis of annual maximum rainfall, analysis of extreme floods, analysis of regional earthquake 

records, analysis of high level ozone concentrations, analysis of extreme maximum and minimum 

temperatures, analysis of ocean climate, analysis of air pollution data. 

medicine 

characteristics of Alzheimer’s disease,testing multiple gene interactions. 

For more details on GEV distribution, its theory and further applications, we refer to 

see(Leadbetter et al. (1987), Embrechts et al. (1997), Castillo et al. (2005), and Resnick 

(2008)). 

However, the GEV distribution has been misused in too many areas; Buishand (1991), Tolikas 

and Gettinby (2009), So, there is a need for generalizations of the GEV distribution. 


Double- Bounded probability density function (DB-PDF) 

Kumaraswamy (1980)proposed a two parameters distribution with double-bounded 

support known as DB-PDF and later so-called The Kumaraswamy distribution. 

FKum(z) = 1 − 

� 

1 − 

fKum(z) = 

1 

a b 

(d − c) 

� �a�b z − c 

; 

d − c 

� �a−1 � � �a�b−1 z − c 

z − c 

1 − 

. 

d − c 

d − c 

Making the linear transformation X = Z−c 

. We can obtain the standard Kumaraswamy 

b−c 

distribution. The pdf of the latter 

FKum(x, a, b) = 1 − (1 − x a ) b ; x ∈ (0, 1) 

fKum(x, a, b) = abx a−1 (1 − x a ) b−1 ; x ∈ (0, 1) 

Kumaraswamy distribution is very similar to the beta distribution, but this formula has 

some advantages over beta distribution. Jones(2009) explored the background of the 

Kum distribution and, more importantly, made clear some similarities and differences 

between the beta and Kum distributions. 



The Kum pdf is unimodal, uniantimodal, increasing, decreasing or constant depending on the 

values of its parameters 










Applications of DB-PDF 

The Kumaraswamy distribution does not seem to be very familiar to statisticians. The 

best example of an application of the Kum distribution is the model of the storage volume 

of reservoir, see Fletcher and Ponnambalam (1996). 

In hydrology, it has been received considerable interest see, for example, Ponnambalam et 

al. (2001) and Ganji et al. (2006). According to Nadarajah (2008), many papers in the 

hydrological literature have used this distribution because it is deemed as a “better 

alternative” to the beta distribution, Koutsoyiannis and Xanthopoulos (1989). 


Kum-GEV distribution 

By combining the works of Kumaraswamy (1980) and Jones (2009), Cordeiro and de Castro 

(2011) defined the cumulative distribution function of the Kum G (Kum-G) distribution by 

F (x) = 1 − {1 − G(x) a } b , 

where a > 0 and b > 0 are two additional parameters. The probability density function 

corresponding to (2) has a very simple form 

f (x) = ab g(x) G(x) a−1 {1 − G(x) a } b−1 . 

Here, we combine between the Kum distribution and GEV distribution. 

cdf and pdf of the Kum-GEV distribution: 

where u = (1 + ξ( x−µ 

σ ))−1/ξ . 

F (x) = 1 − {1 − exp(−au)} b , 

f (x) = σ −1 a b u 1+ξ exp(−au) {1 − exp(−au)} b−1 . (2) 



The Kum-GEV distribution can be reduced to some sub-models for some special values of the 

shape parameters. 

f (x) = σ −1 a b u 1+ξ exp(−au) {1 − exp(−au)} b−1 . 

Parameters 

Distribution a b µ σ ξ 

Gumbel 1 1 - - → 0 

Frechét 1 1 - - > 0 

Weibull 1 1 - - < 0 

GEVD 1 1 - - - 

KumG - - - - → 0 

KumW - - - - < 0 

Table: Some sub-models of the Kum-GEV distribution. 






Parameters 


Gumbel 1 1 - - → 0 







Parameters 


Gumbel 1 1 - - → 0 

Frechét 1 1 - - > 0 







Parameters 


Gumbel 1 1 - - → 0 

Frechét 1 1 - - > 0 

Weibull 1 1 - - < 0 







Parameters 


Gumbel 1 1 - - → 0 

Frechét 1 1 - - > 0 

Weibull 1 1 - - < 0 








Parameters 


Gumbel 1 1 - - → 0 

Frechét 1 1 - - > 0 

Weibull 1 1 - - < 0 


KumG - - - - → 0 







Parameters 


Gumbel 1 1 - - → 0 

Frechét 1 1 - - > 0 

Weibull 1 1 - - < 0 


KumG - - - - → 0 

KumW - - - - < 0 



PDF of the Kum-GEV distribution 

The pdf of the Kum-GEV distribution for some parameter values: 

PDF 

PDF 

0.0 0.2 0.4 0.6 0.8 

0.00 0.05 0.10 0.15 

ξ = − 0.5 

−2 −1 0 1 2 

x 

ξ = 0.5 

2 3 4 5 6 

x 

PDF 

PDF 

0.0 0.2 0.4 0.6 

0.00 0.10 0.20 0.30 

ξ = 0 

−2 −1 0 1 2 

x 

ξ = 1 

1 2 3 4 5 

Figure: Plots of (2) for µ = 0, σ = 1, ξ = −0.5, 0, 0.5, 1, (a, b) = (0.5, 0.5) (solid curve), (a, b) = (0.5, 1) (curve of 

dashes), (a, b) = (0.5, 3) (curve of dots) and (a, b) = (3, 3) (curve of dots and dashes). 

S.Eljabri (University of Manchester) Kum-GEV Distribution MRSc, Sep 28,2012 21 / 39 

x

Hazard rate function 

h(x) = (abu 1+ξ exp(−au))/(σ(1 − exp(−au))), 

HRF 

HRF 

0 200 400 600 

0.2 0.6 1.0 1.4 

ξ = − 0.5 

−2 −1 0 1 2 

x 

ξ = 0.5 

2 3 4 5 6 

x 

HRF 

HRF 

0.0 1.0 2.0 3.0 

0.2 0.6 1.0 

ξ = 0 

−2 −1 0 1 2 

x 

ξ = 1 

1 2 3 4 5 

Figure: Plots of (3) for µ = 0, σ = 1, ξ = −0.5, 0, 0.5, 1, (a, b) = (0.5, 0.5) (solid curve), (a, b) = (0.5, 1) (curve of 

dashes), (a, b) = (0.5, 3) (curve of dots) and (a, b) = (3, 3) (curve of dots and dashes). 

S.Eljabri (University of Manchester) Kum-GEV Distribution MRSc, Sep 28,2012 22 / 39 

x

Simulation Study 

The assessment is based on a simulation study: 

1 generate ten thousand samples of size n from (2). The inversion method is used to 

generate samples, i.e variates of the KumGEV distribution are generated. 

2 compute the maximum likelihood estimates for the ten thousand samples, say 

(�ai , � bi , �µi , �σi , � ξi ) for i = 1, 2, . . . , 10000. 

3 compute the biases and mean squared errors given by 

for h = a, b, µ, σ, ξ. 

biash(n) = 

MSE h(n) = 

1 

10000 

1 

10000 

10000 � 

i=1 

10000 � 

i=1 

� � 

�hi − h 

� �2 �hi − h 

We repeat these steps for n = 10, 20, . . . , 1000 with a = 3, b = 3, µ = 0, σ = 1 and ξ = 0.5, so 

computing biasa(n), biasb(n), biasµ(n), biasσ(n), biasξ(n) and MSE a(n), MSE b(n), MSE µ(n), 

MSE σ(n), MSE ξ(n) for n = 10, 20, . . . , 1000. 


Figure: biasa(n) (top left), bias b(n) (top right), biasµ(n) (middle left), biasσ(n) (middle right) and bias ξ(n) (bottom left) 

versus n = 10, 20, . . . , 1000. 


Figure: MSE a(n) (top left), MSE b(n) (top right), MSE µ(n) (middle left), MSE σ(n) (middle right) and MSE ξ(n) 

(bottom left) versus n = 10, 20, . . . , 1000. 


Applications 

We use the annual rainfall maxima in millimetres from 1938 to 1972 at Uccle, Belgium, over the 

duration of one day. This data set is contained as part of the evd contributed package in the R 

package (R Development Core Team, 2011). The data are due to Sneyers (1977). 

Table: MLEs of the model parameters for the annual rainfall maxima data, the corresponding SEs (given in parentheses) and 

the statistics AIC, BIC and AICc. 

Model a b µ σ ξ AIC BIC AICc 

KumGEV 1 0.1358 20.5080 2.5521 -0.0277 277.98 284.2 279.6 

(-) (0.0233) (1.0703) (0.0599) (0.0021) 

GEV 1 1 28.3824 9.0291 0.2316 279.81 284.5 280.8 

(-) (-) (1.9204) (1.5793) (0.2133) 


Table: The Anderson-Darling and Cramr-von Mises statistics for the annual rainfall maxima data. 

Model Anderson-Darling Cramér-von Mises 

KumGEV 0.1695225 0.02260322 

GEV 0.3105545 0.04270863 


Figure: Fitted probability density functions, (1) (solid curve) and (2) (broken curve), for annual daily rainfall maxima from 

Uccle, Belgium. 








Figure: Probability plots for the fits of (1) (the black plus character) and (2) (the red star character) for annual daily 

rainfall maxima from Uccle, Belgium. 


Figure: Quantile plots for the fits of (1) (the black plus character) and (2) (the red star character) for annual daily rainfall 

maxima from Uccle, Belgium. 


Figure: Return levels for annual daily rainfall maxima from Uccle, Belgium for the fits of (1) (in red) and (2) (in black). 


Figure Return levels for annual daily rainfall maxima from Uccle, Belgium and their 95 percent confidence intervals for the 

fits of (1) (in red) and (2) (in black). 


Bibliography 

Buishand, T. A. (1991). Extreme rainfall estimation by combining data from several sites. Hydrological Sciences, 36, 

345-365. 

Castillo, E., Hadi, A. S., Balakrishnan, N. and Sarabia, J. M. (2005). Extreme Value and Related Models with 

Applications in Engineering and Science. John Wiley and Sons, Hoboken, New Jersey. 

Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Springer 

Verlag, Berlin. 

Fletcher, S. C., and Ponnambalam, K. ”Estimation of Reservoir Yield and Storage Distribution using Moments Analysis.” 

Journal of Hydrology, 182(1-4), 259-275, (1996). 

Ganji, A., Ponnambalam, K. and Khalili, D. (2006). Grain yield reliability analysis with crop water demand uncertainty. 

Stochastic Environmental Research and Risk Assessment, 20, 259-277. 

Jenkinson, A.F. ”The frequency distribution of the annual maximum (or minimum) values of meteorological elements.” 

Quarterly Journal of the Royal Meteorology Society 87, 145-158, (1955). 

Jones, M. C. (2009). Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages. Statistical 

Methodology, 6, 70-91. 

Koutsoyiannis, D. and Xanthopoulos, T. (1989). On the parametric approach to unit hydrograph identification. Water 

Resources Management, 3, 107-128. 


Kumaraswamy, P. (1980). Generalized probability density-function for double-bounded random-processes. Journal of 

Hydrology, 46, 79-88. 

Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1987). Extremes and Related Properties of Random Sequences and 

Processes. Springer Verlag, New York. 

Ponnambalam, K., Seifi, A. and Vlach, J. (2001). Probabilistic design of systems with general distributions of 

parameters. International Journal of Circuit Theory and Applications, 29, 527-536. 

R Development Core Team (2011). A Language and Environment for Statistical Computing. R Foundation for Statistical 

Computing. Vienna, Austria. 

Resnick, S. I. (2008). Extreme Values, Regular Variation, and Point Processes. Springer Verlag, New York. 

Tolikas, K. and Gettinby, G. D. (2009). Modelling the distribution of the extreme share returns in Singapore. Journal of 

Empirical Finance, 16, 254-263. 

Zwiers, F. W., Zhang, X. and Feng, Y. (2011). Anthropogenic influence on long return period daily temperature 

extremes at regional scales. Journal of Climate, 24, 881-892. 


Thank you

The Kumaraswamy GEV distribution - University of Manchester

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?