TRANSMUTED KUMARASWAMY DISTRIBUTION

Khan, Muhammad Shuaib; King, Robert; Hudson, Irene Lena Article TRANSMUTED KUMARASWAMY DISTRIBUTION Statistics in Transition New Series Provided in Cooperation with: Polish Statistical Association Suggested Citation: Khan, Muhammad Shuaib; King, Robert; Hudson, Irene Lena (2016) : TRANSMUTED KUMARASWAMY DISTRIBUTION, Statistics in Transition New Series, ISSN 2450-0291, Exeley, New York, NY, Vol. 17, Iss. 2, pp. 183-210, https://doi.org/10.21307/stattrans-2016-013 This Version is available at: https://hdl.handle.net/10419/207806 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ STATISTICS IN TRANSITION new series, June 2016 183 STATISTICS IN TRANSITION new series, June 2016 Vol. 17, No. 2, pp. 183–210 TRANSMUTED KUMARASWAMY DISTRIBUTION Muhammad Shuaib Khan 1 , Robert King 2 , Irene Lena Hudson 3 ABSTRACT The Kumaraswamy distribution is the most widely applied statistical distribution in hydrological problems and many natural phenomena. We propose a generalization of the Kumaraswamy distribution referred to as the transmuted Kumaraswamy (𝑇𝐾𝑤) distribution. The new transmuted distribution is developed using the quadratic rank transmutation map studied by Shaw et al. (2009). A comprehensive account of the mathematical properties of the new distribution is provided. Explicit expressions are derived for the moments, moment generating function, entropy, mean deviation, Bonferroni and Lorenz curves, and formulated moments for order statistics. The 𝑇𝐾𝑤 distribution parameters are estimated by using the method of maximum likelihood. Monte Carlo simulation is performed in order to investigate the performance of MLEs. The flood data and HIV/ AIDS data applications illustrate the usefulness of the proposed model. Key words: Kumaraswamy distribution, moments, order statistics, parameter estimation, maximum likelihood estimation. 1. Introduction The Kumaraswamy probability distribution was originally proposed by Poondi Kumaraswamy (1980) for double bounded random processes for hydrological applications. The Kumaraswamy double bounded distribution denoted by 𝐾𝑤(𝛼,𝜃) distribution is a family of continuous probability distributions defined on the interval [0,1] with cumulative distribution function given by 𝐺𝐾𝑤(𝑥;𝛼,𝜃)=1−(1−𝑥𝛼)𝜃 , (1) and probability density function (pdf ) corresponding to (1) given by 𝑔𝐾𝑤(𝑥;𝛼,𝜃)=𝛼𝜃𝑥𝛼−1(1−𝑥𝛼)𝜃−1 , (2) 1 School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia. E-mail: Muhammad[email protected]. 2 School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia. E-mail: [email protected]. 3 School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia. E-mails: [email protected]du.au, [email protected]. 184 M. S. Khan, R. King, I. L. Hudson: Transmuted Kumaraswamy … where α>0 and θ>0 are the shape parameters. The Kw probability density function has the same basic properties as the beta distribution. According to Jones (2009) and Cordeiro et al. (2010, 2012) it depends in the same way as the beta distribution on the values of its parameters: it is unimodal for 𝛼>1 and 𝜃>1; uniantimodal for 𝛼<1 and 𝜃<1; increasing for 𝛼>1 and 𝜃≤1; decreasing for 𝛼≤1 and 𝜃>1 and constant for 𝛼=𝜃=1 . Jones (2009) investigated not only properties of the Kumaraswamy distribution but also some similarities and differences between the beta and Kw distributions. According to Jones (2009) the Kumaraswamy distribution has several advantages over the beta distribution, such as a simple normalizing constant, simple explicit formulae for the distribution and quantile functions which do not involve any special functions, a simple formula for random variable generation, explicit formulae for L-moments and simpler formulae for moments of order statistics. On the other hand, the beta distribution has simpler formulae for moments and the moment generating function, a oneparameter sub-family of symmetric distributions, simpler moment estimation and more ways of generating the distribution through physical processes. The Kw distribution is applicable to a number of hydrological problems and many natural phenomena whose process values are bounded on both sides. Cordeiro and de Castro (2009) studied a new class of the Kumaraswamy generalized distributions (denoted by the Kw–G distribution) based on the Kumaraswamy distribution (denoted by Kw distribution). They derived almost all formulas for the probability characteristics of the Kw–G distribution. In hydrology and related areas, the Kw distribution has received considerable interest, see Cordeiro et al. (2010, 2012), Fletcher and Ponnambalam (1996), Ganji et al. (2006), Ponnambalam et al. (2001), Sundar and Subbiah (1989) and Seifi et al. (2000). According to Nadarajah (2008), many papers in the hydrological literature have used this distribution because it is deemed to be a “better alternative” to the beta distribution; see, for example, Koutsoyiannis and Xanthopoulos (1989). This article introduces a new three-parameter distribution which is a generalized two-parameter Kumaraswamy distribution, called the transmuted Kumaraswamy distribution and denoted by 𝑇𝐾𝑤. Recently the generalization of parametric models by transforming an appropriate model, into a more general model, by adding a shape parameter has been intensively studied for many different families of lifetime distributions. Aryal and Tsokos et al. (2009, 2011) considered the following transmuted extreme value distributions: the transmuted Gumbel distribution to model climate data, the transmuted Weibull distribution and their applications to analyse real data sets. Recently Khan et al. (2013 a, b, c) developed the transmuted modified Weibull distribution, the transmuted generalized inverse Weibull distribution and the transmuted generalized exponential distribution. More recently Khan et al. (2014) studied the characteristics of the transmuted Inverse Weibull distribution. Ashour et al. (2013), Elbatal et al. (2013) and Aryal (2013) studied the transmuted Lomax distribution, the transmuted quasi Lindley distribution and the transmuted loglogistic distribution with a discussion on some properties of this family. The most STATISTICS IN TRANSITION new series, June 2016 185 recent families of the transmuted Rayleigh distribution, the transmuted generalized Rayleigh distribution and the transmuted Lindley distribution are derived in studies of Merovici (2013 a, b) & (2014). More recently Ahmad et al. (2015) also studied the transmuted Kumaraswamy distribution and discussed some mathematical results. Using the quadratic rank transmutation map proposed by Shaw et al. (2009), we develop the three-parameter 𝑇𝐾𝑤 distribution. According to this approach a random variable X is said to have a transmuted distribution if its cumulative distribution function (cdf) satisfies the relationship 𝐹(𝑥)=(1+𝜆)𝐺(𝑥)−𝜆[𝐺(𝑥)]2, |𝜆|≤1 (3) and 𝑓(𝑥)=𝑔(𝑥)[(1+𝜆)−2𝜆𝐺(𝑥)], (4) where 𝐺(𝑥) is the cdf of the base distribution, 𝑔(𝑥) and 𝑓(𝑥) are the corresponding probability density functions (pdf) associated with 𝐺(𝑥) and 𝐹(𝑥), respectively. It is important to note that at λ=0 we have the distribution of the base random variable. The paper is organized as follows. In Section 2, we present the analytical shapes of the probability density and hazard functions of the model under study. A range of mathematical properties are considered in Section 3, specifically we demonstrate the quantile functions, moment estimation and moment generating function. Maximum likelihood estimates (MLEs) of the unknown parameters are discussed in Section 4. Entropy and mean deviations are derived in Section 5 and 6. The probability density function (pdf) of order statistics and their moments are derived in Section 7. In Section 8 we evaluate the performance of MLEs using Simulation. Two applications of the 𝑇𝐾𝑤 distribution to the flood data and HIV/ AIDS data are illustrated in Section 9. In Section 10, concluding remarks are addressed. 2. Transmuted Kumaraswamy distribution A random variable X is said to have transmuted 𝐾𝑤 probability distribution denoted by 𝑇𝐾𝑤(𝑥;𝛼,𝜃,𝜆) with parameters 𝛼,𝜃>0 and −1≤λ≤1, 𝑥∈(0,1), if its pdf and cdf are given by 𝑓𝑇𝐾𝑤(𝑥;𝛼,𝜃,𝜆)=𝛼𝜃𝑥𝛼−1(1−𝑥𝛼)𝜃−1{1−𝜆+2𝜆(1−𝑥𝛼)𝜃}, (5) and 𝐹𝑇𝐾𝑤(𝑥;𝛼,𝜃,𝜆)=[1−(1−𝑥𝛼)𝜃][1+𝜆(1−𝑥𝛼)𝜃], (6) where α and θ are the shape parameters and λ the transmuting parameter, representing the different patterns of the subject distribution. The 𝑇𝐾𝑤 distribution approaches the 𝐾𝑤 distribution when the transmuted parameter λ=0. 186 M. S. Khan, R. King, I. L. Hudson: Transmuted Kumaraswamy … If 𝑋 has 𝑇𝐾𝑤(x;α,θ,λ)distribution, then the reliability function (RF), hazard function and cumulative hazard function corresponding to (5) are given by 𝑅𝑇𝐾𝑤(𝑥;𝛼,𝜃,𝜆)=1−[1−(1−𝑥𝛼)𝜃][1+𝜆(1−𝑥𝛼)𝜃], (7) ℎ𝑇𝐾𝑤(𝑥;𝛼,𝜃,𝜆)=𝛼𝜃𝑥𝛼−1(1−𝑥𝛼)𝜃−1{1−𝜆+2𝜆(1−𝑥𝛼)𝜃} 1−[1−(1−𝑥𝛼)𝜃][1+𝜆(1−𝑥𝛼)𝜃], (8) and 𝐻𝑇𝐾𝑤(𝑥;𝛼,𝜃,𝜆)=∫𝛼𝜃𝑥𝛼−1(1−𝑥𝛼)𝜃−1{1−𝜆+2𝜆(1−𝑥𝛼)𝜃} 1−[1−(1−𝑥𝛼)𝜃][1+𝜆(1−𝑥𝛼)𝜃]𝑑𝑥 𝑥 0, 𝐻𝑇𝐾𝑤(𝑥;𝛼,𝜃,𝜆)=−ln|1−[1−(1−𝑥𝛼)𝜃][1+𝜆(1−𝑥𝛼)𝜃]|. (9) Figure 1 shows some possible shapes of probability density function of the 𝑇𝐾𝑤 distribution for selected values of the parameters α,θ and λ, and the hazard function of the 𝑇𝐾𝑤 distribution for the same value of the parameters, respectively. Figure 1. Plots of the 𝑇𝐾𝑤 PDF & HF for some parameter values x f(x) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 =0.5,=0.5,=-0.2 =3,=2,=-0.5 =2,=3,=0 =3,=3,=0.5 =2,=5,=1 x f(x) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 =1,=1,=0 =0.8,=1.5,=-1 =1,=2,=-0.5 =3,=3,=0.5 =2,=2,=-1 x h(x) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 =0.5,=0.5,=-0.2 =3,=2,=-0.5 =2,=3,=0 =3,=3,=0.5 x h(x) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 =0.5,=0.5,=-0.1 =2,=3,=-0.5 =3,=2,=0 =3,=3,=0.5 =5,=2,=1 STATISTICS IN TRANSITION new series, June 2016 187 Figure 1 also illustrates the 𝑇𝐾𝑤 instantaneous failure rates. These failure rates are defined with different choices of parameters. For all choices of parameters the distribution has a monotonically increasing and decreasing behaviour of hazard rates. The transmuted beta and 𝑇𝐾𝑤 distributions share their main special cases. T-Beta (𝑎,1,𝜆) and 𝑇𝐾𝑤(𝛼,1,𝜆) distributions are both power function distributions. The 𝑇𝐾𝑤 distribution approaches the transmuted uniform distribution 𝑎=𝜃=1, 𝜆≤1, uniform distribution for 𝑎=𝜃=1, 𝜆=0 . 3. Moments and quantiles This section presents expressions for the moments, moment generating function and quantiles of the 𝑇𝐾𝑤 distribution. Theorem 1: If 𝑋 has the 𝑇𝑘𝑤(𝑥;α,𝜃,λ) distribution with |λ|≤1, then the 𝑘𝑡ℎ moment of 𝑋 is given as follows 𝐸(𝑋k)=(1−λ)θβ(kα+1,θ)+2λθβ(kα+1,2θ). Proof: Let X have a 𝑇𝑘𝑤 distribution, then the 𝑘𝑡ℎ moment of X is given as 𝐸(𝑋k)=∫𝛼𝜃𝑥𝑘+𝛼−1(1−𝑥𝛼)𝜃−1{1−𝜆+2𝜆(1−𝑥𝛼)𝜃}𝑑𝑥 1 0, 𝐸(𝑋k)=(1−λ)∫𝛼𝜃𝑥𝑘+𝛼−1(1−𝑥𝛼)𝜃−1𝑑𝑥 1 0 +2λ∫𝛼𝜃𝑥𝑘+𝛼−1(1−𝑥𝛼)2𝜃−1 1 0𝑑𝑥. Finally, we obtain the 𝑘𝑡ℎ moment of the 𝑇𝑘𝑤 distribution as 𝐸(𝑋k)=(1−λ)θ ψ1,k+2λθ ψ2,k, (10) where ψ𝒿,k is introduced for simplicity as ψ𝒿,k=β(kα+1,𝒿θ), 𝒿=1,2. The expressions for the expected value and variance are E(𝑋)=(1−λ)θ ψ1,1+2λθ ψ2,1, (11) 188 M. S. Khan, R. King, I. L. Hudson: Transmuted Kumaraswamy … and 𝑉𝑎𝑟(𝑋)=(1−λ)θ ψ1,2+2λθ ψ2,2−{(1−λ)θ ψ1,1+2λθ ψ2,1}2. (12) The coefficient of variation, skewness and kurtosis measures can now be calculated using the following relationships 𝐶𝑉(𝑋)=√𝑉𝑎𝑟(𝑋) E(𝑋), Skewness(𝑋)=E(X−E(𝑋))3 [𝑉𝑎𝑟(𝑋)]32, and Kurtosis(𝑋)=E(X−E(𝑋))4 [𝑉𝑎𝑟(𝑋)]2. The mean, variance and coefficient of variation can be obtained using equations (11) and (12). The relationship between α and the mean is shown in Figure 2. Figure 2.𝛼 vs mean  Mean 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 =-1 =-0.5 =0 =0.5 =1 STATISTICS IN TRANSITION new series, June 2016 189 (𝑎)α vs 𝑉𝑎𝑟𝑇𝐾𝑤 (b) α vs 𝐶𝑉𝑇𝐾𝑤 (𝑐) α vs 𝐶𝑆𝑇𝐾𝑤 (d) α vs 𝐶𝐾𝑇𝐾𝑤 Figure 3. Plots of the 𝑇𝐾𝑤 distribution α 𝑣𝑠 coefficients The relationship between α and the variance is represented in Figure 3a. It is clear that, as α increases, the variance of the subject distribution also increases. The relationship between α and the coefficient of variation (𝐶𝑉𝑇𝐾𝑤) is shown in Figure 3b, which shows that as the parameter α increases, the coefficient of variation of the subject distribution is decreasing. Graphical representation of skewness (𝐶𝑆𝑇𝐾𝑤) and kurtosis (𝐶𝐾𝑇𝐾𝑤) when 𝜃=3 and λ=−1,−0.5, 0,0.5,1 as a function of α are illustrated in Figures 3c and 3d, respectively. This shows the coefficients of skewness and kurtosis have a negative relationship with α.  Var 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 =-1 =-0.5 =0 =0.5 =1  CV 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 =-1 =-0.5 =0 =0.5 =1  CS 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 =-1 =-0.5 =0 =0.5 =1  CK 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 =-1 =-0.5 =0 =0.5 =1 190 M. S. Khan, R. King, I. L. Hudson: Transmuted Kumaraswamy … Theorem 2: If 𝑋 has the 𝑇𝑘𝑤(𝑥;α,𝜃,λ) distribution with |λ|≤1, then the moment generating function of 𝑋 say M𝑥(t) is given as follows M𝑥(t)=(1−λ)∑tm m!θβ(m α+1,θ)+2λ∑tm m!θβ(m α+1,2θ). ∞ m=0 ∞ m=0 Proof: Let X has a 𝑇𝑘𝑤 distribution, then the moment generating function of X is given as M𝑥(t)=∫𝛼𝜃exp(𝑡𝑥)𝑥𝛼−1(1−𝑥𝛼)𝜃−1{1−𝜆+2𝜆(1−𝑥𝛼)𝜃}𝑑𝑥 1 0. Using the Taylor series of function 𝑒𝑡𝑥 reduces the above to M𝑥(t)=(1−λ)∑tm m! ∞ m=0 ∫𝛼𝜃𝑥𝑚+𝛼−1(1−𝑥𝛼)𝜃−1𝑑𝑥 1 0+ 2λ∑tm m! ∞ m=0 ∫𝛼𝜃𝑥𝑚+𝛼−1(1−𝑥𝛼)2𝜃−1 1 0𝑑𝑥. By solving the above integral we obtain M𝑥(t)=(1−λ)∑tm m!θβ(m α+1,θ)+2λ∑tm m!θβ(m α+1,2θ) ∞ m=0 , ∞ m=0 (13) which completes the proof. Theorem 3: The qth quantile 𝑥q of the 𝑇𝑘𝑤 random variable is given by 𝑥q=[1−{1−(1+λ)−√(1+λ)2−4λq 2λ}1θ]1 α, 0< 𝑞 <1. (14) Proof: The qth quantile 𝑥q of the 𝑇𝑘𝑤 distribution is defined as 𝑞=P(𝑋≤𝑥q)=F(𝑥q), 𝑥q≥0. Using the distribution function of the 𝑇𝑘𝑤 distribution we have 𝑞=F(𝑥q)=(1+𝜆)[1−(1−𝑥𝑞𝛼)𝜃]−𝜆[1−(1−𝑥𝑞𝛼)𝜃]2, that is 𝜆[1−(1−𝑥𝑞𝛼)𝜃]2−(1+𝜆)[1−(1−𝑥𝑞𝛼)𝜃]+𝑞=0. Consider this as a quadratic in 1−(1−𝑥𝑞𝛼)𝜃as ∆=1+(2−4𝑞)𝜆+𝜆2. STATISTICS IN TRANSITION new series, June 2016 197 ×𝛼𝜃𝑥𝛼−1(1−𝑥𝛼)𝜃−1{1−𝜆+2𝜆(1−𝑥𝛼)𝜃}, (24) and 𝑓𝑛:𝑛(𝑥)=n{[1−(1−𝑥𝛼)𝜃][1+𝜆(1−𝑥𝛼)𝜃]}n−1 ×𝛼𝜃𝑥𝛼−1(1−𝑥𝛼)𝜃−1{1−𝜆+2𝜆(1−𝑥𝛼)𝜃}. (25) Theorem 5: The probability density function and the 𝑘𝑡ℎ moment of rth order statistic 𝑋𝑟:𝑛 of the random sample X from the 𝑇𝐾𝑤 distribution are given by 𝑓𝑟:𝑛(𝑥)=n(n−1 r−1)∑∑∑𝑤𝑘,ℓ,𝓂,𝜆 ∞ 𝓂=0 ∞ ℓ=0 n−r k=0 {(1−λ)τα,θ,𝓂,λ (1)+2λτα,θ,𝓂,λ (2)}. μk(r:n)=n(n−1 r−1)∑∑∑𝑤𝑘,ℓ,𝓂,𝜆 ∞ 𝓂=0 ∞ ℓ=0 n−r k=0 [(1−λ)θβ{kα+1,θ(𝓂+1)} +2λθβ{kα+1,θ(𝓂+2)}]. Proof: From Balakrishnan and Nagaraja (1992), the pdf of rth order statistics is given by 𝑓𝑟:𝑛(𝑥)=[F(𝑥)]r−1[1−F(𝑥)]n−rf(𝑥) B(r,n−r+1) , (26) where B(.,.) is the beta function. Using (26) the pdf of 𝑥(𝑟) is given by 𝑓𝑟:𝑛(𝑥)=1 B(r,n−r+1)∑(n−r k)(−1)k(F(𝑥))r+k−1f(𝑥). n−r k=0 (27) Substituting (5) and (6) in (27) we obtain 𝑓𝑟:𝑛(𝑥)=n(n−1 r−1)∑(n−r k)(−1)k{[1−(1−𝑥𝛼)𝜃][1+𝜆(1−𝑥𝛼)𝜃]}r+k−1 n−r k=0 ×𝛼𝜃𝑥𝛼−1(1−𝑥𝛼)𝜃−1{1−𝜆+2𝜆(1−𝑥𝛼)𝜃}. The above expression reduces to the pdf of the rth order statistic as 𝑓𝑟:𝑛(𝑥)=n(n−1 r−1)∑∑∑𝑤𝑘,ℓ,𝓂,𝜆 ∞ 𝓂=0 ∞ ℓ=0 n−r k=0 {(1−λ)τα,θ,𝓂,λ (1)+2λτα,θ,𝓂,λ (2)}, (28) where 𝑤𝑘,ℓ,𝓂,𝜆=(n−r k)(r+k−1 ℓ)(r+k+ℓ−1 𝓂)(−1)k+ℓ+𝓂(λ 1+λ)ℓ(1+λ)r+k−1 τα,θ,𝓂,λ (g)=𝛼𝜃𝑥𝛼−1(1−𝑥𝛼)𝜃(𝓂+g)−1, 𝑔=1,2. 198 M. S. Khan, R. King, I. L. Hudson: Transmuted Kumaraswamy … Using (28) the 𝑘𝑡ℎ moment of rth order statistic of 𝑥(𝑟) is given by μk(r:n)=n(n−1 r−1)∑∑∑𝑤𝑘,ℓ,𝓂,𝜆 ∞ 𝓂=0 [(1−λ)∫𝛼𝜃𝑥𝑘+𝛼−1(1−𝑥𝛼)𝜃(𝓂+1)−1 1 0 ∞ ℓ=0 n−r k=0 𝑑𝑥 +2λ∫𝛼𝜃𝑥𝑘+𝛼−1(1−𝑥𝛼)𝜃(𝓂+2)−1 1 0𝑑𝑥]. Therefore, by solving the above integral the 𝑘𝑡ℎ moment of the rth order statistic of the 𝑇𝐾𝑤 distribution can be obtained as μk(r:n)=n(n−1 r−1)∑∑∑𝑤𝑘,ℓ,𝓂,𝜆 ∞ 𝓂=0 ∞ ℓ=0 n−r k=0 [(1−λ)θβ{kα+1,θ(𝓂+1)} +2λθβ{kα+1,θ(𝓂+2)}], (29) which completes the proof. Theorem 6: The probability density function and the 𝑘𝑡ℎ moment of the median order statistic of a random sample X from the TKw distribution are given by 𝑔(𝑥)=(2m+1)! m!m! ∑∑∑𝑤𝑘,ℓ,𝑚,𝓃,𝜆 ∞ 𝓃=0 ∞ ℓ=0 m k=0 {(1−λ)ℰα,θ,𝓃,λ+2λ𝒥α,θ,𝓃,λ} and μk(r:n)=(2m+1)! m!m! ∑∑∑∑𝑤𝑘,ℓ,𝑚,𝓃,𝜆 ∞ 𝓃=0 ∞ 𝓃=0 ∞ ℓ=0 m k=0 [(1−λ)θβ{kα+1,θ(𝓃+1)} +2λθβ{kα+1,θ(𝓃+2)}]. Proof: Let 𝑋,…,𝑋𝑛 be independently and identically distributed ordered random variables from the 𝑇𝐾𝑤 distribution having median order 𝑋𝑚+1 probability density function given by 𝑔(𝑥)=(2m+1)! m!m! ∑(m k)(−1)k{F(𝑥)}m+kf(𝑥). (30) m k=0 Substituting (5) and (6) in (30) we obtain 𝑔(𝑥)=(2m+1)! m!m! ∑(m k)(−1)k{[1−(1−𝑥𝛼)𝜃][1+𝜆(1−𝑥𝛼)𝜃]}m+k m k=0 ×𝛼𝜃𝑥𝛼−1(1−𝑥𝛼)𝜃−1{1−𝜆+2𝜆(1−𝑥𝛼)𝜃}. STATISTICS IN TRANSITION new series, June 2016 199 The above expression reduces to the pdf of the median as 𝑔(𝑥)=(2m+1)! m!m! ∑∑∑𝑤𝑘,ℓ,𝑚,𝓃,𝜆 ∞ 𝓃=0 ∞ ℓ=0 m k=0 {(1−λ)𝒥α,θ,𝓃,λ (1)+2λ𝒥α,θ,𝓃,λ (2)}, (31) where 𝑤𝑘,ℓ,𝓂,𝓃,𝜆=(m k)(m+k ℓ)(m+k+ℓ 𝓃)(−1)k+ℓ+𝓃(λ 1+λ)ℓ(1+λ)m+k 𝒥α,θ,𝓃,λ (ℎ)=𝛼𝜃𝑥𝛼−1(1−𝑥𝛼)𝜃(𝑛+ℎ)−1, ℎ=1,2. Using (31) the 𝑘𝑡ℎ moment of the rth median 𝑥(𝑟) given by μk(r:n)=(2m+1)! m!m! ∑∑∑∑𝑤𝑘,ℓ,𝑚,𝓃,𝜆 ∞ 𝓃=0 ∞ 𝓃=0 ∞ ℓ=0 m k=0 [(1−λ)∫𝛼𝜃𝑥𝑘+𝛼−1(1−𝑥𝛼)𝜃(𝑛+1)−1 1 0𝑑𝑥 +2λ∫𝛼𝜃𝑥𝑘+𝛼−1(1−𝑥𝛼)𝜃(𝑛+2)−1𝑑𝑥 1 0]. Therefore, the above integral reduces to the 𝑘𝑡ℎ moment of the median as follows μk(r:n)=(2m+1)! m!m! ∑∑∑∑𝑤𝑘,ℓ,𝑚,𝓃,𝜆 ∞ 𝓃=0 ∞ 𝓃=0 ∞ ℓ=0 m k=0 [(1−λ)θβ{kα+1,θ(𝓃+1)} +2λθβ{kα+1,θ(𝓃+2)}], (32) which completes the proof. 8. Simulation This section evaluates the performance of the MLEs for the three parameters α,θ and λ of the 𝑇𝐾𝑤 distribution by using Monte Carlo simulation. The simulation of the 𝑇𝐾𝑤 distribution can be performed by using (14). The samples of the 𝑇𝐾𝑤 distribution were generated for different sizes n= 25,50,75,100,200,300,400,500 for fixed choice of parameters α =3,θ=3 and 𝜆=0.5. The estimates of the unknown parameters has been obtained by using BFGS method to minimize the total log-likelihood function. The estimated values of the parameters 𝛼,𝜃,𝜆 with their corresponding standard error, bias and mean square error (MSE) are displayed in Table 1. The plot in Figure 6 evaluates the overall performance of the 𝑇𝐾𝑤 distribution for simulated data sets that show the exact densities and histogram for some selected values of parameters. 200 M. S. Khan, R. King, I. L. Hudson: Transmuted Kumaraswamy … Table 1. Mean, standard Error, Bias and MSE of the 𝑇𝐾𝑤 distribution n Parameter Mean S.E Bias MSE α 3.1119 0.6526 0.1119 0.4384 25 θ 3.2504 1.3394 0.2504 1.8566 λ 0.2605 0.6013 -0.2395 0.4189 α 2.9603 1.0269 -0.0397 1.0561 50 θ 3.6296 0.8641 0.6296 1.1430 λ -0.2737 0.8336 -0.7737 1.2935 α 3.2675 0.3535 0.2675 0.1965 75 θ 3.1488 0.9566 0.1488 0.9372 λ 0.6362 0.3023 0.1362 0.1099 α 2.6158 0.4246 -0.3842 0.3278 100 θ 3.1376 0.5942 0.1376 0.3720 λ 0.0164 0.5084 -0.4836 0.4923 α 2.9212 0.2045 -0.0788 0.0480 200 θ 3.0396 0.8866 0.0396 0.7876 λ 0.4882 0.4256 -0.0118 0.1812 α 3.0556 0.1983 0.0556 0.0424 300 θ 3.6425 0.6270 0.6425 0.8059 λ 0.3285 0.3121 -0.1715 0.1268 α 2.8835 0.1679 -0.1165 0.0417 400 θ 3.0600 0.4560 0.0600 0.2115 λ 0.3212 0.2736 -0.1788 0.1068 α 2.9709 0.1579 -0.0291 0.0257 500 θ 3.2659 0.4845 0.2659 0.3054 λ 0.3331 0.2789 -0.1669 0.1056 STATISTICS IN TRANSITION new series, June 2016 201 Figure 6. Plots of the 𝑇𝐾𝑤 densities for simulated data sets: (a) 𝛼=3,𝜃=3,𝜆=1 and (b) 𝛼=0.5,𝜃=1,𝜆=0.5. (a) x Density 0.0 0.2 0.4 0.6 0.8 0.0 0.5 1.0 1.5 2.0 2.5 (b) x Density 0.0 0.2 0.4 0.6 0.8 1.0 012345 202 M. S. Khan, R. King, I. L. Hudson: Transmuted Kumaraswamy … 9. Applications In this section we provide two data analyses in order to assess the goodness- of-fit of the 𝑇𝐾𝑤 distribution. The first data set is from Dumonceaux and Antle, [32]; with respect to the flood data with 20 observations 0.265, 0.269, 0.297, 0.315, 0.3235, 0.338, 0.379, 0.379, 0.392, 0.402, 0.412, 0.416, 0.418, 0.423, 0.449, 0.484, 0.494, 0.613, 0.654, 0.74. The summary statistics for the 𝑇𝐾𝑤 distribution are given in Table 2. The MLEs and the values of the maximized loglikelihoods for the 𝑇𝐾𝑤 and Kw distributions are given in Table 3. Table 2. Summary Statistics for the 𝑇𝐾𝑤 and Kw distributions for flood data Distribution Median Coefficient of Quartile Deviation Bowley’s skewness Moors(ℳ) kurtosis 𝑇𝐾𝑤 0.4207 4.7328 -0.0048 1.2212 𝐾𝑤 0.4268 4.4817 -0.0176 1.2019 In order to compare the distributions, we consider the Kolmogorov-Smirnov (K-S) test, Cramér-von Mises and Anderson-Darling goodness-of-fit statistics for the flood data. Table 3 gives the MLEs of the unknown parameters and the corresponding standard errors of the 𝑇𝐾𝑤 and 𝐾𝑤 distributions. These results indicate that the 𝑇𝐾𝑤 distribution provides an adequate fit for the flood data. Table 3. MLEs of the unknown Parameters for the flood data with the corresponding standard errors in parenthesis and the goodness-of-fit measures, The K-S test, Cramér-von Mises and Anderson-Darling goodness-of-fit. Distribution Parameter Estimates K-S test 𝒲 𝒜  ˆ  ˆ  ˆ 𝑇𝐾𝑤 3.7252 (0.6489) 10.9575 (6.0334) 0.6143 (0.3752) 0.1930 0.1409 0.8408 𝐾𝑤 3.3631 (0.6033) 11.7882 (5.3589) - 0.2109 0.1658 0.9722 STATISTICS IN TRANSITION new series, June 2016 203 (a) Estimated pdf (b) Estimated Survival function Figure 7. Estimated densities and survival functions of TKw and Kw models fitted to flood data (a) 𝑇𝐾𝑤 (b) 𝐾𝑤 Figure. 8 P-P Plots of the (a) 𝑇𝐾𝑤 and (b) 𝐾𝑤 models fitted to flood data. x Density 0.2 0.3 0.4 0.5 0.6 0.7 0.8 01234 TKw Kw 0.3 0.4 0.5 0.6 0.7 0.0 0.2 0.4 0.6 0.8 1.0 x Survival probability Kaplan-Meier TKw Kw 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 F(x) Empirical cumulative distribution 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 F(x) Empirical cumulative distribution 204 M. S. Khan, R. King, I. L. Hudson: Transmuted Kumaraswamy … (a) ℒ for α (b) ℒ for θ (c) ℒ for 𝜆 Figure 9. The profile of the log-likelihood function for α,θ and λ for flood data Based on the plots of the estimated 𝑇𝐾𝑤 and 𝐾𝑤 densities the relative histogram of the flood data suggests that the fit of the proposed model performs better than the baseline distribution shown in Figure 7(a). Furthermore, the empirical survival function and the fitted survival functions are plotted in Figure 7(b). By comparing the fitted models of these two distribution we have supporting evidence that the 𝑇𝐾𝑤 distribution provides a good fit for flood data. We have supporting evidence that the Kolmogorov-Smirnov (K-S) distance between the empirical and fitted 𝑇𝐾𝑤 distribution is smaller than the 𝐾𝑤 distribution. Table 3 also indicates that the Cramér-von Mises test statistics and Anderson-Darling goodness statistics have the smallest values for the 𝑇𝐾𝑤 distribution for the flood data with regard to the 𝐾𝑤 distribution. Based on these three goodness-of-fit measures we conclude that the 𝑇𝐾𝑤 distribution provides a better fit than the 𝐾𝑤 lifetime distribution. Figure 8 displays the P-P Plots of the 𝑇𝐾𝑤 distribution and  The log-likelihood function 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 10 10.5 11 11.5 12 12.5 13 13.5 14 =10.9575; =0.6143  The log-likelihood function 46810 12 14 16 18 20 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 =3.7252; =0.6143  The log-likelihood function 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10.5 11 11.5 12 12.5 13 13.5 14 =3.7252; =10.9575 STATISTICS IN TRANSITION new series, June 2016 205 𝐾𝑤 distribution. The P-P Plot distance between the ab-line and the fitted model of the 𝑇𝐾𝑤 distribution and 𝐾𝑤 distribution are very similar and close to the baseline model. Figure 9 illustrates the profile of the log-likelihood function for the 𝑇𝐾𝑤 distribution with parameters 𝛼,𝜃 and 𝜆 fitted to the flood data and exhibits the unique maximum for these parameters. Based on these results we conclude that the 𝑇𝐾𝑤 tends to provide a relatively better fit than the 𝐾𝑤 distribution for the flood data. The second data set has been collected from Joint United Nations programme on HIV/ AIDS (UNAIDS), for Infants born to HIV+ women receiving a virological test for HIV within 2 months of birth. The data set consists of 906 observations for the years 2009-13 and is freely available online at http://data.un.org/Data.aspx?d=UNAIDS&f=inID%3a41. The summary statistics for the 𝑇𝐾𝑤 and 𝐾𝑤 distributions for HIV data are given in Table 4. Table 4. Summary Statistics for the 𝑇𝐾𝑤 and Kw distributions for HIV data Distribution Median Coefficient of Quartile Deviation Bowley’s skewness Moors(ℳ) kurtosis 𝑇𝐾𝑤 0.2002 1.3272 0.2754 1.1491 𝐾𝑤 0.2146 1.3137 0.2529 1.0884 The MLEs and the values of the maximized log-likelihoods for the 𝑇𝐾𝑤 and Kw distributions for HIV data are given in Table 5, with the MLEs of the unknown parameters and the corresponding standard errors of the 𝑇𝐾𝑤 and 𝐾𝑤 distributions. The standard error estimates obtained using the observed information matrix appear to be smaller than the parameter estimates. In order to compare the distributions, we consider the AIC (Akaike Information Criterion) for the HIV/ AIDS data. These results indicate that the 𝑇𝐾𝑤 distribution provides better fit for the HIV/ AIDS data. Furthermore, we applied the LR statistics in order to verify which model fits better for the HIV data. The hypotheses to be tested are 𝐻0:𝜔=(α,𝜃,λ)T versus 𝐻𝐴: 𝐻0 is not true, and the LR statistics reduces to 𝛬=2{𝑙(𝜔)−𝑙(𝜔)}=52.8088, where 𝜔 is the MLE of 𝜔 under 𝐻0. The null hypothesis is rejected as the p-value=1.7048E-12 < 𝛼=0.05. Table 5. MLEs of the unknown Parameters for the HIV/ AIDS data with the corresponding standard errors in parenthesis and the goodness-of-fit measure AIC Distribution Parameter Estimates AIC  ˆ  ˆ  ˆ 𝑇𝐾𝑤 0.6724 (0.0251) 1.1222 (0.0837) 0.5491 (0.0882) -633.605 𝐾𝑤 0.6096 (0.0237) 1.3961 (0.0629) - -607.201 206 M. S. Khan, R. King, I. L. Hudson: Transmuted Kumaraswamy … (a) Estimated pdf (b) Estimated Survival function Figure 10. Estimated densities and survival functions of TKw and Kw models fitted to HIV data (a) 𝑇𝐾𝑤 (b) 𝐾𝑤 Figure 11. P-P Plots of the (a) 𝑇𝐾𝑤 and (b) 𝐾𝑤 models fitted to HIV data Finally, in order to assess whether the proposed model is appropriate for HIV data we display the visualization of the estimated 𝑇𝐾𝑤 and 𝐾𝑤 densities and the relative histogram for the HIV/ AIDS data suggests that the fit of the proposed model performs better than the baseline distribution shown in Figure 10(a). Furthermore, the plots of the empirical survival function and the fitted survival functions are shown in Figure 10(b). Both figures suggest that the 𝑇𝐾𝑤 x Density 0.0 0.2 0.4 0.6 0.8 1.0 01234 TKw Kw 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 x Survival probability Kaplan-Meier TKw Kw 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 F(x) Empirical cumulative distribution 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 F(x) Empirical cumulative distribution