Research Article Journal of Volume 11:1, 2020 DOI: 10.37421/jbmbs.2020.11.436 Biometrics & Biostatistics ISSN: 2155-6180 Open Access The Kumaraswamy-Rani Distribution and Its Applications Zitouni Mouna* Laboratory of Probability and Statistics. Sfax University. B.P. 1171, Tunisia Abstract In this paper, a new distribution called the Kumaraswamy-Rani (KR) distribution, as a Special model from the class of Kumaraswamy Generalized (KW-G) distributions, is introduced. Its statistical properties are explored. Estimation parameters based on maximum likelihood are obtained. We have illustrated the performances of the proposed distribution by some simulation studies. Finally, the significance of the KR distribution in terms of modeling real data set has been highlighted before it has been compared to some one parameter lifetime distributions. Keywords: Fixed point method • Kumaraswamy distribution • Maximum likelihood estimation • Newton-raphson method • Rani distribution Introduction The KR Distribution Modeling and analysis of lifetime data is an important aspect of The Kumaraswamy-Generalized distribution statistical work in such wide variety of scientific and technological fields as medicine, engineering, insurance and finance... The cumulative density function (cdf) of the Kumaraswamy-Generalized (Kum-Generalized) distribution proposed by Cordeiro et al. [8] is given by Therefore, the construction of the distributions is the most important −− ab work from a probabilistic point of view and recently many researchers are Fx( ) = 1 (1 ( Gx ( )) ) , (1) concentrated in this area [1-3]. Where a>0, b>0 are shape parameters and G is the cdf of a continuous random variable X. Consquently, the Kumaraswamy-Generalized More that, varying lifetime distributions and their generalizations have distribution has the probability density function (pdf) specified by been created as exponential, Lindley, gamma, lognormal, Weibull, Sujatha, Akash, Rani. Further, A two-parameter family of distributions on called f()= x abg () x G () x(a−− 1) (1− G ()) x ab( 1) , (2) Kumaraswamy distribution is explored. It was introduced by Kumaraswamy ∂Gx() [4]. It has the flexibility in modeling lifetime data. where, gx( )= . ∂x The latter exhibits many similarities to the beta distribution and a According to Cordeiro et al. [8], using the binomial expansion for + number of advantages in terms of tractability which has been discussed by b ∈ , equation (2) can be rewritten as Jones [5]. Among these advantages, we mention: a simple explicit formula ∞ ai(+− 1) 1 for its distribution function, its quantile function. L-moments and moments f()=() x g x∑ wGi () x , (3) i=0 of order statistics [6]. In this direction, another new one parameter lifetime distribution has attracted the attention of statisticians and whetted then where the binomial coefficient wi is defined for all real numbers with, interest named Rani distribution. It was proposed by Rama Shanker and it b −1 has important statistical properties [7]. For the motivating properties of the − i wi = ( 1) ab i Kumaraswamy and Rani distributions, we shall introduce a new probability distribution based on the Kumaraswamy distribution The Rani distribution and the Rani distribution. The proposed distribution is called the The Rani distribution is proposed by Rama Shanker [7]. It is defined Kumaraswamy-Rani (KR) distribution with twos positive parameters over the support x>0 and it has the pdf given by: denoted b and β. In this study some statistical properties of this distribution θ 5 are derived to show their applicability in the real lifetime data analysis. The fx( )= (θ + x4 ) e−θ x , (4) rest of the paper is organized as follows. In Section 2, the KR distribution θ 5 + 24 is identified in addition, some important mathematical properties of the new where θ is the scale parameter of the Rani distribution denoted R(θ). The distribution are presented. In Section 3, maximum likelihood estimation latter is a mixture of exponential (θ) and gamma (5,θ) with mixing proportion is discussed. In section 4, some simulation studies are reported. A real θ 5 . Furthermore, the corresponding cumulative distribution function lifetime data set are used in Section 5 to illustrate the usefulness of the KR θ 5 + 24 distribution. Finally, our work is crowned with conclusion. (cdf) is given by: θθxx(33+ 4 θ 22 x ++ 12 θ x 24) −+ −θ x (5) *Address for Correspondence: Zitouni Mouna, Laboratory of Probability and Fx( )=1 1 5 e . Statistics. Sfax University. B.P. 1171, Tunisia, Tel: +216 74 244 423; E-mail: θ + 24 [email protected] The KR distribution Copyright: © 2020 Mouna Z. This is an open-access article distributed under the In this subsection, we give the new distribution called KR distribution. terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author It is formulated using the Rani (θ>0) and the Kumaraswamy (ab = 1, > 0) and source are credited. distributions. Received 21 January 2020; Accepted 06 February 2020; Published 13 February The KR density for x>0 is expressed as 2020 Mouna Z. J Biom Biostat, Volume 11:1, 2020 (b− 1) 5 33 22 5 θ −−θθθθxx(+ 4 θ x ++ 12 θ x 24) θ ++4 xx θ 4 −θ x fx( )= b55 ( x ) e 1 e . (6) θ + θθ++ b5 () xe 24 24 fx() θ + 24 Hx( )= = 33 22 . The parameters vector of the KR distribution is Θ =(b ,β ). Moreover, 1− Fx () θθxx(+ 4 θ x ++ 12 θ x 24) −θ x 1+ 5 e using equation (3), equation (6) can be defined as: θ + 24 i θ5 ∞ θθ33+ θ 22 ++ θ 4 −−θθxxxx( 4 x 12 x 24) Now, we shall introduce the following Theorem about the inverse fx( )= (θ +x ) e wi 1−+ 1 e , (7) θθ55++∑ 24 i=0 24 cumulative KR distribution. The latter characterizes the proposed b −1 distribution. −1 − i Theorem 2.1: θ where wi = ( 1) bi. Let Y KR( b , ), then xF= () y is the zero of the polynomial equation given by Remark 1: Let X KR( b ,θ ). If b=1, then XR ()θ (Figure 1). θθ44x+4, 33 x + Ax 2 ++ Bx C (9) We can find easily the cumulative distribtion function (c.d.f) of the KR where, distribution θθ771 b A =− +−− (1 (1µθ )b )( + 122 ), θθxx(33+ 4 θ 22 x ++ 12 θ x 24) Fx( ) = 1−+ 1 e−θ x , x> 0. (8) 22 5 θ + 24 1 B = (1−− (1µ )b )( θ66 + 24 θθ ) − , Using the c.d.f in order to obtain the Survival function (Figure 2): b 1 33 22 5 b θθxx(+ 4 θ x ++ 12 θ x 24) −θ +θµ −− Sx( )=1−+ Fx ( )= 1 e x C = (24 )(1 (1 ) ) 5 θ + 24 and µ ∈]0,1[ . and the Hazard function which is an important quantity characterizing life phenomena is given by Proof: In order to get the zero of equation (9), we shall introduce the following equation: Fx( ) = 1−− (1 Gx ( ))b =µµ , ∈]0,1[. 0.5 So, b=0.4, theta=1.5 0.45 b=0.4, theta=2 θθxx(33+ 4 θ 22 x ++ 12 θ x 24) 1 b=0.6, theta=1.5 −+ −θ x −−µ b 1 1 5 e = 1 (1 ) . 0.4 θ + 24 0.35 We have also, 1 5 33 22 0.3 θ++24 θθxx ( + 4 θ x + 12 θ x + 24) −−µ b −θ x (1 ) e 5 = 0. f(x) θ + 24 0.25 In the following, we get, 0.2 1 θ5++24 θθxx (33 + 4 θ 22 x + 12 θ x + 24) θ x − µ b 0.15 e (1 ) = 5 . θ + 24 0.1 It is to be noted that the limited development to order 2 of exponential 0.05 function at 0 can be written as: 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 θ x ()θ x x ex=1++θ . 2 Figure 1: The KR densities of selected values of b and θ... Then, we obtain: ()θx2 1 θ5++24 θθxx (33 + 4 θ 22 x + 12 θ x + 24) ++θµ −b (10) 1 x ((1 ) ) = 5 . 2 θ + 24 1 Applying some mathematical treatments on equation (10) yields, b=8, theta=2 0.9 1 7 b=6, theta=8 44 33 2 b θ 2 b=5, theta=6 θθx+4 xx + −− ((1 µ ) )( + 12 θ ) 0.8 2 11 0.7 +(1 −µθ )bb (65 + 24 θ )x + (24 + θ )(1 −− (1 µ ) ) = 0. 0.6 0.5 Proving that 0.4 θθ44x+4 33 x + Ax 2 ++ Bx C = 0, 0.3 where, 0.2 1 7 b θ 2 0.1 A =−− ((1µθ ) )( + 12 ), 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 B = ((1−+µθ )b )(6 24 θ ) Figure 2: The cumulative distribution functions of selected values of b and θ.. and Page 2 of 4 Mouna Z. J Biom Biostat, Volume 11:1, 2020 1 5 C = (24+θµ )(1 −− (1 )b ) 4.5 which completes this proof. 4 3.5 Parameters Estimations 3 In this section, the maximum likelihood method for the parameters 2.5 estimation of the KR distribution is derived. For this reason, a brief f(x) description of the method for the fitting of the KR parameters is presented. 2 The likelihood function for the parameters is given by 1.5 (b− 1) n 5 33 22 θ −−θθxxθθxx(+ 4 θ x ++ 12 θ x 24) θ ++4 iiii i i L=∏ b55 () xei 1 e . θθ++24 24 (11) 1 i=1 The log-likelihood is formulated as: 0.5 n 5 θ −θ 0 ++θ 4 xi 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 log( L ) = n log( b )∑ log5 (xi ) e i=1 θ + 24 x n 33 22 Figure 3: Histogram of random samples from a KR distribution and a curve of θθxx(+ 4 θ x ++ 12 θ x 24) −θ x +−(be 1)log 1 +ii i i i .