Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.6, No.7, 2016 Double Weighted Weibull Distribution Properties and Application Aamir Saghir, Muhammad Saleem Department of Mathematics, Mirpur University of Science and Technology (MUST), Mirpur AJK. This paper offering a new weighted distribution known as the Double Weighted Weibull Distribution (DWWD). The statistical properties of the (DWWD) are derived and discussed, including the mean, variance, coefficient of variation, moments, mode, reliability function, hazard function and the reverse hazard function. Also the parameters of this distribution are estimated by the maximum likelihood estimation method. The plots of survival function, hazard function and reverse hazard function of (DWWD) are also presented. The worth of the distribution has been demonstrated by applying it to real life data. Keywords:Weighted distribution, Double Weighted distribution, Weibull distribution, Reliability estimation. 1. Introduction In the literature the Weibull distribution attract the most of the researchers due to its wide range applications. Different generalization of the Weibull distribution are available in the literature as Merovci and Elbatal (2015) developed the Weibull-Rayleigh distribution and demonstrated its application using lifetime data. Almalki and Yuan (2013) presented the new modified Weibull distribution by combining the Weibull and the modified Weibull distribution in a serial system.Pal M. et. All (1993) introduce the Exponentiated Weibull distribution. Al-Saleh and Agarwal (2006) proposed another extended version of the Weibull distribution. Xie and Lai (1996) developed the additive Weibull distribution withbathtub shaped hazard function obtained as the sum of two hazard functions. Teimouri and Gupta (2013) studied the three-parameter Weibull distribution.Nasiru (2015) introduced another weighted Weibull distribution from azzalini’s family. Gokarna et al. (2011) presented the transmuted Weibull distribution and discussed its various properties. For another generalizations of Weibull distribution see (PalakornSeenoi et al. (2014), Jing (2010), and Kishore and Tanusree(2011)).The pdf and cdf of the Weibull distribution are given as: 28 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.6, No.7, 2016 f(x; λ) = λxλ−1e−xλ x ≥ 0, λ > 0 (1) F(x; λ) = 1−e−xλλ > 0 (2) The plot of thepdf of the Weibull distribution aregiven in figure 1 given below. Now we are giving another generalization of the Weibull distribution known as double weighted Weibull distributionDWWD. The double weighted distribution and length-biased distributions are the types of weighted distribution proposed by Fisher (1934) and Rao (1965). The weighted distribution has useful application in medicine, ecology and reliability etc. There is a lot of literature on the weighted distribution as Das K.K and Roy, T.D. (2011) introduced the Applicability of length biased weighted generalized Rayleigh distribution,NareeratNanuwong and WinaiBodhisuwan (2014) hosted the length-biased Beta- Pareto (LBBP) distribution and compared with Beta-Pareto (BP) and Length-Biased Pareto (LBP) distributions.. For further important results of weighted distribution you can see also Oluyede and George (2002), Ghitany and Al-Mutairi (2008), Ahmed et al. (2013),Oluyede and Pararai (2012), Oluyede and Terbeche (2007). The Concept of double weighted distribution first time introduced by Al-Khadim and Hantoosh (2013), apply it on the exponential distribution and derive the statistical properties for the double weighted exponential distribution.Rishwan(2013) introduce the Characterization and Estimation of Double Weighted Rayleigh Distribution.Al-khadim and Hantoosh (2014) proposed the double weighted inverse Weibull distribution and deliberate its statistical properties of inverse Weibull distribution. All these studies agreed that the double weighted distribution has very efficient and effectual role in the modelling of weighted distributions. The definition of double weighted distribution introduced by Al-khadim and Hantoosh given in next section. 29 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.6, No.7, 2016 Figure 1. Represent the graph of the Weibull distribution for λ = 2,3,4 2. Materials and Methods 2.1. Double Weighted Distribution The double weighted distribution (DWD) proposed by Al-Khadim and Hantoosh (2013) is given by: w(x) f(x) F(cx) f (x; c) = , x ≥ 0,c > 0 (3) w WD ∞ Where W =∫ w(x)f(x)F(cx)dx (4) D 0 And first weight is w(x) and second weight isF(cx) 2.2. Double Weighted Weibull Distribution Using the first weight function w(x) = xand the pdf and cdf of Weibull distribution given in (1) and (2) in equation (4) then: W =∫∞w(x)f(x)F(cx)dx = ∫∞λxλe−xλ(1−e−cλxλ)dx D 0 0 1 W =∫∞λxλe−xλdx−∫∞λxλe(1−cλ)xλdx = Γ(1+1)− Γ(1+λ) D 0 0 λ 1+1 (1+cλ) λ 1 1+ ((1+cλ) λ−1) 1 W = Γ(1+ ) (5) D λ 1+1 (1+cλ) λ 30 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.6, No.7, 2016 Using (1) (2) and (5) in (3) and considering w(x) = x then pdf of the double weighted exponential distribution is given by: 1 λ(1+cλ)1+λxλe−xλ(1−e−cλxλ) f (x;c,λ) = x ≥ 0,c > 0 λ > 0 (6) w 1 Γ(1+1)((1+cλ)1+λ−1) λ (a) (b) Figure 2 (a) and (b) represent the plot of the probability density function of DWWD for various choice of parameters cand λ. Graph of pdf indicate that peak of probability density curve increases when values of c and λ increases. 2.3. The cumulative density function (CDF) The cumulative density function of (DWWD) is given by: x F (x;c,λ) = ∫ f (t;c,λ)dt w 0 w 1 1+ F (x;c,λ) = (1+cλ) λ ∫xλtλe−tλ(1−e−cλtλ)dt w Γ(1+1)((1+cλ)1+λ1−1) 0 λ F (x; c,λ) = (1+cλ)1+λ1 (γ(1+1,xλ)− γ(1+λ1 , xλ(1+cλ))) (7) w Γ(1+1)((1+cλ)1+λ1−1) λ (1+cλ)1+λ1 λ Where x ≥ 0,c > 0 λ > 0 The graph for CDF of DWWD 31 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.6, No.7, 2016 (a) (b) Figure3 (a) and (b) represent the plot of the cumulative density function of DWWD for various choice of parameters cand λ. 3. Transformed double weighted Weibull distribution Put xλ = θyλθ > 0 in (6) then transformed pdf is given by: dx 1 = θλ dy dx Since f (y;c,λ,θ) = f (x; c,λ) w w dy λ(1+cλ)1+λ1θ1+λ1yλe−θyλ(1−e−θcλyλ) f (y;c,λ,θ) = y ≥ 0,c > 0 λ > 0(8) w 1 Γ(1+1)((1+cλ)1+λ−1) λ 4. Sub modals derived from DWWD 1) Put λ = 1 in (8) then: (1+c)2θ2e−θy(1−e−θcy) (1+c)2θ2e−θy(1−e−θcy) f (y;c,θ) = = x ≥ 0,c > 0 λ > 0(9) w Γ(2)((1+c)2−1) ((1+c)2−1) Which is double weighted exponential distribution (DWED) proposed by Al-Khadim and Hantoosh (2013) 1 2) Put λ = 2 and θ = in (8) then resulting pdf is given by: 2α2 3 y2 c2y2 (1+c2)2y2e−2α2(1−e−2α2) 2 f (y;c,α) = √ x ≥ 0,c > 0 λ > 0 (10) w π 3 α3((1+cλ)2−1) Which is Double Weighted Raleigh Distribution proposed by Rishwan (2013). 32 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.6, No.7, 2016 5. Reliability analysis 5.1. Reliability function R(x) The reliability function or survival function of DWWD is given as: R (x;c,λ) = 1−F (x; c,λ) w w R (x;c,λ) = 1− (1+cλ)1+λ1 (γ(1+1,xλ)− γ(1+λ1 , xλ(1+cλ))) (11) w Γ(1+1)((1+cλ)1+λ1−1) λ (1+cλ)1+λ1 λ Where x ≥ 0,c > 0 λ > 0 (a) (b) Figure4 (a) and (b) represent the plot of the survival function of DWWD for various choice of parameters cand λ. 5.2. Hazard Function H(x) The Hazard function or survival function of DWWD is given as: H (x; c,λ) = fw(x; c,λ) w Rw(x; c,λ) 1 λ(1+cλ)1+λxλe−xλ(1−e−cλxλ) H (x; c,λ) = (12) w 1 1 γ(1+1 , xλ(1+cλ)) Γ(1+1)((1+cλ)1+λ−1)−(1+cλ)1+λ(γ(1+1,xλ)− λ ) λ λ 1+1 (1+cλ) λ Where x ≥ 0,c > 0 λ > 0 33 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.6, No.7, 2016 (a) (b) Figure5 (a) and (b) represent the plot of the Hazard function of DWWD for various choice of parameters cand λ. 5.3. Reverse Hazard function 𝛗(𝐱) The reverse hazard function or survival function of DWWD is given as: φ (x; c,λ) = fw(x; c,λ) w Fw(x; c,λ) 1 λ(1+cλ)1+λxλe−xλ(1−e−cλxλ)) φ (x; c,λ) = (13) w 1 γ(1+1 , xλ(1+cλ)) (1+cλ)1+λ(γ(1+1,xλ)− λ ) λ 1+1 (1+cλ) λ Where x ≥ 0,c > 0 λ > 0 (a) (b) Figure6 (a) and (b) represent the plot of the reverse Hazard function of DWWD for various choice of parameters cand λ. 34 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.6, No.7, 2016 6. Asymptotic behaviours The Asymptotic behaviours of the DWWD can be explained by studying function given in (6) defined over the positive real line [0, ∞) and the behaviour of its derivative as follows:The limits of the pdf given in (6) is given by: 1 λ(1+cλ)1+λxλe−xλ(1−e−cλxλ) limf (x; c,λ) = lim = 0 x→0 w x→0 Γ(1+1)((1+cλ)1+λ1 −1) λ 1 λ(1+cλ)1+λxλe−xλ(1−e−cλxλ) lim f (x; c,λ) = lim = 0 x→∞ w x→∞ Γ(1+1)((1+cλ)1+λ1 −1) λ Sincelimxλ = 0 ,lime−xλ = 0 and lim (1−e−cλxλ) = 1 x→0 x→∞ x→∞ From these limits, we conclude that pdf of DWWD has one mode say x as given by: 0 The pdf of the DWWD is given by 1 λ(1+cλ)1+λxλe−xλ(1−e−cλxλ) f (x; c,λ) = x ≥ 0,c > 0 λ > 0 w 1 Γ(1+1)((1+cλ)1+λ−1) λ Taking logarithm of the pdf of DWWD 1 1 logf (x; c,λ)= logλ+(1+ )log(1+cλ)+λlogx−xλ+log(1−e−cλxλ)−log(Γ(1+ )) w λ λ 1 1+ −log((1+cλ) λ−1) ∂ logf (x; c,λ) = λ−λxλ−1+λcλxλ−1e−cλxλ (14) ∂x w x 1−e−cλxλ The mode of the DWRD is obtained by solving the following non-linear equation with respect to x. λ−λxλ−1+λcλxλ−1e−cλxλ = 0 (15) x 1−e−cλxλ 35 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.6, No.7, 2016 The mode of Double Weighted Weibull Distribution (DWWD)can be calculated by solving above nonlinear equation. 7. Order Statistics The order statistics have great importance in life testing and reliability analysis. Let 𝑋 𝑋 ,𝑋 ,……….,𝑋 be random variables and its ordered values is denoted as 1, 2 3 𝑛 𝑋 𝑋 ,𝑋 ,……….,𝑋 . The pdf of order statistics is obtained using the below function 1, 2 3 𝑛 𝑓 (𝑥) = 𝑛! 𝑓(𝑥)[𝐹(𝑥)]𝑠−1[1−𝐹(𝑥)]𝑛−𝑠 (16) 𝑠:𝑛, (s−1)!(n−s)! To obtain the smallest value in random sample of size n put 𝑠 = 1in (16) then the pdf of smallest order statistics is given by 𝑓 (𝑥) = 𝑛𝑓(𝑥)[1−𝐹(𝑥)]𝑛−1 1:𝑛, For the DWWD 𝑓 (𝑥)=𝑛λ(1+cλ)1+λ1xλe−xλ(1−e−cλxλ)[1− (1+cλ)1+λ1 (γ(1+1,xλ)− γ(1+λ1 , xλ(1+cλ)))]𝑛−1(17) 1:𝑛, Γ(1+1)((1+cλ)1+λ1−1) Γ(1+1)((1+cλ)1+λ1−1) λ (1+cλ)1+λ1 λ λ Where x ≥ 0,c > 0 λ > 0 To obtain the largest value in random sample of size n put 𝑠 =n in 16 then the pdf of order statistics is given by 𝑓 (𝑥) = 𝑛𝑓(𝑥)[𝐹(𝑥)]𝑛−1 𝑛:𝑛, For the DWWD 𝑓 (𝑥)=𝑛λ(1+cλ)1+λ1xλe−xλ(1−e−cλxλ)[ (1+cλ)1+λ1 (γ(1+1,xλ)− γ(1+λ1 , xλ(1+cλ)))]𝑛−1(18) 𝑛:𝑛, Γ(1+1)((1+cλ)1+λ1−1) Γ(1+1)((1+cλ)1+λ1−1) λ (1+cλ)1+λ1 λ λ Where x ≥ 0,c > 0 λ > 0 8. Moment of DWWD The kth Moment of DWWD can be calculated as: 1 1+ E(xk) = (1+cλ) λ ∫∞λxk+λe−xλ(1−e−cλxλ)dx k = 1,2,3,…… Γ(1+1)((1+cλ)1+λ1−1) 0 λ 36 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.6, No.7, 2016 1 1+ E(xk) = (1+cλ) λ (∫∞λxk+λe−xλdx−∫∞λxk+λe(1+cλ)xλdx) Γ(1+1)((1+cλ)1+λ1−1) 0 0 λ 1 1+ k+1 E(xk) = (1+cλ) λ (Γ(1+k+1)− Γ(1+ λ ) ) Γ(1+1)((1+cλ)1+λ1−1) λ (1+cλ)1+k+λ1 λ E(xk) = (1+cλ)1+λ1Γ(1+k+λ1) ((1+cλ)1+k+λ1−1) 1 k+1 Γ(1+1)((1+cλ)1+λ−1) (1+cλ)1+ λ λ k+1 k+1 1+ E(xk) = Γ(1+ λ ) ((1+cλ) λ −1) 1 k Γ(1+1)((1+cλ)1+λ−1) (1+cλ)λ λ E(xk) = Γ(1+k+λ1) ((1+cλ)1+λ1 − 1 ) 1 k Γ(1+1)((1+cλ)1+λ−1) (1+cλ)λ λ 1 k 1+ Let ∈= (1+cλ) λ and ∈ = (1+cλ)λ k = 1,2,3,…… k k+1 E(xk) = Γ(1+ λ ) (∈ − 1) (19) Γ(1+1)(∈−1) ∈k λ 8.1. Mean 2 Γ(1+ ) 1 μ = λ (∈ − ) (20) Γ(1+1)(∈−1) ∈1 λ 1 1 1+ Where ∈ = (1+cλ)λand ∈= (1+cλ) λ 1 8.2. The variance 2 3 2 σ2 = Γ(1+λ) (∈ − 1)−( Γ(1+λ) (∈ − 1)) (21) Γ(1+1)(∈−1) ∈2 Γ(1+1)(∈−1) ∈1 λ λ 1 2 1 1+ Where ∈ = (1+cλ)λ, ∈ = (1+cλ)λ and ∈= (1+cλ) λ 1 2 37