Weighted Lomax distribution

Transcription

1 Kilany SpringerPlus 0165:186 DOI /s RESEARCH Open Access Weighted Loma distribution N. M. Kilany * *Correspondence: neveenkilany@hotmail.com Faculty of Science, Menoufia University, Shebin El Kom, Egypt Abstract The Loma distribution Pareto Type-II is widely applicable in reliability and life testing problems in engineering as well as in survival analysis as an alternative distribution. In this paper, Weighted Loma distribution is proposed and studied. The density function and its behavior, moments, hazard and survival functions, mean residual life and reversed failure rate, etreme values distributions and order statistics are derived and studied. The parameters of this distribution are estimated by the method of moments and the maimum likelihood estimation method and the observed information matri is derived. Moreover, simulation schemes are derived. Finally, an application of the model to a real data set is presented and compared with some other well-known distributions. Keywords: Loma distribution, Weighted distribution, Reliability measures, Method of moments estimation, Maimum likelihood estimation method Introduction Weighted distribution theory gives a unified approach to dealing with model specification and data interpretation problems. Weighted distributions occur frequently in studies related to reliability, survival analysis, analysis of family data, biomedicine, ecology and several other areas, see Stene 1981 and Oluyede and George 00. Many authors have presented important results on weighted distributions, Rao 1965 introduced a unified concept of weighted distribution and identified various sampling situations that can modeled by weighted distributions. These situations occur when the recorded observations can not be considered as a random sample from the original distributions. This mean that sometimes it is not possible to work with a truly random sample from population of interest. Zelen 1974 introduced weighted distribution to represent what broadly perceived as a length-biased sampling. Patil and Ord 1976 studied a size biased sampling and related invariant weighted distributions. Statistical applications of weighted distributions related to human population and ecology can be found in Patil and Rao Gupta and Tripathi 1996 studied the weighted version of the bivariate logarithmic series distribution, which has applications in many fields such as: ecology, social and behavioral sciences. To present the concept of a weighted distribution, suppose that X is a nonnegative random variable with its probability density function p.d.f. f, then the p.d.f. of the weight random variable X w is given by f w = wf Ew, 0 1 The Authors 016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Kilany SpringerPlus 0165:186 Page of 18 where w is a nonnegative weight function and Ew = 0 wf d, 0 < Ew <. The random variable X w is called the weight version of X and its distribution is related to X and is called the weighted distribution with weight function w. Note that, the weight function w in Eq. 1 gave different practical eamples: such as; when w = β, β>0, then the resulting distribution is called a size biased version of X and the p.d.f. of a size random variable X s is defined as f s = β f E β, 0, α>0 In Eq. when β = 1, then the weight function w = and the resulting distribution is called a length-biased distribution and the p.d.f. of a length biased random variable X L is taken the following form: f L = f, 0, α>0 E where EX = µ is the mean of the original distribution [to know more details about different forms of a weight function, see Rao 1985 and Hewa 011]. When an investigator records an observation by nature according to certain stochastic model, the recorded observation will not have the original distribution unless every observation is given an equal chance of being recorded. For eample, suppose that the original observation 0 comes from a distribution with p.d.f f 0 0 ; θ 1, where θ 1 is a parameter vector, and that observation is recording according to re-weighted by weighted function w; θ > 0, θ is a new parameter vector, then comes from a distribution with p.d.f. f = Aw; θ f 0 0 ; θ 1 where A is a normalized constant. The main aim of this paper is to provide another etension of the Loma distribution. So, the Weighted-Loma WL for short distribution is proposed to offer a more fleible model for modelling data in several areas such as lifetime analysis, engineering and biomedical sciences. The objectives of the research are to study some structural properties of the proposed distribution. Loma 1987 proposed Loma distribution Pareto Type-II distribution, and used it for the analysis of the business failure lifetime data. Loma distribution often used in business, economics, and actuarial modeling. It is essentially a Pareto distribution that has been shifted so that its support begins at zero. A random variable X is said to be distributed as Loma with two parameters α shape parameter and scale parameter if it has p.d.f., f L ; α, = α α y + α+1, 0, α, > 0 3 The corresponding cumulative distribution function c.d.f. given by [ α F L ; α, = ] 4

3 Kilany SpringerPlus 0165:186 Page 3 of 18 The trend of parameters induction to the baseline distribution has received increased attention in recent years to eplore properties and for efficient estimation of the parameters. In the literature, some etensions of Loma distribution are available such as the eponentiated Loma by Abdul-Moniem and Abdel-Hameed 01, Marshall Olkin etended-loma by Ghitany et al. 007, Gupta et al. 010, Beta-Loma BL, Kumaraswamy Loma, McDonald-Loma by Lemonte and Cordeiro 013, Gamma-Loma by Cordeiro et al. 015, the generalized transmuted Loma by Nofal et al. 016 and the transmuted Weibull Loma by Afify et al In this paper, the WL distribution is proposed with p.d.f. f = A β 1 f L ; α, where A is a normalized constant and f L ; α, is the p.d.f. of Loma distribution. Using Eq. 3, the WL considered in this paper has p.d.f. Ŵα + 11+α β β 1 f = Γ 1 + α βγ β + α+1, 0, > 0, α>0, 0 <β<α+1 5 where Ŵ is the complete gamma function. Note that, when β = 1, the weighted Loma distribution reduces to the Loma distribution. The corresponding c.d.f. is F = Ŵα + 1 β β F 1 α + 1, β, β + 1; 6 βŵβŵ1 + α β where F 1 a, b, c; z is the hypergeometric function. This paper is organized as follows: In Distributional properties section the distributional properties of the proposed distribution are derived and studied. Section Parameter estimation discusses the estimation problem using the method of moment and the maimum likelihood estimates of the model parameters. The order statistics and the limiting distribution of the etreme values are derived in Order statistics and etreme values section. Section Simulation study includes the simulation study for WL distribution. Finally, a real life application is illustrated in Conclusion section. Distributional properties Shapes of probability density function The behavior of p.d.f. of the WL distribution f at = 0 and =, respectively, is given by, if β<1 α f 0 =, if β = 1, f = 0 0, if β>1 The following theorem describes the shape of p.d.f. of the WL distribution. Theorem 1 The p.d.f. f of the WL distribution is i Decreasing if {0 β 1, α 0, > 0}. ii Unimodal if {1 β α + 1, α 0, 0}.

4 Kilany SpringerPlus 0165:186 Page 4 of 18 Proof The first derivative of f is given by f = g + f where g = α + β + β 1 i If β = 1, then g = α + 1 < 0. Hence f < 0 which implies that f is decreasing. If β <1, then g < 0 for all α + 1 β, 0. Hence f is decreasing. ii If {1 β α + 1, α 0, 0}, f = 0 iff g = 0 which occurs at the point Since 0 = f = β 1 + α β + α β f 0 < 0, Hence f has a local maimum at 0. The behavior of WL distribution density can be illustrated as in the Fig. 1. Hazard rate function From 5 and 6, the survival and the hazard or failure rate functions are obtained by S = 1 F = 1 Ŵα + 1 β β F 1 α + 1, β, β + 1; βŵβŵ1 + α β h = f S = Ŵα + 1β α+1 β 1 + α 1 β β Ŵ1 + α βŵβ β Ŵ1 + α F 1 α + 1, β, β + 1; According to Glaser 1980, in order to study the behavior of h, the behavior of η is eamined where η = d d ln f. The following theorem shows the shapes of the hazard rate function of the WL distribution. Theorem Hazard rate function of the WL distribution is i Decreasing if {0 β 1, α 0, > 0}. ii Upside-down shape if {1 β α + 1, α 0, 0}. Proof Since η = d d ln f = 1 β + α It follows that η = 1 β α + 1 +, η = 1 β 3 + α i For 0 <β 1, η < 0 for all α + 1 β, 0, i.e. η is decreasing. Hence, h is also decreasing. ii For 1 <β<α+ 1, η = 0 occurs at two points

5 Kilany SpringerPlus 0165:186 Page 5 of =0.7, =1.5, = =1, =4, = =5, =6, = =0.5, =1,0. < < 1 =5, =3,5 < < 0 Fig. 1 Plots of the WL p.d.f. for some parameter values Since, 1 = β 1 + β 1α + 1, = β 1 β 1α + 1 α + β α + β η = α α β4 β 1 β 1 α + 1 β 3 3 < 0, 1α + 1 β 1 + α + 1 Then η has a local minimum at and therefore the hazard function has a local minimum at then upside-down shaped. Plots of the WL hazard function at different parameter values are displayed in Fig.. Mean residual life function and reversed failure rate In life testing situations, the remaining lifetimes of a unit of age 0 until the time of failure is known as the residual life. In other words, if X is the life of a unit, then the conditional random variable [X /X > ] is called the residual life RL. Gupta and Gupta 1983 showed that the mean residual life function and also the ratio of any two consecutive moments of residual life characterize the distribution. The mean residual life MRL function of WL distribution is given by:

6 Kilany SpringerPlus 0165:186 Page 6 of =7, =9, = Fig. Plots of the Hazard function of WL distribution MRL = µ 1 = EX /X > = 1 yf y dy S β α+β α+1 B ; α β, α Ŵα + 1 = + β β Ŵ1 + α βŵβ β Ŵα + 1 F 1 α + 1, β, β + 1; where B; a, b is the incomplete beta function. The following two Lemmas are useful to determining the shape of mean residual life function µ. Lemma Bryson and Siddique 1969 Let X be a non-negative continuous random variable with hazard rate function h and mean residual life function µ. If h is increasing decreasing, then µ is increasing decreasing. Lemma 3 Gupta and Akman 1995 Let X be a non-negative continuous random variable with p.d.f. f, hazard rate function h and mean residual life function µ. If h has bathtub upside-down bathtub shaped and f0µ0 > 1= 1, then µhas upsidedown bathtub bathtub shape. Using Lemmas and 3, the following theorem shows the shape of the mean residual life function µ of the WL distribution. Theorem 3 The mean residual life function µ of the WL distribution is increasing bathtub shaped if 0 <β 1 1 <β<α+ 1 for all α >0, > 0. Proof Since h is decreasing for 0 <β 1, then µ is increasing. Moreover, since h is upside-down shaped for 1 <β<α+ 1 and f0µ0 0, µ is bathtub shaped. Figure 3 illustrates the plot of mean residual life function of WL distribution at different parameter values. The reversed residual life can be defined as the conditional random variable [ X/X ] which denotes the time elapsed from the failure of a component given that its life is less than or equal to, 0. This random variable is called also the inactivity time or time since failure.

7 Kilany SpringerPlus 0165:186 Page 7 of =5, = 15, = Fig. 3 Plots of the mean residual life of WL distribution In addition, in reliability, it is well known that the mean reversed residual life and ratio of two consecutive moments of reversed residual life characterize the distribution uniquely; for more details see Kundu and Nanda 010 and Nanda et al The reversed failure for WL Distribution is derived as follows: Rh = f F = β 1+α + 1 α F 1 α + 1, β, β + 1; The rth moment of the reversed residual life is obtained by: m r = E X r / X = 1 F Thus the mean of the reversed residual life is: m 1 = The second moment of the reversed residual life is: where, F 1 a, b, c; z = F 1 a,b,c;z Γ c is the regularized hypergeometric function. Moments and associated measures F 1 α + 1, β, β + 1; β + 1 F 1 α + 1, β, β + 1; The first four moments about the origin of WL distribution 0 r t r 1 Ftdt m = B ; + β, α 1 B ; β, α + + F α, β,+ β, F α, β,1+ β, µ 1 = β α β µ = ββ α βα β µ 3 = ββ + 1β α β 1 + α βα β 7 8 9

8 Kilany SpringerPlus 0165:186 Page 8 of 18 µ 4 = 4 Ŵ 3 + α βŵβ + 4 Ŵ1 + α βŵβ The central moments about the mean are given by µ = µ 3 = µ 4 = αβ 1 + α βα β, αβα + β 3 + α β 1 + α βα β 3, 3αβ α + ββ + β + α + β α β + α β 1 + α βα β 4. Hence; the mean µ and variance σ of WL distribution are µ = β α β and σ αβ =. 1+α βα β The coefficients of skewness β 1, kurtosis β and variation CV of WL distribution are given by: β 1 = Ŵα β 3 Ŵ1 + β 3 3Ŵ 1 + α βŵα βŵ1 + α βŵβŵ1 + βŵ + β + Ŵ + α βŵ1 + α β Ŵβ Ŵ3 + β Ŵα β Ŵ1 + β 3 + Ŵ 1 + α βŵ1 + α βŵβŵ + β β = 1 Ŵα β Ŵ1 + β Ŵ 1 + α βŵ1 + α βŵβŵ + β [ Ŵα βŵ1 + β 3Ŵα β 3 Ŵ1 + β 3 + 6Ŵ 1 + α βŵα βŵ1 + α βŵβŵ1 + βŵ + β ] 4Ŵ + α βŵ1 + α β Ŵβ Ŵ3 + β + Ŵ 3 + α βŵ1 + α β 3 Ŵβ 3 Ŵ4 + β α β CV = Ŵ1+β + α βŵβŵ+β 1+α β α β Ŵ1 + β Moreover, the moment generating M t and characteristic functions φ t are M t = πcscπα β φ t = 1+ t + e t t α α α 1 + Ŵ α, t 1+t 1 +, it 1+α β 1+α β Ŵ1+α 1 F 1 1+α,+α β, it Ŵ[β] 1 F 1 β, α + β; it Ŵ1 + α β1 + θ θ it, where Ŵa, z is the incomplete gamma function, and Cscz is the cosecant of z.

9 Kilany SpringerPlus 0165:186 Page 9 of 18 Lorenz and Bonferroni curves The Bonferroni, Lorenz curves and Gini inde have many applications in economics and ecology to describe inequality distribution of wealth or size like a reliability, medicine and insurance. For WL distribution, Lorenz and Bonferroni curves are: L F [F] = 1+β 1 β Ŵ1 + α F α,1+ β,+ β; Ŵα β and B F F = α + βb ; 1 + β, α βb ; β, α.the scaled total time on test transform of a distribution function is given by α β β β Ŵ1+α F 11+α,β,+β, Ŵα β S F [Ft] = β Hence, Gini inde is obtained from the relationship G = 1 C F ; where C F = 1 0 S F [Ft]f tdt. Entropies An entropy is a measure of variation or uncertainty of a random variable X. The theory of entropy has been successfully used in a wide diversity of applications and has been used for the characterization of numerous standard probability distributions. Two popular entropy measures are the Shannon and Rényi entropies. The Rényi entropy of a random variable with p.d.f. f is defined as I δ = 1 1 δ log f δ d, δ>0, δ =1 For WL distribution, Rényi entropy is derived as I δ = 1 Ŵα + 1 α β+1 δ β 1 δ 1 δ log Ŵα β + 1Ŵβ 0 + α+1 d = log + δ1 δ 1 Ŵα + 1 log Ŵα β + 1Ŵβ Ŵ 1 + α βδŵ1 + β 1δ + 1 γ 1 log Ŵδ + αδ The Shannon entropy of a random variable X is defined by S H = Using the WL density, 0 0 f logf d Ŵα + 1α β+1 S H = Ŵα β + 1Ŵβ 0 β 1 Ŵα + 11+α β log β 1 + α 1 d α+1 + Ŵ1 + α βŵβ Ŵα + 1α β+1 = log Ŵα β + 1Ŵβ β 1 log ψα β ψβ + α + 1 H α H α β + log Ŵα β + 1Ŵβ = log + α + 1H α + β α H α β β 1γ + ψβ Ŵα + 1

10 Kilany SpringerPlus 0165:186 Page 10 of 18 where H n is the n-th harmonic number, γ is the Euler Mascheroni constant and ψz is the digamma function. Reliability parameter In the contet of reliability, the stress strength model describes the life of a component that has a random strength Y that is subjected to a random stress X. The component fails at the instant that the stress applied to it eceeds the strength, and the component will function satisfactorily whenever X < Y. Hence, R = PrX < Y is a measure of component reliability. It has many applications especially in the areas of engineering and stress strength models. The reliability parameter for WL distribution is given by R = Pr X < Y = α 1 β Ŵα Ŵα 1 β 1 + 1Ŵβ 1 α β+1 Ŵα + 1 y Ŵα β + 1Ŵβ β 1 1 y β α 1+1 α ddy +1 y + = β 1 1 Ŵα Ŵα 1 β α β+1 Ŵα + 1 Ŵα β + 1Ŵβ y β 1+β 1 α ˆF +1 1 α 1 + 1, β 1, β 1 + 1; y dy 0 y + 1 {[ / 1 α β +1 Ŵα + 1Ŵ + α 1 + α β 1 β Ŵ 1 α + β 1 + β = α β + 1Ŵβ 1 Ŵβ α 1 β 1 + 1Ŵα β + 1 p F q {1 + α,1+ α β,+ α 1 + α β 1 β }, { + α β,+ α β 1 β }; ] 1 [ / 1 β 1 Ŵα 1 + 1Ŵ1 + α β 1 β Ŵβ 1 + β + α β + 1Ŵβ 1 Ŵβ α 1 β 1 + 1Ŵα β + 1 p F q {1 + α 1, β 1, β 1 + β }, {1 + β 1, α + β 1 + β }; ]} 1 where pf q { } { } a 1, a,..., a p, b1, b,..., b 1 ; z is the generalized hypergeometric function. Parameter estimation In this section, we consider the method of moments and the maimum likelihood techniques is considered to estimate the involved parameters of WL distribution. In addition, the interval estimation is discussed via Fisher information matri. Method of moments estimates The method of moment estimation MME consists of equating the first three moments of the population 7, 8 and 9 to the corresponding moments of the sample. The system of equations that needed is: β α β = m 1 10 ββ α βα β = m 11 ββ + 1β α β 1 + α βα β = m 3 1

11 Kilany SpringerPlus 0165:186 Page 11 of 18 By solving the Eqs. 10, 11 and 1, the parameter estimates are ˆ = m 1m m 1 m 3 + m m 3 m 1 m m + m 1 m 3 m 1 ˆα = m m1 m m 3 m m 1m 3 m 1 m m + m 1 m 3 m1 m + m 1 m m3 m 1 m ˆβ = m 1 m 3 m 1 m + m 1 m. m3 Maimum likelihood estimates Let 1,,..., n be a random sample of size n from the WL distribution, the maimum likelihood estimates MLEs of the parameters are obtained by direct maimization of the log-likelihood function which is given by: ln L;, α, β = nln[ŵα + 1] + nα β + 1ln[] nln[ŵα β + 1] n n nln[ŵβ] + β 1 ln [ i ] α + 1 ln[ i + ] It follows that the maimum likelihood estimators MLEs; ˆ, ˆα and ˆβ are the simultaneous solutions of the equations: n1 + α β n α i + = 0 nln[] + nψα + 1 nψα β + 1 nln[] + nψα β + 1 nψβ + For interval estimation of the parameter vector Θ =, α, β T ; the epected fisher information matri I = [ I ij ], i, j = 1,, 3 is derived as follows: [ ] I 11 = E ln f πcscπα β where ψ z is the trigamma function. n ln[ + i ] = 0 n ln[ i ] = 0 α 1+α β +α β 1+α 1 α+β α β [ β β Ŵ1+α 1+αŴβ + + F 1 Ŵ α+β = Ŵ3 + α β [ ] I = E ln f = ψ α ψ α β + 1; α [ ] I 33 = E ln f = ψ α β ψ β; β [ ] β I 1 = E ln f = α α + 1 ; [ ] I 13 = E ln f = 1 β ; [ ] I 3 = E ln f = ψ α β + 1; α β 1,β, 1 α+β; 1 α+β β ] ;

12 Kilany SpringerPlus 0165:186 Page 1 of 18 Under regularity conditions, Bahadur 1964 showed that as n, n ˆΘ is asymptotically normal 3-variate with vector mean zero and covariance matri I 1. The asymptotic variances and covariance of the elements of ˆΘ are given by: V ˆα = I I 33 I3, V ˆβ = I 11I 33 I13, V ˆθ Cov ˆα, ˆθ n = I 11I I1 n, Cov ˆα, ˆβ n = I 1I 3 I 13 I, Cov ˆβ, ˆθ n = I 13I 3 I 1 I 33, n = I 13I 1 I 11 I 3. n where = det I. The corresponding asymptotic 1001 α% confidence intervals are ˆ ± ci 1/ ; where c is the appropriate z critical value. Order statistics and etreme values The distribution of etreme values plays an important role in statistical applications. In this section the probability and cumulative function of order statistics are introduced and the limiting distribution minimum and the maimum arising from the WL model can be derived. Probability and cumulative function of order statistics Suppose X 1, X,, X n is a random sample from WL distribution. Let X 1:n < X :n < < X n:n denote the corresponding order statistics. The probability density function and the cumulative distribution function of the jth order statistic, say Y = X j:n, are given by n! f Y y = F j 1 y {1 F y } n j f y j 1! n j! 1+α y + 1 α n! = yŵ j Ŵ 1 j + n Ŵβ F α, β,1+ β; y [ y β β Ŵ1 + α F α, β,1+ β; y ] j Ŵ1 + α β [ 1 yβ β Ŵ1 + α F α, β,1+ β; y Ŵ1 + α β ] n j and F Y y = n m=j n m F m y [ 1 F y ] n m Ŵ1 + n F 1 1, j n, +j; = Ŵ 1 j + n [ y β β Ŵ1 + α F 1 + α, β,1+ β; y ] j Ŵ1 + α β [ 1 yβ β Ŵ1 + α F 1 + α, β,1+ β, y Ŵ1 + α β 1 y 1 β β Ŵ1+α β Ŵ1+α F y 11+α,β,1+β; ] n j

13 Kilany SpringerPlus 0165:186 Page 13 of 18 Limiting distributions of etreme values Let m n = X 1:n = min[x 1,X,...,X n ] and M n = X n:n = ma[x 1,X,...,X n ] arising from WL distribution. The limiting distributions of X 1:n and X n:n can be derived in the following theorem. Theorem 3 Let m n and M n be the minimum and the maimum of a random sample from the WL distribution, respectively. Then i lim n p mn a n b n = 1 ep β ; > 0 ii lim n p Mn c n d n = ep 1+α β ; > 0 1 where a n = 0, b n = F 1 1n, c n = 0 and d n = 1. F 1 1 n 1 Proof i Using L Hospital rule, we have F F ε lim ε 0 + F F ε = lim Fε ε 0 + Fε = lim f ε = β. ε 0 + f ε Therefore by Theorem of Arnold et al. 199, the minimal domain of attraction of the WL distribution is the Weibull distribution, proving part i. ii Using L Hospital rule, we have 1 Ft lim t 1 Ft = lim f t = 1+α β t f t Therefore, by Theorem 1.6. and Corollary in Leadbetter et al. 1987, the maimal domain of attraction of the WL distribution is the Fréchet distribution, proving part ii. Simulation study The equation F u = 0, where u is an observation from the uniform distribution 0,1 and F is cumulative distribution function of WL distribution, is used to carry out the simulation study by generating random samples follow WL distribution. The simulation eperiment was repeated 1000 times each with sample sizes; 0, 40, 70, 100 for, α, β = 0.03,, 0.5 and 0.01,.5, 1. The following measures are computed: i Average bias of ˆ, ˆα and ˆβ of the parameters, α and β are respectively; 1 N 1 N 1 N ˆ, ˆα α and ˆβ β N N N ii The Mean square error MSE of ˆ, ˆα and ˆβ of the parameters, α and β are respectively; 1 N, 1 N 1 N ˆ ˆα α and ˆβ β N N N Table 1 presents the average bias and the MSE of the estimates. The values of the bias are seen to be small,positive and the values of the MSEs decreases while the sample size increases.

14 Kilany SpringerPlus 0165:186 Page 14 of 18 Table 1 Bias and MSE for the parameters, α, β α β n Bias MSE Bias α MSE α Bias β MSE β Application The considered a dataset corresponding to remission times in months of a random sample of 18 bladder cancer patients given in Lee and Wang 003. The data are given as follows: 0.08,.09, 3.48, 4.87, 6.94, 8.66, 13.11, 3.63, 0.0,.3, 3.5, 4.98, 6.97, 9.0, 13.9, 0.40,.6, 3.57, 5.06, 7.09, 9., 13.80, 5.74, 0.50,.46, 3.64, 5.09, 7.6, 9.47, 14.4, 5.8, 0.51,.54, 3.70, 5.17, 7.8, 9.74, 14.76, 6.31, 0.81,.6, 3.8, 5.3, 7.3, 10.06, 14.77, 3.15,.64, 3.88, 5.3, 7.39, 10.34, 14.83, 34.6, 0.90,.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05,.69, 4.3, 5.41, 7.6, 10.75, 16.6, 43.01, 1.19,.75, 4.6, 5.41, 7.63, 17.1, 46.1, 1.6,.83, 4.33, 5.49, 7.66, 11.5, 17.14, 79.05, 1.35,.87, 5.6, 7.87, 11.64, 17.36, 1.40, 3.0, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.6, 11.98, 19.13, 1.76, 3.5, 4.50, 6.5, 8.37, 1.0,.0, 3.31, 4.51, 6.54, 8.53, 1.03, 0.8,.0, 3.36, 6.76, 1.07, 1.73,.07, 3.36, 6.93, 8.65, 1.63,.69. We have fitted the WL distribution to the dataset using MLE, and compared the proposed WL distribution with, gamma- Loma GL, Kumaraswamy-LomaKwL, transmuted eponentiated-loma TrEL, Weibull-Loma WL, McDonald-Loma McL, Beta-LomaBL, etended Poisson- Loma EPL, eponential-loma EL and Loma distributions. The c.d.f.s of these models are given as follow: The McLoma McL density function with five parameters α,, β, a and b introduced by Lemonte and Cordeiro 013 is epressed as f McL = βαα + α+1 α a 1 α β b B aβ 1, b , + + where > 0; α,, β, b, a, > 0. Evidently, the McL density function does not involve any complicated function, and it includes several distributions as special sub-models not previously considered in the literature. In fact, the Loma distribution with parameters α and is clearly a basic eemplar for a = β = 1 and b = 0. Beta Loma BL and Kumaraswamy Loma KwL distributions are new models which arise for β = 1 and a = β, respectively. For b = 0 and β = 1, it leads to a new distribution referred to as the Eponentiated Loma EL distribution. The McL distribution allows for greater fleibility of its tails and can be widely applied in many areas. The c.d.f. corresponding to McL density function is given by F McL = I aβ {1 α + α } 1 β, b + 1

15 Kilany SpringerPlus 0165:186 Page 15 of 18 The Eponential Loma EL distribution introduced by El-Bassiouny et al. 015 with c.d.f. α F EL = 1 e β +,, α,, β>0 The gamma-loma GL distribution introduced by Cordeiro et al. 015 based on a versatile and fleible gamma generator proposed by Zagrafos and Balakrishnan 009 using Stacy s generalized gamma distribution and record value theory. The c.d.f. of GL distribution is given by ]] F GL = Ŵ[ a, αlog [ 1 + Ŵ[a], > 0, α,, a > 0 The transmuted Eponentiated-Loma TrEL distribution introduced by Ashour and Eltehiwy 013 with c.d.f. F = α β 1 + γ γ α β, where > 0; α,, β, γ>0 The Weibull Loma WL distribution introduced by Tahir et al. 015 with c.d.f. a1+ F WL = 1 e α 1 b, > 0, a, b, α, > 0 The etended Poisson-Loma EPL distribution introduced by Al-Zahrani et al. 015 introduced with c.d.f. F EPL = α e β1 1+ α, > 0; β 0, α, > 0 The model selection is carried out using the Akaike information criterion AIC, the Bayesian information criterion BIC, the Hannan-Quinn information criterion HQIC and the consistent Akaike information criteria CAIC defined by: AIC = lˆθ + q BIC = lˆθ + qlogn HQIC = lˆθ + qlog logn qn CAIC = lˆθ + n q 1 where lˆθ denotes the log-likelihood function evaluated at the maimum likelihood estimates for parameters θ, q is the number of parameters, and n is the sample size. Table provide the MLEs of the model parameters. The model with minimum AIC or BIC, CAICand HQIC value is chosen as the best model to fit the data. From Table 3,

16 Kilany SpringerPlus 0165:186 Page 16 of 18 Table MLEs for bladder cancer data Distribution MLEs ˆ ˆα ˆβ ˆγ â ˆb 1. WL GL KwL TrEL WL McL BL Loma EPL EL Table 3 The Measures AIC, BIC, HQIC, CAIC for bladder cancer data Distribution Measures Log L AIC BIC HQIC CAIC 1. WL GL KwL Tr EL WL McL BL Loma EPL EL Table 4 Goodness-of-fit tests for bladder cancer data Distribution Statistics Wn A n Un L n 1. WL EL GL TrEL WL BL Loma EPL EL we note that the WL model gives the lowest values for the AIC, BIC, HQIC and CAIC statistics among all fitted models. So, the WL model could be chosen as the best model comparable GL, KwL, TrEL, WL, McL, BL, EPL, EL and Loma distributions.

17 Kilany SpringerPlus 0165:186 Page 17 of 18 Now, the formal goodness-of-fit tests are applied in order to verify which distribution fits better to these data. The Cramér von Mises W n, Anderson Darling A n, Watson Un and Liao-Shimokawa L n tests statistics are considered. For further details, the reader is refereed to Chen and Balakrishnan In general, the smaller the values of the Wn, A n, U n and L n, the better the fit to the data. The values of the statistics Wn, A n, Un and L n are given in Table 4. Based on these statistics, the WL model fits the bladder cancer data better than TrEL, WL, EPL, EXL and Loma models and gives values close to GL and BL it can be concluded. Conclusion In this paper, WL distribution is proposed. A mathematical treatment of the proposed distribution including eplicit formulas for the density and hazard functions, moments, order statistics have been provided. The estimation of the parameters has been approached by maimum likelihood and method of moments and the observed information matri is obtained. The usefulness of the new distribution is illustrated in an analysis of Bladder cancer data. The results indicate that the WL distribution applicable and more fleible than other etensions of Loma distribution. Competing interests The author declares that he has no competing interests. Received: 7 July 016 Accepted: 6 October 016 References Abdul-Moniem IB, Abdel-Hameed HF 01 On eponentiated Loma distribution. Int J Math Arch 3: Afify AZ, Nofal ZM, Yousof HM, Gebaly YM, Butt NS 015 The transmuted Weibull Loma distribution: properties and application. Pak J Stat Oper Res 11: Al-Zahrani B 015 An etended Poisson Loma distribution. Adv Math Sci J 4:79 89 Arnold BC, Balakrishnan N, Nagaraja HN 199 A first course in order statistics. Wiley, New York Ashour SK, Eltehiwy MA 013 Transmuted eponentiated loma distribution. Aust J Basic Appl Sci 77: Bahadur RR 1964 On Fisher s bound for asymptotic variances. Ann Math Stat 354: Bryson MC, Siddique MM 1969 Some criteria for aging. J Am Stat Assoc 64: Chen G, Balakrishnan N 1995 A general purpose approimate goodness-of-fit test. J Qual Technol 7: Cordeiro GM, Ortega EMM, Popović BV 015 The gamma-loma Distribution. J Stat Comput Simul 85: El-Bassiouny AH, Abdo NF, Shahen HS 015 Eponential Loma distribution. Int J Comput Appl 1113:4 9 Ghitany ME, AL-Awadhi FA, Alkhalfan LA 007 Marshall Olkin etended Loma distribution and its applications to censored data. Commun Stat Theory Methods 3610: Glaser RE 1980 Bathtub and related failure rate characterizations. J Am Stat Assoc 75: Gupta RC, Akman O 1995 Mean residual life function for certain types of non-monotonic ageing. Commun Stat Stoch Models 11:19 5 Gupta PL, Gupta RC 1983 On the moments of residual life in reliability and some characterization results. Commun Stat Theory Methods 14: Gupta AK, Tripathi RC 1996 Weighted bivariate logarithmic series distributions. Commun Stat Theory Methods 5: Gupta RC, Ghitany ME, Al-Mutairi DK 010 Estimation of reliability from Marshall Olkin etended Loma distributions. J Stat Comput Simul 80: Hewa AP 011 Statistical properties of weighted generalized gamma distribution. Master dissertation, Statesboro, Georgia Kundu C, Nanda AK 010 Some reliability properties of the inactivity time. Commun Stat Theory Methods 39: Leadbetter MR, Lindgren G, Rootzén H 1987 Etremes and related properties of random sequences and processes. Springer, New York Lee ET, Wang JW 003 Statistical methods for survival data analysis, 3rd edn. Wiley, NewYork Lemonte AJ, Cordeiro GM 013 An etended Loma distribution. Statistics :013 Loma KS 1987 Business failures: another eample of the analysis of failure data. J Am Stat Assoc 49: Nanda AK, Singh H, Misra N, Paul P 003 Reliability properties of reversed residual lifetime. Commun Stat Theory Methods 310:031 04

18 Kilany SpringerPlus 0165:186 Page 18 of 18 Nofal ZM, Afify AZ, Yousof HM, Cordeiro, GM 016 The generalized transmuted-g family of distributions. Commun Stat Theory Methods accepted Oluyede BO, George EO 00 On stochastic inequalities and comparisons of reliability measures for weighted distributions. Math Probl Eng 8:1 13 Patil GP, Ord JK 1976 On size biased sampling and related form invariant weighted distribution. Indian J Stat 39:48 61 Patil GP, Rao GR 1978 Weighted distributions and size biased sampling with applications to wildlife populations and human families. Biometrics 34: Rao CR 1965 On discrete distributions arising out of methods of a ascertainment. In: Patil GP ed Classical and contagious discrete distributions. Pergamon Press and Statistical Publishing Society, Calcutta, pp Rao CR 1985 Weighted distributions arising out of methods of ascertainment. In: Atkinson AC, Fienberg SE eds A celebration of statistics. Springer, New York, pp Stene J 1981 Probability distributions arising from the ascertainment and the analysis of data on human families and other groups. Stat Distrib Sci Work Appl Phys Soc Life Sci 6:33 44 Tahir MH, Cordeiro GM, Mansoor M, Zubair M 015 The Weibull Loma distribution: properties and applications. Hacettepe J Math Stat 44: Zelen M 1974 Problems in cell kinetics and the early detection of disease. In: Proschan F, Sering RJ eds Reliability and biometry. SIAM, Philadelphia Zografos K, Balakrishnan N 009 On families of beta- and generalized gamma generated distributions and associated inference. Stat Methodol 6:344 36