Biometrics & Biostatistics International Journal Research Article Open Access The Weibull-Inverse Lomax (WIL) distribution with Application on Bladder Cancer Abstract Volume 8 Issue 6 - 2019 In this paper, we introduced a four-parameter probability model called Weibull-Inverse Jamilu Yunusa Falgore,1 Sani Ibrahim Lomax distribution with decreasing, increasing and bathtub hazard rate function. The WIL 1 2 distribution density function is J-shaped, positively skewed, and J-shaped in reverse. Some Doguwa, Audu Isah 1Department of Statistics, Ahmadu Bello University, Nigeria of the mentioned distribution’s statistical characteristics are provided including moments, 2Department of Statistics, Federal University of Technology, order statistics, entropy, mean, variance, moment generating function, and quantile function. Nigeria The method of maximum likelihood estimation was used to estimate the parameters of the model. The distribution’s importance is proved by its implementation to the bladder cancer Correspondence: Falgore Jamilu Yunusa, Department of data set. Goodness-of-fit of this distribution by various techniques demonstrates that the Statistics, Ahmadu Bello University, Zaria-Nigeria, WIL distribution is empirically better for lifetime application. Email Keywords: entropy, moments, moment generating functions, Weibull-Inverse lomax Received: November 14, 2019 | Published: December 05, distribution 2019 Introduction The Inverse-Lomax distribution and Weibull G family Real life datasets can be fitted by utilizing a lot of existing The probability density function (pdf) and cumulative distribution statistical distributions. However, most of these real-world datasets function (cdf) of ILD are given by the following equations as define does not follow these existing statistical distributions. Hence, the by Yadav et al.16 as: −+(1λ ) need to propose/develop new distributions that could describes some γλ γ of these situations better and can also provide a better flexibility in gx( ;γλ , )= 1+ (1) x2 x the modelling of real-world data sets compared with the baseline −λ distributions. As a result of this fact, researchers have developed many γ γλ + statistical families of distributions and study most of their properties. Gx( ; , )= 1 (2) x These include: The Transmuted Weibull lomax distribution by Afify where x > 0, γ, λ > 0 are scale and shape parameters respectively. et al.,1 kumaraswamy Marshal-olkin family by Afify et al.,2 Lomax generator by Cordeiro et al.,3 the weibull exponential by Oguntunde Let g( x;ηη) and G( x;) denote the probability density function et al.,4 kumaraswamy-pareto by Bourguignon et al.,5 weibull-G family (pdf) and cumulative distribution function (cdf) of a baseline with by Bourguignon et al.,6 the weibull-dagum distribution by Tahir et al.,7 parameter vector η and also consider the Weibull Cumulative Density the generalized transmuted-G family of distributions by Nofal et al.,8 Function (cdf) Fx( ) =−− 1 exp( dxp ) ( forx >0) with positive among others. Recently, a lot of extensions of distributions have been parameters d and p . Based on this cdf, Bourguignon et al.6 replaced proposed and studied based on the Weibull-G family of distributions the argument x by Gx( ;ηη) /1− Gx( ; ) , and defined the cdf and the by Bourguignon et al.,6 Among them is the Tahir, Merovci, Afify, pdf of the Weibull-G family by: 9-12,4 Yousof, and Oguntunde et al. to mention but few. β Gx(;η ) Inverse Lomax distribution is a member of Beta-type distribution. Fx( ;αβη , , ) = 1− exp − α,x∈ℑ⊆ℜ ;αβ , > 0, (3) Gx(;η ) Other members of the family include Dagum, lomax, Fisk or log- Logistics, Singh maddala, generalized beta distributions of the second where G(x;η) is any baseline cdf which depends on a parameter kind among others, as in Kleiber et al.13 If a random variable say Z has vector η. β a Lomax distribution, then has an Inverse Lomax distribution (ILD). Gx(;ηη )β −1 Gx(; ) α β η αβ η −α It has been utilized to get the Lorenz ordering relationship among f(; x , , )=gx (; )β +1 exp (4) Gx(;η ) Gx(;η ) ordered statistics;.14 Apart from this, it has also many applications in economics, actuarial sciences, and stochastic modeling Kleiber C, The interpretation of the family of distribution above as in Cooray21 13,14 Kotz S & Kleiber C have applied this model on geophysical data, is as follows. Let L be a lifetime random variable having a continuous specifically on the sizes of land fires in California state of US. Rahman cdf G(x;η), then the odds ratio that an individual (component) 15 et al. have discussed the estimation and prediction challenges for following the lifetime L will fail(die) at time x is Gx( ;ηη) /1− Gx( ; ) 16 the inverse Lomax distribution via Bayesian approach. Yadav et al. Let’s consider that the variability of this odds is represented by the have used this distribution for reliability estimation based on Type random variable X and assume that it follows Weibull model with β as II censored observations. Some details about the Inverse Lomax shape and α as scale, then distribution including its applications are available in Reyad and Othman, Falgore et al., Maxwell et al., and Hassan and Mohamed.17-23 Gx(;η ) Pr(L≤≤ x ) = Pr X =Fx ( ;αβη , , ), Gx(;η ) The main aim of this paper is to provide an extension of the Inverse-Lomax distribution using the Weibull-G generator by as given in equation 1 b. Bourguignon et al.6 Therefore, we propose the Weibull-Inverse lomax (WIL) distribution by adding two extra shape parameters to the A random variable X with density function in 1a is denoted by Inverse Lomax distribution. X Weib G(α,β,η). The extra parameters induced by the Weibull ∼ Submit Manuscript | http://medcraveonline.com Biom Biostat Int J. 2019;8(6):195‒202. 195 ©2019 Falgore et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Copyright: The Weibull-Inverse Lomax (WIL) distribution with Application on Bladder Cancer ©2019 Falgore et al. 196 −λββ −1 γ −λ generator are sought just to increase the flexibity of the distribution. 1+ γ −+(1λ ) 1+ If β = 1, it corresponds to the Exponential-Generator by Gupta and γλ γ x x 22 fx( ;α , β , γ , λ ) = αβ 1+−expα Kundu. 21−λ βλ+− x x γ γ 11−+ 11−+ In this context, we proposed and study the WIL distribution based x x on equations 1a and 1 b. (11) The Weibull Inverse Lomax (WIL) Distribution Let E be the last term of equation 11, by expanding E using power By inserting equation 1 in equation 1b yields the WIL cdf below: series −λ iβ λ −β γ γ βαλγ− α −+ 1+ Fx(;,,,)=1exp 1 1 (5) ∞ − iiα x x ( 1) E = ∑ β −λ i i=0 i! γ The pdf corresponding to 5 is given by 11−+ λβ− x γ (1 ) −β −+(1λλ ) 1+ αβγλ γ x γ By inserting this expansion in equation 11 and after some algebra, fx( ;αβ , γ , λ ) = 1+ expα 1−+ 1 21−λ β + we have x xxγ 11−+ −β (i ++11) x + −λβ[ (i +− 1)] 1 −λ ∞ (− 1)iiαγ1 γ fx( ;α , β , γ , λ )=∑ βγλ 1+ 1−+ 1 where α > 0, β > 0 and λ are the shape parameters while γ is the i=0 ix! x scale parameter. Henceforth, we denote a random variable X having pdf 6 by X WIL(α, β, γ, λ).The hazard function h(x), cumulative −λ −β (i ++11) γ hazard function H(x), survival function s(x),and reversed hazard rate letUi =11−+ r(x) of X are∼ given by x λβ(1− ) γ After simplifications and some algebra, we have −+λ + (1 ) 1 −λβ[ (ij ++− 1)] 1 αβγλ γ x γ hx( ;αβγλ , , , )= 1+ + 21−λ β + ∞ 1 x x γ (7) x 11−+ fx( ;α , β , γ , λ ) = γλ[ β (i++ 1) j] ∑ Uij, x2 x ij,=0 (12) ii+1 λ −β (− 1) αβ Γ( β (ij + 1) ++ 1) γ Where U = Hx( ;αβγλ , , , )= α 1−+ 1 (8) ij, ij! !Γ (ββ ( i ++ 1) 1) ( i ++ 1) j x Equation 12 reduces to ∞ λ −β γ fx( ;αβγλ , , , )= ∑Uij, hβ ( i++ 1) j() x (13) sx(;,,,)=expαβγλ α 1−+ 1 (9) x ij,=0 Where −λβ[ (ij ++− 1)] 1 −2 γ γ λβ(1− ) hβ ++( x ) =γλ[ β ( i++ 1) jx] 1 + −β (ij 1) −+(1λλ ) 1+ x αβγλ γ γ + x α −+ 211 β + exp 1 1 is the Inverse x xx−λ γ 11−+ Lomax density. By integrating Equation 13, the cdf of X can be x αβγλ given in the mixture form rx( ; , , , )= −β λ ∞ γ αβγλ 1− expα 1 −+ 1 Fx( ; , , , )= ∑Uij, Hβ ( i++ 1) j() x (14) x ij,=0 Equation 13 above is the major result of this section. (10) Statistical Properties Shapes of the Weibull-Inverse Lomax PDF, CDF, hazard and survival functions Quantile function and median The graphs below shows the shapes of the WIL density at various Quantile function is used in drawing a sample from a particular selected parameter values (Figure 1). distribution function. The quantile function of the WIL distribution is the inverse of 5 and is given by Mixture representation γ Qu( )= 1 (15) By inserting equations 2 and 1 in 1a we have −1 λ log(1− u ) β 11−− α Citation: Falgore JY, Doguwa SI, Isah A. The Weibull-Inverse Lomax (WIL) distribution with Application on Bladder Cancer. Biom Biostat Int J. 2019;8(6):195‒202. DOI: 10.15406/bbij.2019.08.00289 Copyright: The Weibull-Inverse Lomax (WIL) distribution with Application on Bladder Cancer ©2019 Falgore et al.