A New Class of Bivariate Distributions with Lindley Conditional Hazard Functions

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1 ISSN Joural of Statistics Volume 22, pp A New Class of Bivariate Distributios with Lidley Coditioal Hazard Fuctios Mohamed Gharib 1 ad Bahady Ibrahim Mohammed 2 Abstract I this paper, we itroduce a ew class of Bivariate Distributios such that both the Coditioal Hazard Fuctios are Lidley. Some properties of the ew class are studied. Estimatio of the parameters of the ew class is discussed usig the Maximum Likelihood ad Pseudo-likelihood methods. Further, a set of real data is used to compare the results obtaied by these two methods of Estimatio. Fially, a compariso is give betwee the fittig of a real set of data to the ew class ad to the Bivariate class of Expoetial Coditioals (BEC). Keywords Bivariate distributio, Coditioal Hazard Fuctio, Fuctioal equatio, Lidley distributio, Bivariate expoetial coditioals 1. Itroductio The problem of costructig ew classes of Bivariate Distributios with fixed margials or coditioals has received a obvious attetio durig the last few years (Arold et al., 1999, 2001; Balakrisha ad Lai, 2009 ad Kotz et al., 2000). This problem arises specially i modelig, because oce oe moves away from Bivariate (Multivariate) Normality, there few Bivariate (Multivariate) Distributios flexible eough to fit correlated data. I this sese, the study of Bivariate Distributios whe both coditioals belog to some specified parametric families has received special attetio. Balakrisha et al. (2010) itroduced a ew techique for costructig Bivariate Cotiuous Distributios with specified Coditioal Hazard Fuctios. 1 Departmet of Mathematics, Faculty of Sciece, Ai Shams Uiversity, Abbassia, Cairo, Egypt. 2 Departmet of Mathematics, Faculty of Sciece, Al- Azhar Uiversity, Nasr city, Cairo, Egypt.

2 194 Mohamed Gharib ad Bahady Ibrahim Mohammed Geerally, the use of Bivariate Cotiuous Distributios with specified Hazard Coditioals, i modelig, makes the possibility of studyig ad specifyig several reliability properties icludig Survival ad Hazard Bivariate fuctios, the Clayto-Oakes measure, coditioal desities ad Hazard Rate fuctios of the margial ad coditioal Distributios. Added to these the possibility of studyig the reliability properties for the series ad parallel systems with compoet lifetimes havig these particular depedece models (Navarro ad Sarabia, 2011). I the preset paper, a ew class of Bivariate Distributios with Lidley Coditioal Hazard Fuctios is give. A absolutely cotiuous radom variable is said to have Lidley Distributio if its desity fuctio (Lidley 1958, 1965) is, f x = θ2 1 + x exp θx, x > 0, θ > 0 (1.1) θ+1 The correspodig cumulative distributio fuctio is, 1+θ 1+x F x = 1 exp θx, x > 0, θ > 0 (1.2) θ+1 ad the Hazard Fuctio is, λ x = θ2 1+x, x > 0, θ > 0 (1.3) 1+θ 1+x I some practical situatios, Lidley Mixed models provide better fit compared with other models (Ghitayet al., (2008); Hossei ad Noriszura, (2010); Mahmoudi ad Zakerzadeh, (2010) ad Sele ad Gamze (2014)). For the properties of the Lidley Distributio, see Ghitayet al., (2007). I the sequel, we will eed the followig two relatios betwee Survival F(x), desity f(x) ad Hazard λ(x) fuctios: x F(x) = exp λ u du (1.4) ad x f(x) = λ(x)exp λ u du (1.5) 2. The Bivariate Lidley Coditioal Hazard ew class Suppose, X > 0, Y > 0 be two radom variables ad that the correspodig Coditioal Hazard Fuctios be Lidley Distributed ad specified as: λ 1 x y = θ 1 2 y 1+x (2.1) 1+θ 1 y 1+x ad

3 A New Class of Bivariate Distributios with Lidley 195 Coditioal Hazard Fuctios λ 2 y x = θ 2 x 1+y 1+θ 2 x 1+y where θ 1 y ad θ₂(x) are fuctios of y ad x, respectively. (2.2) A Bivariate Distributio satisfyig equatio (2.1) ad (2.2) will be called a Bivariate Lidley Coditioal Hazard (BLCH) Distributio. From equatio (2.1) ad (2.2), usig equatio (1.5), the coditioal desity fuctios are, f X Y x y = θ y 1 + x 1 + θ 1 y 1 + x exp θ 1 y 1 + u 1 + θ 1 y 1 + u ad f Y X (y x) = θ x 1 + y 1 + θ 2 x 1 + y exp θ 2 x 1 + u 1 + θ 2 x 1 + u With f X (x) ad f Y (y) deotig the margial desities, the idetity f X Y x y f Y y = f Y X y x f X x yields the followig relatio f Y y = f X x 2 2 θ 1 y 1 + x 1 + θ 1 y 1 + x exp θ 1 y 1 + u 1 + θ 1 y 1 + u 2 2 θ 2 x 1 + y 1 + θ 2 x 1 + y exp θ 2 x 1 + u 1 + θ 2 x 1 + u 0 x 0 y 0 the, after achievig the existig itegrals, f Y y θ 1 2 y θ 1 y +1 1 y x = f X x θ 2 x θ 2 x y exp θ 2 x y (2.3) 0 y x du du du du Deotig y = log f Y y m x = log f X x θ2 1 y (θ 1 y +1)(1+y) θ2 x (θ 2 x +1)(1+x) (2.4) The, upo takig logarithms i equatio (2.3) ad usig equatio (2.4), we get: y θ 1 y x = m x θ 2 x y

4 196 Mohamed Gharib ad Bahady Ibrahim Mohammed which is a fuctioal equatio of the form f k x g k y = 0 k=1 It s solutio is give by Aczel (1966), as: θ 1 y = β + y, θ 2 x = α + x (2.5) Substitute equatio (2.5) i (2.3), we get: (β + y)2 f Y y 1 + x exp β + y x (β + y) + 1 ( α + x)2 = f X x 1 + y exp α + x y ( α + x) + 1 Equatig joit probability distributio fuctio ad rearrage x ad y terms to get for all x ad y exp β x (α x)2 f X x (1 α + x)(1 + x) = f exp α x (β + y)2 Y y (1 + β + y)(1 + y) Hece, each side is a costat N α, β, (provided that β = α) ad hece the margial is, f X x = N α, β, 1 (1 α + x)(1 + x) (α x) 2 exp β x x > 0 Put the itegral of this probability distributio fuctio over its space to get the costat N α, β, = exp αβ αβ 3 exp αβ β + α β + +αβ α β β Ei αβ where Ei(z)is the Expoetial Itegral fuctio, Ei z = z exp t t dt the f X,Y x, y = N α, β, x 1 + y exp αy β + y x (2.6) x > 0, y > 0, α < 0, β > 0, 0

5 A New Class of Bivariate Distributios with Lidley Coditioal Hazard Fuctios Equatio (2.6) describes the complete class of BLCH Distributio that has the three parameters α < 0 (itesity parameter for Y), β > 0 (itesity parameter forx) ad 0 (iteractio or depedece parameter). Figure 1 ad 2 shows the three dimesioal curve of the BLCH Distributio give by equatio (2.6) for the special cases for α, β, ad. The specific forms of the Coditioal Hazard Fuctios for the equatio (2.6) are λ 1 x y = β+y 2 1+x, x > 0, y > 0,β > 0, 0 (2.7) 1+ β+y 1+x λ 2 y x = α+x 2 1+y, x > 0, y > 0, α < 0, 0 (2.8) 1+ α+x 1+y The compatibility of equatio (2.7) ad (2.8) (Balakrishaet al., 2010) guaratees the existece of the equatio (2.6). 3. Mai properties of BLCH First of all, direct calculatios usig equatio (2.6) gives, N α,β, 1 E XY = exp αβ exp αβ α 2 β β β 2 + α β 2 + αβ 5 β3+ αβα2ββ +22 2β++αβ2+2 β4+eiαβ (3.1) 197 For β =, equatio (3.1) reduces to 1 N α,, E XY = α 6 + 2α αei α exp α 1 2αEi α exp α which for α = agai reduces to 2 E XY = 2 N 1,, 2 3 exp 2 Ei 2 ad hece E(XY) is a decreasig fuctio i Coditioal desities, margials ad momets: The coditioal desities of the equatio (2.6) are f X Y x y = β + y x exp β + y x β + y + 1 x > 0, y > 0, α < 0, β > 0, 0 (3.3)

6 198 Mohamed Gharib ad Bahady Ibrahim Mohammed ad 2 α + x f Y X y x = 1 + y exp α + x y α + x + 1 x > 0, y > 0, α < 0, β > 0, 0 (3.4) i.e. X Y = y~lidley β + y Y X = x~lidley α + x We otice that for β = α the Coditioal Distributios are idetical. Now, usig equatio (3.3) the coditioal momets are E X k Y = y = 1 + k + β + y Γ 1 + k β + y k k 0, 1 + β + y which are ratioal fuctios i the coditioal variable. For k = 1, we have, β + y + 2 E X Y = y = β + y 1 + β + y We otice that this Regressio fuctio is oliear ad decreasig for all oegative values of the coditioed variable with curves both similar to that give i Figure 3. Now, the margial desities correspodig to equatio (2.6) are f X x = f Y y = N α, β, x 1 α + x α + x 2 N α, β, y 1 + β + y β + y 2 exp βx exp αy x > 0, α < 0, β > 0, 0, y > 0, α < 0, β > 0, 0. We otice that for β = α the radom variablesx ad Y are idetically distributed. Usig these desities we ca calculate directly the momets of X(Y).

7 A New Class of Bivariate Distributios with Lidley Coditioal Hazard Fuctios For example: E X = ad E Y = N α, β, 1 exp αβ β 2 4 exp αβ αβ β + + β β β + β 2 α 2 β 2 α 2 β + Ei αβ N α, β, 1 exp αβ α 2 4 exp αβ α 2 αβ β + β 2 + β 2 + For = 0 we otice that E(XY) = E(X)E(Y). α 2 β + 2 α 1 β + Ei αβ 3.2. Totally egative of order two: Accordig to Hollad ad Wag (1987a, 1987b) a joit desity fuctio f X,Y x, y of two cotiuous radom variables X ad Y is said to be totally egative of order 2 (TN₂) iff f x, y < 0 where 2 f x 1, x 2 = l f x 1 x X1,X 2 x 1, x 2 (3.5) 2 which is called the Local Depedece fuctio. Theorem: The BLCH Distributio give by equatio (2.6) is TN₂. Proof: From equatio (2.6) ad (3.5), we get: f x, y =, Thus, f X,Y (x, y) is TN₂ sice > 0. Shaked (1977) proved that if f X,Y (x, y) is TN₂, the the Coditioal Hazard rate of (X Y=y) is icreasig iy. A similar property holds for Y X = x. The property that the BLCH equatio (2.6) is TN₂ explais the mootoicity of the Hazard Rate fuctio of the coditioal distributio of X Y = y (Y X = x) as a fuctio of y(x) as it is clear from equatio (2.7) ad (2.8). 199

8 200 Mohamed Gharib ad Bahady Ibrahim Mohammed Remarks: It follows from the previous theorem that the two X ad Y joitly distributed by equatio (2.6) are egatively correlated. It is easy to see that the coditioal desities i equatio (3.3) ad (3.4) have the same local depedece fuctios as the joit desity f X,Y (x,y). i.e., fx Y x, y = fy X x, y = f x, y. f x, y = 0, iff = 0. Therefore X ad Y are idepedet iff = 0 (Balakrisha ad Lai, 2009). 4. Estimatio of the Parameters of BLCH Assume that (x i, y i ) (i = 1,..., ) is a radom sample from the BLCH give by equatio (2.6) Maximum Likelihood Estimatio (MLE) of α, β ad : The Log-likelihood fuctio for the assumed sample is, l α, β, = log N α, β, + log (1 + x i )(1 + y i ) +α y i β Differetiatig ad settig the partial derivatives equal to zero, we get the Likelihood equatios as: N (α,β, ) α = 1 N(α,β,) N (α,β, ) β = 1 N(α,β,) N (α,β, ) = 1 N(α,β,) x i x i y i y i (4.1) x i (4.2) x i y i (4.3) Now, choosig iitial values for α ad β the searchig for a value for to make equatio (4.3) holds. The, usig the value obtaied for with the chose iitial value for β we search for a value for α that makes equatio (4.1) holds. Fially, we use the obtaied searched values for α ad to search a value for β that makes equatio (4.2) holds.

9 A New Class of Bivariate Distributios with Lidley Coditioal Hazard Fuctios 4.2. Maximum Pseudo-likelihood Estimatio (MPLE): A alterative estimatio techique is MPLE, which was itroduced by Besag (1975, 1977) ad Arold ad Strauss (1988b). The techique ivolves Pseudo-likelihood fuctio that does ot ivolve the ormalizig costat equatio (2.6). To estimate the parameters α, β ad, accordig to this method, we have to maximize the fuctio f X Y (x i y i )f Y X (y i x i ) Whe the samplig distributio is the BLCH equatio (2.6), the required coditioal desities are give by equatio (3.3), (3.4) ad therefore we have Log-Pseudo-likelihood fuctio i the form logpl α, β, = log 1 + x i + 2 log β + y i log 1 + β + y i β + y i x i + log 1 + y i +2 log ( α + x i ) log α + x i + 1 ( α + x i )y i. Differetiatig with respect to α, β, ad, respectively, we get the Pseudolikelihood equatios as: log PL α,β, α log PL (α,β,) β log PL α, β, 1 + α+x i 1 (β+y i ) = y i 2 = x i + 2 = 2 y i β + y i 1 α+x i +1 1 (4.5) (4.6) β+y i +1 y i β + y i + 1 x i x i ( α+x i ) ( α+x i +1) +2 (4.7) The Pseudo-likelihood estimates of α, β ad ca be obtaied by solvig log PL (α,β,) log PL (α,β,) = 0, = 0, = 0, α α α i the same maer as it is explaied i Sectio (4.1). 5. Applicatio log PL (α,β,) The followig data has bee obtaied from Adrews ad Herzberg (1985). Usig these data we shall estimate the parameters α, β ad of the equatio (2.6). The data icludes the observatios o patiets havig bladder tumors whe they 2 x i y i 201

10 202 Mohamed Gharib ad Bahady Ibrahim Mohammed etered the trial. These tumors were removed ad patiets were give a treatmet called placebo pills. At subsequet follow-up visits, ay tumors foud were removed, ad treatmet was cotiued. The variables observed are X, time (i moth) to first recurrece of a tumor, ad Y, time (i moth) to secod recurrece of a tumor i Table 1. For the give data, we have, y i = 418, x i = 217, x i y i = These umerical results are used to obtai each of the MLE (usig equatio (4.1)- (4.3)), ad the MPLE (usig equatio (4.5)-(4.7)). Table 2 cotais, the true (iitial) values of α, β,, their MLE, MPLE ad their Mea Squared Errors. The BEC Distributio discussed extesively by Arold ad Strauss (1988a) has the followig joit desity: f X,Y x, y = exp m 11 m 12 y m 21 x + m 22 xy x > 0, y > 0, m₁₂ > 0, m₂₁ > 0, m₂₂ 0, where m 11 is the ormalizig costat. For the give data, the MLE of m₁₂, m₂₁, m₂₂ are give by: m 12 = , m 21 = , m₂₂ = The Likelihood Ratio Test (LRT) will be used to test the ull hypothesis H₀: Distributio of the data is BEC. Uder H₀, the Likelihood Ratio statistic: X L = 2 l α, β, l m 12, m 21, m 22 which has a asymptotic Chi-square Distributio with 1 degree of freedom. H₀ is ² rejected at a sigificace level α if X L > χ 1.α. I additio, for model selectio, we use the AIC ad BIC, defied as: AIC = log likelihood 2p BIC = log likelihood p 2 log() where p is the umber of parameters i the model ad is the sample size.

11 A New Class of Bivariate Distributios with Lidley Coditioal Hazard Fuctios The model with higher AIC (BIC) is the oe that better fits the data. Table 3 gives a compariso betwee the Log-likelihood, AIC ad BIC for BEC Distributio ad BLCH Distributio for the give data i Table 3. For the give data, the Likelihood Ratio statistic is, ² X L = > χ = 3.84, the, we caot accept the ull hypothesis, i. e., the LRT rejects the assumptio that the BEC model is suitable for the give data. 6. Coclusio I this study, a Bivariate Lidley Distributio is itroduced by specified Coditioal Hazard Fuctios. I additio, the parameters of BLCH via the method of MLE ad MPLE are show. Based o the results, Table 2 the MLE are close to the MPLE. Therefore, the computatioally simpler method of MPLE seems to be a attractive alterative, to the more difficult method of MLE sice the coditioal distributios ad hece the MPLE are ot based o the ormalizig costat. Furthermore, Table 3 shows that the BLCH Distributio is better to fit the give data compared with the BEC Distributio. 203 Ackowledgemet The authors are greatly idebted to the referees ad the editors for their costructive commets. Table 1: Data o time of first recurrece (X i ), ad secod recurrece (Y i ) of bladder tumor for patiets udergoig placebo pills treatmet. Patiet X i Y i Patiet X i Y i Patiet X i Y i

12 204 Mohamed Gharib ad Bahady Ibrahim Mohammed Table 2: Estimatio of parameters of the Bivariate equatio (2.6) True value MLE MSE MPLE MSE α= β= = Table 3: Selectio criteria for BEC ad BLCH to the give data Class Log-likelihood AIC BIC BEC BLCH Figure 1: BLCH with α = 1, β, = 2 Figure 2: BLCH with α = 10, β = 20, = 50 E(..) Figure 3: The Regressio curve x(y)

13 A New Class of Bivariate Distributios with Lidley Coditioal Hazard Fuctios Refereces 1. Aczel, J. (1966). Lectures o Fuctioal Equatios ad Their Applicatios. Academic Press, New York. 2. Adrews, D. F. ad Herzberg, A. M. (1985). Data: A Collectio of Problems from May Fields for the Studet ad Research Worker. Spriger-Verlag, New York. 3. Arold, B. C., Castillo, E. ad Sarabia, J. M. (1999). Coditioal Specificatio of Statistical Models. Spriger Series i Statistics, Spriger Verlag, New York. 4. Arold, B. C., Castillo, E. ad Sarabia, J. M. (2001). Coditioally specified distributio: A itroductio. Statistical Sciece, 16, Arold, B. ad Strauss, D. (1988a). Bivariate Distributios with Expoetial coditioals. Joural of the America Statistical Associatio, 83, Arold, B. ad Strauss, D. (1988b). Pseudo-likelihood Estimatio. Sakhya, Series B, 53, Balakrisha, N., Castillo, E. ad Sarabia, J. M. (2010). Bivariate cotiuous distributios with specified Coditioal Hazard Fuctios. Commuicatios i Statistics: Theory ad Methods, 39, Balakrisha, N. ad Lai, C. D. (2009). Cotiuous Bivariate Distributios. Secod Editio. Spriger, New York. 9. Besag, J. E. (1975). Statistical aalysis of o-lattice data. The Statisticia, 24, Besag, J. E. (1977). Efficiecy of pseudo likelihood estimators for simple Gaussia fields. Biometrica, 64, Ghitay, M. E., Al-Mutairi, D. K. ad Nadaraja, S. (2008). Zero-trucated Poisso Lidley distributio ad its applicatio. Mathematics ad Computers i Simulatio, 79, Ghitay, M. E., Atieh, B. ad Nadarajah, S. (2007). Lidley distributio ad its applicatio. Mathematics ad Computers i Simulatio, 78, Hollad, P. W. ad Wag, Y. J. (1987a). Regioal depedece for cotiuous Bivariate desities. Commuicatios i Statistics: Theory ad Methods, 16, Hollad, P. W. ad Wag, Y. J. (1987b)/ Depedece fuctios for cotiuous Bivariate desities. Commuicatios i Statistics: Theory ad Methods, 16, Hossei, Z. ad Noriszura, I. (2010). Negative biomial-lidley distributio ad its applicatio. Joural of Mathematics ad Statistics, 6,

14 206 Mohamed Gharib ad Bahady Ibrahim Mohammed 16. Kotz, S., Balakrisha, N. ad Johso, N. L. (2000). Cotiuous Multivariate Distributios, Vol. 1: Models ad Applicatios. Joh Wiley ad Sos, New York. 17. Lidley, D. V. (1958). Fiducial distributios ad Baye s theorem. Joural of the Royal Statistical Society, Series B, 20, Lidley, D. V. (1965). Itroductio to Probability ad Statistics from a Bayesia Viewpoit, Part II: Iferece. Cambridge Uiversity Press, New York. 19. Mahmoudi, E. ad Zakerzadeh, H. (2010). Geeralized Poisso-Lidley Distributio. Commuicatios i Statistics: Theory ad Methods, 10, Navarro, J. ad Sarabia, J. M. (2011). Reliability properties of Bivariate coditioal proportioal hazard rate models. Joural of Multivariate Aalysis, 74(2), doi: /j.jmva Sele, C. ad Gamze, O. K. (2014). A New Customer Lifetime Duratio Distributio: The Kumaraswamy Lidley Distributio. Iteratioal Joural of Trade, Ecoomics ad Fiace, 5, Shaked, M. (1977). A family of cocepts of depedece for Bivariate Distributios. Joural of the America Statistical Associatio, 72,