On a new absolutely continuous bivariate generalized exponential distribution

Transcription

1 Statistical Methods and Applications anuscript No. (will be inserted by the editor On a new absolutely continuous bivariate generalized exponential distribution S. M. Mirhosseini M. Aini D. Kundu A. Dolati Received: date / Accepted: date Abstract In this paper we studied a three-paraeter absolutely continuous bivariate distribution whose arginals are generalized exponential distributions. The proposed three-paraeter bivariate distribution can be used quite effectively as an alternative to the Block and Basu bivariate exponential distribution. The oint probability density function, the oint cuulative distribution function and its associated copula have siple fors. We derive different properties of this new distribution. The axiu likelihood estiators of the unknown paraeters can be obtained by solving siultaneously three non-linear equations. We propose to use EM algorith to copute the axiu likelihood estiators, which can be ipleented quite conveniently. One data set has been analyzed for illustrative purposes. Finally we propose soe generalization of the proposed odel. Keywords Generalized exponential Bivariate exponential distribution Dependence Measure of association EM algorith Matheatics Subect Classification (2 MSC Priary 62E5 MSC Secondary 62H INTRODUCTION The two-paraeter generalized exponential (GE distribution proposed by Gupta and Kundu [] has been used quite successfully to analyze lifetie data and it has received soe attention in recent years; see, e.g.[3, 5, S. M. Mirhosseini Departent of Statistics, Faculty of Matheatical Science, Ferdowsi University of Mashhad,P.O. Box , Mashhad, Iran M. Aini Departent of Statistics, Faculty of Matheatical Science, Ferdowsi University of Mashhad,P.O. Box , Mashhad, Iran D. Kundu Departent of Matheatics and Statistics, Indian Institute of Technology Kanpur, Pin 286, India A. Dolati Departent of Statistics, Yazd University, Yazd, , Iran Tel.: Fax: E-ail: adolati@yazd.ac.ir

2 2 S. M. Mirhosseini et al 6]. A two-paraeter GE distribution with the shape paraeter α and the scale paraeter λ has the following cuulative distribution function (CDF ( F GE (x;α,λ = e λx α. ( Fro now on a GE with the CDF ( will be denoted by GE(α, λ. When α =, it becoes an exponential distribution with paraeter λ, and we will denote it by Exp(λ. It is well known that the probability density function (PDF and the hazard function (HF of a GE distribution can take different shapes. If the shape paraeter is less than or equal to one, the PDF and the HF are decreasing functions. When the shape paraeter is greater than one, then PDF is a uniodal, and the corresponding HF is an increasing function. For soe recent developents on the GE distribution, and for its different applications, the readers are referred to [, 8] and the references cited therein. In any reliability and life testing experients, bivariate data arise quite naturally. Aong the different bivariate lifetie odel, Marshall-Olkin bivariate exponential odel [22] is the ost popular one. It is a singular distribution, that is a pair (X,Y distributed as Marshall-Olkin odel has this property that P(X = Y >. Block and Basu [5] introduced a three-paraeter absolutely continuous bivariate exponential distribution by reoving the singular part of the Marshall-Olkin odel. Although Block and Basu bivariate exponential distribution does not have exponential arginals, but its arginals have decreasing PDFs and HFs. Several authors have been studied applications of the Block and Basu odel; see, e.g [2]. A wide survey on different bivariate distributions can be found in [4,7]. Recently, different bivariate exponential and bivariate generalized exponential (BGE distributions have been proposed and studied in the literature; see, e.g. [9 2, 24, 28, 32]. Most of these odels are singular distributions, and they can be used if there are ties in the data. It is not trivial to extend the univariate GE distribution to the bivariate or the ultivariate case. According to Joe [3], Section 4., a paraetric faily of distributions should satisfy four desirable properties: (a There should exist an interpretation like a ixture or other stochastic representation. (b The argins should belong to the sae paraetric faily and nuerical evaluation should be possible. (c The bivariate dependence between the argins should be described by a paraeter and cover a wide range of dependence. (d The distribution and density functions should preferably have a closed-for representation; at least nuerical evaluation should be possible. The ai of this paper is to study a new absolutely continuous bivariate generalized exponential distribution, whose arginals are univariate GE distributions and fulfill all of the properties (a (d. This new three-paraeter BGE distribution is obtained using the distribution of iniu order statistics of two independent saples of ordinary exponential distribution when the saple size is rando; see, e.g, [7]. All oint or arginal distributions and density functions, the corresponding oents, the copula and its density have siple analytic representations that can be easily eployed in applications. Moreover, the proposed distribution has positive quadrant dependent property which iplies that the easures of association such as Pearson s oent correlation, Kendall s tau and Spearan s rho for this distribution vary between and, which is a suitable range of dependence for applications. The axiu likelihood estiators (MLEs can be obtained by siultaneously solving three non-linear equations. The MLEs cannot be obtained in explicit fors. For known shape paraeter the MLEs of the scale

3 On a new absolutely continuous bivariate generalized exponential distribution 3 paraeters can be obtained by using the EM algorith. In the proposed EM algorith, at each E-step the axiization can be perfored explicitly. Hence the ipleentation of the EM algorith is quite siple in this case. Finally, the shape paraeter can be estiated by axiizing the profile log-likelihood function. The rest of the paper is organized as follows. We discuss the derivation of the BGE distribution in Section 2. In Section 3 we study its different properties. Statistical inference is provided in Section 4. The analysis of a real data set is presented in Section 5. Finally we propose soe generalizations and conclude the paper in Section 6. 2 BGE DISTRIBUTION: PDF AND CDF Following the idea given in [7], let {X,X 2,...} and {Y,Y 2,...} be two sequences of utually independent and identically distributed (i.i.d. rando variables. It is assued that for k {,2,3,...}, X k Exp(λ and Y k Exp(λ 2. Consider a sequence of independent Bernoulli trials in which the kth trial has probability of α k, < α, k {,2,3,...} and let N be the trial nuber on which the first success occurs. The discrete rando variable N has the probability ass function ( P(N = n = ( α α (... α α 2 n n ; n =,2,..., and its probability generating function is given by, see, e.g., [26] g(t = E(t N = ( t α ; t [,]. (2 Let U = in(x,...,x N and V = in(y,...,y N. The oint survival function of (U,V is then F(u,v = P{U u,v v} = n= [P(X i up(y i v] n P(N = n = g (e (λ u+λ 2 v = { e (λ u+λ 2 v } α. (3 Since F(u,v = F (u F 2 (v + F(u,v, the associated oint distribution function is given by F(u,v = ( e λu α + ( e λ2v α { e (λ u+λ 2 v } α, u,v,λ,λ 2 >, < α. (4 Fro (4, by taking v or u, it follows that U GE(α,λ and V GE(α,λ 2. (5 Therefore, it follows that the arginals of (4 are GE distributions. A pair (U,V distributed as (4 is said to have BGE with paraeters α, λ and λ 2, and it will be denoted by BGE(α,λ,λ 2. The oint density function of BGE is given by f (u,v = λ λ 2 αe (λ u+λ 2 v { αe (λ u+λ 2 v }{ e (λ u+λ 2 v } α 2. (6

4 4 S. M. Mirhosseini et al It can be easily seen that the oint PDF of BGE is a decreasing function in both arguents, and it resebles the oint PDF of the Block and Basu bivariate exponential distribution. Plots of the CDF and PDF of the BGE distribution are provided in Fig.. (a (b (c (d Fig. Plots of the oint CDF (left panels and PDF (right panels of BGE distribution. In (a, (b (α,λ,λ 2 = (.,, and in (c, (d (α,λ,λ 2 = (.9,,. Since < e (λ u+λ 2 v <, for u,v >, by using the binoial series expansion we have ( ( e (λ u+λ 2 α v α = ( e (λ u+λ 2 v, and the oint survival function (3 can be rewritten as ( α F(u,v = = = ( + e (λ u+λ 2 v.

5 On a new absolutely continuous bivariate generalized exponential distribution 5 We observe that the infinite series is suable, differentiable and hence by differentiating with respect to u and v we get f (u,v = λ λ 2 = ( α ( + 2 e (λ u+λ 2 v. (7 3 PROPERTIES 3. MOMENTS & CONDITIONAL MOMENTS First we provide expressions for the oint oent generating function (gf and Pearson s correlation coefficient of the BGE odel. Proposition If the rando vector (U,V BGE(λ,λ 2,α then (a the oint.g.f. of (U,V, for t < λ and s < λ 2, is given by ( α M U,V (t,s = ( + λ λ 2 2 ( λ t( λ 2 s, (8 = (b the Pearson s correlation coefficient of (U,V is given by ( α ( + Corr(U,V = 2 γ(α, 2, (9 κ(α, ( = where γ(α,β = Ψ(α + β Ψ(β, and κ(α,β = Ψ (β Ψ (α + β, is the digaa function, Ψ (. is its derivative and Ψ(t = d lnγ (t, dt is the gaa function [9]. Γ (t = x t e x dx, t >, Proof Starting fro M U,V (t,s = E(e tu+sv, using (7 we have ( α M U,V (t,s = λ λ 2 = ( + 2 e ( λ tx ( λ 2 sy dxdy. Since the quantity inside the suation is absolutely integrable, interchanging the suation and integration we have the required result. For part (b using (7, direct calculation shows that the product oent could be obtained as E(UV = Since U GE(α,λ and V GE(α,λ 2, fro [], using ( α ( + = λ λ 2 2. E(U = Ψ(α + Ψ( λ, E(V = Ψ(α + Ψ( λ 2

6 6 S. M. Mirhosseini et al and Var(U = Ψ ( Ψ (α + λ 2, Var(V = Ψ ( Ψ (α + λ2 2 the expression for correlation coefficient is iediate. The following result gives the conditional oents of BGE odel. Proposition 2 If the rando vector (U,V BGE(λ,λ 2,α, then the conditional expectation of V given U, is given by E(V U = u = λ λ 2 h U (u, where h U (u = f U (u/ F U (u is hazard rate function of U. Proof The conditional density function of (V U = u is given by ( α f V U=u (v = Thus, E(V U = u = = = = λ 2 αe λ u ( e λ u α x f X2 X =u(xdx α( e λ u α = αλ 2 ( e λ u α ( α = ( e λ u α αλ 2 e λ u ( e λ u α = λ λ 2 F U (u f U (u. = ( + 2 e ( λ u+λ 2 v. ( + e ( λ u ( α ( + e λ ( u ( λ 2 xe λ2x dx It is known that (see, e.g., [] the hazard rate function of univariate GE distribution with the shape paraeter α, has a decreasing hazard function if α. Therefore, we have the following result. Corollary If the rando vector (U,V BGE(λ,λ 2,α, then E(V U = u, is an increasing function in u. The following results will be useful for developent of the EM algorith. If (U,V and N are sae as defined before, then the oint PDF of U, V and N can be written as f (u,v,n = n 2 λ λ 2 e n(λ u+λ 2 v P(N = n; u >,v >,n =,2,... ( Hence the conditional probability of N = n, given U and V can be written as P(N = n U = u,v = v = Kn 2 e na P(N = n; n =,2,..., ( where A = λ u + λ 2 v, and K = αe A { αe A }{ e A } α 2.

7 On a new absolutely continuous bivariate generalized exponential distribution 7 Proposition 3 If (U,V BGE(λ,λ 2,α, and N is sae as defined before, then Proof If we denote E(N U = u,v = v = Kαe A ( e A α 3 { e 2A (α 2 6α e A (3α + 2 }. h(t = E ( te A N, then it follows that n 3 e A P(N = n = h ( + 3h ( + h (. n= Now the result follows by using the fact that fro (2 h(t = ( te A α ; t [,]. 3.2 STRESS-STRENGTH PARAMETER & DISTRIBUTION OF THE MINIMUM The stress-strength paraeter, R = P(U < V, is useful for data analysis purposes. The following result gives a convenient for for the stress-strength paraeter of BGE odel. Proposition 4 If (U,V BGE(λ,λ 2,α, then Proof Fro (7 we have P(U < V = = = y = = P(U < V = λ 2 λ + λ 2. f (x,ydxdy ( α y ( + 2 λ λ 2 e (λ x+λ 2 y dxdy ( α ( + λ 2 e λ2y ( e λy dy = λ 2 λ + λ 2 = ( α ( + which copletes the proof. = λ 2 λ + λ 2, Proposition 5 If (U,V BGE(λ,λ 2,α, then W = in(u,v GE(α,λ + λ 2. Proof P(W w = P(U w,v w which copletes the proof. = n= [P(X i wp(y i w] n P(N = n = g (e (λ +λ 2 w = { } e (λ α +λ 2 w

8 8 S. M. Mirhosseini et al 3.3 RÉNYI ENTROPY The entropy of a rando vector (U,V with the oint pdf f (x,y is a easure of variation of the uncertainty. Rényi entropy [27] is defined by for β >, where I R (β = log{t (β}, β T (β = Proposition 6 Suppose that (U,V BGE(λ,λ 2,α. Then f β (x,ydxdy. T (β = = k= ( β β(α 2 ( k α +β (λ λ 2 β ( + k + β 2. (2 Proof Fro (6 and substituting the transforations x = e λu and y = e λ2v, we have T (β = α β (λ λ 2 β (xy β ( β ( (α 2β αxy xy dxdy. By applying the binoial series expansion to the integrand and integrating with respect to x and y, we have the required result. 3.4 DEPENDENCE PROPERTIES In what follows we discuss the dependence properties of the BGE distribution through its associated copula. In view of Sklar s Theore [33], solving the equation C α {F (x,f 2 (y} = F(x,y, for the function C α : [,] [,] [,] yields the underlying copula associated with the pair (U,V having the BGE distribution defined by (4 as C α (u,v = u + v { ( u α ( v α } α = u + v uv{u α + v α } α, (3 for all u,v (, and < α. The theory and applications of copulas are well docuented in [25]. The survival copula Ĉ α (u,v = u + v +C α ( u, v, associated to C α is given by Ĉ α (u,v = ( u( v{( u α + ( v α } α, (4 for all u,v (, and < α. The copula (3 is not a eber of the Archiedean faily of copulas [25], but its associated survival copula (4 turns to be an Archiedean copula with the strict generator ϕ(t = ln( ( t α (see, Exaple 3 in [7]. Recall that for two copulas C and C 2, we say that C 2 is ore concordant than C (written C c C 2 if C (u,v C 2 (u,v, or equivalently Ĉ (u,v Ĉ 2 (u,v, for all u,v (,. A pair (X,X 2 with the copula C is positively quadrant dependent (written PQD if Π c C, where Π(u,v = uv is the product copula [25].

9 On a new absolutely continuous bivariate generalized exponential distribution 9 The dependence properties of the BGE distribution depend only on the paraeter α. The following result provides the dependence ordering of the BGE faily of distributions with respect to the paraeter α. Proposition 7 The copula C α defined by (3 is negatively ordered with respect to α; that is for α,α 2 (,], such that α α 2, we have C α2 c C α. Proof Since Ĉ α is an Archiedean copula with the generator ϕ α (t = ln( ( t α, and ϕ α2 ϕ α (t = ln( ( e t α 2 α is non-decreasing in t, for α α 2, in view of Theore in [25], we have Ĉ α2 c Ĉ α, or equivalently, C α2 c C α. Corollary 2 Suppose that (U,V BGE(λ,λ 2,α. Then (U,V is PQD. Proof As a consequence of Proposition 7, for α we have that Π(u,v = C (u,v C α (u,v for all u,v (,. Reark Since the BGE distribution defined by (4 has the PQD property, it is suitable to describe the positive dependence of a rando pair (U,V. However, it is very siple to consider a distribution to describe a negative dependence. It suffices to consider the copula C given by C α(u,v = u C α (u, v. It is obvious that the properties of a copula C α can be obtained in a siple way fro the corresponding properties of a copula C α ; see, [25] for detail. Let (U,V and (U,V be two continuous rando vectors with the sae univariate arginals and the respective oint density functions f and g. The pair (U,V is said to be ore positive likelihood ratio dependent (PLRD than the pair (U,V, denoted by (U,V PLRD (U,V, if f (x,y f (x 2,y 2 g(x,y 2 g(x 2,y f (x,y 2 f (x 2,y g(x,y g(x 2,y 2, (5 whenever x x 2 and x 2 y 2 [3]. When U and V are independent, then the pair (U,V is said to be PLRD and the condition (5 reduces to f (x,y f (x 2,y 2 f (x,y 2 f (x 2,y. Holland and Wang [2] showed that a sufficient condition for PLRD in the case of continuous rando variables is that 2 x y ln f (x,y. Proposition 8 Suppose that (U,V BGE(λ,λ 2,α. Then (U,V is PLRD. Proof Let (U,V be a rando vector with the oint distribution function G u (x,y = x α + y α (x + y xy α, x,y [,], < α. Let g u (.,. be the density function associated with G u (.,.. Since 2 x y lngu (x,y, for all x,y [,], < α, by the result of Holland and Wang [2] the pair (U,V is PLRD, i.e., (U,V PLRD (U,V, where U and V are independent with the sae univariate arginal distribution as U and V. For i =,2, let ϕ i (t be the increasing apping t λ i ln( t. Since the pair (U,V has the sae oint distribution as the pair (ϕ (U,ϕ 2 (V, in view of Theore 9.D.2 in [3], we have the required result. In the following we discuss the tail dependence properties of the BGE distribution. For a pair (U,V with the copula D, the lower (resp, upper tail dependence coefficient, λ L (resp, λ U is defined by [3,25] and λ L (U,V = li u + λ U (U,V = 2 li u D(u, u, (6 u D(u,u. (7 u

10 S. M. Mirhosseini et al Proposition 9 Suppose that (U,V BGE(λ,λ 2,α. Then λ L (U,V =, λ U (U,V =. Proof By taking into account (6, fro (3, the lower tail dependence coefficient of (U,V can be expressed as u(2 (2 u α α λ L (U,V = li u + =. u By taking into account (7, the upper tail dependence coefficient of (U,V can be expressed as u(2 (2 u α α λ U (U,V = 2 li u =. u In the rest of this section we provide expressions for soe well-known easures of association for a vector (U,V having BGE distribution. The population version of two of the ost coon nonparaetric easures of association between the coponents of a continuous rando pair (U,V are Kendall s tau (τ and Spearan s rho (ρ which depend only on the copula C of the pair (U,V, and are given by τ(c = 4 C(u,vdC(u,v, (8 and See [25] for detail. ρ(c = 2 C(u, vdudv 3. (9 Proposition Suppose that (U,V BGE(λ,λ 2,α. Then and τ(u,v = + 4αB(2,2α (Ψ(2 Ψ(2α +, ρ(u,v = 9 2α 2 =( ( α [B(,α] 2, where B(a,b = xa ( x b dx, denotes the beta function. Proof First recall that for a copula based easure of association κ, satisfying Scarsini s axios [3] and any pair of continuous rando variables (T, S with associated copula C, one has κ(t, S = κ( T, S, or equivalently, κ(c = κ(ĉ. Now the result follows fro Proposition 9 in [7]. Reark 2 Note that as a consequence of the PQD property of the BGE distribution, for (U,V BGE(λ,λ 2,α, we have Corr(U,V, τ(u,v and ρ(u,v. We provide the values of the Kendall s tau and the Spearan s rho for BGE distribution in Table. It is iediate that as α increases, both Kendall s tau and Spearan s rho decrease to, as it should be. Moreover, it is observed that ρ s(α τ(α = 3 as α, although we could not proved it theoretically. 2

11 On a new absolutely continuous bivariate generalized exponential distribution ρ α ρ(α τ(α s(α τ(α Table Kendall s tau and Spearan s rho associated with the faily of distributions (4 for different values of α (,]. 4 STATISTICAL INFERENCE 4. MAXIMUM LIKELIHOOD ESTIMATION In this section we discuss the MLEs of the paraeters of BGE distribution, based on a rando saple of size, naely {(u,v,...,(u,v } fro BGE(λ,λ 2,α. The log-likelihood function becoes ( ( l(θ = ln(αλ λ 2 v i + (α 2 ln e (λ u i +λ 2 v i + i= λ u i + λ 2 i= i= ( ln αe (λ u i +λ 2 v i where θ = (α,λ,λ 2. The axiu likelihood estiates can be obtained by axiizing (2 with respect to the unknown paraeters. The three noral equations becoe; l(θ α = ( α + ln e (λ u i +λ 2 v i e (λ x i +λ 2 y i i= i= αe (λ u i +λ 2 v i =, l(θ = u i e λ λ u i + (α 2 (λ u i +λ 2 v i i= i= e (λ u i +λ 2 v i + α u i e (λ u i +λ 2 v i i= αe (λ u i +λ 2 v i =, l(θ = v i e λ 2 λ 2 v i + (α 2 (λ u i +λ 2 v i i= e (λ u i +λ 2 v i + α v i e (λ u i +λ 2 v i i= αe (λ u i +λ 2 v i =. i= Note that the Newton-Raphson ethod or other optiization routine ay be used to axiize (2. To avoid that we propose to use the profile likelihood ethod to copute the MLEs of the unknown paraeters. For fixed α, the MLEs of λ and λ 2 can be obtained by axiizing the profile log-likelihood function with respect to λ and λ 2. We use EM algorith to copute the MLEs of λ and λ 2 for a given α, and finally we axiize the profile log-likelihood function of α, to copute the MLE of α. For ipleenting the EM algorith, we treat the proble as a issing value proble. Suppose along with (u,v, we observe the associated N value also. Therefore, the coplete observations are as follows: {(u,v,n,...,(u,v,n }. Based on the coplete saple, the log-likelihood function without the additive constant becoes i= (2 l c (λ,λ 2 = lnλ + lnλ 2 λ n i u i λ 2 i v i. (2 i= i=n

12 2 S. M. Mirhosseini et al For fixed α, the MLEs of λ and λ 2 becoe λ (α = i= n and λ 2 (α = iu i i= n. (22 iv i Now we are in a position to provide the EM algorith. For fixed α, at the E-step of the EM algorith, we construct the pseudo log-likelihood function at the k-th iterate by replacing the true value of N by its expected value. It takes the following for; l s (λ,λ 2 λ (k (k (α,λ 2 (α = lnλ + lnλ 2 λ u i E(N u i,v i,λ (k (k (α,λ 2 (α,α i= λ 2 i= v i E(N u i,v i,λ (k (k (α,λ 2 (α,α. (23 Here λ (k (k (α and λ 2 (α, are the values of λ and λ 2, respectively at the k-th iterate and E(N u i,v i,λ (k (k (α,λ 2 (α,α can be obtained using Proposition 4. Hence at the M-step, the axiization can be easily perfored to obtain λ (k+ (α and λ (k+ 2 (α as where C i = E(N u i,v i,λ (k λ (k+ (α = i= C iu i and λ (k+ 2 (α = i= C iv i, (24 (k (α,λ 2 (α,α. Continue the iteration until the convergence is et. Let us denote these estiates as λ (α and λ 2 (α. Finally axiize the profile log-likelihood function of α, to obtain the MLE of α, say α. Therefore, the MLEs of α, λ and λ 2 becoe α, λ ( α and λ 2 ( α, respectively. It can be easily seen that the density function of the BGE distribution satisfies all required conditions for the MLEs to be consistent and asyptotically norally distributed. We have the following result. Proposition If θ is the MLE of θ, then n(θ ˆθ d N 3 (,I. (25 Here d eans convergence in distribution and N 3 (,I, denotes the 3-variate noral distribution with ean vector and the covariance atrix I, and the atrix I is the Fisher inforation atrix; the eleents of the atrix I are presented in the Appendix. 4.2 TESTING OF HYPOTHESES We perfor the following two testing of hypotheses probles. PROBLEM : Testing whether the two arginals have the sae distributions or not, can be carried out as follows: H : λ = λ 2 versus H : λ λ 2. (26 In this case the MLE of λ = λ = λ 2 can be obtained along the sae line as before. The pseudo log-likelihood function becoes; l s (λ λ (k (α = 2lnλ λ i= (u i + v i E(N u i,v i,λ (k (α,α, (27

13 On a new absolutely continuous bivariate generalized exponential distribution 3 where λ (k denotes the estiate of λ at the k-th iteration. If D i = E(N u i,v i,λ (k (α, then λ (k+ (α = 2 i= D i(u i + v i. (28 Siilarly, as before the estiate of α can be obtained by axiizing the profile log-likelihood function of α. Now if we denote the estiates of α and λ as α and λ, respectively, then using the standard likelihood ratio, which has the asyptotic distribution as follows: ( 2 l( α, λ, λ 2 l( α, λ, λ χ 2. PROBLEM 2: If we want to test whether the two coponents are independent or not, the following test can be perfored: Under the null hypothesis the MLEs of λ and λ 2 can be obtained as H : α = versus H : α. (29 λ = i= u i and λ 2 = i= v i. (3 In this case, since α is in the boundary under H, the standard results do not apply. But using Theore 3 of Self and Liang [29], it follows that ( 2 l( α, λ, λ 2 l(, λ, λ χ2. (3 5 DATA ANALYSIS In this section we provide the analysis of a real data set fro McGilchrist and Aisbett [23]. The data has been obtained fro an experient, where M individuals are observed and ties between recurrence of a particular type of event are recorded. In this study the recurrence tie of infection in kidney patients who are using a portable dialysis achine are recorded. The infection occurs at the point of infection of the catheter, and when it occurs, the catheter has to be reoved, the infection cleared up and then the catheter reinserted. Recurrence ties are ties fro infection until next infection. For each patient two such recurrence ties are given; naely first recurrence tie (FRT and second recurrence tie (SRT. The data for 23 patients are reported in Table 2 Before, progressing further, we obtain soe basic statistics of the first recurrence tie and second recurrence tie, and they are reported in Table 3. It is clear fro Table 3 that both FRT and SRT have very long right tail. To get an idea about the shape of the epirical hazard function of the arginals, we provide the scaled TTT plots of FRT and SRT in Figure 2, as suggested by Aarset []. This plot provides an idea of the shape of the hazard function of a distribution. It has been shown in [] that the scaled TTT transfor is convex (concave if the hazard rate is decreasing (increasing. For this data set, it indicates that both of variables have decreasing epirical hazard functions. Note that fro Table 3 the Spearan s rho and Kendall s tau for FRT and SRT data are given by.266 and.84, respectively. This observation deonstrates an obvious positive dependence between involved data. This observation shows that the proposed BGE ay be used for analyzing this bivariate data set.

14 4 S. M. Mirhosseini et al No. FRT SRT No. FRT SRT Table 2 Kidney infection data of 23 patients. The patient No., first recurrence tie (FRT, second recurrence tie (SRT are reported. Statistics FRT SRT Miniu 2 8 st Quartile 3 26 Median 34 4 Mean st Quartile Maxiu Standard deviation Pearson s corr..9 Kendall s tau.84 Spearan s rho.266 rho/tau.449 Table 3 Descriptive statistics of the data vector. We divide all the data points by ainly for coputational purposes, it is not going to affect in the statistical inference. To get initial estiates of λ and λ 2, we fit exponential distribution to the arginals, and obtain the initial estiates of λ and λ 2 as.93 and.86. For each α, we use these initial estiates to start the EM algorith. The profile log-likelihood function of α becoes a uniodal function, and it is reported in Figure 3. Finally axiizing the profile log-likelihood function of α, we obtain the MLEs of α, λ and λ 2, and they are α =.799, λ =.7559 and λ 2 =.655, and the corresponding log-likelihood value is The associated 95% confidence intervals are.799.6, and , respectively. To see whether the GE distribution fits the arginal data or not, we copute the Kologorov- Sirnov (KS distances of GE(.799,.7559 and GE(.799,.655 to the epirical CDF of FRT and SRT data, respectively. It is observed that the KS distance between GE(.799,.7559 and epirical CDF of FRT is.73 and the associated p value is Siilarly, the KS distance between GE(.799,.655 and epirical CDF of SRT is.689 and the associated p value is Therefore, the GE distribution can be used to fit the arginals reasonably well, for the above data set.

15 On a new absolutely continuous bivariate generalized exponential distribution (a (b Fig. 2 Scaled TTT plots of (a FRT and (b SRT. 49 Profile log likelihood function α Fig. 3 Profile log-likelihood function of α. We perfor the following two testing of hypotheses probles; naely (26 and (29. In the first case (26, the value of the test statistic is.22. Since the p =.6466, we do not reect the null hypothesis. In the second case the value of test statistic is In this case p =.74, hence we reect the null hypothesis. Note that fro Table we see that the population versions of Kendall s tau and Spearan s rho for BGE distribution with estiated paraeter α =.7 are given by.285 and.94, respectively. These values are close to the Spearan s rho and Kendall s tau between FRT and SRT, given in Table 3. This observation also shows that the proposed BGE ay be used for analyzing this bivariate data set. The natural question that arises here is whether the BGE odel fits these bivariate data or not. In the next section we provide a copula goodness-of-fit test.

16 6 S. M. Mirhosseini et al For coparison purposes we have fitted (i the three paraeters Block and Basu [5] bivariate exponential odel with the pdf f (x,y = λ λ(λ 2 +λ 3 λ +λ 2 e λ x (λ 2 +λ 3 y, x < y λ 2 λ(λ +λ 3 λ +λ 2 e (λ +λ 3 x λ 2 y, x > y where λ,λ 2,λ 3 > and λ = λ +λ 2 +λ 3 and (ii the five-paraeter absolutely continuous bivariate generalized exponential distribution of the for F(x,y = [( e λ x α + ( e λ 2x α 2 ] α, which is constructed based on the Clayton s copula [25] proposed in Kundu and Gupta [2], to this data set. (32 Model Estiated paraeters Log-likelihood BEG ˆα =.799 ˆλ =.7559 ˆλ2 = Block and Basu odel ˆλ =.9297 ˆλ2 =.8586 ˆλ3 = Kundu and Gupta odel ˆα =.562 ˆα =.46 ˆλ =.68 ˆα 2 =.5392 ˆλ2 = Table 4 The MLEs and the values of Loglikelihood The MLEs and corresponding log-likelihood values are given in Table 4. Therefore, based on the loglikelihood values, we can say that the proposed BGE odel provides a better fit than Block and Basu [5] bivariate exponential odel and is coparable with the Kundu and Gupta [2] odel for this data set. 5. A COPULA GOODNESS-OF-FIT TEST Once a odel has been stated and estiated the natural question is to check whether the initial odel assuptions are realistic. In other words we are faced with the so-called goodness-of-fit proble. As an advantage of the Sklar s Theore [25] the arginal distributions and the copula can be chosen independently of one another. The univariate GE distribution provide adequate descriptions of the FRT and SRT data, individually. Since the copula of the BGE odel has a closed and siple for, one can also try to perfor a copula goodness-of-fit test. A review and coparison of goodness-of-fit procedures is given in [8]. Let (x,y,...,(x n,y n be observations fro a rando vector (X,Y. When dealing with bivariate data, the ost natural way of checking the adequacy of a copula odel would be to copare the fitted copula and the epirical copula (see, e.g, [6] of data defined by C n ( i n, n = nuber of pairs (x,y in the saple with x x (i,y y (, n where x (i and y (, i, n, denote order statistics fro the saple. Figure 4 shows graph of the pairs (x i,y i and (u i,v i, i =,...,23, for FRT and SRT data. Fro siulations provided in [8] a good cobination of power and conceptual siplicity is provided by the Craer-von Mises statistic: S n = n i= (C α (u i,v i C n (u i,v i 2,

17 On a new absolutely continuous bivariate generalized exponential distribution 7 where C α is the fitted copula and u i = rank of x i aong x,...,x n n + and v i = rank ofy i aong y,...,y n. n + This statistic easures how close the fitted copula is fro the epirical copula of data. The P-value of the test is coputed using a paraetric bootstrap procedure described in Appendix A of [8]. To this end, we applied this procedure to both the FRT and SRT data. The bootstrap values S,...,S of the Craer-von Mises test statistic are generated and we found the proportion of these values that are larger than S n =.394 as P-value.3. Thus we ay conclude that the copula C α defined by (3 with the association paraeter α =.799 perfors a good fit for this data set. SRT noralized ranks (SRT FRT noralized ranks (FRT (a (b Fig. 4 Pairs of observations (a and pairs of noralised ranks (b for FRT and SRT data. 6 CONCLUSIONS In this paper we studied a bivariate absolutely continuous generalized exponential distribution, whose arginals are generalized exponential distributions. It has three paraeters and the arginals have decreasing hazard functions. Therefore, the proposed odel can be used as an alternative to the quite popular Block and Basu bivariate exponential odel. We derive different properties of the proposed odel, and also provide the EM algorith for coputation of the MLEs of the unknown paraeters. We have analyzed one real data set, and it is observed that the proposed odel provides a good fit to the data set. Now we briefly discuss different generalizations of the proposed odel. ( Although we have developed the ethodology for the bivariate case, along the sae line ultivariate generalized exponential distribution also can be defined. Several properties can be obtained even for the ultivariate case also. (2 Instead of taking

18 8 S. M. Mirhosseini et al exponential distribution for X and Y, it is possible to develop the ethodology when they follow Weibull distribution. In this case we can obtain bivariate/ ultivariate exponentiated Weibull distribution as an alternative to the already-existent bivariate Weibull odels; see, e.g [4]. It will be interesting to develop different properties of this ore flexible distributions. More work is needed along these directions. ACKNOWLEDGEMENTS: The authors would like to thank the referees for their helpful coents which iproved significantly the earlier draft of the paper. APPENDIX. FISHER INFORMATION MATRIX Fro (2 we have 2 l(θ α 2 2 l(θ λ 2 2 l(θ λ 2 2 = n n [ α 2 e (λ ] x i +λ 2 y i 2 i= αe (λ, x i +λ 2 y i = n λ 2 = n λ 2 2 (α 2 (α 2 n i= n i= x 2 i e (λ x i +λ 2 y i x 2 i e (λ x i +λ 2 y i n ( e (λ x i +λ 2 y i 2 α i= ( αe (λ x i +λ 2 y i 2, y 2 i e (λ x i +λ 2 y i y 2 i e (λ x i +λ 2 y i n ( e (λ x i +λ 2 y i 2 α i= ( αe (λ x i +λ 2 y i 2, n i= 2 l(θ x i e (λ x i +λ 2 y i n = α λ ( e (λ x i +λ 2 y i 2 + x i e (λ x i +λ 2 y i i= ( αe (λ x i +λ 2 y i 2, 2 n l(θ y i e = α λ 2 (λ x i +λ 2 y i n i= ( e (λ x i +λ 2 y i 2 + y i e (λ x i +λ 2 y i i= ( αe (λ x i +λ 2 y i 2, 2 n l(θ x i y i e = (α 2 λ λ 2 (λ x i +λ 2 y i n ( e (λ x i +λ 2 y i 2 α x i y i e (λ x i +λ 2 y i i= ( αe (λ x i +λ 2 y i 2. i= The Fisher inforation is I(θ = [I(θ i ], where I i (θ = E 2 l(θ θ i θ, and θ = (θ,θ 2,θ 3 = (α,λ,λ 2. We shall now present the exact expressions of I i (θ, for i =,2,3. Direct calculations show that ( 2 l(θ I = E α 2 ( 2 l(θ I 22 = E λ 2 = n α 2 = n λ 2 ( + = k= ( k α +3( α 2 k ( + k ( + 2α(α 2A + αa 2, ( 2 l(θ I 33 = E λ2 2 = n λ2 2 ( + 2α(α 2A + αa 2 ( 2 l(θ I 2 = E α λ ( 2 l(θ I 3 = E α λ 2 ( 2 l(θ I 23 = E λ λ 2 = nα = nα λ (B + B 2, (B + B 2, λ 2 = nα ((α 2C + αc 2, λ λ 2,

19 On a new absolutely continuous bivariate generalized exponential distribution 9 where A = A 2 = ( ( α 4 ( ( α ( + 3 4, ( α 2 ( α k (k , = = k= B = B 2 = C = C 2 = = ( ( α 3 ( + ( +( α 2 α k ( + k + 2 3, = k= = ( ( α 4 ( ( ( α 2 α k ( + k = k= ( α ( ( α ( + 3 4,, References. Aarset, M.V. (987. How to identify a bathtub hazard rate?, IEEE Transactions on Reliability, 36, Achcar, J. A., Coelho-Barros, E. A. and Mazucheli, J. (23. Block and Basu bivariate lifetie distribution in the presence of cure fraction, Journal of Applied Statistics, 4, Asgharzadeh, A. (28. Approxiate MLE for the scaled generalized exponential distribution under progressive type-ii censoring, Journal of the Korean Statistical Society, 38, Balakrishnan, N. and Lai, C. D. (29. Continuous Bivariate Distributions, second edition. Springer, New York. 5. Block, H. and Basu, A. P. (974. A continuous bivariate exponential extension, Journal of the Aerican Statistical Association, 69, Deheuvels, P. (979. La fonction de dépendance epirique et ses propriétés Un test non paraétrique díndépendance, Acadéie Royale de Belgique Bulletin de la Classe des Sciences 5e Série, 65, Dolati, A., Aini, M. and Mirhosseini, S. M. (24. Dependence properties of bivariate distributions with proportional (reversed hazards arginals, Metrika, 77, Genest, C., Reillard, B. and Beaudoin, D. (28. Goodness-of-fit tests for copulas: A review and a power study. Insurance: Matheatics and Econoics, 42, Gradshteyn, I. S. and Ryzhik, I. M. (2. Table of Integrals, Series and Products, sixth ed., Acadeic Press, San Diego.. Gupta, R. D. and Kundu, D. (999. Generalized exponential distributions, Austral. New Zealand J. Statist, 4, Gupta, R. D. and Kundu, D. (27. Generalized exponential distribution; existing ethods and soe recent developents, J. Statistical Planning and Inference, 37, Holland, P. W. and Wang, Y. J. (987. Dependence functions for continous bivariates densities, Coun. Statist. Theory and Methods, 6:3, Joe, H. (997. Multivariate Models and Dependence Concepts, Chapan & Hall, London. 4. Jose, K. K., M., Ristíc, M. M. and Joseph, A. (2. Marshall Olkin bivariate Weibull distributions and processes, Statistical Papers, 52, Kannan, N., Kundu, D, Nair, P. and Tripathi, R. C. (2. The generalized exponential cure rate odel with covariates, Journal of Applied Statistics, 37:, Ki, N. (24. Approxiate MLE for the scale paraeter of the generalized exponential distribution under rando censoring, Journal of the Korean Statistical Society, 43:, 9 3.

20 2 S. M. Mirhosseini et al 7. Kotz, S., Balakrishnan, N. and Johnson, N.L. (2. Continuous ultivariate distributions, Vol., second ed., Wiley: New York. 8. Kundu, D. and Gupta, R. D. (28. Generalized exponential distribution: Bayesian estiations, Coputational Statistics and Data Analysis, 52, Kundu, D. and Gupta, R. D. (29. Bivariate generalized exponential distribution, J. Multivariate Analysis,, Kundu, D. and Gupta, R. D. (2. Modified Sarhan Balakrishnan singular bivariate distribution, J. Statistical Planning and Inference, 4, Kundu, D. and Gupta, R. D. (2. Absolute continuous bivariate generalized exponential distribution, AStA Adv. Stat. Anal., 95, Marshall, A. W. and Olkin, I. (967. A ultivariate exponential distribution, Journal of the Aerican Statistical Association, 62, McGilchrist, C. A. and Aisbett, C. W. (99. Regression with frailty in survival analysis, Bioetrics, 47, Mohsin, M., Kazianka, H., Pilz, J. and Gebhardt, A. (24. A new bivariate exponential distribution for odeling oderately negative dependence, Stat. Methods Appl, 23, Nelsen, R. B. (26. An Introduction to Copulas. Second Edition, Springer, New York. 26. Pillai, R. N. and Jayakuar, K. (995. Discrete Mittag Leffler distributions, Statistics and Probability Letters, 23, Rényi, A. (96. On easures of entropy and inforation, in: Proceedings of the 4th Berkeley Syposiu on Matheatical Statistics and Probability, vol. I, University of California Press, Berkeley, pp Sarhan, A. and Balakrishnan, N. (27. A new class of bivariate distributions and its ixture, Journal of Multivariate Analysis, 98, Self, S.G. and Linag, K. L. (987. Asyptotic properties of the axiu likelihood estiators, and likelihood ratio test under nonstandard conditions, Journal of the Aerican Statistical Association, 82, Scarsini, S. (984. On easures of concordance. Stochastica, 8, Shaked, M. and Shanthikuar, G. (27. Stochastic Orders, Springer, New York. 32. Shoaee, S. and Khorra, E. (22. A new absolute continuous bivariate generalized exponential distribution, J. Statistical Planning and Inference, 42, Sklar, A. (959. Fonctions de répartition à n diensions et leurs arges. Publ. Inst. Statist. Univ. Paris 8,