On the construction of bivariate exponential distributions with an arbitrary correlation coefficient

Transcription

1 Downoaded from orbit.dtu.d on: Sep 23, 208 On the construction of bivariate exponentia distributions with an arbitrary correation coefficient Badt, Mogens; Niesen, Bo Friis Pubication date: 2008 Document Version Pubisher's PDF, aso nown as Version of record Lin bac to DTU Orbit Citation (APA: Badt, M., & Niesen, B. F. (2008. On the construction of bivariate exponentia distributions with an arbitrary correation coefficient. Kgs. Lyngby: Technica University of Denmar, DTU Informatics, Buiding 32. (D T U Compute. Technica Report; No Genera rights Copyright and mora rights for the pubications made accessibe in the pubic porta are retained by the authors and/or other copyright owners and it is a condition of accessing pubications that users recognise and abide by the ega requirements associated with these rights. Users may downoad and print one copy of any pubication from the pubic porta for the purpose of private study or research. You may not further distribute the materia or use it for any profit-maing activity or commercia gain You may freey distribute the URL identifying the pubication in the pubic porta If you beieve that this document breaches copyright pease contact us providing detais, and we wi remove access to the wor immediatey and investigate your caim.

2 On the construction of bivariate exponentia distributions with an arbitrary correation coefficient Mogens Badt and Bo Friis Niesen October, 2008 Abstract In this paper we use a concept of mutivariate phase type distributions to define a cass of bivariate exponentia distributions. This cass has the foowing three appeaing properties. Firsty, we may construct a pair of exponentiay distributed random variabes with any feasibe correation coefficient (aso negative. Secondy, the cass satisfies that any inear combination (projection of the margina random variabes is a phase type distributions, The atter property is potentiay important for the deveopment hypothesis testing in inear modes. Thirdy, it is very easy to simuate the exponentia random vectors. Keywords: Matrix exponentia; phase type distribution; bivariate exponentia distribution; arbitrary correation AMS subject cassification: 62H05; 60E0. Introduction This paper deas with the construction of bivariate exponentia distributions being arbitrariy correated. We use a construction based on a cass of mu- Institute for Appied Mathematics and Systems, Nationa University of Mexico, Mexico City, badt@stats.iimas.unam.mx Informatics and Mathematica Modeing, Technica University of Denmar, Kgs. Lyngby, bfn@imm.dtu.d

3 tivariate Phase type distributions. We represent an exponentia distribution as a higher order phase type distribution and then in (correate two such phase type distributions to obtain a mutivariate phase type distribution. In this way we are abe to construct a cass of bivariate exponentia distributions with any feasibe correation coefficient and which furthermore has the property that any inear combination of the correated exponentia distributions is a phase type distribution. The atter cosure property is appeaing both from an appications point of view as we as for statistica considerations where hypothesis concerning inear sub spaces are we defined within the cass of phase type distributions. Aso, the simuation of bivariate exponentia distributions becomes an amost trivia matter. A number of bivariate distributions with exponentia marginas have been discussed in the iterature. The cassica distributions suggested by Kibbe (94 (sometimes referred to as the Moran Downton distribution and Marsha & Oin (967 are restricted to the case of non negative correation. Other distributions ie the Farie Gumbe Morgenstern distributions see e.g. Kotz, Baarishnan & Johnson (2000 p.353 and Raftery s distribution see e.g. Kotz, Baarishnan & Johnson (2000 p.377 can have both positive and negative correation but ony in a imited range. These distributions are a characterized by having a joint Lapace transform, which is rationa. Sti other distributions of more compicated structure have been suggested, in particuar Gumbe s type I and type III bivariate exponentias ( Kotz, Baarishnan & Johnson (2000 pp but again with some restrictions on the form of the correation. For a somewhat contrived exception, see the discussion foowing the description of Raftery s discussion pp in Kotz, Baarishnan & Johnson (2000. In section 2 we provide the necessary bacground on univariate Phase type distributions. In section 3 we introduce a cass of mutivariate Phase type distributions with particuar address to cross moments and correation. In section 4 we finay construct the bivariate exponentia distributions and show that the cass is sufficienty rich to contain arbitrariy (feasibe correated exponentia variabes. To our nowedge this is the first construction that maes it possibe to obtain negativey correated exponentia variabes in a way that can be easiy used for further modeing and anaysis. The construction cannot immediatey be generaized to higher dimensions. In section 5 we present our concusions. 2

4 2 Phase type distributions Let {X t } t 0 denote a Marov process on a finite state space E {, 2,..., d, d+ }. We assume that the states, 2,..., d are transient whie the state d + is absorbing. Then {X t } t 0 has an intensity matrix (infinitesima generator Λ on the form Λ ( T t 0 0 where T is a d d matrix of intensities corresponding to the transient states and t is a d dimensiona coumn vector. The matrix T is often referred to as the sub intensity matrix and t as the exit (rate vector. We assume that the Marov process has an initia distribution (π, 0 with π (π,..., π d, represented as a row vector, concentrated on {, 2,..., d}, i.e. π i IP(X 0 i and d i π i. Since the rows of an intensity matrix must sum to 0 we have that t T e, where e is the d dimensiona coumn vector of s. Let τ inf{t 0 X t d + } denote the time of absorption in state d +. Then we say that τ has a phase type distribution with representation (π, T and write τ PH(π, T, Neuts (975. The density of τ is given by f τ (x π exp(t xt, where exp( denotes the matrix exponentia. The distribution function is F τ (x π exp(t xe whie its n th moment is given by n!( T n e. For an overview on phase type distributions, their properties and use in ris theory we refer to e.g. Badt (994. Phase type distributions are mathematicay tractabe in the sense that probabiistic reasoning using the underying Marov process structure provides a powerfu framewor for estabishing exact, and in many cases even expicit, soutions to compicated modes. Any distribution on R + may be approximated arbitrariy cose by a phase type distribution by increasing the number of phases (i.e. the number of transient states in the underying Marov process. Phase type distributions are, however, not adequate for modeing heavy taied phenomena, since their tais are decreasing exponentiay fast (ight taied. 3 Correated phase type distributions The construction of correated phase type random variabes is based on a cass of mutivariate phase type distributions originay introduced by Kua-, 3

5 rni (989. Consider a Marov jump process {X t } t 0. Let r i : E R +, i,..., m be m different mappings and define new random variabes Y,..., Y m by Y i τ 0 r i (X t dt. Then Y i is constructed by adding up rewards earned in the different states up to absorption. The Y i s are phase type distributed which can be seen easiy by adjusting the intensities adequatey, see Kuarni (989 for detais. Now define the d m matrix K { ij } {r j (i} and et Y (Y,..., Y m. Then (see Badt & Niesen (2008 a, Y PH ( π, (Ka T, for a non zero a R n +, where (v is the diagona matrix with vector v as diagona and, denotes the usua inner product in R n. Thus the Lapace transform of a, Y is given by L a,y (s π ( si (Ka T (Ka ( T e π ( s (KaT I t, where I denotes the identity matrix of appropriate dimension. Since L a,y (s IE (exp( s a, Y, it is readiy seen that L a,y ( is the joint Lapace transform of Y at a (a,..., a m. Theorem 3. The mutidimensiona Lapace transform of Y is given by where s (s,..., s m. L Y (s,..., s m π ( (KsT I t, There are no expicit formuae, in genera, for the joint density or distribution function constructed in this way. Transform inversion is a difficut matter in higher dimensions and often not feasibe due to the appearance of essentia singuarities. Kuarni (989 provides a neat method for cacuating the conditiona surviva function given X 0 i through a system of inear partia differentia equations, which is a matter of routine to sove for numericay. 4

6 Exampe 3. Let X, X 2, X 3 exp( be exponentiay distributed with intensity. Define Y X + X 3 and Y 2 X 2 + X 3. Then Y (Y, Y 2 has a mutivariate phase type distribution with π ( 0 0, 0 0 T 0 and K The variabes Y and Y 2 are obviousy positivey correated. In a more genera context there has been numerous definitions of mutivariate exponentia and Gamma distributions. The exampe above is an exampe of two random variabes which are marginay Erang distributed. Their joint distribution is a mutivariate phase type distribution in the sense of Kuarni (989. Most bivariate gamma constructions, as presented by e.g. Kotz, Baarishnan & Johnson (2000 have positive correation or ony imited negative correation. An exception is Schmeiser and La s Bivariate Gamma distribution, but this construction depends on using the inverse cumuative distribution function, which is not easy to treat neither numericay nor anayticay. We now turn to the construction of two phase type distributed random variabes which may be either positivey or negativey correated. Assume that the mutivariate phase type distribution we aim to construct have margina distributions PH(α, S and PH(β, T. Then consider the sub intensity matrix U and reward matrix K given by U ( S D 0 T and K ( e 0 0 e. ( Here the dimension of e equas the corresponding dimensions of S and T respectivey. Thus Y is the time the Marov jump process underying the phase type distribution with sub intensity matrix U spends in states corresponding to S and Y 2 is the time the process spends in the states corresponding to the states in T. The matrix D defines the rates of the transition from the S states to the T states. The joint distribution of Y (Y, Y 2 is readiy seen to be given by the density f Y (y, y 2 αe Sy De T y 2 t. 5

7 The corresponding joint Lapace transform is thus given by L Y (s, s 2 α (s S I D (s 2 I T t. (2 The margina distributions can be obtained by this formua as f Y2 (y 2 0 πe Sy De T y 2 tdy α( S De T y 2 t. A sufficient condition for Y 2 PH(β, T is hence that β α( S D. Integrating out the other variabe resuts in the condition De s, which is equivaent to u Ue being zero on the entries corresponding to the S indices. Theorem 3.2 The joint moments are given by IE (Y n Y n 2 2 n!n 2!α( S n D( T n 2 t. (3 In particuar, the covariance between Y and Y 2 is Cov(Y, Y 2 α( S (( S D eβ( T e. Proof. Foows immediatey differentiating (2. In deriving the formua for the covariance we assume that β α( S D. Q.E.D. Exampe 3.2 Let α β ( and S T and et U and K be defined as in (. Then the conditions β αs D and De s impies that the ony possibe form of D is the matrix D

8 This corresponds to the margina distributions being independent. Thus the ony possibe mutivariate phase type distribution with margina 4 dimensiona Erang distribution which can be represented by an eight dimensiona U matrix is the one which consists of two independent 4 dimensiona Erang distributions. In order to incorporate a non trivia correation structure it wi be necessary to represent the same distribution in a matrix U of higher dimension. 4 Construction of bivariate exponentia distributions The exampe from the previous section aso appies to the case of a bivariate exponentia distribution, i.e. the ony bivariate exponentia distribution which can be obtained using a two dimensiona U matrix is the trivia one consisting of two independent exponentiay distributed random variabes. The foowing emmas sha provide a method for introducing correation between the variabes in the construction using the U matrix. The foowing resuts are we nown and easiy proven. Lemma 4. If X PH(π, T and T e λe for some constant λ > 0, then X is exponentiay distributed with intensity λ. Lemma 4.2 Let π (π, π 2,..., π n be any initia vector and λ λ λ λ 2 λ 2 λ... 0 T , λ where it is assumed that λ < λ i for a i. Then PH(π, T is equivaent to an exponentia distribution with rate λ. We sha denote this specia form of the matrix T by E(λ, where λ (λ,..., λ n, λ. Another method for constructing an exponentia distribution from a higher order phase type distribution is by time reversing the construction above. 7

9 We refer to Andersen, Neuts & Niesen (2004 and Ramaswami (990 for detais on time reversing genera phase type distributions. However, for the case where π ( the time reversed is particuary simpe Lemma 4.3 Let α (α,..., α n with α i λ n i λ n i+ j λ j λ λ j and S λ λ λ n λ n λ, (4 where λ < λ i for a i. Then PH(α, S represents an exponentia distribution with rate λ. We sha refer to the matrix (4 as Er(λ, where λ (λ, λ n,..., λ. We appy these Lemmas to the construction of bivariate exponentia distributions with an arbitrary correation coefficient. Let Y and Y 2 be exponentia distributions with intensities λ and µ respectivey. For a given n N we may write Y PH(π, E(λ and Y 2 PH(α, Er(µ, (5 where λ (nλ, (n λ,..., λ, µ (µ, 2µ,..., nµ, π (, 0,..., 0, and α ( n, n,..., n. Now we construct the joint density by appropriatey specifying the matrix D λp, and etting P {p ij }. Then the Lapace transform is given by L Y (s, s 2 n n i i j j p ij p ij i n n + s + n + n i+ s + n j n j s 2 + s 2 +. We sha mae use of the foowing partia fraction expansion. 8

10 Lemma 4.4 n j s + j n! ( j (j!(n!( j! s +. Proof. The formua hods for n j. Assume that it is vaid for n. Then n+ j n s + s + j ( j j j j n + s + n + n! (j!(n!( j! ( j s + n! ( j (j!(n!( j! s + (n +!( j (j!(n!( j!(n + (n +!( j (j!(n +!( j! s + (n +! + s + n + (j! j n + s + n + n + s + n + ( + j (n +!( j!. [ s + s + n + Using induction the atter sum is then seen to be equa to ( n+ j /(n + j!. Q.E.D. Theorem 4. The joint density for Y (Y, Y 2 is given by ] f(y, y 2 c λe λy µe µy 2, with c ( + (n+ n ( n ( n in+ j p ij ( i j ( n i ( j (6. 9

11 Proof. n i j p ij n i+ n n n j n! (n i!(n!( (n + i! n! ( j (j!(n!( j! t + in+ j ( (n+ i s + n!n! ( (n+ ( (n!(n! s + t + in+ j in+ j p ij ( n ( i j (n i!( (n + i!(j!( j! ( n ( (n+ ( s + t + ( i j (!(! p ij (n i!( (n + i!(j!( j! ( ( n n ( (n+ ( s + t + from which the resut foows. Coroary 4. If p ij δ i j, p ij ( i j ( n i c {n + + } ( + (n+ n If p ij δ i+j (n+, c ( + n ( n ( n Proof. The inner sum of (6 reduces to in+ ( n i ( i ( n ( n ( + 2 min (, and ( n+ 0 j ( j ( + 2 n ( j. ( j Q.E.D..,

12 which again can be rewritten into + n h0 ( h ( h + n min (, and ( n+ h0 ( h ( h, respectivey, from which the resut foows. Appying (3 to the current setting yieds ( IE(Y Y 2 i λµ n i j ( n + j p ij. Q.E.D. The maxima negative correation between two positive random variabes is obtained by an antithetic construction. Thus in the case of the exponentia distribution V, V 2 exp(, we obtain maximum negative correation by etting V og(u and V 2 og( U, where U is uniformy distributed over [0, ]. The maximum negative correation coefficient between two exponentia distributions exp(λ and exp(µ respectivey is hence π 2 /6 as obtained noting that og(u og( udu 2 0 π2 /6, see e.g. Kotz, Baarishnan & Johnson (2000 p Theorem 4.2 Let p ij δ i j and et Y (n and Y (n 2 denote the exponentiay distributed random variabes (5 based on the dimension n N. Then ( corr Y (n, Y (n 2 i i π2 2 6 as n, i.e. the maximum negative correation. Proof. Without oss of generaity we may assume that λ µ. Since p ij δ i j we get that ( IE Y (n Y (n 2 n ( i i ( n + i.

13 By interchanging the order of summation we see that ( IE Y (n Y (n 2 n n n n ( + n + + n + n+ ( + + n + n + n + n 2 n+ n+ + n n n+ n+ {n n + } {n n + } n+. n+ n+ n+ We then consider the difference between the (n + th and n th order of the 2

14 ast term. This amounts to n+ n+ n+2 n+ n+ n+ n+2 n+ n+2 n+ n+ n + (n n + (n (n n + (n + 2. n+ n+ n+ n+2 + n + + n + n + 2(n + (n + (n + (n + ( n + Hence the resut foows noting that n n 2 n + n + + n + π2 6. Q.E.D. Pease notice that the joint distribution of ( og(u, og( U is not a mutivariate phase type distribution itsef. The maxima positive correation between two positive random variabes is of course. We now prove that the we can reach this maximum correation in the imit if we specify P to be the diagona matrix where the diagona is the running from ower eft to the upper right corner of P. Theorem 4.3 Let p ij δ i+j (n+ and et Y (n and Y (n 2 denote the exponentiay distributed random variabes (5 based on the dimension n N. Then ( corr Y (n, Y (n 2 n i.e. the maximum positive correation. 3 as n,

15 Proof. ( IE Y (n Y (n 2 n n n n n n ( i ( n + j j ( i ( n + n+ i ( 2 n+ i {min(, n + i} i i i i 2 n from which the resut foows. min(, + n, n δ i+j (n+ Q.E.D. Remar 4. If p ij n we get independent random variabes Y and Y 2. Finay we construct a two dimensiona exponentia distribution with arbitrary correation coefficient. Theorem 4.4 Let ρ ( π2,. Then we can construct a two dimensiona 6 exponentia vector (Y, Y 2 with correation coefficient ρ in the foowing way. If ρ > 0 we choose n N such that ρ max corr(y (n, Y (n 2 > ρ and P ρ {δ i+j n } i,j + ρ ρ max ρ max n ee, and if ρ < 0 we choose n N such that ρ min corr(y (n, Y (n 2 < ρ and P ρ {δ i j } i,j + ρ ρ min ρ min n ee. Proof. Foows from additivity of the first cross moment. 4 Q.E.D.

16 5 Concusions We have constructed a cass of bivariate exponentia distributions which may match any feasibe correation. The construction is based on one singe underying Marov jump process which maes it a trivia matter to simuate the bivariate reaizations. Any inear combination of the margina exponentia variabes is Phase type distributed with a nown representation. We have outined three different choices of D corresponding to most negativey correated, independent and most positivey correated cases. There exists an ampe space for further specifications of D which may resut in aternative sub casses of interest. We have shown, however, that with the three somehow extreme cases we may construct variabes with any feasibe correation coefficient by appropriate mixing. Acnowedgments The authors are gratefu for the support by Otto Mønsteds foundation, Danish research counci for technoogy and production sciences grant no and Sistema Naciona de Investigadores, grant References Andersen, A. T.; Neuts, M. F. & Niesen, B. F. (2004. On the Time Reversa of Marovian Arriva Processes. Stochastic Modes, 20(3: Badt, M. (994. A review on phase type distributions and their use in ris theory. Astin Buetin, 35(:45 6. Badt, M. & Niesen, B. F. (2008. Mutivariate matrix exponentia distributions: a characterization. submitted. Kibbe, W. F. (94. A two variate gamma type distribution. Sanhy a, 5: Kotz, S.; Baarishnan, N. & Johnson, N. L. (2000. Continuous Mutivariate Distributions. John Wiey and Sons. 5

17 Kuarni, V. G. (989. A new cass of mutivariate phase type distributions. Operations Research, 37(:5 58. Marsha, A. W. & Oin, I. (967. A Mutivariate Exponentia Distribution. Journa of the American Statistica Association, 62(37: Neuts, M. F. (975. Probabiity distrubutions of phase type. In Liber Amicorum Professor Emeritus H. Forin, pages , Department of Mathematics, University of Louvian, Begium. Ramaswami, V. (990. A Duaity Theorem for the Matrix Paradigms in Queueing Theory. Commun. Statist. -Stochastic Modes, 6(:5 6. 6