Convolutions of multivariate phase-type distributions

Insurance: Mathematics and Economics 48 (2011) 374–377

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Convolutions of multivariate phase-type distributions Jasmin Berdel, Christian Hipp ∗ Karslruhe Insitute of Technology, Germany

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Article history: Received July 2009 Received in revised form January 2011 Accepted 15 January 2011 Keywords: Phasetype distributions Dependence models Convolution

abstract This paper is concerned with multivariate phase-type distributions introduced by Assaf et al. (1984). We show that the sum of two independent bivariate vectors each with a bivariate phase-type distribution is again bivariate phase-type and that this is no longer true for higher dimensions. Further, we show that the distribution of the sum over different components of a vector with multivariate phase-type distribution is not necessarily multivariate phase-type either, if the dimension of the components is two or larger. © 2011 Elsevier B.V. All rights reserved.

1. Introduction and summary Assaf et al. (1984) developed a multivariate extension of univariate phase-type distributions introduced by Neuts (1975). They show that their class of multivariate phase-type (MPH) distributions fulfils most of the properties of univariate phase-type distributions. In particular, the class of MPH distributions is closed under conjunctions, finite mixtures, and under the formation of coherent systems. The purpose of this paper is to answer the question whether the class of Assaf’s multivariate phase-type distributions is closed under finite convolutions. We show that the class of bivariate phase-type distributions has this property. We construct two phase-type distributions of dimension three and show that their convolution is not phase-type by applying a result of O’Cinneide (1990) on the moment generating function of an MPH distribution. Thus, classes of MPH distributions of dimension three or higher are not closed under finite convolutions. It was shown by Kulkarni (1989) and Cai and Li (2005) that the distribution of the sum over all univariate components of a vector with multivariate phase-type distribution is again of phasetype. We show that this is no longer true if the dimension of the components is two or larger. Let M = (M (t ))t ≥0 be an absorbing continuous-time Markov process with finite state space S = {1, . . . , n, 0}, eventually absorbing state 0, initial distribution (α, 1 − α 1), and intensity matrix

Λ=

∗



A 0



−A1 0

,

Corresponding author. E-mail address: [email protected] (C. Hipp).

0167-6687/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2011.01.004

with regular nontrivial intensity n × n matrix A. Let S1 , . . . , Sm be nonempty stochastically closed subsets of S, i.e. M (t ) ∈ Si implies M (τ )∈ Si for all τ > t. Without loss of generality we may assume that i=1,...,m Si = {0}. Let Xi , i = 1, . . . , m, be the hitting times of these closed sets: Xi := inf{t ≥ 0 : M (t ) ∈ Si },

i = 1, . . . , m.

The distribution of X := (X1 , . . . , Xm )T is called multivariate phasetype (MPH) with representation (α, A, S , S1 , . . . , Sm ). We write X ∼ (α, A, S , S1 , . . . , Sm ) to indicate that (α, A, S , S1 , . . . , Sm ) is (one) representation of X or its distribution, respectively. Assaf et al. (1984) assume that α 1 = 1, i.e. P{X = 0} = 0. For the results in this paper, this assumption is not needed. 2. Convolutions of bivariate phase-type distributions In this section we prove the following Theorem 2.1. The class of bivariate phase-type distributions is closed under finite convolutions. Proof. Let X and Y be two stochastically independent bivariate random vectors with X ∼ (α, A, K , K1 , K2 ) and Y ∼ (β, B, L, L1 , L2 ) satisfying K1 ∩ K2 = {0} and L1 ∩ L2 = {0}. We construct (π , Q , S , S1 , S2 ) as a bivariate phase-type representation for the random vector V = (V1 , V2 )T := X + Y = (X1 + Y1 , X2 + Y2 )T with the following components: S := K × L,

S1 := K1 × L1 ,

S2 := K2 × L2

(1)

with (0, 0) as an eventually absorbing state, initial distribution π given by

π(k,l) := αk βl ,

(k, l) ∈ S

(2)

J. Berdel, C. Hipp / Insurance: Mathematics and Economics 48 (2011) 374–377

and transition intensity matrix Q with entries defined for (k′ , l′ ) ∈ S via

q(k,l),(k′ ,l′ )

1{l = l′ }a ′ , (k, l) ∈ (K ∪ K )c × L, 1 2 kk  1{k = k′ }b ′ , (k, l) ∈ (K ∪ K ) × (L ∪ L )c ,  1 2 1 2 ll   1{l = l′ }akk′ , := (3)   (k, l) ∈ (K1 \ {0} × L1 ) ∪ (K2 \ {0} × L2 ),   λ(kl),(k′ l′ ) , (k, l) ∈ (K1 × L2 \ {0}) ∪ (K2 × L1 \ {0}).

Here λ(kl),(k′ l′ ) are the intensities of the product Markov process (MX , MY ), where MX and MY are the underlying independent Markov processes for the representations of X and Y , respectively, which are given by

λ(kl),(k′ l′ ) = 1{l = l′ }akk′ + 1{k = k′ }bll′ ,

(k, l), (k′ , l′ ) ∈ S .

(4)

Apparently, Q is a nontrivial intensity matrix. We show that the corresponding Markov process M (t ) = (M1 (t ), M2 (t )) is eventually absorbed in state (0, 0), and that the vector W = (W1 , W2 ) of the first hitting times of M on K1 × L1 and K2 × L2 , respectively, is equal — in distribution — to V for which we use the notation W =d V . To illustrate the construction, we consider the behavior of M (t ) piecewise on the different sets in (3) by applying the strong Markov property for each of the corresponding hitting times. We start the process at (k, l) with k ̸∈ K1 ∪ K2

l ̸∈ L1 ∪ L2 .

and

(5)

Then M (t ) =d (MXk (t ), l), up to t = τ1 = inf{s : M1 (s) ∈ K1 ∪ K2 }, where MXk (t ) is an independent copy of the process MX (t ) which is started in k. Let k1 = M1 (τ1 ). Then, given M (s), s ≤ τ1 , M (τ1 + t ) = ( ,

l d k1 MY

(t )),

t ≤ τ2 ,

where τ2 = inf{s : M2 (s) ∈ L1 ∪ L2 } − τ1 . Let l1 = M2 (τ1 + τ2 ) and assume that 0 ̸= k1 ∈ K1

0 ̸= l1 ∈ L1 .

and

(6)

Then, given M (s), s ≤ τ1 + τ2 , k

M (τ1 + τ2 + t ) =d (MX1 (t ), l1 ),

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3. Counterexamples We shall use the characterization of moment generating functions of an MPH distribution of O’Cinneide (1990). For this we need several notations. Let p be a polynomial on Cm . At any point t ∈ Cm , p has a representation p(s) =

−

···

nm ∈N

−−

a(t , n1 , n2 , . . . , nm )

n2 ∈N n1 ∈N

m ∏ (si − ti )ni , i=1

where the coefficients a(t , n1 , n2 , . . . , nm ) are themselves polynomials by ω(t ) := ∑min t. The order of the zero of p in t is given m min i=1 ni : a(t , n1 , n2 , . . . , nm ) ̸= 0 for t ∈ C . Now consider an affine subspace A = A [J , λ] of Cm defined by

 A [J , λ] :=



s = (s1 , . . . , sm ) ∈ C : T

m

−

si = λ

(8)

i∈J

for a set of indices J ⊂ {1, . . . , m} and a constant λ ∈ C. Then ω is constant on A almost everywhere (with respect to the Lebesguemeasure on A). The order of p on A is defined as this common value and will be denoted by ordp (A). Let q be an m-dimensional rational function on Cm with representation q(s) =

p1 (s) p2 (s)

,

s ∈ Cm

for two m-dimensional polynomials p1 and p2 on Cm . The order of the pole of q on A is defined by ordq (A) := ordp2 (A) − ordp1 (A). If ordq (A) > 0 holds, q has a pole of order ordq (A) in almost every t ∈ A. We now can formulate the following theorem (O’Cinneide, 1990, Theorem 3): Theorem 3.1. Let Q be the moment generating function of an m-dimensional MPH distribution. Let I1 , I2 ⊂ {1, . . . , m} be two nonempty sets of indices with I1 ̸⊂ I2 , I2 ̸⊂ I1 and I1 ∩ I2 ̸= ∅. Then for λ1 , λ2 ∈ C ordQ (A [I1 , λ1 ] ∩ A [I2 , λ2 ])

t ≤ τ3 ,

where τ3 = inf{s : M1 (s) = 0} − τ1 − τ2 . Finally, given M (s), s ≤ τ1 + τ2 + τ3 , l

M (τ1 + τ2 + τ3 + t ) = (0, MY1 (t ))

  ≤ max ordQ (A [I1 , λ1 ]) , ordQ (A [I2 , λ2 ]) ,   with A Ij , λj for j = 1, 2 defined as in (8). In particular, an MPH distribution of dimension n cannot have a p(s) moment generating function of the form Q (s) = q(s) for s =

until absorption in (0, 0) which happens at τ1 + τ2 + τ3 + τ4 , say, with probability 1. We see that

(s1 , . . . , sn )T with    q(s) = λ1 − (si + sj ) λ2 − (sj + sk ) r (s)

(τ1 + τ2 , τ1 + τ2 + τ3 + τ4 ) =d (X1 + Y1 , X2 + Y2 )

for pairwise different indices i, j, k ∈ {1, . . . , n} and polynomials p and r with p(s)r (s) ̸= 0 almost everywhere on {s ∈ Rn : si + sj = λ1 , sj + sk = λ2 }. We now construct two phase-type distributions of dimension three whose convolution is not MPH. Let X = (X1 , X2 , X3 )T ∼ (π , A, K , K1 , K2 , K3 ) with K = {1, 2, 3, 0}, K1 = {2, 3, 0}, K2 = {3, 0}, K3 = {0}, π = (1, 0, 0) and

or W =d V . The case 0 ̸= k1 ∈ K2 and 0 ̸= l1 ∈ L2 is symmetric to case (6). Now consider the case 0 ̸= k1 ∈ K1

0 ̸= l1 ∈ L2 .

and

(7)

Then, given M (s), s ≤ τ1 + τ2 , k

l

M (τ1 + τ2 + t ) =d (MX1 (t ), MY1 (t )), l



t ≥ 0.

0 0

A= k

Let τ3 = inf{s : MY1 (s) = 0} and τ4 = inf{s : MX1 (s) = 0}. Then W1 = τ1 + τ2 + τ4 , W2 = τ1 + τ3 + τ2 , and

(W1 , W2 ) =d (X1 + Y1 , X2 + Y2 ) = V . The case 0 ̸= k1 ∈ K1 and 0 ̸= l1 ∈ L2 is symmetric to the case (7). If k1 = 0 or l1 = 0, then the argument is simpler and shorter. The same is true for the cases k ∈ K1 ∪ K2 or/and l ∈ L1 ∪ L2 . 

−α

α −β 0

0



β −γ

.

Let Y = (Y1 , Y2 , Y3 )T ∼ (θ , B, L, L1 , L2 , L3 ) be stochastically independent of X with L = {1, 2, 3, 0}, L1 = {3, 0}, L2 = {2, 3, 0}, L3 = {0}, θ = (1, 0, 0) and for some b ̸= β

 B=

−a 0 0

a −b 0

0 b −c

 .

376

J. Berdel, C. Hipp / Insurance: Mathematics and Economics 48 (2011) 374–377

Let MX and MY be the underlying independent Markov processes for the representations of X and Y , respectively. Let σi for i = 1, 2, 3 be the waiting times of MX

σ1 := inf{t : MX (t ) ∈ K1 ∪ K2 ∪ K3 }, σ2 := inf{t : MX (t ) ∈ (K1 ∩ K2 ) ∪ (K1 ∩ K3 ) ∪ (K2 ∩ K3 )} − σ1 , σ3 := inf{t : MX (t ) = 0} − σ1 − σ2 ,

Proof. For simplicity, we restrict the proof to the bivariate case. The general result can be obtained by induction over m. τ1 and τ2 are the waiting times of M as before. Hence, τ1 and τ1 +τ2 = inf{t : M (t ) ∈ S1 ∪ S2 } are finite stopping times of M. By using the strong Markov property we get, that for i1 ∈ S1 ∪ S2 , given M (τ1 ) = i1 , the waiting times τ1 and τ2 are conditionally independent. Thus, we get

and τi for i = 1, 2, 3 the corresponding waiting times of MY . As the embedded Markov chains both equal the sequence (1, 2, 3, 0, 0, . . .) with probability 1, it follows that σ1 , σ2 , σ3 , τ1 , τ2 , τ3 are stochastically independent and exponentially distributed with parameter α, β, γ , a, b, c, respectively. X and Y can be represented by these waiting times

MGFτ (s) = E es1 τ1 +s2 τ2

 σ1 σ1 + σ2 , = X = σ1 + σ2 + σ3     τ1 + τ2 Y1 τ1 . Y = Y2 = τ1 + τ2 + τ3 Y3

  × E es2 τ2 |M (τ1 ) = i1    for s ∈ D := s ∈ R2 : MGFτ (s) < ∞ . It holds D = {J 1 (i1 ) × J 2 (i1 ) : i1 ∈ H } where H := {i1 ∈ S1 ∪ S2 : P (M (τ1 ) = i1 ) > 0} and J i (i1 ) := {si ∈ R : E [esi τi |M (τ1 ) = i1 ] < ∞}. Moreover E [esi τi |M (τ1 ) = i1 ] is a rational function and J i (i1 ) = (−∞, ci (i1 )) with ci (i1 ) ∈ R¯ for every i1 ∈ H and i = 1, 2, see

X1 X2 X3

 



We now consider the convolution of X and Y T

The moment generating function of V is, with s = (s1 , s2 , s3 )T Q (s) := E [exp (s1 V1 + s2 V2 + s3 V3 )]

    = E e(s1 +s2 +s3 )σ1 E e(s1 +s2 +s3 )τ1  (s +s )σ   (s +s )τ   s σ   s τ  ×E e 2 3 2 E e 1 3 2 E e 3 3 E e 3 3 α a β = α − (s1 + s2 + s3 ) a − (s1 + s2 + s3 ) β − (s2 + s3 ) γ b c × b − ( s1 + s3 ) γ − s3 c − s3

  = 2 > max ordQ (A [I1 , λ1 ]) , ordQ (A [I2 , λ2 ]) = 1. As this is inconsistent with the conclusion of Theorem 3.1, V is not MPH. Thus, we have the following Theorem 3.2. Convolutions of MPH distributions of dimension m ≥ 3 are not necessarily MPH. Theorem 3.1 can apparently not be applied to bivariate distributions. By similar computations as of O’Cinneide (1990), we derive another result on the moment generating function of the spacings of a multivariate phase-type distribution which we shall use later on. Theorem 3.3. Let a random vector V = (V1 , . . . , Vm ) have an mdimensional phase-type distribution and

τ := (τ1 , . . . , τm )T := spacings(V ) := (V(1) , V(2) − V(1) , . . . , V(m) − V(m−1) )T . Then the moment generating function MGFτ of τ has a representation K − k=1

ak

m ∏

(l)

qk (sl ),

=

for s = (s1 , . . . , sm ) ∈ Rm

(9)

P (M (τ1 ) = i1 )E es1 τ1 +s2 τ2 |M (τ1 ) = i1





−

P (M (τ1 ) = i1 )E es1 τ1 |M (τ1 ) = i1





i1 ∈S1 ∪S2



We now construct a phase-type distribution of dimension four, such that the sum of two bivariate marginal distributions is not MPH. Let X = (X1 , X2 , X3 , X4 )T ∼ (α, A, K , K1 , K2 , K3 , K4 ) with K = {1, 2, 3, 4, 0}, K1 = {2, 3, 4, 0}, K2 = {3, 4, 0}, K3 = {4, 0}, K4 = {0}, α = (1, 0, 0, 0) and

−α1

 0 A= 0 0

ordQ (A [I1 , λ1 ] ∩ A [I2 , λ2 ])



i1 ∈S1 ∪S2



for s1 + s2 + s3 < min{α, a}, s2 + s3 < β, s1 + s3 < b, s3 < min{γ , c }. Let I1 = {2, 3}, I2 = {1, 3}, λ1 = β and λ2 = b. Clearly, I1 ̸⊂ I2 , I2 ̸⊂ I1 and I1 ∩ I2 ̸= ∅ hold. We get the following relation

MGFτ (s) =

=

Assaf et al. (1984).

V = (V1 , V2 , V3 ) := (X1 + Y1 , X2 + Y2 , X3 + Y3 ) . T



−

α1 −α2 0 0

0

α2 −α3 0



0 0 

α3  −α4

.

Let σi for i = 1, 2, 3, 4 be the waiting times of the underlying Markov process MX . As the embedded Markov process equals the sequence (1, 2, 3, 4, 0, 0, . . .) with probability 1, it follows that σi are stochastically independent with σi ∼ Exp(αi ). We now consider the sum of the two random vectors (X1 , X3 )T and (X2 , X4 )T , i.e. V := (X1 + X2 , X3 + X4 )T . The moment generating function of τ := (τ1 , τ2 )T := spacings(V ) is, with s = (s1 , s2 )T MGFτ (s) := E [exp (s1 τ1 + s2 τ2 )]

        = E e(2s1 +2s2 )σ1 E e(s1 +2s2 )σ2 E e2s2 σ3 E es2 σ4 α1 α2 α3 α4 = α1 − (2s1 + 2s2 ) α2 − (s1 + 2s2 ) α − 2s2 α4 − s2

for s ∈ D := {(s1 , s2 )T ∈ R2 : 2s1 + 2s2 < α1 , s1 + 2s2 < α2 , s1 + s2 < α3 , s2 < α4 } and MGFτ (s) = ∞ on R2 \ D . As this is a contradiction to the conclusion of Theorem 3.3, we have the following Theorem 3.4. The distribution of the sum over different components of a vector with MPH distribution is not necessarily MPH, if the dimension of the components is two or larger. As marginal distributions of MPH distributions are again MPH, see Assaf et al. (1984), Theorem 3.4 indicates that the distribution of the sum of dependent random vectors each with MPH distribution is not necessarily MPH. 4. Related classes of multivariate distributions

i=1

(l)

with K ∈ N, constants ak ∈ R and rational functions qk for l = 1, . . . , m and k = 1, . . . , K . Thus, MGFτ is finite on an m-dimensional ¯ m. interval (−∞, c ) with c ∈ R

There exist other extensions of phase-type distributions to the multivariate case that are closed under finite convolutions. Kulkarni (1989) developed a class of multivariate phase∗ -type distributions (MPH ∗ ) based on the idea of the total accumulated

J. Berdel, C. Hipp / Insurance: Mathematics and Economics 48 (2011) 374–377

reward. He also considers an absorbing continuous-time Markov process. Additionally, each state has m different reward rates. A reward for a state is defined as the product of a reward rate with the occupation time of the Markov process in this state. Kulkarni calls the distribution of the vector of the m different total accumulated rewards multivariate phase∗ -type. Bladt and Nielsen (2010a) introduced a new class of multivariate phase-type distributions, MVPH, and a new class of multivariate matrix exponential distributions, MVME. The distribution of an m-dimensional random vector X is MVPH, if aT X has a (univariate) phase-type distribution for every non-negative vector a ∈ Rm . The distribution of an m-dimensional random vector X is MVME, if aT X has a matrix-exponential distribution for every non-negative vector a ∈ Rm , i.e. has a rational Laplace Transform (see Asmussen and O’Cinneide, 2000). In abbreviated notation the following inclusions hold: MPH ( MPH ∗ ⊂ MVPH ⊂ MVME. The list of subclasses, extensions and other variations does not stop here. Bladt and Nielsen constructed bivariate exponential distributions as a subclass of MPH ∗ with arbitrary correlation coefficient (Bladt and Nielsen, 2010b) and another class of multivariate matrix exponential distributions MME ∗ ⊂ MVME (Bladt and Nielsen, 2010a). Assaf et al. (1984) constructed for classes C of real functions from [0, ∞)k to [0, ∞) for some k ∈ N the following distributions. A random vector X respectively its distribution is said to be C -generated if there exist k independent random variables Yj ∈ PH and functions gi ∈ C satisfying Xi = gi (Y ). A random vector X respectively its distribution is said to be in the C -closure if g (X ) ∈ PH for all g ∈ C . Explicitly they considered the classes C1 , the minimum of a subset of the components, C2 , the sums of subsets of the components, and C3 , the coherent life functions. The corresponding Ci -closure is denoted with Ci and the class of Ci -generated distributions with Gi , for i = 1, 2, 3. They showed G3 ⊂ G1 ⊂ MPH ⊂ C1 ⊂ C3 , but MPH ∗ ̸⊆ C3 (O’Cinneide, 1990). Considering the moment generating function of the spacings of (X1 , X2 , X1 + X2 ), for X1 , X2

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stochastically independent and exponentially distributed, we get G2 ̸⊆ MPH, but G2 ⊂ MPH ∗ ⊂ MVME ⊂ C2 and C1 ̸⊆ C2 (Kulkarni, 1989; Bladt and Nielsen, 2010a). 5. Concluding remarks Within this paper, we showed that the class of bivariate phasetype distributions is closed under finite convolutions, but that classes of MPH distributions of higher dimension do not possess this property. Moreover, we showed that the distribution of the sum of multidimensional components of a vector with MPH distribution is not necessarily MPH. In the meantime there exists a variety of extensions of phasetype and matrix exponential distributions to the multivariate case, each with their specific properties of which we only stated some and refer to the literature for the remainder. More research on properties as well as constructions of related new classes may follow. References Asmussen, S., O’Cinneide, C.A., 2000. Matrix-exponential distributions. In: Encyclopedia of Statistical Sciences. John Wiley & Sons, Inc., pp. 435–440. Assaf, D., Langberg, N.A., Savits, T.H., Shaked, M., 1984. Multivariate phase-type distributions. Operations Research 32, 688–702. Bladt, M., Nielsen, B.F., 2010a. Multivariate matrix-exponential distributions. Stochastic Models 26 (1), 1–26. Bladt, M., Nielsen, B.F., 2010b. On the construction of bivariate exponential distributions with an arbitrary correlation coefficient. Stochastic Models 26 (2), 295–308. Cai, J., Li, H., 2005. Multivariate risk model of phase type. Insurance: Mathematics and Economics 36, 137–152. Kulkarni, V.G., 1989. A new class of multivariate phase type distributions. Operations Research 37 (1), 151–158. Neuts, M.F., Probability distributions of phase type. Liber Amicorum professor Emeritus H. Florin. Belgium. Dept. of Mathematics. University of Louvain. Belgium, 1975, pp. 173–206. O’Cinneide, C.A., 1990. On the limitations of multivariate phase-type families. Operations Research 38 (3), 519–526.