Distributions with rational transforms Joint work with Mogens Bladt - PowerPoint PPT Presentation

Distributions with rational transforms Joint work with Mogens Bladt New Frontiers in Applied Probability honouring Søren Asmussen Sandbjerg Castle 2/8 2011

Outline Outline • Moment recursions • Kulkarni’s multivariate phase–type distributions (MPH ∗ ) • Multivariate definition and main theorem (MVME) • Some constructions • Classification of multivariate gamma distributions • Distributions on the reals: uni- and multivariate • The killing of a conjecture • Further work

Laplace transform from moments Laplace transform from moments • An m –dimensional ME distribution is uniquely determined from its first 2 m − 1 moments. • Solve for f i and g i in the first 2 m − 1 moment equations. • g i obtained from certain Hankel matrices. i µ i = M i � ( − 1) j f m − j ψ i − j µ i = i = 0 , 1 , ... , i ! j =0 g m and ψ i = � i − 1 j =0 ( − 1) j ψ i − 1 − j g m − 1 − j 1 where ψ 0 = g m • Moments of higher order are given recursively by m − 1 � g i ( − 1) m + i µ i + j for j ≥ 0 . µ m + j = i =0

Kulkarni’s multivariate phase–type Kulkarni’s multivariate phase–type distributions – MPH ∗ distributions – MPH ∗ • n different reward rates for each state of T given by R m N k � � X j = R ij Z ik . i =1 k =1 ⋄ Here N k is the number of visits to state k before absorption and Z ik are the k ’th sojourn in state j • Partial differential equations for joint survival function. • Joint Laplace-Stieltjes transform � − 1 e . ( − T ) − 1 ∆ ( R s ) + I � H ( s ) = γ • Includes previous work by Assaf defining the class MPH.

Joint transform and moments Joint transform and moments E ( � n i =1 Y r i Theorem 1 The cross–moments I i ) , where Y follows an MME ∗ distribution with representation ( γ , T, R ) , and where r i ∈ N , are given by r ! r � � ( − T ) − 1 ∆( r σ ℓ ( i ) ) e . γ i =1 ℓ =1 Here r = � n i =1 r i , r j is the j th column of R and σ ℓ is one of the r ! possible ordered permutations of the derivatives, with σ ℓ ( i ) being the value among 1 . . . n at the i ’th position of that permutation.

General definition of multivariate matrix General definition of multivariate matrix exponential distributions exponential distributions Definition 1 A non–negative random vector X = ( X 1 , ..., X n ) of dimension n is said to have multivariate matrix–exponential distribution (MVME) if the joint Laplace transform L ( s ) = E [exp( − < X , s > )] is a multi–dimensional rational function, that is, a fraction between two multi–dimensional polynomials. Here < · , · > denotes the inner product in R n with s = ( s 1 , . . . , s n ) ′ . Our main theorem characterizes the class of MVME. Theorem 2 A vector X = ( X 1 , . . . , X n ) follows a multivariate matrix–exponential distribution if and only if < X , a > = � n i =1 a i X i has a univariate matrix–exponential distribution for all non–negative vectors a � = 0 .

Outline of proof Outline of proof � e − < X , s > � • Only if part: Suppose E is rational in s . Then � e − s< X , a > � � e − < X ,s a > � consider E = E that is obviously rational in s . • If part: Suppose < X , a > has ME representation ( β ( a ) , D ( a ) , d ( a )) for all a > 0 . ⋄ The dimension of D is bounded by some integer m . ⋄ Using the moment relations we express the coefficients f i ( a ) and g i ( a ) of the Laplace transform in terms of certain determinants of the moments. ⋄ The j th moment is a sum of j th order monomials in the components of a . ⋄ We conclude that f i and g i are rational in a .

The transform is of a particular simple form The transform is of a particular simple form Lemma 1 If � X , a � is MVME distributed then we may write its Laplace transform for � X , a � as f 1 ( a ) s m − 1 + ˜ f 2 ( a ) s m − 2 + ... + ˜ ˜ f m − 1 ( a ) s + 1 g m − 1 ( a ) s + 1 , g 0 ( a ) s m + ˜ g 1 ( a ) s m − 1 + .... + ˜ ˜ where the terms ˜ f i ( a ) and ˜ g i ( a ) are sums of n –dimensional monomials in a of degree m − i and m is the common order except a set of measure zero.

Farlie Gumbel Morgenstern construction Farlie Gumbel Morgenstern construction Consider F ( x 1 , x 2 ) = F 1 ( x 1 ) F 2 ( x 2 ) (1 + ρ (1 − F 1 ( x 1 )) (1 − F 2 ( x 2 ))) , where F i are univariate cumulative distribution functions. This expression can be rewritten as 1 + ρ F 1 ,M ( x 1 ) F 2 ,M ( x 2 ) + 1 − ρ F ( x 1 , x 2 ) = F 1 ,M ( x 1 ) F 2 ,m ( x 2 ) 4 4 1 − ρ F 1 ,m ( x 1 ) F 2 ,M ( x 2 ) + 1 + ρ + F 1 ,m ( x 1 ) F 2 ,m ( x 2 ) , 4 4 where F i,m ( x ) = 1 − (1 − F i ( x )) 2 and F i,M ( x ) = F 2 i ( x ) i.e. the distribution of minimum and maximum respectively of two F i distributed independent random variables.

Theorem 3 The bivariate Farlie-Gumbel-Morgenstern distribution formed from two matrix-exponential distributions is in MME ∗ . An MME ∗ representation is ( γ 1 ⊗ γ 1 , 0 , 0 , 0 )   1 − ρ 1+ ρ 1 S 1 ⊕ S 1 2 ( s 1 ⊕ s 1 ) 4 ( s 1 ⊕ s 1 ) e ˜ 4 ( s 1 ⊕ s 1 ) e ˜ γ 2 ,M γ 2 ,m   1+ ρ 1 − ρ 2 s 1 ˜ 2 s 1 ˜ 0 S 1 γ 2 ,M γ 2 ,m       ∆ − 1 ∆ − 1 0 0 2 ,M S ′ 2 ∆ 2 ,M 2 ,M ( s 2 ⊕ s ′ 2 )∆ 2 ,m     ˜ 0 0 0 S 2 ,m with 2 α 2 ( − S 2 ) − 1 , π 2 = µ − 1 α 2 = µ − 1 ˜ 2 π 2 ◦ s 2 , � µ 2 ,m � π 2 ,m , 1 − µ 2 ,m 2 ,m ( α 2 ⊗ α 2 ) ( − S 2 ⊕ S 2 ) − 1 , π 2 ,M = π 2 ,m = µ − 1 π 2 , µ 2 ,M µ 2 ,M α 2 ,M = ( µ 2 ,M ) − 1 ( 0 , π 2 ,M ◦ s 2 ) α 2 ,m = ( µ 2 ,m ) − 1 π ( m ) ˜ ◦ ( s 2 ⊕ s 2 ) , ˜ 2

1 1 µ f ( x ) µ f ( x ) • Suppose f ( x ) is (univariate) ME • Then f ( x ) is (proportional to) an MME ∗ density • For n = 2 we get       � α ( − C ) − 1 − C �  C  e 0  , , 0 ,    µ 0 C e 0 • Not always the most interesting representation • Joint distribution of age and residual life time in equilibrium renewal process. Closely related to size–biased distributions • The result can be generalized to apply for the n th order moment distributions, but we have no probabilistic interpretation at this point.

Bi and multivariate exponentials and Bi and multivariate exponentials and gammas gammas • A multitude of various definitions • Most of these have rational joint Laplace transform for integer shape parameter • Many of these are in MPH and most are in MPH ∗ • The MME ∗ provides a framework for categorization

Moran and Downton’s Bivariate Exponential Moran and Downton’s Bivariate Exponential The MME ∗ representation of this distribution is γ ( a ) = ( α 1 , α 2 )     − λ 1 λ 1 (1 − p 1 )  1 0 T = R = .    λ 2 (1 − p 2 ) − λ 2 0 1 ∞ ( λ 1 (1 − p 1 ) x 1 λ 2 (1 − p 2 ) x 2 ) i − 1 � f ( x 1 , x 2 ) = λ 1 λ 2 p 2 e − ( λ 1 x 1 + λ 2 x 2 ) . (( i − 1)!) 2 i =1 with (slightly more general) Laplace transform ( α 1 s 2 λ 1 p 1 λ 2 + α 2 s 1 λ 1 λ 2 p 2 ) + λ 1 λ 2 (1 − (1 − p 1 )(1 − p 2 )) . s 1 s 2 + ( s 2 λ 1 + s 1 λ 2 ) + λ 1 λ 2 (1 − (1 − p 1 )(1 − p 2 ))

Cheriyan-Ramabhadran’s Bivariate Gamma Cheriyan-Ramabhadran’s Bivariate Gamma With MME ∗ representation γ = (1 , 0 , . . . , 0) , the matrix T is an ( m 0 + m 1 + m 2 ) × ( m 0 + m 1 + m 2 ) matrix of Erlang structure   − λ 0 λ . . .   e m 0 e m 0   0 − λ 0 . . .     T = R = , .   e m 1 0 . . . .. .. .   . . . . . .   . . . .     e m 2 0   0 0 − λ . . . The density is given by f ( x 1 , x 2 ) = � min ( x 1 ,x 2 ) e − x 1 − x 2 x m 0 − 1 ( x 1 − x ) m 1 − 1 ( x 2 − x ) m 2 − 1 e x d x ( m 0 − 1)!( m 1 − 1)!( m 2 − 1)! 0

Dussauchoy-Berland’s bivariate gamma Dussauchoy-Berland’s bivariate gamma γ = (1 , 0 , 0 , 0) and     − λ 1 0 0 λ 1 1 ρ � 2   � � �   1 − λ 2 1 − λ 2 0 − λ 1 2 ρλ 2  λ 1  1 ρ    ρλ 1 ρλ 1  T = , R = .       1 0   0 0 − λ 2 λ 2         1 0 0 0 0 − λ 2 • X 1 − ρX 2 and X 2 are independent with LST � l 1 � � l 2 � λ 1 + ρs 1 λ 2 , λ 1 + ρs 1 + s 2 λ 2 + s 1 in MME ∗ for positive integer values of l 1 and l 2 . An MME ∗ representation, (even in MPH) for l 1 = l 2 = 2 and ρλ 1 ≥ λ 2 is

Bivariate exponentials with arbitray Bivariate exponentials with arbitray correlations correlations • Can be seen as a generalization of Farlie–Gumbel–Morgenstern distributions. • Mixtures of combinations of order statistics • A distribution can be seen as the average the distribution of its order statistics • Eksplicit form of generator   − 2 λ λ p 11 λ p 12 λ   0 − λ p 21 λ p 22 λ       0 0 − µ µ     0 0 0 − 2 µ