MIXED POISSON DISTRIBUTIONS ASSOCIATED WITH HAZARD FUNCTIONS OF EXPONENTIAL MIXTURES

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1 MIXED POISSON DISTRIBUTIONS ASSOCIATED WITH HAZARD FUNCTIONS OF EXPONENTIAL MIXTURES Moses W. Wakoli and Joseph A. M. Ottieno 2 Abstra The hazard funion of an exponential mixture charaerizes an infinitely divisible mixed Poisson distribution which is also a compound Poisson distribution. Given the hazard funion, the probability generating funions (pgf) of the compound Poisson distribution and its independent and identically distributed (iid) random variables are derived.the recursive forms of the distributions are also given. Hofmann hazard funion has been discussed and re-parameterized. The recursive form of the distribution of the iid random variables for the Hofmann distribution follows Panjer s model. Key words: Mixed Poisson distributions, Laplace transform, exponential mixtures, complete monotocity, infinite divisibility, compound Poisson distribution, Hofmann distributions, Panjer s model. INTRODUCTION Motivated by Walhin and Paris (999, 2002), this paper shows that there is a link between Poisson and exponential mixtures. Specifically mixed Poisson distributions, in the form of Laplace transform, can be expressed in terms of hazard funions of exponential mixtures. The hazard funion of an exponential mixture is completely monotone, and thus the mixing distribution is infinitely divisible through Laplace transform. A Poisson mixture with an infinitely divisible mixing distribution is infinitely divisible too. Further, an infinitely divisible mixed Poisson distribution is a compound Poisson distribution. Technical University of Kenya 2 University of Nairobi 209

2 To obtain the pgf of the iid random variables of a compound Poisson distribution, the pgf of a mixed Poisson distribution which is infinitely divisible and expressed in terms of the hazard funion of the exponential mixture is equated to the pgf of the compound Poisson distribution. By differentiation method, the pmf of the iid random variables is obtained. The compound Poisson distribution is also obtained recursively in terms of the pmf of the iid random variables and the hazard funion of the exponential mixture. The models developed have been applied to a class of mixed Poisson distribution known as Hofmann distributions. In this case both the differentiation method and the power series expansion have been used to obtain the distribution of the iid random variables. The paper has the following seions: In seion 2 mixed Poisson distribution is expressed in terms of Laplace transform. In seion 3 the mixed Poisson distribution is expressed in terms of hazard funion of exponential mixture. Seion 4 discusses infinite divisibility in relation to complete monotocity and Laplace transform leading to infinite divisible mixed Poisson distribution. Seion 5 derives compound Poisson distribution in terms of pgfs and recursive form. Seion 6 deals with applications of the models derived to various cases of Hoffmann distributions. Hoffman hazard funion has been re-parameterized in seion 7 and concluding remarks are in seion 8. 2 Mixed Poisson Distribution in Terms of Laplace Transform Let where Z(t) is a discrete random variable. A mixed Poisson distribution is given as p n (t) 0 p n (t) prob(z(t) n) (2.) λt (λt)n e n! g(λ)dλ n 0,, 2,... (2.2) λt (λt)n E(e ) n! ( )n t n E(( ) n λ n e λt ) n! 20

3 When n 0, we have p 0 (t) E(e λt ) L Λ (t) (2.3) the Laplace transform of the mixing distribution g(λ) Differentiating p 0 (t), n times we get p n (t) ( ) n tn p (n) 0 (t) E[( )n (λ) n e λt ] ( ) n tn n! p(n) 0 (t) n! L(n) Λ (t) n 0,, 2, 3,... (2.4) which is the mixed Poisson distribution in terms of the Laplace transform of the mixing distribution. 3 Mixed Poisson Distribution in Terms of the Hazard Funion of the Exponential Mixture The pdf of an exponential mixture is defined by f(t) where g(λ) is the mixing distribution. The corresponding survival funion is S(t) λ e λ t g(λ) dλ (3.) S(t λ) g(λ) dλ e λt L Λ (t) g(λ)dλ Remark The survival funion of an exponential mixture is the Laplace transform of the mixing distribution. 2

4 The hazard funion of the exponential mixture is given by h(t) f(t) S(t) S(t) L (t) L(t) ds dt (3.2) Let θ(t) In( L Λ (t) ) (3.3) θ (t) L (t) L(t) h(t) (3.4) p 0 (t) L Λ (t)) e In L Λ(t) e In( L Λ (t) ) e θ(t) (3.5) where p 0 (t) e t 0 h(x) dx (3.6) and t θ(t) 0 h(x) dx (3.7) is the cumulative or integrated hazard funion. Using (3.6) and (3.7), p 0 (t) can be obtained and then use formula (3.5) to derive p n (t). 22

5 The pgf of the mixed Poisson distribution is given by H(s, t) n0 n p n (t) s n [ 0 e λt λt (λt)n e n! n0 (λts) n n! e λt e λts g(λ) dλ e λt( s) g(λ) dλ L Λ (t ts) e In L Λ (t ts) g(λ) dλ] S n g(λ) dλ H(s, t) e θ(t ts) (3.8) which is the survival funion of the exponential mixture at time t ts. Remark 2 Given a hazard funion of an exponential mixture, the pgf of the mixed Poisson distribution is the survival funion of the exponential mixture at time t ts. The mean and variance of the mixed Poisson distribution can hence be obtained as follows: H (s, t) d ds H(s, t) tθ (t ts) H(s, t) H (s, t) t 2 [(θ (t ts)) 2 θ (t ts)] H(s, t)] then H(, t), H (, t) t θ (0) and H (, t) t 2 [θ (0)] 2 θ (0)] E[Z(t) H (, t) t θ (0) V ar[z(t)] H (, t) + H (, t) (H (, t)) 2 (3.9) t θ (0) t 2 θ (0) 23

6 4 Infinite Divisibility Definition : Infinite Divisibility (Karlis and Xekalaki, 2005) A random variable X is said to have an infinitely divisible distribution, if its charaeristic funion Ψ(t) can be written in the form: Ψ(t) [Ψ n (t)] n where Ψ n (t) is a charaeristic funion for any n, 2, 3,... In other words a distribution is infinitely divisible if it can be written as the distribution of the sum of an arbitrary number n of independently and identically distributed random variables. Definition 2 : Completely Monotonicity A funion Ψ on [0, ] is completely monotone if it possesses derivatives Ψ (n) of all orders and ( ) n Ψ (n) (t) 0, t > 0 The link between infinite divisibility and complete monotonicity is given in the following propositions. Proposition : (Feller, 97, Vol II, Chaer XIII) The funion ω is the Laplace transform of an infinitely divisible probability distribution iff ω e Ψ where Ψ has a completely monotone derivative and Ψ(0) 0 Remark 3 : and In our situation p 0 (t) L Λ (t) e θ(t) (4.) L λ (0) L Λ (t) e θ(t) and (4.2) θ(0) 0 The derivative of θ(t) is θ (t) h(t) which is the hazard funion of an exponential mixture. Hesselager et. al. (998) have stated the following theorem: A distribution with a completely monotone hazard funion is a mixed exponential distribution. 24

7 Remark 4 The theorem implies that a hazard funion of an exponential mixture is completely monotone. According to proposition, θ (t) h(t) is completely monotone and θ(0) 0 Therefore p 0 (t) L Λ (t) is the Laplace transform of an infinitely divisible distribution. Thus the mixing distribution, g(λ) is infinitely divisible. Proposition 2 (Maceda, 948) If in a Poisson mixture the mixing distribution is infinitely divisible, the resulting mixture is also infinitely divisible. Proposition 3 (Feller, 968; Ospina and Gerbes, 987) Any discrete infinitely divisible distribution can arise as a compound Poisson distribution. From the above discussion we have the following: Theorem Mixed Poisson distributions expressed in terms of Laplace transforms can also be expressed in terms of hazard funions of exponential mixtures. Such Poisson mixtures are infinitely divisible and hence are compound Poisson distributions. 5 Compound Poisson Distribution Let Z(t) Z N(t) X + X X N(t) (5.) where X is are iid random variables and N(t), is also a random variable independent of X i s Then Z N(t) is said to have a compound Poisson distribution 5. Compound Poisson Distribution in terms of pgf Let H(s, t) E[s Z N(t) ] p n (t) s n n0 the pgf of Z N(t)) 25

8 and F (s, t) E(s N(t) ) f j s j j0 the pgf of N(t) G(s, t) E(s X i ) g x (t) s x x0 the pgf of X i It can be proved that If N(t) is Poisson with parameter θ(t), then H(s, t) F[G(s, t)] (5.2) H(s, t) e θ(t)[ G(s,t)] (5.3) 5.2 The Distribution of iid Random Variables of the Compound Poisson Distribution Since an infinitely divisible mixed Poisson distribution is also a compound Poisson distribution, we equate their pgfs given in (3.9) and (5.3) i.e e θ(t t s) e θ(t)[ G(s,t)] G(s, t) θ(t t s) θ(t) (5.4) which is the probability generating funion of iid random variables expressed in terms of the cumulative hazard funion of the exponential mixture. The corresponding probability mass funion obtained by differentiation method is given by: g x (t) x! g 0 (t) 0 d x ds x G(s, t) s0 for x, 2, 3,... (5.5) 5.3 Compound Poisson Distribution in Recursive Form By differentiating equation (5.3), H (s, t) θ(t) G (s, t) H(s, t) (5.6) 26

9 where G (s, t) d G(s, t) ds By definition H(s, t) p n (t) s n H (s, t) n p n (t) s n (5.7) n0 n G(s, t) g x (t) s x G (s, t) x g x (t) s x (5.8) x0 x By comparing (5.6) to (5.7) we get n p n (t) s n θ(t) G (s, t) H(s, t) n θ(t) [ x g x (t) s x ] p n (t) s n x [θ(t) n n p n (t) θ(t) n x x n0 n x g x (t) p n x (t)] s n n p n (t) s n x g x (t) p n x (t) for n, 2, 3,... (5.9) P ut x i + (n + ) p n+ (t) θ(t) n (i + ) g i+ (t) p n i (t) for n 0,, 2... i0 (5.0) Equation (5.0) can be used to obtain p n (t) iteratively and can also be used to obtain the mean and the variance of the mixed Poisson distribution. 5.4 Panjer s Recursive Model Let p n (a + b n p n ); n, 2, 3,... where a and b are real numbers. This recursive model is known as Panjer s recursive model of class zero denoted by (a, b, 0) class of distributions. We can extend it to p n (a + b n p n ); n 2, 3,... 27

10 which is Panjer s (a, b, ) class. In general, we have p n (a + b n p n ; n k +, k + 2,... which is Panjer s (a, b, k) class for k 0,, 2,... Some probability mass funions in this paper take Panjer s recursive form. Remark 5 In auarial literature, this recursive relation is due to the work of Panjer (98). In statistical literature, however, this relation had been published by Katz (965) based on his PhD dissertation of Hofmann Hazard Funion A class of mixed Poisson distributions known as Hofmann distribution has been described by Walhin and Paris (999) as: and p 0 (t) e θ(t) (6.) where θ (t) p n (t) ( ) n tn n! pn 0 (t) for n, 2,... (6.2) p ( + ) a for p > 0, c > 0 and a 0 (6.3) and θ(0) 0 (6.4) Remark 6 Clearly θ (t) is a hazard funion of an exponential mixture. We shall refer to it as Hofmann hazard funion of an exponential mixture; i.e θ (t) h(t) We will now consider various cases of a p ( + ) a for p > 0, c > 0 and a 0 (6.5) 28

11 6. When a 0 θ (t) h(t) p, a constant (6.6) which is the hazard funion of the exponential distribution with parameter p Therefore ( ) n h (n) (t) 0 dn dt n h(t) ( )n h n (t) 0 Therefore h(t) is completely monotone. The cumulative hazard funion is therefore t θ(t) p 0 dx implying that Therefore, θ (t ts) p t ( s) θ(0) 0, θ (0) p and θ (0) 0 p 0 (t) e θ(t) e p (n) 0 (t) ( )n p n e p n (t) ( ) n tn n! p(n) 0 (t) p n (t) e () n n! which is a Poisson distribution with parameter. θ (t ts) H(s, t) e (6.7) e p t ( s) which is the pgf of Poisson distribution with parameter Therefore E[Z(t)] and Var(Z N(t) ) Since θ(0) 0 and h(t) θ (t) is completely monotone, then p 0 (t) is a Laplace transform of an infinitely divisible mixing distribution. 29

12 The corresponding infinitely divisible mixed Poisson distribution is a compound Poisson distribution whose iid random variables have pgf given by implying that the pmf is G(s, t) s ( s) g x (t) for x g x (t) 0 for x (6.8) The compound Poisson distribution in recursive form is By iteration n p n (t) θ(t) n x x g x (t) p n x (t) n p n (t) θ(t) p n (t) for n, 2, 3... F or n p (t) θ(t) p 0 (t) F or n 2 p 2 (t) (θ(t))2 2! F or n 3 p 3 (t) (θ(t))3 3! p 0 (t) p 0 (t) (6.9) Moments By induion, p n (t) e () n n! Sum equation (6.9) over n; i.e. E(Z(t) n n for n 0,, 2... n p n (t) n p n (t) E(Z(t) E(Z(t) θ(t) Multiply (6.9) by n and then sum the result over n; i.e. E(Z(t)) 2 n 2 p n (t) n E(Z(t)) 2 () 2 + V ar(z(t)) E(Z(t)) 2 (E(Z(t)) 2 220

13 6.2 When a and c p h(t) θ (t) p + which is a hazard funion of Pareto (exponential-exponential) distribution, with parameters p and p. Therefore ( ) n h (n) (t) ( ) n p n+ ( + ) (n+) dn dt n h(t) ( )n h n (t) 0 Therefore h(t) is completely monotone. The cumulative hazard funion is Implying that, Therefore, Therefore, t θ(t) p 0 p + px dx In( + ) θ(t ts) In( + s) θ(0) 0, θ (0) p and θ (0) p 2 p 0 (t) e In(+) + p (n) 0 (t) ( )n n! p n ( + ) n p n (t) ( + )n + n 0,, 2, 3,... which is a geometric(poisson-exponential) distribution with parameter The pgf is given by H(s, t) ( + + s) which is the pgf of a geometric distribution with parameter + Therefore, E[Z(t) and Var(Z(t)) + () 2 + Since θ(0) 0 and h(t) θ (t) is completely monotone, then p 0 (t) is a Laplace transform of an infinitely divisible mixing distribution. 22

14 The corresponding infinitely divisible mixed Poisson distribution is a compound Poisson distribution whose iid random variables have pgf given by By power series expansion, G(s, t) G(s, t) x In( + s)) In ( + ) ( + )x x In ( ( + )x g x (t) x In ( + ) + ) sx which is logarithmic series distribution with parameter +. By the differentiation method, we have G x (x )! (s, t) In( + ) ( + )x ( + s) x ( + )x g x (t) x In( g x (t) g x (t) x x + ) x, ( x ) g x (t) (a + b x ) g x (t) for x 2, 3, 4... which is Panjer s recursive model with a + and b + The compound Poisson distribution in recursive form is: n p n (t) In( + ) n p n (t) n x n ( + )x p n x (t) x x ( + )x x In( + ) p n x(t) for x 2, 3, 4,... F or n, p (t) p + p 0(t) F or n 2, t p 2 (t) ( + t )2 p 0 (t) (6.0) 222

15 In general, p n (t) ( + )n + n 0,, 2,... which is the geometric distribution with parameter + Moments Sum the recursive relation (6.0) over n; thus n n p n (t) E(Z(t)) Next, multiply the recursive relation (6.0) by n + and then sum the result over n; thus E(Z(t)) 2 () () 2 and Var(Z(t)) + () When a c θ (t) h(t) p + t which is a hazard funion of Pareto (exponential-gamma) distribution, with parameters p and. Therefore, ( ) n h (n) (t) ( ) n p n ( + ) (n+) dn dt n h(t) ( )n h n (t) 0 Therefore h(t) is completely monotone. The cumulative hazard funion is implying that, t θ(t) p 0 + x dx p In( + t) θ(t ts) p In( + t ts) 223

16 Therefore θ(0) 0, θ (0) p and θ (0) p p 0 (t) e θ(t) p In(+t) e ( + t) p p (n) 0 (t) (p + n )!) ( )n n! ( + t) p n n! (p )! p n (t) ( ) n tn (p + n )!) n! ( )n n! ( + t) p n n! (p )! ( ) p + n t p n (t) ( n + t )n ( + t )p n, 2, 3,... which is a negative binomial (Poisson-gamma) distribution with parameters p and +t The pgf is given by H(s, t) e θ(t ts) p In (+t ts) e ( + t ts )p ( +t t +t s)p which is the pgf of a negative binomial distribution with parameters p and +t E[Z(t)] t θ (0) V ar(z(t)) t θ (0) t 2 θ (0) tp + t 2 p + 2 Since θ(0) 0 and h(t) θ (t) is completely monotone, then p 0 (t) is a Laplace transform of an infinitely divisible mixing distribution. The corresponding infinitely divisible mixed Poisson distribution is a compound Poisson distribution whose iid random variables have pgf given by G(s, t) In( + t ts)) In ( + t) 224

17 By power series expansion, In[( + t)( t +t G(s, t) s)] In( + t) t ( +t )x In ( +t x ) s x x ( t +t )x x In ( g x (t) x t +t ) sx ( t +t )x x In ( t x, 2, t ), which is logarithmic series distribution with parameter +t. By the differentiation method, we have G x (x )! (s, t) In( + t) ( t + t )x ( t + t s) x g x (t) x! d x G(s, t) ds x s0 ( t +t )x x In( + t) ( t +t )x x In +t g x (t) g x (t) x x t + t t + t ( x ) x, 2... g x (t) (a + b x ) g x (t) for x 2, 3, 4.. which is Panjer s recursive model with a t + t and b t + t The compound Poisson distribution in recursive form is: n t n p n (t) p ( + t )x p n x (t); n, 2, 3,... (6.) x t F or n, p (t) p + t p 0(t) ( ) p + t F or n 2, p 2 (t) ( 2 + t )2 p 0 (t) ( ) p + 2 t F or n 3, p 3 (t) ( 3 + t )3 p 0 (t) 225

18 By induion, ( ) p + n p n (t) n t ( + t )n ( + t )p, n 0., 2,... (6.2) which is the negative binomial distribution with parameters p and +t Moments Sum equation (6.) over n n p n (t) p n n E(Z(t)) n t ( + t )x p n x (t) x Next, multiply the recursive relation (6.) by n and then sum the result over n; thus 6.4 When a E(Z(t)) 2 n n 2 p n (t)] V ar(z(t)) + 2 θ (t) h(t) p + which is a hazard funion of Pareto (exponential-gamma) distribution, with parameters p and c. Therefore ( ) n h (n) (t) ( ) n p (c) n ( + ) (n+) ) dn dt n h(t) ( )n h n (t) 0 Therefore h(t) is completely monotone. The cumulative hazard funion is Implying that, t θ(t) p p c 0 + dx In( + ) θ(t ts) p c In( + s) 226

19 Therefore, θ(0) 0, θ (0) p and θ (0) pc p 0 (t) ( + ) p c p (n) 0 (t) ( )n c n Γ( p c + n) Γ( p c ) ( + ) p c n ( ) n c n n! ( p c + n )!) n! ( p c )! ( + ) p c n p n (t) ( ) n tn n! ( )n c n n! ( p c + n )!) n! ( p c )! ( + ) p c n ( p p n (t) c + n ) ( n + )n ( + ) p c for n 0,, 2,... which is a negative binomial (Poisson-gamma) distribution with parameters p c and + The pgf is given by H(s, t) e θ(t ts) e p c In (+ s) ( + s ) p c ( + c + s)p which is the pgf of a negative binomial distribution with parameters p c and + E[Z(t) t θ (0) V ar(z(t)) t θ (0) t 2 θ (0) + p 2 Since θ(0) 0 and h(t) θ (t) is completely monotone, then p 0 (t) is a Laplace transform of an infinitely divisible mixing distribution. The corresponding infinitely divisible mixed Poisson distribution is a compound Poisson distribution whose iid random variables have pgf given by In( + s)) G(s, t) In ( + ) 227

20 By power series expansion In[( + )( + G(s, t) s)] In( + ) ( + )x In ( + x ) s x x ( + )x x In ( x ( + )x g x (t) x In ( + ) sx + ) for x 0,, 2,... which is logarithmic series distribution with parameter +. By the differentiation method, we have G x (x )! (s, t) In( + ) ( + )x ( + s) x g x (t) x! d x G(s, t) ds x s0 ( + )x x In + g x (t) g x (t) x x + + ( x ) x, 2... g x (t) (a + b x ) g x (t) for x 2, 3, 4.. which is Panjer s recursive model with a + and b + The compound Poisson distribution in recursive form is: n p n (t) p c n ( + )x p n x (t) for n, 2, 3,... (6.3) x F or n, p (t) p c + p 0(t) F or n 2, p 2 (t) 2 ( + )2 [ p c (p c + )p 0(t) ( p F or n 3, p 3 (t) c + 2 ) ( 3 + )3 p 0 (t) ( p c + ) 2 ( + )2 p 0 (t) 228

21 By induion, ( p p n (t) c + n ) n ( p c + n ) n ( + )n p 0 (t) ( + )n ( + ) p c for n, 2, 3,... which is the negative binomial distribution with parameters p c and + Moments Sum equation (6.3) over n n p n (t) p c n E(Z(t)) n n ( + )x p n x (t) x Next, multiply the recursive relation (6.3) by n + and then sum the result over n; thus E(Z(t)) 2 n n 2 p n (t) () p 2 V ar(z(t)) + p When a 2 h(t) θ (t) p ( + ) 2 for p > 0, and c > 0 which is a hazard funion of the exponential-inverse Gaussian distribution. Therefore ( ) n h (n) n (2n ) (t) ( ) 2 dn dt n h(t) ( )n h n (t) 0 (2n 3) 2 Therefore h(t) is completely monotone. t θ(t) p 0 (2n 5) 2 ( + cx) 2 dx 2p c [( + ) 2 ] (2n+) ( + )

22 implying that θ(t ts) 2p c [( + s) 2 ] Therefore, Therefore, θ(0) 0, θ (0) p and θ (0) 2 pc p 0 (t) e 2p c [(+) 2 ] n p n 0 (t) ( ) n p n k0 p n (t) ( ) n tn n! ( )n p n where ()n n! The pgf is Therefore n k0 (n + k)! (n k)!k! ( c 4p )k ( + ) (n+k) 2 ] p 0 (t) n k0 (n + k)! (n k)!k! ( c 4p )k ( + ) (n+k) 2 ] p 0 (t) (n + k)! (n k)!k! ( c 4p )k ( + ) (n+k) 2 ] p 0 (t) for n, 2, 3,... p 0 (t) e 2p c [(+) 2 ] H(s, t) e θ(t ts) e 2p c [(+ s) 2 ] E(Z(t)) t θ (0) and Var(Z(t)) t θ (0) t 2 θ (0) + 2 pc t2 Since θ(0) 0 and h(t) θ (t) is completely monotone, then p 0 (t) is a Laplace transform of an infinitely divisible mixing distribution. The corresponding infinitely divisible mixed Poisson distribution is a compound Poisson distribution whose iid random variables have pgf given by G(s, t) ( + s) 2 ( + ) 2 ( + ) 2 ( + s) 2 ( + ) 2 230

23 By power series expansion ( + ) 2 G(s, t) ( + ) 2 [ ( + s) 2 ] G(s, t) ( + ) 2 2 ( + ) 2 x! Γ( 2 ) Γ(x 2 ) ( + )x s x g x (t) ( + ) 2 ( + ) 2 ( + ) 2 θ(t) By differentiation method x 2 Γ(x 2 ) x! Γ( 2 ) ( + )x x, 2, 3,... Γ(x 2 ) x! Γ( 2 ) ( + )x x, 2, 3,... G (x) (s, t) Γ[x 2 ] ( 2 ) ()x ( + s) (2x ) 2 Γ( 2 ) [( + ) 2 ] g x (t) ( + ) 2 θ(t) g x (t) g x (t) Γ(x 2 ) x! ( 3 2x ) Γ[x 2 ] x! Γ( 2 ) ( + )x for x, 2, 3... (x )! ( + )x Γ[x 2 ] ( x 2, 3, which is in Panjer s recursive form, where a n p n (t) ( + ) 2 n x Replace n by n + to get + )x 2 + and b Γ[x 2 ] (x )!! Γ( 2 ) ( + )x p n x (t) for n, 2, 3,... (n + )p n+ (t) ( + ) 2 Let x i + (n + )p n+ (t) ( + ) 2 n+ x n i0 Γ[x 2 ] (x )! Γ( 2 ) ( + )x p n+ x (t) Γ[i + 2 ] i! Γ( 2 ) ( + )i p n i (t) n, 2,... (6.4) By iteration, F or n 0, p (t) ( + ) 2 p0 (t F or n, p 2 (t) ()2 2! [( + ) ! c 4p ( + ) 3 2 ] p0 (t) 23

24 For n 2, p 3 (t) ( + ) 2 [ () 3 3! 2 k0 (2 + k)! (2 k)!k! ( c 4p )k ( + ) (3+k) 2 + () 3 3! ()4 4! () ! 3 2 ( + ) k0 3 () 2 ( + )2 ( + ) () ! (3 + 0)! [ (3 0)!0! ( c 4p )0 ( + ) (4+0) ( + k)! ( k)!k! ( c 4p )k ( + ) (2+k) () 3 ( + )3 ] p 0 (t)] (3 + )! (3 )!! ( c 4p ) ( + ) (4+) 2 + (3 + 2)! (3 2)!2! ( c p )2 ( + ) (4+2) (3 + 3)! 2 + (3 3)!3! ( c p )3 ( + ) (4+3) 2 ] p 0 (t) p 4 (t) ()4 4! 3 k0 In general therefore, we have (3 + k)! (3 k)!k! ( c 4p )k ( + ) (4+k) 2 p 0 (t) p n (t) ()n n! n k0 (n + k)! (n k)!k! ( c 4p )k ( + ) (n+k) 2 p 0 (t) for n, 2, 3... where, p 0 (t) e 2p c [(+) 2 ] Moments Sum the recursive relation (6.4) over n; thus (n + ) p n+ (t) E(Z(t) n0 Next, multiply the recursive relation (6.4) by n + and then sum the result over n; thus (n + ) 2 p n+ (t) E[Z(t)] 2 n0 V ar(z(t)) + () pc t2 2 pc t

25 Using notations by Willmot (986), let Then we get t, c 2β p µ and s z p n (t) p 0 (µ) n n! where, n k0 (n + k)! (n k)!k! ( β 2µ )k ( + 2β) (n+k) 2 p 0 (t) for n, 2, 3... The pgf H(s, t) is given by The mean and variance are p 0 (t) e µ β [(+2β) 2 ] P(z) e µ β [( 2β(z )) 2 ] E(N) E(Z(t)) µ; Var(N) Var(Z(t)) µ( + β) The pgf, G(s, t) of the iid random variables is given by Q(z) [( 2β(z ))] 2 ( + 2β) 2 ( + 2β) 2 The pmf g x (t) of the iid random variables is given by q n 2 Γ(n 2 ) ( + 2β) 2 ( 2β +2β )n n!γ( 2 ) [( + 2β) for n, 2, 3,... 2 ] which satisfies Panjer s recursive model with a 2β + 2β Using notations of Sankara (968), let t, n r p ( + c) 2 and b 3β + 2β b c + c and i k The recursive formula for the Poisson-inverse Gaussian distribution becomes r Γ(k + 2 (r + )p r+ (t) a ) r ( k!γ( k0 2 ) ( b) k p r k a 2 + k ) ( b) k p k r k k0 r (2k )(2k 3)...(5.3.) ( b k a )k p r k for r 0,, 2,... k! k0 233

26 6.6 When a 2 h(t) θ (t) p ( + ) 2 for p > 0, and c > 0 which shall be referred to as Polya-Aeppli hazard funion. Therefore ( ) n h (n) (t) ( ) n (n + )!pc n ( + ) (n+2) dn dt n h(t) ( )n h n (t) Therefore h(t) is completely monotone. ( ) n (n + )!pc n ( + ) (n+2) 0 implying that Therefore t θ(t) p θ(t) 0 + θ(t ts) ( + cx) 2 dx s s The pgf is Therefore θ(0) 0, θ (0) p and θ (0) 2pc p 0 (t) e + H(s, t) e θ(t ts) e s s E(Z(t)) t θ (0) and Var(Z(t)) t θ (0) t 2 θ (0) + 2 pc t 2 Since θ(0) 0 and h(t) θ (t) is completely monotone, then p 0 (t) is a Laplace transform of an infinitely divisible mixing distribution. The corresponding infinitely divisible mixed Poisson distribution is a compound Poisson distribution whose iid random variables have pgf given by s G(s, t) s s ( + )( + s + s + s) + 234

27 which is the pgf of zero-truncated (shifted) geometric distribution. By power series expansion G(s, t) + s x0 ( + )x s x g x (t) + ( + )x x, 2, 3,... and g 0 (t) 0 g x (t) g x (t) ( ( + ) + )x ( ) x 2 () x 2, 3,... + ( ) for x 2, 3,... x which is Panjer s form with a + and b 0 The compound Poisson distribution in the recursive form is given by (n + )p (n+) (t) ( + ) 2 n (i + ) ( + )i p n i (t) (6.5) i0 By iteration, n 0, p (t) n, p 2 (t) [ 2 n 2, 3 p 3 (t) ( + ) 2 [ 2 ( + ) 2 p 0(t) ( + ) 2 () 2 + ( + ) 4 ( + ) 3 ] p 0(t) 2 (i + ) ( + )i p 2 i (t) i0 ( + ) 2 [p 2(t) + + p (t) + ( + )2 p 0 (t)] () 2 ( + ) 4 p 0(t) + p 3 (t) [ 6 n 3, 4 p 4 (t) p 4 (t) ( + ) 5 ( + ) 3 p 0(t) + 2 ( + ) 3 p 0(t) + ( + )2 p 0 (t)] () 3 ( + ) 6 + ()2 ()2 + ( + ) 5 ( + ) 4 ] p 0(t) 4 j ( + ) 2 ( ) 4 j 3 (i + ) ( + )i p 3 i (t) i0 [ j! ( + )j () 4 j ] p 0 (t) 235

28 p 4 (t) ( + )4 p 0 (t) In general, the pattern is where p n (t) ( + )n p 0 (t) 4 j n j This is the Polya-Aeppli distribution. 6.7 When a h(t) θ (t) ( ) 4 j ( ) n j p 0 (t) e + ( ( + ) )j j! ( ( + ) )j j! lim a p 2 ( + c 2 t) a for p > 0, and c > 0 lim a p 2 k0 lim a p 2 p 2 k0 k0 a ( a ) ( a 2)...[ a (k )] (c 2 t) k k! ( ) k a k ( + a ) ( + 2 (k ) a )...[( + a )] (c 2 t) k k! lim a ( ac) k (t) k k! Let b ac as a h(t) θ (t) p 2 k0 p 2 e bt ( bt) k which is the hazard funion for Gompertz distribution. Therefore, ( ) n Therefore h(t) is completely monotone. The cumulative hazard funion is k! h (n) (t) ( ) n b n p 2 e bt dn dt n h(t) ( )n h n (t) 0 θ(t) p 2 t 0 e bx dx p b [ e bt ] 236

29 implying that θ(t ts) p b [ e bt( s) ] Therefore θ(0) 0, θ (0) p and θ (0) b p Since Gompertz distribution is an exponential mixture, the survival funion is the Laplace transform of the mixing distribution and hence, p 0 (t) e p b [e bt ] e p b j0 p (n) 0 (t) j0 [ p b e bt ] j j! j0 p n (t) ( ) n tn n! p n (t) j0 j0 e btj [e p b [ p b ]j j! ] ( bj) n e btj p j ( bj) n e btj p j j0 ( ) n tn n! ( bj)n e btj p j (btj) n n! e btj e p b [ p b ]j ; j 0,, 2... j! which is Poisson mixture of Poisson distribution also known as Neyman Type A distribution. The pgf is Therefore H(s, t) e θ(t ts) e p b [ e bt( s) ] E(Z(t)) t θ (0) and Var(Z(t)) t θ (0) t 2 θ (0) + pb t 2 Since θ(0) 0 and h(t) θ (t) is completely monotone, then p 0 (t) is a Laplace transform of an infinitely divisible mixing distribution. 237

30 The corresponding infinitely divisible mixed Poisson distribution is a compound Poisson distribution whose iid random variables have pgf given by G(s, t) g x (t) e bt ( s) e bt e bt ( s) e bt e bt e bt e bt [ebts ] e bt e bt [ e bt x0 (bt) x x! (bt) x s x ] x! x, 2... which is zero-truncated Poisson distribution with parameter bt g x (t) g x (t) 0 + bt x x, 2,... and it is in Panjer s recursive form, where a 0 and b bt Remark 7. Whereas Klugman et. al. (2008) have stated that the distribution of the iid random variables is Poisson, in this study, the distribution is zero-truncated Poisson distribution. The difference is due to the assumion that N is Poisson with a constant parameter λ instead of θ(t), which is a cumulative hazard funion. The recursive formula for the compound Poisson distribution is given by: (n + ) p n+ (t) p b [ e bt ] e bt n i0 n (i + ) i0 (bt) i 6.8 When a 0 and a i! e bt (bt) i+ (i + )! p n i(t) p n i (t) n 0,, 2,... h(t) θ (t) p ( + ) a for p > 0, c > 0 and a > 0 but a h (n) (t) θ (n) (t) ( ) n a(a + )a + 3)...(a + n )c n p ( + ) a n+ 238

31 Therefore h(t) is completely monotone. t θ(t) p 0 ( + cx) a dx p c( a) [( + ) a ] implying that and also θ(t ts) p c( a) [( + s) a ] θ(0) 0 θ (0) p and θ (0) acp and p 0 (t) e H(s, t) e θ(t ts) p c( a) [(+) a ] e p c( a) [(+ s) a ] Therefore E(Z(t)) t θ (0) and Var(Z(t)) t θ (0) t 2 θ (0) + ac 2 Since θ(0) 0 and h(t) θ (t) is completely monotone, then p 0 (t) is a Laplace transform of an infinitely divisible mixing distribution. The corresponding infinitely divisible mixed Poisson distribution is a compound Poisson distribution whose iid random variables have pgf given by G(s, t) + s)] a ( + ) a ( + ) a ( + s) a ( + ) a ( + ) a ( + ) a ( ) ( ) a x g x (t) a x! and g 0 (t) 0 Γ(a + x ) Γ(a) x ( + )x s x ( + ) a ( + )x ( + ) a for x, 2, 3,... The recursive formula of the compound Poisson distribution is therefore ( + ) a n x Γ(a + x ) (x )! Γ(a) ( + )x p n x (t) for x, 2, 3,

32 Therefore by replacing n by n +, we have (n + ) p (n+) (t) ( + ) a Put x i + we get n+ x (n + ) p (n+) (t) ( + ) a) + Γ(a + x ) (x )! Γ(a) ( + )x p n+ x (t) for x 0,, 2, 3,... n i0 as given by Walhin and Paris (2002) Moments To obtain the first moment, sum (6.6) over n to get (n + ) p n+ (t) E[Z(t)] n0 Γ(a + i) i! Γ(a) ( + )i p n i (t) for n 0,, 2, 3,... ( + ) a ( + ) a Next, multiply (6.6) by n + and then sum the result over n, to get E[Z(t)] 2 (n + ) 2 p n+ (t) n0 [( + ) + a] V ar(z(t)) + ac 2 (6.6) 7 Parameterization of Hofmann Hazard Funion This seion identifies the hazard funion of an exponential mixture which accommodates the extended truncated negative binomial (ETNB) distribution as the distribution of the iid random variables for the compound Poisson distribution. 240

33 7. Construing ETNB Distribution ( ) ( q(t)) r r ( q(t)) k k k0 ( ) r + ( q(t)) k k k theref ore ( ) r + k (q(t)) k k P rob(x k) ( q(t)) r k, 2, 3,... is a probability mass funion called zero-truncated negative binomial distribution with parameter q(t). It is also called the extended truncated negative binomial (ETNB) distribution according to Klugman et. al. (2008) because the parameter r can extend below zero. Let q(t), c > 0, t > 0. + Then Prob(X k) ( ) r + k k ( + ) r ( + )k The probability generating funion is given by: G(s) k p k s k ( ) r + k ( k + s)k ( + ) r k ( ) r ( k k + s)k ( + ) r G(s) ( + s) r ) ( + ) r k, 2, 3,... 24

34 7.2 Identifying the Hazard Funion Let us parameterize Hofmann hazard funion by putting a r +. Thus h(t) and hence p ( + ) r+ p > 0, c > 0 and r θ(t) p rc [ ( + ) r ], θ(0) 0 θ(t t s) p rc [ ( + s) r ] For an infinitely mixed Poisson distribution [ ( + s) r G(s, t) ( + ) r ( + s) r ( + ) r which is the pgf of ETNB distribution. ( ) r x x ( + ) r ( ) r + x g(x) x [( + ) r ] ( + s)x ] ( + )x x, 2, 3,... which is the pmf of a zero-truncated negative binomial distribution or extended truncated negative binomial distribution with parameters g x (t) g x (t) g x (t) g x (t) and this is Panjer s model ( ) r + x x ( ) r + x 2 x r + x + x r + x + x + [ + r x ] a + b x ( + )x ( + )x x 2, 3,

35 8 Concluding Remarks The hazard funion of an exponential mixture charaerizes an infinitely divisible mixed Poisson distribution which is also a compound Poisson distribution. Hofmann hazard funion is a good illustration of the theory. For further research other classes of hazard funions should be considered, in particular those based on compound Poisson distributions with continuous iid random variables. Hazard funions expressed in terms of the modified Bessel funion of the third kind would be of interest. 243

36 References [] Feller, W. An Introduion to probability Theory and Its Applications Vol I, 3rd Edition, John Wiley and Sons (968):. [2] Feller, W. An Introduion to probability Theory and Its Applications. Vol II. John Wiley and Sons (97):. [3] Hesselager, O., Wang, S. and Willmot, G. E. Exponential and Scale Mixtures and Equilibrium Distributions Scandinavian Auarial Journal, 2, (998): [4] Katz, L. (945) Charaeristics of frequency funions defined by firstorder difference equations Dissertation (PhD), University of Michigan, Ann Arbor. [5] Katz, L. (965) Unified treatment of a broad class of discrete probability distributions In Patil, G. P. (ed) Classical and Contagious Discrete Distributions; pp Pergamon Press, Oxford. [6] Klugman, S.A: Panjer, H. H. and Willmot G. E. Loss Models: From Data to Decisions, 3rd Edition, John Wiley and Sons (2008): [7] Maceda, E.C. On The Compound and Generalized Poisson Distributions. Annals of Mathematical Statistics, 9 (948):, [8] Ospina,V and Gerber, M.U A Simple Proof of Feller s Charaerizations of the Compound Poisson Distribution. Insurance Mathematics and Economics, 6, (987): [9] Panjer, H. Recursive Evaluation of a Family of Compound DistributionsASTIN Bulletin, 8, (98): [0] Sankara, N. Mixtures by the Inverse Gaussian Distribution Sankya, Series B, Volume 30, pp [] Walhin, J.F and Paris J. Using Mixed Poisson Processes in Conneion with Bonus Malus Systems Astin Bulletin, Vol 29, No., (999): pp [2] Walhin, J.F and Paris J. A general family of over-dispersed probability laws Belgian Auarial Bulletin, Vol 2, No., (2002): pp 8. [3] Willmot, G. Mixed Compound Poisson Distributions ASTIN Bulletin Vol.6, (986): pages S59 S

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