A general recursive formula for the discrete stable and Linnik distributions and a family of extensions Hélène Cossette, David Landriault and Étienne Marceau École d’actuariat (Université Laval) July 2001 Abstract The purpose of this paper is to present a general method to compute recursively the probability mass function of the discrete stable, discrete Linnik and discrete Mittag-Le• er distribution. The recursive computation method is based on the rep- resentation of these distributions as compound distributions and on the Panjer al- gorithm (see Panjer (1981), Klugman et al. (1998) or Rolski et al. (1999)). Another distribution is also introduced and various extensions are proposed. Keywords: Discrete Linnik distribution; Discrete stable distribution; Discrete Mittag- Le• er distribution; Discrete Sibuya distribution; Discrete compound distribution; Panjer Algorithm. Correspondance to: Etienne Marceau, École d’Actuariat, Pavillon Alexandre-Vachon (Lo- cal 1620), Université Laval, Québec, Québec, Canada, G1K 7P4. email: [email protected] *This research was funded by individual operating grants from the Natural Sciences and Engineering Research Council of Canada, the Fonds pour la formation de chercheurs et l’aide à la recherche du Gouvernement du Québec and by a joint grant from the Chaire en Assurance L’Industrielle-Alliance (Université Laval). 1 1 Introduction The discrete stable, discrete Linnik and discrete Mittag-Le• er distributions have been studied separately by di¤erent authors, namely by Christoph and Schreiber (1998), Pakes (1995) and Pillai and Jayakumar (1995). For these three distributions, Devroye (1993) has suggested simulation methods and Rémillard and Theoderescu (2000) has proposed estimation methods and examined their tail behavior. The discrete stable, discrete Linnik and discrete Mittag-Le• er distributions can be represented as compound distributions from a common family. This property allows to compute recursively the probability mass function of these distributions. The recursive computation is based on the Panjer algorithm (Panjer and Willmot (1992), Klugman et al. (1998) or Rolski et al. (1999)). From their probability generating function, we can …nd an expression for their probability mass function. However, these are not very useful because they are not easily applicable. With the Panjer algorithm, it becomes simple to compute exactly the probability mass function of these distributions. Based on the representation of the stable, Linnik and Mittag-Le• er distribution as a compound distribution from a common family, we introduce a distribution which belongs to this same family. We also present, due to the possible compound representation, a variety of distributions which are members of an extended family. The Panjer algorithm used to compute the mass probability function for the stable, Linnik and Mittag-Le• er distribution can be generalized to evaluate the probability mass function of these new distributions. A numerical example is provided in the last section to compare the behavior of di¤erent distributions previously studied. 2 Description of the discrete stable and Linnik distri- butions 2.1 Discrete stable distribution The discrete stable distribution was introduced by Steutel and Van Harn (1979). The properties of the discrete stable distribution can be found in Devroye (1993), Christoph and Schreiber (1998) and Rémillard and Theodorescu (2000). Let N be a random variable (r.v.) with a discrete stable distribution with parameters (0; 1] and > 0 2 N Stable (; ) : Its probability generating function (p.g.f.) PN (t) is given by PN (t) = exp (1 t) ; t 1 : (1) f g j j Clearly, if = 1; then (1) becomes PN (t) = exp (1 t) (2) f g and from (2), it follows that N has a Poisson distribution with mean N P oisson () : For that reason, we can consider the discrete stable distribution as an extension of the Poisson distribution. Observe also that if Ni Stable (i; ) 2 for i = 1; :::; m; then N1 + ::: + Nm Stable (1 + ::: + m; ) ; (see Steutel and Van Harn (1979)). As it is explained in Christoph and Schreiber (1998), the following explicit expression for the probability mass function (p.m.f.) for the discrete stable distribution is obtained from (1) j j 1 j ( 1) () Pr (N = k) = ( 1)k ; (3) k j! j=0 X for k = 0; 1; 2; :::. The sum in (3) is absolutely convergent. However, one can express this distribution as a compound Poisson distribution. This representation will be used to recursively compute the exact values of the p.m.f. of the discrete stable distribution. 2.2 Discrete Linnik distribution The properties of the discrete Linnik distribution are examined in Devroye (1990, 1993), Pakes (1995) and Rémillard and Theoderescu (2000). Assume that N follows a discrete Linnik distribution with parameters (0; 1] ; > 0; > 0; 2 N Linnik( ; ; ) with p.g.f. PN (t) given by 1 PN (t) = ; t 1: (4) (1 + (1 t) ) j j If = 1, then (4) becomes 1 P (t) = ; (5) N (1 + (1 t)) which corresponds to the p.g.f. of the negative binomial distribution with parameters and N NegBin( ; ): We can therefore consider the discrete Linnik distribution as an extension of the negative binomial distribution. Letting = 1 in (4), we obtain 1 P (t) = (6) N 1 + (1 t) which is the p.g.f. of a discrete Mittag-Le• er distribution with parameters and N MittagLeffler( ; ): See Pillai and Jayakumar (1995) for a survey of the properties of the Mittag-Le• er dis- tribution. Also, if = 1 and = 1, (4) becomes 1 P (t) = (7) N (1 + (1 t)) from which we conclude that N has a geometric distribution N Geometric( ): 3 Let be a gamma r.v. with parameters and and N = be a discrete stable r.v. with parameters and : In such a case, it has been shownj in Devroye (1993) that the r.v. N has a Linnik distribution. Its p.m.f. is given by j j 1 k 1 j ( 1) () Pr (N = k) = ( 1) dG () k j! 0 j=0 Z X j k 1 j ( 1) 1 j = ( 1) () dG () k j! j=0 0 X Z j 1 j ( 1) ( + j) = ( 1)k j ; j = 0; 1; 2; :::; (8) k j! ( ) j=0 X given (3) and the following expression for the moments of ( + j) E j = j ; (9) ( ) for j = 0; 1; 2; ::: . The p.m.f. of the Mittag-Le• er distribution is given by (8) with = 1. Pillai and Jayakumar (1995) provides another expression for the p.m.f. of the Mittag-Le• er distribution. However, as for the stable distribution, both the discrete Linnik and the Mittag- Le• er distribution can be represented as a compound negative binomial and a compound geometric distribution respectively. This allows the computation of the exact values of their p.m.f. by a general recursive method. 3 Compound distributions Suppose a r.v. N which can be represented as follows M Z ; M > 0 N = k (10) 2 k=1 0P;M = 0 4 and suppose that M has either a binomial, Poisson, negative binomial or geometric distri- bution. Then N follows either a compound binomial, compound Poisson, compound nega- tive binomial or compound geometric distribution. We assume that Zk; k = 1; 2; ::: forms a sequence of i.i.d. random variables independent of M: Then, wef can express theg p.g.f. of N as follows PN (t) = PM (PZ (t)) : (11) In fact, the discrete stable, discrete Linnik and discrete Mittag-Le• er distribution can be represented as in (10) where the Zk0 s have a Sibuya distribution with parameter Zk Sibuya ( ) ; for (0; 1]. The p.g.f. of the Sibuya distribution with parameter is given by 2 PZ (t) = 1 (1 t) ; (12) for t 1 and the p.m.f. by j j 0; k = 0 P (Z = k) = fZ (k) = ; k = 1 ; (13) 2 (1 ):::(k 1 ) k! ; k = 2; 3; ::: 4 4 where (12) and (13) are given in Devroye (1993), Pillai and Jayakumar (1995) and Christoph and Schreiber (1998). When = 1, the Sibuya distribution is degenerated at 1. Based on (10), we later introduce a discrete distribution which we name the discrete binomial-Sibuya distribution. 3.1 Discrete stable, Linnik and Mittag-Le• er distributions We can represent N as in (10), where M has a Poisson distribution with parameter ; when N follows a discrete stable distribution with parameters (0; 1] and > 0: Using (11) with (2) and (12), the p.g.f. of the discrete stable distribution2 is obtained by PN (t) = PM (PZ (t)) = exp (1 1 + (1 t) ) f g = exp (1 t) : f g The stable distribution can therefore be also described as a compound Poisson distribution with parameters and FZ ; where Z follows a Sibuya distribution. When N follows a discrete Linnik distribution with parameters (0; 1] ; > 0; > 0; we can represent N as in (10) where M follows a negative binomial2 distribution with parameters and . Simple calculations show that (4) results from the substitution of (5) and (12) in (11). Similarly, when N has a Mittag-Le• er distribution with parameters and ; we can represent N as in (10) where M has a geometric distribution with parameter : The p.g.f. (6) is obtained when we replace (7) and (12) in (11). As in the case of the stable distribution, we can represent the Linnik and the Mittag-Le• er distribution as a compound negative binomial and a compound geometric distribution respectively. Remark: The discrete stable, discrete Linnik and discrete Mittag-Le• er distribu- tions can be also named Poisson-Sibuya, negative binomial-Sibuya and geometric-Sibuya distributions respectively. In a previous section, we have shown that we can represent a discrete Linnik distribu- tion as a gamma mixture of a discrete stable distribution. Given the correspondence made above between a Linnik distribution and a compound negative binomial distribution, this follows from the fact that we can represent a negative binomial distribution as a gamma mixture of a Poisson distribution (see e.g.