Compound COM-Poisson Distribution with Binomial Compounding Distribution

Transcription

1 Compound COM-oisson Distribution with Binomial Compounding Distribution V.Saavithri Department of Mathematics Nehru Memorial College Trichy. J.riyadharshini Department of Mathematics Nehru Memorial College Trichy. Z. arvin Banu Department of Mathematics Nehru Memorial College Trichy. February 6, 8 Abstract Conway-Mawell oisson distribution is two parameter oisson distribution and also a generalization of Bernoulli, Geometric distributions. In this paper, the compound COM-oisson distribution with binomial compounding distribution is proposed. Its properties are also derived. The parameters of newly introduced distribution are estimated by the method of profile likelihood estimation. Introduction In 95, oisson - Binomial distribution was discussed by Skellam and fitted by him to quadrat data on the sedge Care flacca. In McGuire et al. this distribution was used to represent variation in the numbers of corn-borer larvae in randomly chosen areas of a field. In 96, Conway & Mawell introduced this distribution in the contet of queuing systems. In 5, Galit Shmueli revived this distribution and used for fitting discrete data. They use the acronym COM-oisson for this distribution. The COM-oisson belongs to the eponential family as well as to the twoparameter power series distributions family. Its even stronger, it is easy to use ISSN: age 33

2 fleible for fitting over and under-dispersed data. In many practical applications, the equidispersion property of the oisson distribution is not observed in the count data at hand, it motivates the search for more fleible models for this type of data. In this paper we consider the COM-oisson distribution and study the mean and variance. In this paper we define the Compound COM-oisson distribution with binomial compounding distribution. Its properties are also derived. This paper is laid out as follows: Section describes the study of COMoisson distribution. In section 3, the Compound COM-oisson distribution with binomial compounding distribution is defined and discussed some of its properties. Section 4 deals with the profile likelihood estimation of the newly introduced distribution. Section 5 concludes this paper. COM-oisson Distribution The probability density function of COM-oisson distribution [7] is ( ) λ, (!) Z(λ, ),,,... () λj for λ > and. j Here the parameter governs the rate of decay of successive ratios of probabilities such that ( ) () ( ) λ where Z(λ, ) The probability generating function of COM-oisson distribution is G (s) Z(λs, ) Z(λ, ) (3) The mean and variance are M ean() G () λzλ (λ, ) Z(λ, ) λ Zλλ (λ, ) λzλ (λ, ) λzλ (λ, ) V ar() + Z(λ, ) Z(λ, ) Z(λ, ) d where Zλ (λ, ) [Z(λ, )], dλ d [Z(λ, )] Zλλ (λ, ) dλ ISSN: age 34

3 3 Compound COM-oisson Distribution with Binomial Compounding Distribution Consider the several events that can happen simultaneously at an instant, we have a cluster (of occurences) at a point. Assume that there are Y independent random variables of the form, and N denotes the sum of these random variables. (ie) N Y Then, the compound COM-oisson binomial model is derived by the following assumptions (i) denotes the number of objects within a cluster and it follows binomial distribution with parameters (n, p) (ie) Binomial(n, p) (ii) Y denotes the number of clusters and it follows COM-oisson distribution with parameters λ and. (ie) Y CM (λ, ) This random variable, N formed by compounding these two random variables and Y gives the Compound COM-oisson Distribution with Binomial Compounding Distribution. Its probability generating function (GF) can be derived as follows. The probability mass function (MF) of is n n ( ) p q,,,,... where p >, q >, p + q Its probability generating function is G (s) E(s ) s ( ) n n s p q n (ps) q n (q + ps)n G (s) (q + ps)n ISSN: (4) age 35

4 Also, the random variable Y having probability mass function in () and the probability generating function is GY (s) Z(λs, ) Z(λ, ) (5) Since i s are iid and independent of Y, probability generating function of the random variable N is given by GN (s) E(sN ) E(s Y ) E(s Y /Y y) (Y y) y [E(s )]y (Y y) y GY (G (s)) (6) Z(λG (s), ) Z(λ, ) j [λ(q + ps)n ] Z(λ, ) j λj [q + ps] Z(λ, ) j Epanding the summation and collecting the coefficient of sm in the above equation we get λj (N m) (p)m q m for m,,... Z(λ, ) m j m/n The probability mass function of N is for m Z(λ, ) (N m) λj m m (p) q Z(λ, ) m for m,,... j m/n (7) The mean and variance are derived as follows j (q + ps)n j [λ(q + ps)n ] GN (s) Z(λ, ) j j (q + ps)n j [λ(q + ps)n ] Z(λ, ) j GN () ISSN: j(λ)j Z(λ, ) j 4 age 36

5 M ean(n ) GN () Zλ (λ, ) Z(λ, ) j j [λ(q + ps)n ] (q + ps)n (8) s Z(λ, ) j n j n j(j i) [λ(q + ps) ] λn(q + ps) p (q + ps)n Z(λ, ) j n j n j [λ(q + ps) ] (n )p(q + ps) + Z(λ, ) j j j j(j )λ jλ + (n )p Z(λ, ) j j GN (s) GN () [Zλλ (λ, ) + (n )pzλ (λ, )] Z(λ, ) V ar(n ) GN () + GN () [GN ()] Zλ (λ, ) Zλ (λ, ) [Zλλ (λ, ) + (n )pzλ (λ, )] + Z(λ, ) Z(λ, ) Z(λ, ) " # [Zλ (λ, )] Zλλ (λ, ) + (np p + )Zλ (λ, ) Z(λ, ) Z(λ, ) " # [Zλ (λ, )] V ar(n ) Zλλ (λ, ) + (np + q)zλ (λ, ) (9) Z(λ, ) Z(λ, ) The ratio between variance and mean is Ratio V ar(n ) M ean(n ) " # [Zλ (λ, )] Zλλ (λ, ) + (np + q)zλ (λ, ) Z(λ, ) Z(λ, ) Zλ (λ, ) Z(λ, ) Zλλ (λ, ) Zλ (λ, ) + (np + q) Zλ (λ, ) Z(λ, ) 4 rofile Likelihood Estimation Let N, N,..., Nm be the samples follows the Compound COM-oisson distribution with binomial compounding distribution with parameters λ >,, n > and p >. ISSN: age 37

6 L m Y (N Ni ) i m Y λj (p)ni ( p)ni N Z(λ, ) i i j Ni /n The log likelihood function is m j λj λ + l logl mlog (p)ni ( p)ni log N i j i j Ni /n () Differentiating equation () partially with respect to λ and equating to zero l λ jλj m j Ni /n λj i j Ni /n jλj (p)ni ( p)ni j m λj Ni ( p)ni (p) Ni j Ni Differentiating equation () partially with respect to p and equating to zero l p Ni λj Ni Ni Ni (p) ( p) m Ni p p j Ni /n j λ Ni ( p)ni i (p) N i j Ni /n Then ma log(λ,, n, p m) ma ma log(λ, n, p m) is calculated. λ,,n,p 5 λ,n,p Conclusion The Compound COM-oisson distribution with binomial compounding distribution is introduced and its properties are derived. Also the parameters of the newly introduced distribution are estimated. This distribution can be applied to Bacterial count data. ISSN: age 38

7 References [] CONSUL.C (989) : Generalized oisson Distributions: roperties and Applications, Marcel Dekker Inc., New York/Basel. [] CONWAY R.W, AND W.L. MAWELL (96): A queuing model with state dependent service rates, Journal of Industrial Engineering,, pp [3] JOHNSON N.L, KOTZ S AND KEM A.W (5): Univariate Discrete Distributions, 3rd edition, Wiley Series in robability and Mathematical Sciences. [4] MACEDA E.C (948) : On the Compound and Generalized oisson Distributions, Annals of Mathematical Statistics, Vol.9, No.3. (sept), pp [5] MCGUIRE J.U, BRINDLEY T.A, and BANCROFT T.A (957) : The distribution of European corn borer yrausta Nubilalis (Hbn.) in field corn, Biometrics, 3, [errata and etensions (958) 4, ]. [9.4, 9.5] [6] MEDHI J (): Stochastic rocesses, nd edition, New Age International () Ltd., ublishers. [7] SHMELI G, MINKA T., KADANE J.B, BORLE S AND BOATWRIGHT (5): A useful distribution for fitting discrete data:revival of the COM-oisson ditribution, J.R.Stat.Soc.Ser.C(Appl. Stat), 54, 7-4. [8] SKELLAM J.G. (95) : Studies in statistical ecology I: Spatial pattern, Biometrika, 39, [9.4, 9.5, 9.8] ISSN: age 39