COM-Poisson Neyman Type A Distribution and its Properties

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1 COM-oisso Neyma Type A Distributio ad its roperties S. Thilagarathiam Departmet of Mathematics Nehru Memorial College Trichy, Tamil Nadu, Idia. rathiam.thilaga93@gmail.com V. Saavithri Departmet of Mathematics Nehru Memorial College Trichy, Tamil Nadu, Idia. saavithriramai@gmail.com R. Seethalakshmi Departmet of Mathematics Sastra Uiversity Thajavur, Tamil Nadu, Idia. seetharaaja@yahoo.co.i Abstract I this paper, COM-oisso Neyma type A distributio is itroduced. COM-oisso Neyma type A distributio is a Compoud COM-oisso distributio with oisso compoudig distributio. It is a geeralizatio of Neyma type A distributio. Its properties are also studied. This distributio is foud to be a better distributio for atural disaster data. Key Words: oisso distributio, Neyma type A distributio, COMoisso Distributio, COM-oisso Neyma type A distributio. Itroductio The COM-oisso distributio is a geeralizatio of oisso distributio. It is the geeralizatio of some well kow discrete distributios like egative biomial, Beroulli ad Geometric distributios. I 962, Coway & Maxwell itroduced this distributio i the cotext of queuig systems. I 2005, Galit Shmueli revived this distributio ad used for fittig discrete data. The COMoisso cosists of a extra parameter, which we deote by ν ad which govers the rate of decay of successive ratios of probabilities such that. ISSN: ( x ) xν ( x) age 8

2 Neyma(939) costructed a statistical model of the distributio of larvae i a uit area of a field by assumig that the variatio i the umber of clusters of eggs per uit area could be represeted by a oisso distributio with parameter, while the umber of larvae developig per clusters of eggs are assumed to have idepedet oisso distributio all with the same parameter π. I 949, Thomas itroduced a two-parameter coutig distributio distict from the Neyma type-a oly i that mother pulses appeared i the fial process alog with daughter pulses. Although she origially referred to this as the double-oisso distributio, it is called the Thomas distributio. I 943, Fellr, derived Neyma s distributios as compoud oisso distributios. This iterpretatio makes them suitable for modellig heterogeeous populatios ad reders them examples of apparet cotagio. Neyma distributio also arise as geeralized oisso if the umber of larvae observed at ay plot are assumed to be hatched from oisso distributed egg masses have some other discrete distributio. I this paper, COM-oisso Neyma type A distributio is itroduced. COM-oisso Neyma type A distributio is a Compoud COM-oisso distributio with oisso compoudig distributio. It is a geeralizatio of Neyma type A distributio. Its properties are also studied. I sectio 2,3, COM-oisso ad Neyma Type A Distributios are defied. I sectio 4, COM-oisso Neyma Type A Distributio is defied ad its mea ad variae are obtaied. arameter estimatios are derived i sectio 5. 2 COM-oisso Distributio The probability desity fuctio of COM-oisso distributio is ( x) where Z(, ν) j ν j0 (j!) x, (x!)ν Z(, ν) x 0,, 2,... for > 0 ad ν 0. The probability geeratig fuctio of COM-oisso distributio is (s) Z(s, ν) Z(, ν). ISSN: age 9

3 3 Neyma Type A Distributio The radom variable has a Neyma type A distributio with parameters ad ν. The probability geeratig fuctio of Neyma type A distributio is ν(s ) G (s) e(e ) The probability mass fuctio of Neyma type A distributio is r{ x} 4 ν x e ν (e ν )j x j, x, 2,... x! j0 j! Com-oisso Neyma Type A Distributio The COM-oisso Neyma type A distributio arises i a model formed by supposig that objects (which are to be coutable) occur i clusters. Suppose there are Y idepedet radom variables of the form, ad N deotes the sum of these radom variables, amely N Y The, the COM-oisso Neyma type A distributio model is derived by supposig that (i) deotes the umber of objects with i a cluster where oi(). (ii) Y deotes the umber of clusters, where Y Com oi(µ, ν). This radom variable N, formed by compoudig i this fashio gives rise to the Com-poisso Neyma type A distributio ad its probability geeratig fuctio ca be derived easily. The probability geeratig fuctio of is kow to be G (s) e(s ) () The probability mass fuctio of Y is (Y y) where µy, y 0,, 2,... (y!)ν, j0 for µ, ν 0 The probability geeratig fuctio of Y is ISSN: age 20

4 y GY (s) E(s ) sy (Y y) y sy y µy (y!)ν sy µy y (y!)ν (sµ)y y (y!)ν GY (s) (2) Z(µs, ν) Sice i s are iid ad idepedet of Y, the robability geeratig fuctio of the radom variable N ca be readily foud as follows GN (s) E(sN ) E s y E s y Y y (Y y) y0 y [E(s)] (Y y) y0 y [G (s)] (Y y) GY (G (s)) y0 Z(µG (s), ν) Z(µe(s ), ν) µe(s ) GN (s) j0 j (3) Now, sice the probability geeratig fuctio of N i (3) ca be expressed as e(s ) j0 ISSN: j e j ejs j0 4 age 2

5 upo collectig the coefficiet of sm i the above series, we fid a explicit expressio for the robability mass fuctio of N as. "m # µi e i (i)m (N m) i (i!)ν m! x (N x) (µe )j (j)x x x! j0 This is the probability mass fuctio of the Com-isso Neyma type A distributio ad deote it by N C N T A(µ,, ν) Special Case Whe ν we have Neyma type A distributio. 5 roperties The mea ad variace ca be calculated from the first ad secod derivatives of the probability geeratig fuctio by settig s G0N (s) e j (j)ejs j0 Mea G0 () j0 j G00N (s) e j (j)2 ejs j0 2 G00N () j0 j 2 V ar(n ) G00 () + G0 () [G0 ()]2 2 V ar(n ) ISSN: j0 j 2 + j0 j 5 j 2 j0 age 22

6 From these,we fid the ratio betwee mea ad variace to be j 2 V ar(n ) j0 M ea j j0 6 j0 j + Maximum Likelihood Estimatio Let x, x2,.., x be the samples follows the COM-oisso Neyma Type A distributio with parameter, µ > 0, ad ν 0. L Y (N xi ) i Y i Y i xi (xi!) (µe )j (j)xi j0 Y j xi (µe ) (j) xi (xi!) i j0 The log likelihood fuctio is l log L log i xi xi! log j0! j + i log j0! j µ j ) (j)xi (e Differetiatig l partially with respect to, µ ad ν ad equatig to zero, we get. ISSN: age 23

7 xi xi xi! i i xi xi! i j0 j j0 log(j!) j0 j0 j ) (e )(j)xi j(e j0 j0 + i j0 j0 + )j (j)xi (e i j j ) (j)xi (e )j (j)xi (e log(j!) j ) (j)xi (e j0 j0 )j (j)xi (e Solvig the above three equatios usig umerical techiques, estimators of, µ ad ν ca be obtaied. Refereces [] Coway, R.W. ad W.L.Maxwell(962). A queuig model with state depedet service rates. Joural of Idustrial Egieerig, 2, pp [2] Shmeli G, Mika T., Kadae J.B, Borle S ad Boatwright (2005): A useful distributio for fittig discrete data:revival of the COM-oisso ditributio, J.R.Stat.Soc.Ser.C(Appl. Stat), 54, [3] Davit,F.N, ad Moore,.G.,(954), Notes o cotagious distributio i plat populatios, aals of Botay, New series, 8, 47-53,[ ]. [4] Ceruschi,F, ad Cstagetto,L.,(946) Chais of rare evets, Aals of Mathematical statistics, 7, 53-6,[9.6.]. [5] Sheto,L.R., ad Bowma,K.O.,(967), Remark o large sample estimators for some discrete distributios, Teechoeetrics, 9, , [9.6.,9.6.4.] [6] Barto,D.E,(957), The Modality of Neyma s Cotagious distributio of Type A, Trabajors de Estadistica, 8, 3-22, [9.6.]. [7] riyadharshii J., ad Saavithri,V., COM-oisso olya-aeppli rocess, aper commuicated to the joural of Statistics ad probability letters. ISSN: age 24

Direction: This test is worth 150 points. You are required to complete this test within 55 minutes.

Direction: This test is worth 150 points. You are required to complete this test within 55 minutes. Term Test 3 (Part A) November 1, 004 Name Math 6 Studet Number Directio: This test is worth 10 poits. You are required to complete this test withi miutes. I order to receive full credit, aswer each problem