A Family of Multivariate Abel Series Distributions. of Order k

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1 Appled Mthemtcl Scences, Vol. 2, 2008, no. 45, A Fmly of Multvrte Abel Seres Dstrbutons of Order k Rupk Gupt & Kshore K. Ds 2 Fculty of Scence & Technology, The Icf Unversty, Agrtl, Trpur, Ind e-ml: rupk_cftech@yhoo.co.uk & 2 Deprtment of Sttstcs, Guht Unversty, Guwht, Assm, Ind e-ml: dskkshore@gml.com Abstrct In ths pper n ttempt s mde to defne the multvrte bel seres dstrbutons (MASDs) of order k. From MASD of order k, new dstrbuton clled the qus multvrte logrthmc seres dstrbuton (QMLSD) of order k s derved. Some well known dstrbutons re lso obtned by new method of dervton. Lmtng dstrbuton of QNMD of order k re studed. Mthemtcs Subect Clssfcton: 62E5; 62E7 Keywords: Multvrte Abel seres dstrbutons of order k, qus multvrte logrthmc seres dstrbuton of order k, qus multnoml dstrbuton of order k, qus negtve multnoml dstrbuton of order k. Introducton In ths study, consderng the multvrte Abel seres dstrbuton of order k, we hve defned multvrte Abel seres dstrbutons of order k. From the MASDs of order k, new dstrbuton clled Qus Multvrte logrthmc seres dstrbuton of order k s obtned. Also vrnt of qus negtve multnoml dstrbuton of order k s studed. Moreover, on usng new method of dervton some well known dstrbutons, vz. qus multnoml dstrbuton of type-i of order k (QMD-I (k)), qus multnoml dstrbuton of type-ii of order k (QMD-II (k)), multple generlzed posson dstrbuton of order k (MGPD (k)) etc. re obtned. A property,.e. the lmtng dstrbuton of the QNMD of order k hs been found out.

2 2240 R. Gupt nd K. K. Ds 2. Multvrte Abel seres dstrbuton of order k nd ts specl cses Let us consder fnte nd postve functon f ( ) of ( ) =,..., mk, where ech ( = () m), = () k s non-negtve nteger. For ny rel z, we hve the multnoml bel seres expnson of order k s, x ( k ) ( ) (, ) = ( ) /! ( ) = xz x = x = =.. () m k f f z x z x d f where the summton s over x, x 2, x 3,..., x k such tht x = x nd ech x ( = () m) ( k ) beng non-negtve nteger nd the fctor d f ( ) = xz / x! s denoted by β ( x, z) s ndependent of, whch s lwys greter thn zero. The domn of = (,..., mk ) s subspce of n mk-dmensonl prmeter spce subect to restrctons 0, f z 0 nd ( xz) 0, f z 0, z belongng to sutble subect of rel numbers. Thus, (2.) cn be wrtten, x ( ) (, ) = β (, ) ( ).. (2) x = x f f z x z x z usng the forml seres expnson (2.), we suggest the followng defnton for the multvrte bel seres dstrbuton of order k (MASD (k)) Defnton A multvrte dscrete dstrbuton of order k, pk ( x) s sd to be MASD (k) fmly, f t hs the followng probblty functon (p.f.), x ( k ) pk ( x) pk ( x;, z) = ( xz) /( k )! d f ( ) / f( ) = xz x = x... (3) where x = x, ech x beng non-negtve nteger, the ( = () m; = () k) nd z re prmeters nd f() re stted n (2.). For z = 0, the probblty functon (2.3) becomes multvrte power seres dstrbuton of order k.

3 Multvrte Abel seres dstrbutons 224 For z = 0 nd k =, the probblty functon (2.3) becomes multvrte Abel seres dstrbuton (Nnd nd Ds, 996) nd the z = 0, t becomes usul multvrte power seres dstrbuton (Ptl, 965). Dervton of some dstrbutons I. Here we derve new dstrbuton from MASD (k), clled the multvrte logrthmc seres dstrbuton of order k. Let us consder Abel seres expnson of order k of the logrthmc seres functon f() gven by, x! x log = x ( ) ;..... (4)! = x x ; x = 0,,...; for m, x > 0,0 < < ( for m& k); < Thus the ssocted MASD-fmly of order k hs the followng probblty functon, x! x pk ( x) = x ( ) ;......(5)! = x x log ; x = 0,,...; for m, x > 0,0 < < ( for m& k); < ~ MLSDk(,..., mk) (Ak, Kubok nd Hrno (984)) For k =, the probblty functon (2.5) becomes usul MLSD wth prmeters, 2, 3,..., m,.e., m x! p x x x x m m = x (,..., m) = ; 0; 0 m m = = x! log = =..... (6) For m =, the p.f. (2.5) becomes the multprmeter logrthmc seres dstrbuton of order k (Phlppou, 988). II. Here we derve new dstrbuton from MASD (k), clled the qusmultvrte logrthmc seres dstrbuton of order k (QMLSD (k)) s follows.

4 2242 R. Gupt nd K. K. Ds Consder the logrthmc seres functon f ( ) log ( ) = mk nd (,..., mk ) expnson of log ( ) of order k s, =, where =. Then the multvrte Abel seres x! x log ( ) = ( x z) x z x! = x where ( x ) x = x s defned n (2.) nd 0 < = mk <. Thus the ssocted MASD fmly of order k hs the followng p. f. x! ( x )! [ log( ) ] x k( ) = x = x x.(7) p x ( xz) xz ; () () x 0, = m, = k& 0 < < x.(8) The p.f. (2.8) s clled the QMLSD (k). If z 0, then the p. f. (2.8) becomes the multnoml logrthmc seres dstrbuton of order k. For k =, the p.f. (2.8) becomes QMLSD (Nnd & Ds, 996) nd then s z 0, t becomes the common multnoml logrthmc seres dstrbuton (Johnson & Kotz, 969, p.303). III. (k)). Next we obtn qus multnoml dstrbuton of type-i of order k (QMD-I Let us consder the smple seres functon f ( ) = ( + b) n, where ( ) = + +. Then the multvrte Abel seres =,..., mk nd... mk expnson of ( + b) n of order k s, n x ( + b) = ( xz) ( b+ xz) x = x x n Hence the correspondng MASD fmly fnds the p.f. of QMD-I (k), n x n x n pk( x) = ( xz) ( b+ xz) /( + b)... (9) x n x

5 Multvrte Abel seres dstrbutons 2243 where, 0 x = x n Suppose, n = x n! n x! x! ( ) ( + b), ech x beng non- negtve nteger nd p = ; = () m& = () k; p 0 = ( + b) ndφ = Usng (2.0) n (2.9), we get z ( + b) n p x p x p p x m k n x ( 0 φ) ( φ).(0) x k( ) = +..() x = = m k where, 0 x n, p = nd = = For k =, we get the p.f. (2.9) s QMD-I (Jnrdn, 975). IV. Now, we derve the multple generlzed posson dstrbuton of order k. Let us consder the exponentl seres functon f ( ) = e, where (,..., mk ) expnson of ( ) = nd = mk. Then the multvrte Abel seres f = e s, x xz e = ( x z) /( x )! e...(2) x = x where x = x s stted n (2.) Thus the correspondng p.f. of the MASD fmly of order k s, m k ( ) ( ) x ( x z) pk( x) = [ xz e / x!]..... (3) = = where x = x nd ech x beng non-negtve nteger. The p.f. (2.3) s known s the MGPD of order k nd for k =, the p.f. (2.3) reduces to MGPD (Jnrdn, 975). V. Fnlly, we derve the qus negtve multnoml dstrbuton of order k. Let us consder the seres functon, f ( ) = ( b ) n, where ( ) = + +. Then the multvrte Abel seres =,..., mk nd... mk expnson of ( b ) n of order k s,

6 2244 R. Gupt nd K. K. Ds Γ n+ x n x n x ( b ) = ( xz) b xz x = x x! Γ( n ) where re gven n (2.). x = x Then the ssocted fmly of the MASD of order k hs the p.f. Γ n+ x n x x n pk( x) = ( xz) b xz ( b ) x! ( n Γ )... (4) ; x 0; = () m nd = () k. Suppose, p = ; = () m& = () k; ( b ) b z Q = ndφ =...(5) ( b ) ( b ) Applyng (2.5) to the p.f. (2.4), we get Γ n+ x n x x pk( x) = P( P xφ) Q xφ..(6) x! ( n Γ ) where P x φ 0 nd Q P = The probblty functons (2.4) nd (2.6) re clled QNMD of order k. If z = 0, then (2.4) nd (2.6) reduces to negtve multnoml dstrbuton of order k. If k =, then (2.4) nd (2.6) reduces to QNMD nd then for z = 0, t reduces to common negtve multnoml dstrbuton (Johnson nd Kotz, 969, p. 292). 3. Propertes of QNMD of order k Lmtng dstrbutons The QNMD of order k, (2.6) wth P ( = () m, = () k), Q, φ nd n tends to multple generlzed posson dstrbuton wth prmeters λ ( = () m, =

7 Multvrte Abel seres dstrbutons 2245 () k) nd ϕ, s n, P 0 nd φ 0, such tht np = λ nd nφ = ϕ. The probblty functon of ths lmtng dstrbuton s gven n (2.3). Acknowledgements One of the uthors, Rupk Gupt s grteful to Dr. R. K. Ptnk, Pro Vce-Chncellor of The Icf Unversty, Trpur, Ind nd Prof. J. J. Kwle, Drector, INEUC for ther constnt encourgements nd motvtons to pursue reserch works. The uthor cknowledges the fnncl support receved from the Icf Unversty, Trpur. Also, the uthors thnks the referees for ther helpful suggestons nd comments. References [] Ak, S. nd Hrno, K. (988). Some chrcterstcs of the bnoml dstrbuton of order k nd relted dstrbutons, Sttstcl Theory nd Dt Anlyss II, (ed. K. Mtust), -222, North Holnd. [2] Ak, S., Kubok, H nd Hrno, K. (984). On dscrete dstrbutons of order k, Ann. Inst.Sttst. Mth., 36, [3] Ak, S. nd Hrno, K. (994). Dstrbutons of numbers of flures nd successes untl the consecutve k successes, Ann. Inst. Sttst. Mth., 46, [4] Chrlmbdes, Ch. A. (986). On dscrete dstrbutons of order k, Ann. Inst. Sttst. Mth., 38, [5] Comtet. L. (974). Advnced Combntorcs, D. Redl Publshng Compny, Inc., Boston, U.S.A [6] Consul, P. C. (974). A smple urn model dependent on predetermned strtegy, Snkhy, B.36, [7] Consul, P.C.nd Jn, G.C.(973).A generlzton of the Posson dstrbuton, Technometrcs, 5(4), [8] Ds, K. K.(Mrch, 993). Some spects of clss of qus bnoml dstrbutons, Assm Sttstcl Revew, 7, [9] Jnrdn, K. G. (975). Mrkov-Poly urn models wth predetermned strteges I, Gurt Sttst. Revew, 2, 7-32.

8 2246 R. Gupt nd K. K. Ds [0] Johnson, N. L. nd Kotz, S. (969). Dscrete Dstrbutons, John Wley nd Sons, Inc., New York. [] Hrno, K. (986). Some propertes of the dstrbutons of order k, Fboncc Numbers nd Ther Applctons (eds.a. N. Phlppou, G. E. Bergum nd A. F. Hordm), 43-53, Redel, Dordrecht. [2] Lng, K. D. (988). On Bnoml dstrbutons of order k, Sttst. Probb. Lett., 6, [3] Nnd, S. B. nd Ds, K. K. A Fmly of the Abel Seres Dstrbutons, Snkhy: The Indn Journl of Sttstcs, 994, Volume 56, Seres B, Pt. 2, pp [4] Nnd, S. B. nd Ds, K. K. A Fmly of the Multvrte Abel Seres Dstrbutons, Snkhy: The Indn Journl of Sttstcs, 996, Volume 58, Seres A, Pt. 2, pp [5] Phlppou, A. N., Georghou, C. nd Phlppou, G. N. (983). A generlzed geometrc dstrbuton nd some of ts propertes, Sttst. Probb. Lett.,,7-75. Receved: Februry 7, 2008