BULLETIN OF MATHEMATICS AND STATISTICS RESEARCH

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1 Vol.6.Iue..8 (July-Set.) KY PUBLICATIONS BULLETIN OF MATHEMATICS AND STATISTICS RESEARCH A Peer Revewed Internatonal Reearch Journal htt: Eal:edtorbor@gal.co RESEARCH ARTICLE A GENERALISED NEGATIVE BINOMIAL DISTRIBUTIONAND ITS IMPORTANT FEATURES Reader n Stattc, R.R.M. Cau, Janakur,Trbhuvan Unverty, Neal Eal- ah.bnod@gal.co ABSTRACT In th aer, generalzaton of Negatve bnoal dtrbuton ha been obtaned by ng generalzed Poon dtrbuton of Conul and Jan (97) wth the two-araeter gaa dtrbuton. It robablty denty functon ha been obtaned. The frt four oent of the dtrbuton have been obtaned and the etaton of t araeter by the ethod of oent ha been dcued. Iortant feature of th dtrbuton have been dcued.th dtrbuton ha been found ore general n nature and wder n coe than the negatve bnoal dtrbuton. The dtrbuton ha been ftted to oe well known data-et havng varance greater than ean and t ha been oberved that the generalzed negatve bnoal dtrbuton gve better ft than the negatve bnoal dtrbuton. Keyword: generalzed Poon dtrbuton, cooundng, negatve bnoal dtrbuton, goodne of ft, oent. INTRODUCTION In the theory of Stattc, gaa dtrbuton lay a very ortant role. It a eber of two- araeter faly of contnuou robablty dtrbuton. The robablty denty functon of gaa dtrbuton can be ereed n ter of the gaa functon araeterzed n ter of a hae araeter '' and cale araeter 'a'. A contnuou rando varable ad to follow two araeter gaa dtrbuton wth araeter and a, f t robablty denty functon gven by a g a ;,. e a = Γ ( ) For and, a > For a =, t reduce to the one araeter gaa dtrbuton and for = t reduce to the eonental dtrbuton. The r th oent about orgn of gaa dtrbuton obtaned a (.) 8

2 Vol.6.Iue..8 (July-Set) Bull.Math.&Stat.Re (ISSN:48-58) ( r) r r = a ; Where (-r) = (+)... (+r+) (.) and the ean and varance of the gaa dtrbuton are a and a reectvely. Gaa dtrbuton ha alo been ued a a ng dtrbuton wth the varou robablty dtrbuton. It well known that a contnuou ture of Poon dtrbuton where the ng dtrbuton a gaa dtrbuton reult n a negatve bnoal dtrbuton. That, we can vew the negatve bnoal a a Poon ( ) dtrbuton, where telf a rando varable, dtrbuted accordng to Gaa (a, ) whch can ybolcally be hown a NBD (, ) = Poon ( ) Gaa (a, ) (.) Where = ( + aa ) Negatve bnoal dtrbuton ha been found to rovde ueful rereentaton n accdent Stattc, brth and death roce. There are of coure tuaton where a good ft not obtanable wth negatve bnoal dtrbuton, and n uch cae t uual to conder the oblty of a ture of dtrbuton or a contagou dtrbuton havng ore than two araeter. In the reent aer, a generalzaton of negatve bnoal dtrbuton ha been obtaned by ng the GPD wth the two araeter gaa dtrbuton. The varou aect of the reultant dtrbuton uch a t oent, etaton of araeter, goodne of ft, have alo been obtaned. GENERALISED NEGATIVE BINOIAL DISTRIBUTION (GNBD) A generalaton of the negatve bnoal dtrbuton can be obtaned by takng the Conul and Jan' GPD n lace of the clacal Poon dtrbuton whch can ybolcally be hown a θ Gaa (a, ) (.) The robablty denty functon of th Gaa ture of GPD obtaned a follow. GPD (, ) P ( + ) ( + θ) a θ e a e =. d (.)! Γ θ a e ( + a) + θ e = + Γ! ( ) d a θ e Γ +! ( ) + a + a = ( θ ) ( ) Γ + a Let a(+a) = and (+a) = q ( ) ( θ ) ( ) θ! +! = e q!!!!.) ( +! )!! q (.)! + θ θ θ = e q + θ e The ereon (.) the robablty denty functon of a gaa ture of GPD. At θ =, t reduce to robablty a functon of negatve bnoal dtrbuton. Hence, the relaton (.) ay be called a a generaled negatve bnoal dtrbuton (GNBD). It can alo be ereed a = = P for ( θ ) ( ) Γ + θ e + q = for =,,, (.4) 8

3 Vol.6.Iue..8 (July-Set) Bull.Math.&Stat.Re (ISSN:48-58) Puttng = n the relaton (.4), we get robablty denty functon of a new generaled for of the geoetrc dtrbuton gven by Mhra (9) a P ( )( θ ) Γ ( + ) θ q for =,,, (.5) e =. MOMENTS OF GNBD The r th oent about orgn of the generaled for (.) obtaned a r ( ) r = E E ( θ) ( + ) + Γ ( + ) Γ( ) r a θ e a e r = d (.) = Obvouly the ereon under bracket the r th oent about orgn of the GPD. Takng r = and ung the ean of GPD for the ereon n bracket, the frt oent about orgn of the GNBD obtaned a a a e = d (.) Γ( ) ( θ ) q.e Mean ( ) = = θ a θ Takng r = n (.) and ung the econd oent about orgn of the GDP, the econd oent about orgn of the GNBD can be obtaned a (.) a a e = + d (.4) Γ( ) ( + ) q = + θ a ( θ) a ( + ) q = + Takng r = n (.) and ung the thrd oent about orgn of the GPD for the ereon n bracket, the thrd oent about orgn of the GNBD can be obtaned a ( + ) a e Γ( ) (.5) a θ = + + d (.6) 5 4 ( θ ) + q + q + + q = + + θ θ θ 5 4 Slarly, takng r = 4 n (.) and ung the fourth oent about orgn of the GPD for the ereon n bracket, the fourth oent about orgn of the GNBD can be obtaned a a e Γ( ) (.7) + 8θ + 6θ 4 a 7+ 8θ 6 4 = d (.8) θ θ θ + 8θ + 6θ q θ q = q q The central oent of the GNBD ha thu been obtaned a (.9) 84

4 Vol.6.Iue..8 (July-Set) Bull.Math.&Stat.Re (ISSN:48-58) = ( θ) ( ) ( θ) q + q q = +. θ q q = + = + ( θ ) + q + q + + q = θ θ θ ( + ) ( θ) q q q q + + on a lttle lfcaton, we get ( + θ ) q q q = + + b = After a lttle lfcaton, we get ( 6+ ) + 8θ + 6θ q θ q = (6+ ) q q Iortant feature of the GNBD (a) Fro ean and varance of GNBD q θ = + ( ) (.) (.) (.) q e.. ( θ ) = > Th gve the nequalty > θ (.) The araeter θ aear to lay an ortant role n elanng the nequalty between ean and varance. At θ =, we have > whch the charactertc of the negatve bnoal dtrbuton. But for non-zero value of θ the lower lt for the varance becoe dfferent fro ean. For < θ <, the lower lt of the varance hgher than the ean, obvouly rovdng an roved lower lt of the varance. Slarly for < θ < the lower lt of the varance lower than the ean. Thu the araeter θ ha caacty to tretch the lower lt of the varance u and down accordng to the oberved dtrbuton. Obvouly for non-negatve value of θ the nequalty between ean and varance elaned n a better way than that elaned by the negatve bnoal dtrbuton. Thu the araeter θ take nto account the nequalty between ean and varance ore cloely than that taken by NBD and o t eected that n all uch cae 85

5 Vol.6.Iue..8 (July-Set) Bull.Math.&Stat.Re (ISSN:48-58) where varance greater than ean, the GNBD would elan oberved data ore cloely than the negatve bnoal dtrbuton. (b) Generaled Logarthc Sere Dtrbuton (GLSD) obtaned by Bnod Kuar Sah() the ltng for of zero-truncated GNBD: Proof: Probablty a functon of the zero-truncated GNBD can be obtaned fro (.) a P ( ) ( ) P =! ( θ ) θ e + q = (.4) Th can be ut n eanded for a =,,,... θ e P ( ) = ( + )( + )...( + ) ( )! ( ) q ( θ ) ( ) ( )!! Takng lt, we get Lt θ Lt Lt Lt P ( ) = e ( ) ( + )( + ) ( + ) q ( θ ) (! )! Lt.e P ( ) ( θ ) θ qe q = (.5) log q! It the ltng for of zero-truncated GNBD. It reduce to the Logarthc ere dtrbuton atθ =, we ter t a a generalzed logarthc ere dtrbuton (GLSD). 4. ESTIMATION OF PARAMETERS The generalzed ture dtrbuton (.) cont of three araeter, andθ and o for etatng thee araeter by the ethod of oent, the frt three oent are requred. The varance ( ) of the GNBD can alo be ereed a q = + θ q Let, = ( θ ) n (4.), we get (4.) a K ( Say) = + = (4.) K.e. a = (4.) Ung the thrd central oent and the ean of the GNBD, we have 86

6 Vol.6.Iue..8 (July-Set) Bull.Math.&Stat.Re (ISSN:48-58) ( θ ) 4 a a K ( Say) = = (4.4) Subttutng the value of θ = and a = K 4 ( ) n (4.4) and after a lttle lfcaton, we get f = K + K K = (4.5) Th a olynoal n of the fourth degree and ay be olved by ung oe teratve ethod uch a the Newton Rahon ethod or the Regula Fal ethod. and. The etate of the araeter, and θ can alo be obtaned by ung P(=), Takng logarth both de of the frt relaton of (.4), we get Log P(=) = log log P =.e. = (4.6) log The ean of the GNBD can alo be wrtten a = a.e. = a (4.7) Equatng the relaton (4.6) and (4.7), we get log P = Let Alo, P = log =.e. log = a log alog P = = K. The relatoncan be wrtten a (4.8) log = (4.9) ak K = = and q = + a K + K + ( ) Subttutng the value of and an (4.9), we get an etatng equaton for a log = K K + K.e. f ( ) K ( K ). Or, K (K - ) = log K (4.) = log =... (4.) K + After relacng the oulaton oent by the correondng ale oent, th equaton ay be olved by the Newton Rahon ethod or Regula Fal ethod for the value of whch gve an etate of θ. Subttutng th value of n (4.) the etate of q and o of can be obtaned. Subttutng thee etate of θ and n the ereon for ean, an etate of can be obtaned. 87

7 Vol.6.Iue..8 (July-Set) Bull.Math.&Stat.Re (ISSN:48-58) 5. GOODNESS OF FIT The GNBD uoed to have t alcaton to all thoe cae where the negatve bnoal dtrbuton ha. The GNBD wa ftted to a nuber of data-et where revouly negatve bnoal dtrbuton wa ued by other. It wa found that the ft gven by the GNBD were cloer than thoe gven by negatve bnoal dtrbuton. Only three of the are beng reorted here. In table, the data et of Greenwood and Yule(9) on the nuber of accdent to 647 woen workng on H.E. Shell durng fve week condered. In table, the Garan data on count of Euroean red te on ale leave and n table, the Bortkewtch' data on the nuber of death caued by hore kck n the Pruan ary have been condered. In the frt two data et the ean le than the varance ndcatng the cae of negatve bnoal dtrbuton and n the thrd data et a ean not gnfcantly dfferent fro the varance, o t the cae of Poon dtrbuton. The oberved and eected frequence accordng to the GNBD have been gven n the table. For quck coaron the eected frequence accordng to the negatve bnoal dtrbuton have alo been gven n the reectve table and and thoe accordng to the Poon dtrbuton n the table. Table : Accdent of 647 woen workng on H.E. Shell durng 5 week Nuber of accdent Oberved Frequency Eected Frequency Negatve Bnoal GNBD Total =.4654 χ = =.699 =.7984 d.f. = ˆ =.47 ˆ = P( χ ) =.5. θˆ =.6499 Table : Count of the nuber of Euroean red te on ale leave Nuber of red te Oberved Frequency leaf Eected Frequency Negatve Bnoal GNBD

8 Vol.6.Iue..8 (July-Set) Bull.Math.&Stat.Re (ISSN:48-58) Total = χ =.9.7 = = d.f. = 4 ˆ =.988 ˆ =.676 P( χ ) = θˆ =.56 Nuber of death Table : Death due to hore-kck n the Pruan Ary Oberved Frequency Eected Frequency Poon GNBD Total.. =.6 χ =.. = =.5956 d.f. = ˆ =.65 ˆ =.9799 P( χ ) =.96.9 θˆ = CONCLUSION Coarng the eected frequence wth the oberved one and alo by ther ch-quare value and -value, t evdent that the GNBD rovde very cloe ft to the data-et. Th dtrbuton gve cloer ft to the dcrete data of negatve bnoal n nature than the clacal negatve bnoal dtrbuton. Generaled Logarthc Sere Dtrbuton (GLSD) obtaned by Bnod Kuar Sah() the ltng for of zero-truncated GNBD. Acknowledgeent The author eree ther grattude to the referee for h valuable coent and uggeton whch roved the qualty of the reearch artcle. The hearty thank goe to Profeor A. Mhra for h treendou gudance and valuable uggeton. REFERENCES []. Conul, P.C., and Jan, G.C. (97a): A generalaton of the Poon dtrbuton, Technoetrc, 5, []. Greenwood, M., and Yule, G.U. (9): An nqury nto the nature of frequency dtrbuton rereentatve of ultle haenng wth artcular reference to the occurrence of ultle attack of deae or of reeated accdent, Journal of the Royal Stattcal Socety, Sere A, 8,

9 Vol.6.Iue..8 (July-Set) Bull.Math.&Stat.Re (ISSN:48-58) []. Mhra, A. (9): A new generalzed for of geoetrc dtrbuton, Internatonal Journal of Agrcultural and Stattcal Scence, Vol.5, No., [4]. Sah, B.K. (): Generalaton of oe countable and contnue ture of Poon dtrbuton and ther alcaton, Ph.D. The, Patna Unverty,