Parameter Estimation and Hypothesis Testing of Two Negative Binomial Distribution Population with Missing Data

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1 Avlble ole wwwsceceeccom Physcs Poce Ieol Cofeece o Mecl Physcs Bomecl ee Pmee smo Hyohess es of wo Neve Boml Dsbuo Poulo wh Mss D Zhwe Zho Collee of MhemcsJl Noml UvesyS Ch Absc Ou uose s o el wh he mee esmo hyohess es o he equly of wo eve boml sbuo oulos wh mss he cossecy symoc omly of he esmos e ove I o ssc o es equly of wo eve sbuos s lm sbuo e obe 0 0 Publshe by lseve by lseve BV Seleco L Seleco /o ee /o evew ee-evew ue esosbly ue esosbly of ICMPB Ieol of me Commee oze] Oe ccess ue CC BY-NC-ND lcese Keywos:mss mxmum lkelhoo esmo hyohess es Iouco he oblem of mss s vey commo my sues fel exemes see Lekowsk Ls & Sehouwe] Km & Cuy] Mss c bs mee esme A vey of mehos hve bee eveloe o esme he ukow mees of ffee moels whe hee exss mss Refeece ] els wh he mee esmo hyohess es o he equly of wo exoel sbuo ue ye ceso smle wh mss Refeece 4] cosee he esmo fo wo boml sbuos wh lly mss xc Lkelhoo Ifeece fo wo xoel Poulos ue Jo ye -II Ceso ws sue by Blksh Rsoul see 5] Moeove Refeece 6] vese he esmo es fo wo oml oulos wh lly mss I hs e Ou uose s o el wh he mee esmo hyohess es o he equly of wo eve boml sbuo oulos wh mss he cossecy symoc omly of he esmos e ove I o ssc o es equly of wo eve sbuos s lm sbuo e ve he es of hs e s oze s follows I Seco II we ouce ob he mee esmo I Seco III we e he lm sbuo of esmo he hyohess es Publshe by lseve BV Seleco /o ee evew ue esosbly of ICMPB Ieol Commee Oe ccess ue CC BY-NC-ND lcese o:006/jho0054

2 476 Zhwe Zho / Physcs Poce cofece evl bou he ffeece of mees wo oulos e scusse Seco IV Mxmum lkelhoo esmo I hs seco we cose he mxmum lkelhoo esmo fo he mee eve boml sbuo oulos wh mss Cose he follow wo eve boml sbuo oulos whose obbly fucos e f x; C x x- fo x 0 ohewse whee e ukow mees Fo hs sbuo we focus o he cse whee some vlues smle of sze my be mss h s we ob he follow comlee obsevos Y fom wo bove oulos Whe s mss Whe Y s mss Fuhemoe we ssume h P P I wh follows we cose he esmo of Bse o he obsevos he lkelhoo fuco c be we - C ] L Hece he lohm of he lkelhoo fuco s ve by Noe h l C l - l ] l L l L Solv he lo-lkelhoo equo we ob I sml wy we hve Y - ]

3 Zhwe Zho / Physcs Poce he lm oey of esmos I hs seco we cose he lm oey of esmos heoem : s s whee " " eoes coveece lmos suely Poof Noe h e eeely ecl sbue vble we hve s Afe smle clculo we e whch comb wh ves Fuhe by we hve s s s heoem : s s whee " " eoes coveece lmos suely By us he sme meho s we use o ove heoem we c ove heoem so we om he oof hee Lemm : Le k k k hs couous l evo If L N0 he k k ] N0 whee j eoe coveece sbuo j k k j heoem : N0 whee Poof Le W We hve { W } s eeely ecl sbue vble W Le W W W W

4 478 Zhwe Zho / Physcs Poce By mulve cel lm heoem we hve 0 N W W We whee Le We hve By smle clculo we hve By Lemm we hve 0 ] N whee heoem 4: 0 N whee By us he sme meho s we use o ove heoem we c ove heoem 4 so we om he oof hee

5 Zhwe Zho / Physcs Poce es he equly of wo oulos cofece evl fo I hs seco we cose he follow hyoheses: 0 : 0 : 0 H H Fs we esblsh es sscs scuss he lm sbuo of es sscs Le By he so le umbe lw we hve s s s heefoe we c ob he follow esul: heoem 5: 0 ] N Pcully ue he-ull hyohess 0 H we hve 0 ] N Poof By Slusky heoem heoem heoem we c ove heoem 4 Le wh follows we scuss he symoc cofece evl of Fo 0< <ssume h ssfes x e Fo ve cofece level by heoem we hve ] P heefoe we ob he cofece evl of :

6 480 Zhwe Zho / Physcs Poce Ackowleme hs wok s suoe by Nol Nul Scece Fouo of Ch No Refeeces ] J M Lekowsk J R L S A Sehouwe "Sees fo he lyss of mue fom smle suvey: he ol mecl ce ulzo exeue suvey" Mecl Ce vol ] JO Km J Cuy "he eme of mss mulve lyss" Socolocl Mehos & Resech vol ] ZW Zho SYW RW L L "Pmee esmo hyohess es of wo exoel oulos ue ye ceso smle wh msse " Joul of Jl Uvesy Scece o vol ] HY LI ZJOU "smo fo wo boml sbuos wh lly mss " Joul of Gsu Lhe Uvesy Nul Sceces vol ] N Blksh Abbs Rsoul "xc lkelhoo feece fo wo exoel oulos ue jo ye -II ceso" Comuol Sscs & D Alyss vol ] YP Lu "smo es fo wo oml oulos wh lly mss " Joul of Nohes Noml Uvesy Scece o vol

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