Inference on Curved Poisson Distribution Using its Statistical Curvature

Transcription

1 Rsarch Joural of Mahacal ad Sascal Sccs ISSN ol Ju 3 Rs. J. Mahacal ad Sascal Sc. Ifrc o Curvd Posso Dsrbuo Usg s Sascal Curvaur Absrac Sal Babulal ad Sadhu Sachaya Dpar of sascs Th Uvrsy of Burdwa Ws Bgal INDIA Avalabl ol a: Rcvd 5 h Aprl 3 rvsd 4 h Ju 3 accpd h Ju 3 Sascal Curvaur of Bradly fro hlps coparg curvd poal faly of dsrbuos wh corrspodg poal faly of dsrbuo. Th aalyss curvd faly has b ad asy by ha cocp. Bu suabl s procdurs ar uavalabl for dscr curvd po fals. I hs papr a suabl s procdur for Curvd Posso dsrbuo s obad Kywords: Sascal curvaur curvd poal faly curvd Posso dsrbuo llhood fuco.. Iroduco Curvd poal faly of dsrbuos plays a pora rol Sascs. Bu cas of frc o such dsrbuos w hav o fac cra obsacls bcaus curvd poal faly dos o hav good sascal proprs as h poal faly. Bradly fro has copard curvd poal faly wh poal faly roducg h cocp of Sascal Curvaur or fro s Curvaur. H has show hs papr ha fals wh sall curvaur joy good sascal proprs of poal faly as poal faly has curvaur ro. Morovr ca b show ha f o as a apl of a dsrbuo fro poal faly ad dals as a curvd poal faly h h sascal curvaur of such curvd poal faly bcos acly qual o ro. So afr fdg h valus of h volvd parar of a curvd poal dsrbuo for whch h sascal curvaur aas sall valus ca b cocludd ha for such valus of h parar h s sasc of cra hypohss s quval o ha of h corrspodg poal faly. I has b foud ha so frc procdurs ar avalabl fragally for couous dsrbuos of curvd poal faly. Bu cas of dscr dsrbuos fro curvd poal faly w cao fd such procdurs for sg purpos. Usg sascal curvaur hs papr was o fd ou a suabl s procdur for Curvd Posso dsrbuo whch h parar of Posso dsrbuo volvs aohr dscr dsrbuo. I hs papr Ufor dsrbuo s a as h volvd dscr dsrbuo. So prlary dscussos rgardg hs opc Dfo. Curvd poal Faly of Dsrbuos: L d hav a dsrbuo P θ θ Θ R q. Suppos P θ has a dsy pf of h for f p θ η θ T Ψ θ h whr > q. Th h faly {P θ θ Θ} s calld curvd poal faly. apl.: A s of dpdly ad dcally dsrbud rado varabls whch follow N θ θ θ s h uow parar volvd h dsrbuo. Ths s a apl of a curvd poal dsrbuo of couous yp. apl.: Suppos ad Y ar wo rado varabls such ha ~ B p f dpdly Y ~ B p whr p s h uow parar volvd boh h dsrbuos. Ths s a apl of a curvd poal dsrbuo of dscr yp. Iraoal Scc Cogrss Assocao 6

2 Rsarch Joural of Mahacal ad Sascal Sccs ISSN ol Ju 3 Rs. J. Mahacal ad Sascal Sc. Dfo. Sascal Curvaur: Th cocp of ahacal curvaur ds o h curvd ls uclda -spac say {η θ θ Θ} whr Θ s h rval of ral l. For ach θ η θ s a vcor whos copo ws drvavs wh rspc o θ s dod by. η θ.. η θ ad η θ η θ θ θ Ths drvavs ar assud o s couously ghborhood of a valu of θ whr s wshd o df h curvaur. L us also suppos ha a syrc o-gav df ar Σ θ s dfd couously θ. L M θ b a ar wh rs dod by ν θ ν θ ν θdfd by..... υ θ υ θ ηθ ' Σθ ηθ ηθ ' Σθ η θ M θ υ θ υ θ η θ ' Σθ ηθ η θ ' Σθ η θ ad l / γ θ θ 3 M / υ θ. Th γ θ s h curvaur of a θ wh rspc o r producσ θ. γ θ gv. h sascal curvaur of a θ s h gorc curvaur of {η θ : θ Θ } a θ wh rspc o h covarac r produc Σ θ as dfd. ad.. Hr wll sad for h faly of dss {f θ : θ Θ} our curvd poal faly. Sascal Curvaur s also ow as fro s Curvaur. apl.3: Th valu of h sascal curvaur of h dsrbuo gv apl. s for all possbl valus 33 of θ. Tha s for all possbl valus of θ h sascal curvaur hav vry sall valus. So hr h s procdur for θ s quval o ha of for poal Noral dsrbuo. apl.4: Th sascal curvaur of h dsrbuo gv apl. s gv h followg fgur: / plo of h curvaur of h curvd poal faly alus of h curvaur Fgur - Fgur - s s ha for p.6 o.85 h valu of h curvaur s sall A suabl s procdur for p h dsrbuo gv apl. has b rd o b foud ou by Sachaya Sadhu ad Babulal Sal 3. p Iraoal Scc Cogrss Assocao 7

3 Rsarch Joural of Mahacal ad Sascal Sccs ISSN ol Ju 3 Rs. J. Mahacal ad Sascal Sc. Iraoal Scc Cogrss Assocao 8 Dfo. 3 Curvd Posso Dsrbuo: A dsrbuo s dfd as Curvd Posso dsrbuo f s of h followg for. Suppos. ar dpd Po varabls ad hav so jo p..f p. I s plcly assud ha ach > wh probably. Th h jo p..f of s N I N I p f log N s of o-gav grs N s of posv grs. Ths papr was o fd ou a suabl s procdur for hs dsrbuo. Sascal Curvaur of Curvd Posso dsrbuo Cosdr h Curvd Posso dsrbuo as gv dfo.3. Th s p.d.f. s gv quao.3. L ar dpdly ad dcally dsrbud..d as U dscr. Now coparg quao.3 wh dfo.. coparg h p.d.f of Curvd Posso dsrbuo wh h for of Curvd poal dsrbuo w g η η log ad 3. T ad T Now Now covarac bw T ad T s o b foud. cov cov cov 3.6 Now ] [

4 Rsarch Joural of Mahacal ad Sascal Sccs ISSN ol Ju 3 Rs. J. Mahacal ad Sascal Sc. 3.8 Fro cov So cov 3. Thrfor fro ad Σ η ad log η & η / / Usg h cocp of sascal curvaur ad h sascal curvaur of Curvd Posso dsrbuo ca b foud. I hs papr sascal curvaur of h Curvd Posso dsrbuo for varous valus of has b draw hrough a prograg sascal sofwar R. Th sascal curvaurs for varous valus of ar gv fgur- fgur-3 ad fgur-4. Plo of h Sascal curvaur of Curvd Posso dsrbuo wh alus of h curvaur Fgur - Iraoal Scc Cogrss Assocao 9

5 Rsarch Joural of Mahacal ad Sascal Sccs ISSN ol Ju 3 Rs. J. Mahacal ad Sascal Sc. Plo of h Sascal curvaur of Curvd Posso dsrbuo wh alus of h curvaur Fgur-3 Plo of h Sascal curvaur of Curvd Posso dsrbuo wh 3 alus of h curvaur Fgur-4 I hs way h sascal curvaur for hs dsrbuo ca b foud ou for dffr valus of whr s ubr of obsrvaos. Fro abov pcurs s s ha for for dffr valus of parar sascal curvaur has dffr valus; cas of ohr valus of for dffr valus of sascal curvaurs hav h sa valu. Bu all cass h curvaurs assg vary sall valus. Ifrc of h Curvd Posso dsrbuo Thr ar s procdurs for Posso dsrbuo whch dos o coa aohr dscr dsrbuo wh s parar. As s procdur for Posso dsrbuo whch coas aohr dscr dsrbuo wh s parar ar o avalabl o rach o ay dcso abou of h abov od dsrbuo a s procdur should b foud ou. Iraoal Scc Cogrss Assocao

6 Rsarch Joural of Mahacal ad Sascal Sccs ISSN ol Ju 3 Rs. J. Mahacal ad Sascal Sc. To ovrco hs probl hs papr would l o fd ou a s procdur o draw a frc abou h ull hypohss. Fro fgur- fgur-3 ad fgur-4 w s ha h valu of h curvaur s sall for all valus of. Now l us dosra how o g h s hod for such dsrbuo. For apl l us wor for gg s for H: vs. H :. Th s procdur s as follows: Hr ~ Posso whr ~ U s ad s ar dpdly ad dcally dsrbud. L as valus 3 ad ~ U. Th h probably ha as a parcular valu s gv by. 4. Cosdrg s probably of ag a parcular valu h jo probably bcos 4. L as a parcular valu ad as a parcular valu.. Hc for a obsrvao h llhood fuco s L ; 4.3 L b h fd valu of ad l b h fd valu of. Th fro 4.3 log L log c 4.4 whr c s a cosa. log L quag 4.5 o w g ˆ L L L ˆ So H wll b rjcd f 4.8 s lss ha C whr C s a cosa. Now fro 4.8 Iraoal Scc Cogrss Assocao

7 Rsarch Joural of Mahacal ad Sascal Sccs ISSN ol Ju 3 Rs. J. Mahacal ad Sascal Sc. Iraoal Scc Cogrss Assocao Hr. So f f h for a fd f f < / ad f f /. Hc fro 4.8 ad 4. H wll b rjcd f 4. < C. Now C < < < f 4.a f 4.b Hr should b< ad should b>. 4. Thrfor for a s α s for varous valus of h valus of ad o b foud ou such ha α 4.3 Now o achv hs o way s o cosdr h followg α 4.4a ad α 4.4b For calculao purpos quals of 4.3a ad 4.3b ar α 4.5a ad α 4.5b Cosdrg codo 4. ad usg quaos 4.5a ad 4.5b h valus of ad hav b foud ou ag α.. For hs purpos hs papr uss a R progra. Th followg abls wll gv h valus of ad fro whch o ca rach o h dcso. Tabls o fd h valus of corrspodg o dffr valus of ad :

8 Rsarch Joural of Mahacal ad Sascal Sccs ISSN ol Ju 3 Rs. J. Mahacal ad Sascal Sc. alus of Tabl- Tabl o fd h valus of corrspodg o dffr valus of ad Wh alus of R.H.S. of 4.5a alus of / for whch L.H.S. s jus > R.H.S. alus of L.H.S. whch s jus > R.H.S. alus of / O ca procd hs way for or valus of ul s lss ha fro 4. alus of Tabl- Tabl o fd h valus of corrspodg o dffr valus of ad Wh alus of R.H.S. of 4.5a alus of / for whch L.H.S. s jus > R.H.S. alus of L.H.S. whch s jus > R.H.S. alus of / O ca procd hs way for or valus of ul s lss ha fro 4. Iraoal Scc Cogrss Assocao 3

9 Rsarch Joural of Mahacal ad Sascal Sccs ISSN ol Ju 3 Rs. J. Mahacal ad Sascal Sc. alus of Tabl-3 Tabl o fd h valus of corrspodg o dffr valus of ad 3 Wh 3 alus of R.H.S. of 4.5a alus of / for whch L.H.S. s jus > R.H.S. alus of L.H.S. whch s jus > R.H.S. alus of / O ca procd hs way for or valus of ul s lss ha fro 4. I a slar ar abls ca b ford for ohr valus of. Tabls o fd h valus of corrspodg o dffr valus of ad : alus of Tabl-4 Tabl o fd h valus of corrspodg o dffr valus of ad Wh alus of R.H.S. of 4.5a alus of / for whch L.H.S. s jus > R.H.S. alus of L.H.S. whch s jus > R.H.S. alus of / I hs cas w ca o procd farhr bcaus for ohr valus of h valus of wll b lss ha. Bu ca o b < by 4.. alus of Tabl-5 Tabl o fd h valus of corrspodg o dffr valus of ad Wh alus of R.H.S. of 4.5a alus of / for whch L.H.S. s jus > R.H.S. alus of L.H.S. whch s jus > R.H.S. alus of / I hs cas w cao procd farhr bcaus for ohr valus of h valus of wll b lss ha. Bu cao b < by 4.. Iraoal Scc Cogrss Assocao 4

10 Rsarch Joural of Mahacal ad Sascal Sccs ISSN ol Ju 3 Rs. J. Mahacal ad Sascal Sc. Tabl-6 Tabl o fd h valus of corrspodg o dffr valus of ad 3 Wh 3 alus alus of R.H.S. of 4.5aalus of / for whch alus of L.H.S. whch s jus > R.H.S.alus of of L.H.S. s jus > R.H.S. / I hs cas w cao procd farhr bcaus for ohr valus of h valus of wll b lss ha. Bu cao b < by 4.. [L.H.S. Lf Had Sd; R.H.S. Rgh Had Sd] Hc fro abls Tabl- o Tabl-6 o ca asly g h valus of ad for dffr valus of ad. Ad hc dcso rgardg 4.3 ca b a. Th followg subsco coas h corrspodg R progra. R progra usd for fdg valus of ad ;alpha. # Tag ; h sa progra wll ru for 3 ad so o. sq4by # For copuao purpos o 4 has b a. Mor valus ca b a. produc for :6 { produc[]facoral[]**alpha/ } produc # produc sybolss h R.H.S of 4.5a ad 4.5b for dffr valus of lada.o /lada.o #Hr / for dffr valus of. ## PROGRAM FOR 4.5a.. TO FIND TH ALUS OF ## #### PROGRAM TO B DON TAKING #### [] sp-lada.o*[]*lada.o*[]^[] # [I hs progra s s h prsso gv udr suao h L.H.S. of 4.5a for ] #### PROGRAM TO B DON MANY TIMS CHANGING TH ALUS OF "" #### s s[]p-lada.o*[]*lada.o*[]^[5] for :3{ []. s[]p-lada.o*[]*lada.o*[]^[5] [][]. } produc[5];[5] su su[]s[] for :3{ su[]su[-]s[] } su ## PROGRAM FOR 4.5b.. TO FIND TH ALUS OF ## #### PROGRAM TO B DON TAKING #### # Hr s quval o as usd 4.5b [] Iraoal Scc Cogrss Assocao 5

11 Rsarch Joural of Mahacal ad Sascal Sccs ISSN ol Ju 3 Rs. J. Mahacal ad Sascal Sc. sp-lada.o*[]*lada.o*[]^[] # f cssary hs l has o b ru ay s for dffr valus of by for loop # [I hs progra s s h prsso gv udr suao h L.H.S. of 4.5a for ] #### PROGRAM TO B DON MANY TIMS CHANGING TH ALUS OF "" #### ss ss[]p-lada.o*[]*lada.o*[]^[5] for : { [][]-. ss[]p-lada.o*[]*lada.o*[]^[5] [][]-. } produc[5];[5] su su[]ss[] for : { su[]su[-]ss[] } su Cocluso I hs papr a s procdur for Curvd Posso dsrbuo has b foud ou. Hr Ufor dsrbuo has b usd as h dsrbuo volvd Posso dsrbuo. O ay us ohr dscr dsrbuos plac of Ufor dsrbuo also ad ply of scops ar hr for dog furhr rsarch. Ths papr hops h s procdur wll hlp o a dcso abou a daa ha follow Curvd Posso dsrbuo. Rfrcs. fro Bradl. Dfg h Curvaur of a Sascal Probl Wh Applcaos o Scod Ordr ffccy. Th Aals of Sascs Das Gupa A. Profssor of Sascs Purdu Uvrsy Nos o poal Faly. hp:// /~dasgupa/pfaly.pdf yar of accss 3. Sadhu Sachaya Sal Babulal Ifrc a Curvd Boal Dsrbuo Iraoal Joural of Sasa ad Mahaa Lha.L. Roao Josph P. Tsg Sascal Hypohss Thrd do. Sprgr 9 5. Krshaoorhy K. Hadboo of Sascal Dsrbuos wh Applcaos Chapa ad Hall/ CRC 6 6. Rchardso Thoas ad Sprs Pr Parrg Ad Scorg Md Acsral Graphs. Tchcal Rpor No. CMU- PHIL Hur Davd. R. Curvd poal Faly Modls for Socal Nwors Pa Sa Uvrsy 6 8. Laga ad Paso 999 ad Laga Laga s Lawyr Daas. 9. Ghosh R.K. May K.C. A Iroduco o Aalyss Dffral Calculus Par 7. Aposol Mahacal Aalyss Addso Wsly publshg copay 973. Clar Davd R. Thayr Charls A. A Prr o h poal Faly of Dsrbuos Call Papr Progra o Grald Lar Modls 4. Bllo Aladr Chrohuov cor Posror Ifrc Curvd poal Falsudr Icrasg Dsos ar v: 94.33v [ah.st] 3. Sascal sofwar usd: R R vrso Copyrgh C Th R Foudao for Sascal Copug ISBN Plafor: 386-pc-gw3/386 3-b Iraoal Scc Cogrss Assocao 6