Generalized Estimators Using Characteristics of Poisson distribution. Prayas Sharma, Hemant K. Verma, Nitesh K. Adichwal and *Rajesh Singh

Transcription

1 Geeralzed Esaors Usg Characerscs of osso dsrbuo raas Shara, Hea K. Vera, Nesh K. Adchwal ad *Rajesh Sgh Depare of Sascs, Baaras Hdu Uvers Varaas(U..), Ida-5 * Corresdg auhor rsghsa@gal.co Absrac I hs arcle, we have prosed a geeralzed class of esaors, eeal class of esaors based o adapo of Shara ad Sgh (5) ad Solak ad Sgh () ad sple dfferece esaor for esag ukow pulao ea case of osso dsrbued pulao sple rado saplg whou replacee. The ressos for ea square errors of he prosed classes of esaors are derved o he frs order of approao. I s show ha he adaped verso of Solak ad Sgh (), eeal class of esaor, s alwas ore effce ha usual esaor, rao, produc, eeal rao ad eeal produc pe esaors ad equal effce o sple dfferece esaor. Moreover, he adaped verso of Shara ad Sgh (5) esaor are alwas ore effce ha all he esaors avalable leraure. I addo, heorecal fdgs are supred b a eprcal sud o show he superor of he cosruced esaors over ohers wh earhquake daa of urke. Ke words: Aular arbue, b-seral, ea square error, sple rado saplg.. Iroduco I he saplg leraure, s well kow ha effcec of he esaor of pulao paraeers of a sud varable ca be creased b he use of aular forao relaed o, whch s hghl correlaed wh sud varable. Several auhors cludg Sgh ad Kuar (), Sgh ad Solak () ad Shara ad Sgh(4a,b) suggesed esaors usg aular forao uder dffere suaos bu whou cosderg he dsrbuos of sud ad aular varae. However soe dsrbuo, lke osso dsrbuo s geerall used for he aural pulao o ress he probabl of a uber of rare eves. Afershocks cosue he greaes proros of shocks a earhquake caalogue ad f afershocks are effecvel cosdered, he ca gve

2 us forao for udersadg he whole ccle of sesc acv. Due o hs reaso esag he uber of afershocks s ra sesolog ad has receved uch aeo rese leraure (see Ozel,). Rao pe esaors ca gve us forao abou he uber of afershocks a specfed rego. Cosequel, hs paper we have prosed a geeral class of eeal esaors ad dfferece esaor for pulao ea usg aular forao fro a osso dsrbued pulao. Here we used he earhquake daa for eprcal sud sce earhquake are rare eves ad geerall follows a osso dsrbuo. Cosder a pulao U u,u,..., u of sze N defable ad dsc us. Le ad be he sud ad aular varables assocaed wh each u j,,... N u j of he pulao respecvel. Assue ha s are kow us ad Y s are ukow us for all he pulao. Supse a rado saple of sze s draw usg sple rado saplg whou replacee (SRSWOR) fro he pulao. Le us assue ha he pare pulao has a osso dsrbuo. We kow ha he aure of he saplg dsrbuo depeds o he aure of he pulao fro whch he rado saple s draw herefore, rado saples whch are draw fro a osso dsrused pulao follows also a osso dsrbuo. Furher les us selec observaos,, (=,...,) fro a osso dsrbued pulao. Usg hs saplg desg we ca defe classcal rao esaor as rao (.) where ad are he saple eas of he sud ad aular varables fro osso dsrbued pulao, respecvel. f we supse ha aular varable has osso dsrbuo wh paraeer, he he values of he paraeers of aular varable are gve b, S, C S respecvel, furher le he sud varable has a osso dsrbuo wh paraeer, he he resso for sud varable ca be wre as Y, S, C S Y. The correlao coeffce bewee he sud varable ad aular varable s obaed b usg rvarae reduco ehod. The rvarae reduco ehod s a appealg ehod for cosrucg bvarae osso dsrbuo (See La (995)). The ehod s o creae a par of depede osso dsrbued

3 rado varable fro hree depede osso dsrbued rado varables. Le k, w ad z are depedel osso dsrbued observaos fro a osso dsrbued pulao he a bvarae osso dsrbuo of sud varable ad aular varable s geeraed b seg k z ad w z for =,,...,. Assug ha he paraeers of k, w ad z are, ad respecvel. The correlao coeffce bewee he sud varable ad aular varable s defed as Cov(, ) S S Here, s resrced o be srcl sve sce, ad are alwas sve. Sce we have draw he observaos, paraeers ad,=,,..., fro a osso dsrbued pulao wh, he we have k z / ad w z /. The covarace bewee ad s defed as Cov, E, E E To fe he bas ad ea square error, le us defe e Y, Y e E e E e V, Ee e Y Cov, Y V Now ressg esaor rao er of e s, we have Y Y e e e e e r The Bas ad MSE of he esaor rao are respecvel, gve b

4 Bas MSE Y rao (.) Y rao (.). Esaors Leraure Koucu ad Ozel () suggesed a eeal rao esaor for esao ukow pulao ea as k (.) The bas ad MSE of he esaor k are respecvel, gve b Bas k (.) 8 Y 4 MSE k Y (.) 4 I case of egave correlao coeffce bewee sud varable ad aular varable Koucu ad Ozel () suggesed eeal produc esaor as k (.4) The bas ad MSE of he esaor k are respecvel, gve b Bas 5 k (.5) 8 Y 4 9 MSE k Y (.6) 4. Adaped fal of esaors

5 Followg Solak ad Sgh (), we prose a class of esaors of pulao ea sple rado saplg usg forao fro a osso dsrbued pulao, as (.) where s suabl chose scalar. Here, we oe ha. For =, ha s usual ubased esaor sple rado saplg.. For =, k, ha s eeal rao esaor.. For =-, k, ha s eeal produc esaor. The prosed class of esaors s a geeralzed esaor class of ubased esaors eeal rao k ad eeal produc esaor k Epressg he esaor equao (.) ers of e s, we have e Y e (.) e Neglecg he ers havg wer greaer ha wo of he above resso, we have e ee Y Ye e (.) 8 Takg ecaos of boh sdes of equao (.), we ge he bas of he esaor o he frs degree of approao, as, Bas Y (.4) 8 Squarg boh sdes of equao (.) ad eglecg ers of e s havg wer greaer ha wo, we have e Y Y e e e (.5) 4 Takg ecaos of boh sdes of above resso, we ge he MSE of he esaor o he frs degree of approao, as

6 Y MSE ( ) (.6) 4 I s eresg o oe here ha f we pu =,,-, ad - equao (.6), we ge respecvel he MSE ressos of he esaors, sple rao, sple produc, k ad k up o he frs order of approao. Dffereag equao (.6) wh respec o ad he equag o zero, we ge he opu value of as * (.7) ug he opu value of o equao (.6), we ge he u MSE of he esaor as Y MSE ( ) (.8) I saplg leraure a auhors suggesed dfferece pe esaors o ge ore effce esaes, ovg alog hs dreco, we have suggesed a dfferece esaor for as R b (.9) Epressg he esaor R equao (.9) ers of e s, we have R Y e be Or Y Ye be R (.) Splfg ad eglecg ers of e s havg wer greaer ha wo ad akg ecaos boh sdes, we have MSE Y b R by (.) Dffereag equao (.) wh respec o b ad equag o zero, we ge he opu value of b as

7 b * Y (.) ug he opu value of b o equao (.), we ge he u MSE of he esaor R as Y MSE ( R ) (.) The MSE resso (.9) s sae as he MSE resso of geeralsed class of eeal esaor defed earler. 4. The Suggesed Geeralsed Class of Esaors We prose a geeralzed fal of esaors for pulao ea of he sud varable Y, as w w w w (4.) where w ad w are suable cosas o be deered such ha MSE of s u, ad are eher real ubers or he fucos of he kow paraeers of aular varables such as coeffce of varao C, skewess, kuross ad correlao coeffce (see Shara ad Sgh ()). I s o be eoed ha () For w, w () For w, w =(,), he class of esaor reduces o he class of esaor as p (4.) =(w,), he class of esaor reduces o he class of esaor as q w (4.)

8 A se of ew esaors geeraed fro (4.) usg suable values of w, w,, ad are lsed Table 4.. Table 4.: Se of esaors geeraed fro he class of esaors Subse of prosed esaor w w (Srvasava, 967) 4-5 w 6 w 7 w w - w aoher se of esaors geeraed fro class of esaor q gve (4.) usg suable values of ad are suarzed able 4. Table 4.: Se of esaors geeraed fro he esaor q Subse of prosed esaor () q w

9 () q w () q w (4) q w (5) q w - (6) q w (7) q w (8) q w (9) q w - Epressg (4.) ers of e s, we have e e ke ke w e w w wy where, k (4.4). Up o he frs order of approao, we have Y w b w Ye ae de ae e w e (4.5), d Y where a k ad d k k

10 Squarg boh sdes of equao (4.5) ad eglecg ers of e s havg wer greaer ha wo, we have Y w d w d Y e a e ae e e ww Y e e ae w (4.6) Takg ecaos boh sdes, we ge he MSE of he esaor approao as o he frs order of MSE ( ) w d w A w B ww C (4.7) where, Y A d a a, B Y a. C, The opu values of w ad w are obaed b zg (4.7) ad s gve b w * d B Ad AB C w * d C (4.8) AB C Subsug he opal values of w ad w equao (4.7) we oba he u MSE of he esaor as d B MSE ( ) d (4.9) AB C ug he values of A, B, C,d ad splfg, we ge he u MSE of esaor MSE ( ) (4.) d as

11 5. Effcec Coparsos Frs we copare he effcec of he prosed esaor p wh Koucu ad Ozel () eeal rao esaor, MSE If MSE k Y 4 4 Y or or 4 (5.) We observe ha he codo lsed (5.) shows ha prosed class of esaors s alwas beer ha he esaor of Koucu ad Ozel (). Ne, we copare he adaped Solak ad Sgh () esaor wh prosed fal of esaors p, MSE MSE k If 4 9 Y Y (5.) 4 Or 4 5 (5.) O solvg we observe ha above codos holds alwas rue. Fall, we are coparg he ea square errors of geeralsed class of esaors ad eeal class of esaors

12 MSE If MSE Y (5.4) d or d (5.5) for he codo gve (5.5) he geeralsed class of esaors perfors well. 6. Eprcal sud Daa Sascs: To llusrae he effcec of prosed esaors he applcao, we cosder he earhquake daa of Turke for he uercal coparsos of he prosed esaors ad esg eeal rao ad produc esaors he sple rado saplg. The daa s obaed fro he daa base of Kadll Observaor, Turke. Earhquake s a uavodable aural dsaser for Turke sce a sgfca ro of urke s subjec o freque desrucve ashocks, her foreshocks ad afershocks sequece. We cosder he ashocks ha occurred 9 ad havg surface wave agudes M S. 5, her foreshocks wh fve das week wh M S. ad afershocks wh oe oh wh M S. 4. I hs area, 9 ashocks wh surface agude M S.5 have occurred bewee 9 ad. The pulao cosss of he desrucve earhquakes. I he pulao daa se he uber of afershocks s a sud varable ad he uber of foreshocks s a aular varable. Noe ha we ake saple sze =. The MSE values of he prosed esaors are copued wh cosderg he dsrbuo of sud ad aular varables. To oba he dsrbuo of hese varables, we f he osso dsrbuo o he earhquake daa se. To oba he for he osso dsrbued daa,

13 Turke s dvded o hree a eoecoc doas based o he eoecoc zoes of Turke. The foreshocks Turke are separaed accordg o hese eoecoc zoes. I hs wa, he paraeers, ad are obaed. Accordg o he goodess of he f es, s obvous ha he osso dsrbuo fs he uber of shocks for Rego wh paraeer =4.8.48, p value.4 ad 8. 4 Rego, ad.., p value.5.4, p value. for for Rego.The he correlao bewee he sud varable ad aular varable s sve. 7 ad ca be oed ha he uber of foreshocks s relaed o he uber of afershocks herefore, rao esaors are preferable hs case. We copue he perce relave effcec (RE) of dffere esaors usg gve resso RE Var MSE Table 6.: RE values of Esaors Esaors REs. r. k.5 k R 6.7

14 997.4 Table 6. ehbs ha he perce relave effcec of he esg ad prosed geeralsed class of esaors usg he osso dsrbued pulao. Fro he above able we aalse ha he usual ubased esaor ad rao esaor r has sae RE s whereas eeal rao ad eeal produc esaors are ore ad less effce ha he sple rao esaor. The eeal produc esaor k s less effce ha all he oher esaors sce he correlao s sve ad hgh. I ca also be see ha he prosed geeralzed eeal esaor ad dfferece esaor R are equall effce bu ore effce ha he rao esaor r, eeal rao ad eeal produc esaors (due o Koucu ad Ozel, ). The geeralsed class of esaors has au RE aog all he esaors cosdered here herefore s bes esaor he sese of havg he leas MSE. Mos of he ebers of he geeralsed class of esaors have approael sae or less effce ha he u ea square error of he esaor of he prosed class of esaors herefore s worhless o eo her ea square errors here hough he ebers are show he able 4. ad 4.. Cocluso I hs sud we have prosed a eeal class of esaors, a dfferece esaor ad a geeralsed class of esaors cosderg he dsrbuo of he sud ad aular varable. The Bas ad ea square errors ressos are derved up o he frs order of approao. I has bee show uder heorecal ad eprcal coparsos ha he wo prosed esaors eeal class of esaors ad dfferece esaor are equall effce bu alwas ore effce ha all oher avalable esaors hs paper. I s also observed ha he geeralsed class of esaors gve seco 4 are alwas ore effce ha he oher esaors usg osso dsrbued pulao. The prosed geeralzed class of esaors s advaageous he sese ha he properes of he esaors, whch are ebers of he prosed class of esaors, ca be easl obaed fro he properes of he prosed geeralzed class. Thus he sud ufes properes of several esaors for pulao ea usg osso dsrbued pulao

15 Refereces. Koucu, N. () : Effce esaors of pulao ea usg aular arbues, Appled Maheacs ad Copuao, 8, Koucu, N. ad Ozel Gaze () : Eeal Esaors usg Characerscs of osso Dsrbuo: A Applcao o Earhquake daa,, 8,9 95. h Ieraoal coferece of Nuercal Aalss ad Appled Maheacs, AI Cof. roc. 558, Ozel G, Ial C (8) The probabl fuco of he coud osso process ad a applcao afershock sequeces Turke, Evroercs 9(): Ozel G. (a) A bvarae coud osso odel for he occurrece of foreshock ad afershock sequeces Turke, Evroercs (7): Ozel, G.(): O Cera roperes of A Class of Bvarae Coud osso Dsrbuos ad a Applcao o Earhquake Daa, Revsa Colobaa de Esadsca, 4(): Shara,. Ad Sgh, R. (): Effce esaor of pulao ea srafed rado saplg usg aular arbue. WASJ 7(), Shara,. ad Sgh, R. (5): Geeralzed class of esaors for pulao eda usg aular forao, Heceepe Joural of Maheacs ad Sascs (I ress) 8. Shara,. ad Sgh, R. (4a): Iproved Rao Tpe Esaors Usg Two Aular Varables uder Secod Order Approao. Maheacal Joural of Ierdscplar Sceces, Vol., No., Shara,. ad Sgh, R. (4b): Iproved Dual o Varace Rao Tpe Esaors for opulao Varace. Chlea Joural of Sascs, 5(), Sgh, R., Cauha,., Sawa, N., ad Saradache, F.,(7): Aular forao ad a pror values cosruco of proved esaors. Reassace Hgh ress.. Sgh, R. ad Kuar, M., (): A oe o rasforaos o aular varable surve saplg. MASA, 6:, Sgh, H.. ad Solak, R.S. (): A ew procedure for varace esao sple rado saplg usg aular forao. Sa. aps. DOI.7/s

16 . Solak, R. S. ad Sgh, H.., (): Iproved esao of pulao ea usg pulao proro of a aular characer. Chlea joural of Sascs. 4 (), Srvasava, S. K. (967): A esaor usg aular forao saple surves. Calcua Sas. Assoc. Bull. 6:.