RATIO ESTIMATORS USING CHARACTERISTICS OF POISSON DISTRIBUTION WITH APPLICATION TO EARTHQUAKE DATA

Size: px

Start display at page:

Download "RATIO ESTIMATORS USING CHARACTERISTICS OF POISSON DISTRIBUTION WITH APPLICATION TO EARTHQUAKE DATA"

Griselda Bailey
6 years ago
Views:

1 The 7 h Ieraoal as of Sascs ad Ecoomcs Prague Sepember 9-0 Absrac RATIO ESTIMATORS USING HARATERISTIS OF POISSON ISTRIBUTION WITH APPLIATION TO EARTHQUAKE ATA Gamze Özel Naural pulaos bolog geecs educao egeerg surace markeg sesmolog socal scece ad survval aalss are eremel large; cosequel samplg mehods have o be coduced for characerzg hose pulaos. Rao esmaors are commol used o oba more effce esmaes for he pulao mea f he sud varable s hghl correlaed wh he aular varable. I s well kow ha he use of he pulao formao of aular varable mproves he precso of he esmae(s of he parameer(s. Rao esmaors are based o a sample whose dsrbuo s o cosdered. However here are suaos whch Posso dsrbued pulao ma be approprae. Ths paper proses geeralzed class of rao esmaors from Posso dsrbued pulao. The mea square error (MSE equaos of prosed esmaors are compared applcao wh usual rao esmaor. B hese comparsos we fd ha rao esmaors usg Posso dsrbuo characerscs as aular varable formao s beer ha usual rao esmaors. The codos are also foud ha prosed esmaors are more effce. The fdgs are supred b umercal llusrao wh earhquake daa of Turke. Ke words: Rao-pe esmaors; Smple radom samplg; Mea square error; Posso dsrbuo; Effcec. JEL ode: Iroduco Smple radom samplg (SRS from a fe pulao has araced much of he researchers ad pracoers workg surves. Rao esmaors are commol used he SRS o oba more effce esmaes for he pulao mea f he sud varable s hghl correlaed wh he aular varable. I s well kow ha he use of he pulao formao of aular varable mproves he precso of he esmae(s of he parameer(s he SRS. Several auhors cludg Ssoda ad wved (98 Upadhaa ad Sgh (999 Kadlar ad g (004 Gupa ad Shabbr ( Koucu ad Kadlar (009 Sgh ad Vshwakarma (00 Shabbr ad Gupa (0 obaed a large 070

2 The 7 h Ieraoal as of Sascs ad Ecoomcs Prague Sepember 9-0 umber of mproved rao esmaors/classes of esmaors for he pulao mea of he sud varable usg aular varable formao he SRS. The problem of esmag he pulao mea or oal he presece of a aular varable has bee wdel dscussed he SRS whou cosderg he dsrbuo. However he Posso dsrbuo s geerall used for he aural pulaos o epress he probabl of a gve umber of rare eves ad here has bee o effor devoed o he developme of rao esmaors for a Posso dsrbued pulao. No-esece of he rao esmaors for he Posso dsrbued sample obsacles usage of hem samplg heor self ad s applcaos (Ozel ad Ial 008. The am of hs sud s o derve ew rao esmaors for he pulao mea from a Posso dsrbued pulao. We also eame he behavor of he esmaors of mea for he rao esmaors he SRS. The earhquake daa s used for he umercal eample sce earhquakes are rare eves ad geerall follows a Posso dsrbuo (Ozel 0a. Suggesed Esmaors for he Posso srbued Populao osder a fe pulao U (u u...u cossg of N defable ad dsc us. Le ad respecvel be he sud ad aular varables assocaed wh each u N ( j... N of he pulao. Assume ha s are kow us ad s are ukow us for all he pulao. Supse ha a sample of sze s seleced accordg o he SRS. The aure of he samplg dsrbuo depeds o he aure of he pulao from whch he radom sample s draw. Le us assume ha he pare pulao has a Posso dsrbuo. Ths meas ha he radom samples whch are draw from a Posso dsrbued pulao follow also a Posso dsrbuo. The le us selec he observaos (... from a Posso dsrbued pulao. B hs wa we sugges ha he followg a geeralzed class of rao esmaors for he pulao mea of he sud varable from Posso dsrbued pulao as Rˆ (a b Rˆ where a b 0 a b a b. Here / / are he sample meas of a b he sud ad aular varables from Posso dsrbued pulao respecvel. Noe ha ( a b are eher cosas or fuco of kow parameers of he pulao such as a or ( ( ad. As meoed before he de of dsperso s used for he frs me samplg heor for he aular varable formao. Le he aular 07 u j

3 The 7 h Ieraoal as of Sascs ad Ecoomcs Prague Sepember 9-0 varable has a Posso dsrbuo wh parameer 0 he S ( ad ( of he aular varable are gve b S ( S S 0 ( ( N 04 ( respecvel where 0 N ( 0 0 / 0 / N 0 N ( 4 ad ( ( N 04 N for he Posso dsrbued pulao. Le he sud varable has a Posso dsrbuo wh parameer ad le he aular varable has a Posso dsrbuo wh paramaer he he coeffce of correlao bewee he sud varable ad aular varable s obaed b he rvarae reduco mehod (Ozel 0b. A bvarae Posso dsrbuo of ad s geeraed b seg m z ad w z.... Assumg ha he parameers of m w ad z are ad he coeffce of he correlao bewee ad equals cov( ( ( ( ( Po ( S S ( ( ( ( where ( [( ( ] ( ( ( ( ( E E ad (. A obvous properes of S E S s ha he correlao s resrced o be srcl sve sce ad 0. Sce we selec he observaos (... from a Posso dsrbued pulao wh parameers ad he we ge (m z / ad (w z /. The covarace of ad s cov( ( ( ( ( ( ( ( where E( ad E(. The MSE of he prosed esmaor ca be foud usg Talor seres mehod defed as 07

4 The 7 h Ieraoal as of Sascs ad Ecoomcs Prague Sepember 9-0 g( g(cd g(cd g( ( ( ( c d where g( Rˆ ad g( R. Eq. ( ca be appled o he geeralzed class of rao esmaors order o oba he MSE equao ad we have MSE(Rˆ a R V( ar cov( V( (4 (a b a R ( ar ( (a b where R. Thus we oba he MSE of he prosed esmaor as a b MSE( (a b MSE(Rˆ a R ( ar ( Sce ( ( ad we have MSE( a R ar. (5 Effcec omparsos The rao esmaors preseed Table wll be compared wh each oher accordg o her MSE equaos he heor. Table Some members of he class of rao esmaor for ad Usual Rao Esmaors a b Rao Esmaors a b ( ( ( ( ( ( PR ( 6 ( ( 07 ( ( ( ( 4 ( ( 5 6 ( 7 ( (

5 The 7 h Ieraoal as of Sascs ad Ecoomcs Prague Sepember 9-0 ( 7 ( 8 PR 9 0 ( ( ( PR 4 PR4 5 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 6 ( ( 7 ( ( ( PR5 PR6 PR7 ( ( ( PR8 ( PR9 ( ( ( 8 ( ( 9 0 ( ( ( ( ( ( ( ( ( ( ( ( ( 4 ( ( ( 5 ( ( ( ( 6 ( ( ( ( ( 7 ( ( ( 8 ( ( 9 ( 0 ( ( ( ( ( p ( 4 ( ( ( 5 ( ( 6 074

6 The 7 h Ieraoal as of Sascs ad Ecoomcs Prague Sepember 9-0 We frs compare... 6 for a wh j for a Table o oba j he effcec comparso as follows: Usg Eq. (5 we ca wre MSE( MSE(. (6 j R R a R ar From hs equal we have ( a ( a 0 where ( a 0. R ( a The we have R 0 ( a( a. I s wre as ( ar. Hece he effcec codo for Eq. (6 s foud as. ( ar 0.Ths codo s alwas sasfed sce R are alwas sve whe a 0 ad 0. Hece we ca fer ha he prosed rao esmaors j are more effce ha he esmaors... 6 usg he j aular varable formao. 5. Numercal Illusrao I he sud we cosder he earhquake daa of Turke for he umercal comparsos of he prosed ad oher rao esmaors he SRS. We cosder mashocks ha occured Turke bewee 900 ad 0 havg surface wave magudes M S 5. 0 her foreshocks wh fve das wh M S. 0 ad afershocks wh oe moh wh M S I hs area 0 mashocks wh surface magude M S 5. 0 have occured bewee 900 ad 0. The pulao cosss of he desrucve earhquakes. I he pulao daa se he umber of afershocks s a sud varable ad he umber of foreshocks s a aular varable. The MSE values of usual rao esmaors r r... 7 ad PR PR9 obaed from Eq. (4 whou cosderg he dsrbuos of he sud ad aular varables. The summar sascs for he pulao are gve. The he MSE values of he prosed esmaors... are compued from Eq. (5 wh cosderg he dsrbuos ( 6 of he sud ad aular varables. Several sudes modeled earhquakes as a Posso dsrbuo (Ozel 0a b. To oba he dsrbuo of hese varables we f Posso are

7 The 7 h Ieraoal as of Sascs ad Ecoomcs Prague Sepember 9-0 dsrbuo o he earhquake daase. The Posso dsrbuo provded a adequae f wh p-value <0.0 ad ch-square value ( for he goodess of f es. Ths meas ha he Posso dsrbuo wh parameer (ear fs he probabl fuco of he aular varable ad he epeced umber of foreshocks of a ma shock appromael equals o seve per ear. Afer obag he frequec dsrbuo of afershocks ad goodess of f es ( p-value= <0.0 s see ha he sud varable has a Posso dsrbuo wh parameer The summar sascs for he Posso dsrbued pulao are gve. To oba for he Posso dsrbued daa Turke s dvded o hree ma eoecoc domas based o he eoecoc zoes of Turke. The foreshocks Turke are separaed accordg o hese eoecoc zoes. B hs wa he parameers ad are obaed. Accordg o he goodess of f es s see ha he Posso dsrbuo fs he umber of shocks for area Rego wh parameer 4.8 ( p - value 0.04 wh parameer ( 0.04 p - value 0.0 for Rego ad. ( 0.0 p - value 0.05 for Rego. The he correlao bewee he sud varable ad aular varable s sve ( 0. 7 ad ca sad ha he umber of foreshocks s relaed o he umber of afershocks. Therefore he rao esmaors ca be used for he esmao of he pulao mea he SRS. The MSE values of he usual rao esmaors r r... 7 ad PR... PR9 are obaed ad he prosed mea esmaors (... 6 are compued usg SRS ad Table. 076

8 The 7 h Ieraoal as of Sascs ad Ecoomcs Prague Sepember 9-0 Table R values ad MSE equaos for he rao esmaors of he Posso dsrbued pulao Rao Esmaors ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( R MSE Equaos ( ( ( ( ( ( ( ( ( ( ( ( ( MSE ( MSE ( ( MSE ( ( ( ( MSE ( 4 ( ( ( MSE ( 5 ( MSE ( 6 ( MSE ( 7 ( ( ( MSE ( 8 ( ( ( MSE ( 9 ( MSE ( 0 ( ( ( ( MSE ( (( ( ( ( ( MSE ( ( ( ( ( ( ( ( ( MSE ( ( ( ( ( ( ( MSE ( 4 (( ( ( ( ( MSE ( 5 (( ( ( ( ( MSE ( 6 ( ( ( ( ( ( ( ( MSE( 7 ( ( ( 077

9 The 7 h Ieraoal as of Sascs ad Ecoomcs Prague Sepember ( ( ( ( 0 ( ( ( ( ( ( 4 ( ( 5 6 ( ( ( ( ( ( MSE ( 8 (( ( MSE ( 9 ( MSE ( 0 ( ( ( MSE ( ( ( ( MSE ( ( MSE ( ( ( MSE ( 4 ( ( ( ( MSE ( 5 ( ( ( ( MSE ( 6 ( ( We use he followg epresso o fd he relave effcec (RE of rao esmaors usg he characerscs of Posso dsrbuo whe compared wh he usual rao esmaors. The he prosed ad usual rao esmaors are compared wh respec o her MSE ad RE values. We foud ha ( he prosed faml rao esmaors usg characerscs of Posso dsrbuo perform beer ha usual faml usual rao esmaors ( he relave effcec of he prosed faml rao esmaors are appromael 56 mes more ha he usual rao esmaors for he Posso dsrbued daa ( he larges ga effcec s observed b usg ( ad wh ( f er-group comparso of for he prosed esmaors s doe for he Posso dsrbued daa. (v he MSE value of he prosed -faml rao esmaor usg ad 078 ogeher s smaller ha he oher usual - faml rao esmaors (v he prosed rao esmaors... have he same value of ( 6 MSE wh... sce 5. However f here s a pulao for dffere dsrbued pulaos de of dsperso wll dffer from. I such a case... 6 eld dffere MSE values from Thus he class of he prosed rao esmaors s o be preferred o usual rao esmaors for he Posso dsrbued pulao he SRS.

10 The 7 h Ieraoal as of Sascs ad Ecoomcs Prague Sepember 9-0 ocluso I hs sud frs we suggesed rao esmaors for he pulao mea usg de of dsperso as a aular varable. The we have developed ew rao esmaors usg characerscs of Posso dsrbued aular varable for he pulao mea SRS ad obaed her MSE equaos. ffere classes of rao esmaors are also prosed usg he aular varable formao wh cosderg he dsrbuo of pulao. B MSE equaos ad RE values he MSE values are compared ad s foud ha he prosed faml esmaors are alwas more effce ha he usual -faml esmaors for he Posso dsrbued earhquake daa. Ths heorecal resul s also supred b a umercal eample based o a earhquake daa of Turke. I he forhcomg sudes we hope o develop ew esmaors for he pulao mea for he Posso dsrbued pulao usg oher samplg mehods. Refereces Gupa S Shabbr J (007 O he use of rasformed aular varable esmag pulao mea. J Sa Pla Iferece 7(5: Gupa S Shabbr J (008 O mproveme esmag he pulao mea smple radom samplg. Joural of Appled Sascs 5(5: Kadlar g H (004 Rao esmaors smple radom samplg. Appled Mahemacs ad ompuao 5: Koucu N Kadlar (009 Rao ad produc esmaors srafed radom samplg. J Sa Pla Iferece 9(8: Ozel G Ial (008 The probabl fuco of he comud Posso process ad a applcao afershock sequeces Turke Evromercs 9(: Ozel G (0a A bvarae comud Posso model for he occurrece of foreshock ad afershock sequeces Turke Evromercs (7: Ozel G (0b O cera properes of a class of bvarae comud Posso dsrbuos ad a applcao o earhquake daa Revsa olombaa Esadsca 4(: Shabbr J Gupa S (0 O esmag he fe pulao mea smple ad srafed radom samplg ommucao Sascs: Theor ad Mehods 40(0: Sgh HP Vshwakarma GK (00 A geeral procedure for esmag he pulao mea srafed samplg usg aular formao. Mero 68(:

11 The 7 h Ieraoal as of Sascs ad Ecoomcs Prague Sepember 9-0 Ssoda BVS wved VK (98 A modfed rao esmaor usg coeffce of varao of aular varable. Joural of he Ida Soce of Agrculural Sascs (: -8. Upadhaa LN Sgh HP (999 Use of rasformed aular varable esmag he fe pulao mea. Bomercal Joural 4(5: oac Gamze Özel Haceepe Uvers eparme of Sascs Akara Turke gamzeozl@haceepe.edu.r 080

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

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