PENALIZED CHI SQUARE DISTANCE FUNCTION IN SURVEY SAMPLING
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1 Jot Stattcal Meetg - Secto o Surve Reearch Metho PEAIZED CHI SQUARE DISTACE FUCTIO I SURE SAMPIG Patrck J. Farrell a Sarer Sgh School of Mathematc a Stattc, Carleto Uvert, 5 Coloel B Drve, Ottaa, Otaro KS 5B6, Caaa. Depratmet of Stattc, St. Clou State Uvert, 70 Fourth Aveue South, St. Clou, M , USA. SUMMAR I the preet vetgato, e propoe a pealze ch uare tace fucto for etmatg the total/mea of a fte populato. The fucto prouce a geeral cla of etmator for the populato total that clue, amog other, the etmator of Searl (964, Sgh a Srvatava (980, a the famou ubae rato etmator of Hartle a Ro (954. Several of the etmator the reultg geeral cla are ot member of the faml bae o the poeer techue of Devlle a Saral (99. The etg gap the GES evelope at Stattc Caaa to tu Searl (964 etmator a ubae etmato through calbrato coul be flle th the help of techolog evelope here. Ke or: Aular formato; Calbrato; Etmato of total/mea; Moel-Ate approach; Rato a regreo tpe etmator; Searl etmator.. ITRODUCTIO Whe aular formato avalable, the mot commol ue etmator of the populato total/mea the geeralze lear regreo (GREG etmator. I hat follo, e coer the mplet cae of GREG, here formato o ol oe aular varable ha bee, a ra th a gve amplg eg, p (.. The cluo probablte π Pr( a π Pr( & are aume to be trctl potve a ko. et (, collecte. Suppoe that from a populato Ω {,,..,,.., } probablt ample ( Ω be a bvarate obervato cotg of the value of the aular varable a the varable of teret for the -th populato elemet. The Horvtz- Thompo (95 etmator of the populato total (. here π are the Horvtz-Thompo eght. The varace of (. ( Θ ( Ω (. : 963 here Θ ( π π π ( D (.A ubae etmator for (. (.3 here D Θ π. Devlle a Saral (99 propoe a e etmator for the total, amg t GREG: (.4 Here the are eght that, for a gve metrc, are a cloe a poble a average ee to the hle repectg the calbrato euato (.5 et be a et of utabl choe eght. Mmzg the ch uare tpe tace fucto ( ubect to (.5, el e eght a (.6 (.7 Partcular choce for el fferet form of the etmator (.4. Subttutg (.7 to (.4, el the geeralze regreo etmator of the populato total β (.8 here β ( Θ ( e e Ω, th appromate varace: (.9 A a etmator for (.9 coere b Saral et al. (989, Devlle a Saral (99 a Saral (996,
2 Jot Stattcal Meetg - Secto o Surve Reearch Metho ( D ( e e (.0 earl ( C (.5 here e β.the etmator (.8 ute geeral a clue fferet etmator a partcular cae. For eample, f the (.8 reuce to the rato etmator tue b Cochra (977 a ( (. here. If, the (.8 reuce to geeral regreo etmator β ( (. Ufortuatel there o choce of that reult (.8 matchg the lear regreo etmator of Hae, Hurtz a Mao (953 gve b hhm β ol ( here ( ( ( a (.3 β ol,. Wu a Stter (00 ae the cotrat o the eght a evelope a e etmator of the populato total A B ol ( here ( ( (.4 B ol a A Bol th a. ote that f the B ol β ol, but for a IPPS cheme (ecept mple raom amplg thout replacemet t the cae that hhm. Wu a Stter (00 mae hhm b eglectg a mall term, A, from ther etmator (euato o. 9, page 87, JASA. I hort, the etmator (.4 recovere b Wu a Stter (00 a pecal cae of the Devlle a Saral (99 etmator obtae b ettg oe aular varable out of p at a fe level. Th emotrate that, for a ueual IPPS amplg cheme, the calbrato techue caot acheve the loer bou for the varace of the tratoal lear regreo etmator. Ug a pror formato o C (, Searl (964, 967 uggete the follog etmator of the populato total 964 th mea uare error gve b ( ( ( C MSE earl (.6 Re (978 tue the properte of (.5 a ha fou t to be ueful f C large a the ample ze mall. A mlar cocluo regarg ample ze a alo reache b Searl (964 a Arholt a Hebert (995. Fa a Herrot (979 have cue the mportace of the Jame- Ste (96 proceure hle etmatg the come for mall place from ceu ata. ote that the etmator ue to Jame-Ste (96 a Searl (964 ca be ho to be from the ame faml. Praa (989 propoe a rato tpe etmator of the populato total uer a SI eg a here ( p r (.7 earl a ubae etmator of hle Ja (987 coere the applcato of Searl (964 etmator to the uual lear regreo etmator ( a α β (.8 ote that t ca be ho that (.8 ha mea uare error: f S S ( ρ, fα β S ρss S MSE( a ( f S ρ (.9,othere f C ( ρ Sgh a Srvatava (980 vetgate a ubae regreo etmato trateg for the populato total,, a: ( ( ( ( (.0 I the preet vetgato, e propoe a pealze ch uare tace fucto that prouce a geeral cla of etmator for the populato total that clue, amog other cue here, the etmator of Searl (964, Sgh a Srvatava (980, a the famou ubae rato etmator of Hartle a Ro (954 gve b ( ( ( HR r here r. (.,
3 Jot Stattcal Meetg - Secto o Surve Reearch Metho. PEAIZED CHI SQUARE DISTACE FUCTIO We ugget here the pealze ch uare tace fucto D ( ( (. here are eght a a potve uatt that reflect a pealt to be ece b the vetgator bae o pror kolege, or the ere for certa level of effcec a ba. Dfferet choce for reult fferet etmator, hle creag reult a ecreae the mea uare error of the etmator; ufortuatel ha the e effect of creag the ba. If 0, the the pealze ch uare tace fucto (. reuce to the Devlle a Saral (99 tace fucto. Coerg the etmator of populato total to be of the form e (. We hall mmze the pealze ch uare tace fucto for fve fferet tuato:. o aular formato avalable.. Calbrato of Devlle a Saral ( Pealze calbrato cotrat. 4. Ubae trateg of Sgh a Srvatava ( Ubae rato etmator b Hartle a Ro (954. We hall alo cu the etmator belogg to each oe of the above tuato. 3. O AUIIAR IFORMATIO AAIABE I the abece of aular formato, mmzg (. th repect to gve D ( 0 hch mple that ( (3. (3. O ubttutg (3. to (., e obta a e etmator of populato total gve b e ( ( (3.3 The pealze etmator e epeet of the choce of, hch cate that Searl(964 etmator a uue etmator t cla. Follog Searl (964, the mea uare error of the pealze etmator 965 ( ( ( MSE (3.4 e o that the effcec of (3.3 relatve to the Horvtz- Thompo etmator gve b RE (3.5 The ba (3.3 gve b B( e { } (3.6 Iteretgl f, the RE but RB( ( e B e ; thu a creae the pealt ma be avatageou that the ga relatve effcec ma outegh the creae ba. If 0 the RE a RB ( e 0. If C, the pealze etmator reuce to Searl (964 etmator. I practce the bet choce of value for the pealt the rage from 0 to. If, the pealze etmator 00% more effcet tha the Horvtz-Thompo etmator th RB ( e 0.5. If 0.5, the relatve effcec 5% th RB ( e 0.. Jame-Ste (96 Etmator If the are aume to be epeet a etcall trbute accorg to a ormal trbuto th mea θ a varace D, the each the obvou etmate of t repectve θ. For k > 3, Jame a Ste (96 efe ( k ' D δ S (3.7 a a etmator of θ, th a {( k D S} { ( k } S. ote that f D S, the the pealze etmator e (3.3 reuce to Jame a Ste (96 etmator. Fa a Herrot (979 Etmator Follog Jame a Ste (96, f (, D θ ~ ( β, A regreo etmator ( ' ' ~ θ a, Fa a Herrot (979 combe the th the rect etmator to form a emprcal Bae etmator of θ a D D A ( θ (3.8 Smlarl, the cove combato of the propoe pealze etmator th the emprcal Bae etmator ll lea to the etmator of Fa a Herrot (979
4 Jot Stattcal Meetg - Secto o Surve Reearch Metho hf e bae (3.9 The etmator (3.9 alo calle a compote etmator a pla a emet role mall area etmato. 4. CAIBRATIO OF DEIE AD SARDA I orer to mmze the pealze ch-uare tace fucto ubect to the calbrato cotrat of Devlle a Saral (99, e coer the agrage fucto ( Settg 0, el ( λ (4. ( (4. o that a pealze etmator of populato total gve b e (4.3 ote that f, the (4.3 reuce to the uual rato etmator of the populato total, amel e ( (4.4 The mea uare error of the etmator (4.3 gve b here ( ( ( MSE (4.5 e ( Θ ( e e a ( ( β Ω teretg to ote that f. It β, the a the relatve effcec alo approache ft; hoever th ma have a erou avere effect o ba. If ko, the a etmator for the mea uare error of (4.3 ( ( ( MS E e (4.6 here ( D ( e e a e pealze etmator of varace of the Devlle a Saral (99 etmator. 966 PRODUCT METHOD OF ESTIMATIO The problem of etmatg the prouct of to varable ell ko he the to varable are egatvel correlate. For eample, a etmate of the force, F, of certa obect gve b F m a, here m a â are the average ma a accelerato. For further etal o prouct etmato, ee Robo (957 a Murth (964. Mmzato of (. ubect to a e calbrato cotrat, efe a (4.7 lea to calbrate eght gve b ( (4.8 a the follog pealze etmator of the populato total ( e (4.9 If, the ( e (4.0 hch ame prouct etmator tue b Murth ( PEAIZED CAIBRATIO COSTRAIT We ugget here a e pealze calbrato cotrat a: ( (5. Mmzato of the pealze ch uare tace fucto (. ubect to (5. lea to the calbrate eght (5. The reultat pealze etmator of the populato total : e hle the mmum mea uare error of (5.3 ( ( ( e (5.3 MSE (5.4 here (. A etmator of the varace of (5.3
5 e E MS (5.5 here e e D. ote that (5.3 mlar to the etmator tue b Ja (987. If, the t reuce to a pealze rato etmator, a e (5.6 hch mlar to the etmator tue b Praa (989. Follog Praa (989, the pealt (5.6 ca be take a. Smlarl, mmzato of (. th repect to the calbrato cotrat lea to e (5.7 hch a e etmator of the populato total. 6. SIGH AD SRIASTAA UBIASED ESTIMATIO STRATEG Here e coer a lghtl fferet pealze tace fucto gve b D ( (6. a ugget a e calbrato cotrat 0 (6. here Ω Ω, o that the agrage fucto λ (6.3 Settg 0 el { } λ (6.4 Subttutg (6.4 (6. gve λ (6.5 o that the mmal eght are (6.6 hch atf the coto of mmal tace f 0 > (6.7 Thu a e pealze etmator of the populato total e (6.8 Uer SRSWOR amplg, a f { } { } a, the (6.8 become e (6.9 hch etcal to the ubae regreo tpe etmator propoe b Sgh a Srvatava (980. ote that { } { } le betee 0 a a hece atfe the coto of mmal tace. 7. HARTE AD ROSS ESTIMATOR I orer to mmze (6. ubect to the Devlle a Saral (99 calbrato cotrat, e coer the agrage fucto λ (7. Settg 0, el (7. o that a pealze etmator of the populato total : e (7.3 Uer SRSWOR amplg, f a { } { } the (7.3 reuce to r e (7.4 hch euvalet to the Hartle a Ro (954 etmator. Jot Stattcal Meetg - Secto o Surve Reearch Metho 967
6 Jot Stattcal Meetg - Secto o Surve Reearch Metho COCUSIO A pealze ch uare tace fucto ha bee propoe that cover a er varet of etmator tha the orgal ch uare fucto trouce b Devlle a Saral (99. The propoe fucto prouce a geeral cla of etmator for the populato total that clue the etmator of Searl (964, Sgh a Srvatava (980, a the famou ubae rato etmator of Hartle a Ro (954, amog other. FURTHER STUD The eteo of oe-meoal pealze ch-uare tace fucto to to-meoal pealze ch-uare tace fucto a tu of reultat etmator o the le of Sgh, Hor a u (998 progre. ote that e have coere ol mplet pealze ch-uare tace fucto, but a oe amog the tace fucto cue b Devlle a Saral (99 ca be pealze, a ll lea to more teretg etmator. ACKOWEDGEMETS Th reearch a upporte b the atural Scece a Egeerg Coucl of Caaa. The eco author a pot-oc at Carleto Uvert urg th vetgato. REFERECES Arholt, A.T. a Hebert, J.. (995. Etmatg the mea th ko coeffcet of varato. Amerca Stattca, 49(4, Cochra, W.G. (977. Samplg Techue, Thr Eto. e ork: Wle. Devlle, J.C. a Saral, C.E.(99.Calbrato etmator urve amplg. J. Amer. Statt. Aoc., 87, Hartle, H.O. a Ro, A (954. Ubae rato etmator. ature, 74, Horvtz, D.G. a Thompo, D.J. (95. A geeralato of amplg thout replacemet from a fte uvere. J. Amer. Statt. Aoc., 47, Ja, R.K. (987. Properte of etmator mple raom amplg ug aular varable. Metro, Praa, B. (989. Some mprove rato tpe etmator of populato mea a rato fte populato ample urve. Commu. Statt. Theor Meth., 8(, Re,.. (979. A tu o the ue of pror kolege o certa populato parameter etmato. Sakha, C, 40, Robo, D.S. (957. Applcato of multvarate polka to the theor of ubae rato-tpe etmato. J. Amer. Statt. Aoc., 5, 5-5. Saral, C.E. (996. Effcet etmator th mple varace ueual probablt amplg. J. Amer. Statt. Aoc., 9, Saral, C.E., Seo, B. a Wretma, J.H. (989. The eghte reual techue for etmatg the varace of the geeral regreo etmator of the fte populato total. Bometrka, 76(3, Searl, D.T. (964. The utlzato of a ko co-effcet of varato the etmato proceure. J. Amer. Stat. Aoc., 59, 5-6. Searl, D.T. (967. A ote o the ue of a appromatel ko co-effcet of varato. Amerca Stattca, (, 0-. Sgh, P. a Srvatava, A. K. (980. Samplg cheme provg ubae regreo etmator. Bometrka, 67, Sgh, S., Hor, S. a u, F. (998. Etmato of varace of geeral regreo etmator : Hgher level calbrato approach. Surve Methoolog, 4, Wu, C. a Stter, R.R. (00. A moel-calbrato approach to ug complete aular formato from urve ata. J. Amer. Statt. Aoc., 96, Patrck ( Et Sarer ( E-mal : pfarrell@math.carleto.ca E-mal: arer@ahoo.com Jame, W. a Ste, C. (96. Etmato th uaratc lo. Proceeg of the Fourth Berkele Smpoum of Mathematcal Stattc a Probablt, ol., Uvert of Calfora Pre, Murth, M.. (964. Prouct metho of etmato. Sakha, 6,
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