A GENERAL CLASS OF ESTIMATORS UNDER MULTI PHASE SAMPLING

Transcription

1 TATITIC IN TRANITION-ew sees Octobe 9 83 TATITIC IN TRANITION-ew sees Octobe 9 Vol. No. pp A GENERAL CLA OF ETIMATOR UNDER MULTI PHAE AMPLING M.. Ahed & Atsu.. Dovlo ABTRACT Ths pape deves the geeal estatos fo fte populato ea usg ultvaate auxla foato ude ultphase saplg. Hee a ube of auxla vaables ae cosdeed each phase ude geeal saplg desg. The popetes of these estatos ae studed ad the esults ae peseted fo sple ado saplg wthout eplaceet RWOR schee. Usg a odfed cost fucto the optu saple szes ae also deved. Ke wods ad Phases: Multvaate auxla foato; Multphase saplg; Cha estatos.. Itoducto Mult-phase saplg schee s ve useful fo estatg fte populato paaetes whe a ube of auxla vaables ae avalable cheapl elated to the suve vaable. The estato of populato ea of a suve vaable ude the patal owledge of the auxla eas has bee cosdeed b Chad 975 Kegea Muejee et al. 987 ad vastava et al. 99. Howeve the esults ae cofed to the use of two auxla vaables ol the ea of oe s ow whle the othe s uow. The cosdeed the two-phase saplg schee ol fo sple ado saplg wthout eplaceet RWOR at both the phases. Ahed 995 ad Tpath & Ahed 995 exteded these esults cosdeg oe tha two auxla vaables two-phase saplg. oe ecet wos elated to ultphase saplg ae Ahed et al. 995&96 Ahed 3 ad Daa et al.4. uppose that... s a fte populato of sze gve ad Y deotes the stud vaable wth populato ea Y Y Y /. M Ahed eal: sahed@squ.edu.o Coespodg autho. Atsu.. Dovlo eal: atsu@squ.edu.o Depatet of Matheatcs ad tatstcs ulta Qaboos Uvest P.O. Box 36 Al-Khoudh PC 3 Muscat ultaate of Oa.

2 84 M.. Ahed Atsu.. Dovlo: A geeal class of uppose that... whee... ae -auxla vaables ad the ae avalable fo -th phase souce... wth odeate cost to estate the populato ea Y of the stud vaable Y. uppose that j s avalable fo all j - whee s a sub-saple daw fo - ude the saplg desg D... The stud vaable Y s obseved fo all a copaatvel sall saple wth odeate cost. uppose that Y s a ubased estate of the populato ea Y ude the saplg desg D at the -th phase. Futhe j s a ubased estate of the populato ea j j j / ude the saplg desg D... at the -th phase. The laout of auxla vaables ad stud vaable fo geealzed ultphase saplg ae gve as follows: ouce ze Auxla Vaables Populato st phase d phase No. of vaables -th phase Total auxla vaables gathe ze No. of vaables

3 TATITIC IN TRANITION-ew sees Octobe 9 85 oe patcula cases of geealzed ultphase saplg ae gve as follows: ad... Cocha 977 Classcal Rato ad Regesso estatos > ad... Cocha 977 Multple Rato ad Regesso estatos ad 3... Cocha 977 Classcal Two-phase o Double aplg Rato ad Regesso estatos > ad 3... Cocha 977 Multvaate Two-phase o Double aplg Rato ad Regesso estatos ad 3... Two-phase o Double aplg Rato ad Regesso estatos wth two auxla vaables > > ad 3... Two-phase o Double aplg Rato ad Regesso estatos wth oe auxla vaables... Multphase Rato ad Regesso estatos wth oe auxla vaables... Multphase Rato ad Regesso estatos wth oe auxla vaables. The poposed estatos ad the popetes Chad 975 Kegea vastava et al. 99 Muejee et al. 987 gh et al. 994 Ahed ad Al 996 Ahed 997 Ahed et al Ahed et al. 994 ad Tpath ad Ahed 995. Daa et al. 4 Ahed et al. 995&996 Ahed 3 Fo estatg Y we a cosde the estato pocedue... G g Y

4 86 M.. Ahed Atsu.. Dovlo: A geeal class of G gy... st d th... >. H Y h. Usg leazato Ahed et al. 995&996 showed that + G H Y T.3 Whee T T... T T ad T T T... T Two specal cases of H as Ahed 3 defed a j j. ; j j j j H Y.4 Aothe estato pocedue based o fxed weghts defed as a j j. j ; j j j j H Y u.5 whee a j ae sutabl chose costats ad u j ae fxed weghts. We assue that j s defed such as to be codtoall ubased fo j.e. the codtoal expectato gve - j j E fo all j... ad....6 Futhe we have the codtoal covaace gve C ad Y j j C.7 j whee E ad C stad fo codtoal expectato ad covaace espectvel at -th phase gve It s alwas possble to defe ubased estates j ad Y povded each... has possble pobablt of selectg fo

5 TATITIC IN TRANITION-ew sees Octobe 9 87 fxed sze saplg schee. If π π... π ; π > fo all π Hee the ae the cluso pobabltes fo whee π π. π. π s the codtoal cluso pobablt of -th ut gve - Defe R Y ad Y Y π.8 j j π j / j jj EE... E C j j ad α E E E C Y. j Defe a b u... j a a... a a dag a a... a b b... b b... dag b b b u u... u u dag u u... u R R... R α α α... α jj R dag jj ; j j... ad. Fo lage saples the cotbuto of the thd ad hghe ode poducts ad cetal oets ca be goed. Tag the expectato both sdes ad afte soe splfcato we have stated the followg theoes. Theoe.: Fo lage saples the bases ad ea squae eos of G H ad H fo estatg the populato ea of the stud vaable Y ae gve espectvel b + M G V Y T T α.9 B H a R R a tace a R R Y + a R a. + M H V Y a R R a α.

6 88 M.. Ahed Atsu.. Dovlo: A geeal class of B H b R R b u taceb R R u Y + b u R α. + M H V Y b u R R u b α.3 Theoe.: The optu choces of T a ad b whch zes MG MH ad MH ae gve espectvel bt α a R α ad b R u α fo all ad fo lage saples the u ea squae eos of G H ad H ae sae ad gve b α α.4 M V Y 3. Results ude RWOR schee Now we wll gve the esults whe the RWOR schee s adopted fo selectg fo all... Defe j fo all j Y Y jj σ U j j j σ Y Y ad σ Y Y j U j j... ad... j j Λ Λ... Λ Λ Λ... Λ Also defe Λ Λ jj σ... Λ Λ Λ φ φ φ φ... σ j ad φ σ j fo all j j... ad... If RWOR s used at all the phases the α φ Λ ad U j

7 TATITIC IN TRANITION-ew sees Octobe 9 89 V Y σ 3. whee a b a b Fo RWOR the bases ad ea squae eos of T ad T ae espectvel gve b B H a R R a Tace a R R Y Λ + Λ 3. a R a σ + Λ φ M H a R R a 3.3 σ + Λ φ M G T T 3.3 B H b R R b u Tace b R R u Y Λ + Λ b u R 3.4 φ M H σ + b u R Λ R u b φ 3.5 Fo RWOR the optu choces of T a ad b whch ze MG MH ad MH ae gve espectvel b T Λ φ a R Λ φ ad b R Λ u φ fo all... Fo lage saples the u ea squae eo of H ad H ae sae ad gve b M H whee ρ... σ ρ ρ 3.6 s ultple coelato coeffcet.

8 9 M.. Ahed Atsu.. Dovlo: A geeal class of 4. electo of optu saple szes fo a fxed cost Fo desgg a saple suve effcetl t s essetal to have soe boad foato about the vaablt the populato ad o the cost of dffeet steps volved cag out the suve. A desg-based easue of the saplg eo the esults of the suve s gve b the ea squae eo of the estato used whch educes to ts vaace the case of a ubased estato. I saple suves the vaace of the estato usuall deceases wth cease saple sze. Futhe vaace ad cost would also deped o the atue of the saplg ut. Hece fo the above saplg schee t becoes ecessa to tae both these aspects to accout avg at the optu saplg ut ad the optu szes whch would povde the axu foato pe ut of cost. We wll choose the values of such that the u ea squaed eo fo fxed cost fucto. uppose C ad C s the pe ut cost fo j auxla vaables j ad the stud vaable Y espectvel. uppose C h s the ovehead cost ad C t s the total cost the a cost fucto a be defed as C C + C + C 4. t h The optu choces of... fo whch ea su of squaes of eos of T ad T 3 ae u fo the cost fucto ae gve b C C ρ... ρ... t h g C 4. g C ad C C ρ... The u ea squae eo s t h ρ σ g M C t C h whee ρ... ρ... + ρ... g C C

9 TATITIC IN TRANITION-ew sees Octobe 9 9 REFERENCE AHMED M.. AND ALI M.A.996. The geeal class of cha estatos fo the poduct of two eas usg double saplg. Joual of tatstcal tudes AHMED M oe estato pocedue usg ultvaate auxla foato saple suves. Upublshed Ph.D thess Depatet of tatstcs & Opeatos Reseach Algah Musl Uvest Algah- Ida. AHMED M The geeal class of cha estatos fo the ato of two eas usg double saplg. Coucato tatstcs-theo ad Methods AHMED M A ote o egesso tpe estatos usg ultple auxla foato. Austala & New Zealad Joual tatstcs AHMED M..3. Geeal cha estatos ude ult phase saplg. Joual of Appled tatstcal cece Vol. No. 4 pp AHMED M.. KHAN.U. AND TRIPATHI T.P.994. Two geeal class of cha ato ad poduct estatos fo a fte populato ea based o twophase saplg ad ultvaate foato. Joual of tatstcal tudes AHMED M.. RAHMAN M.. AND AHMED R.998. The geeal class of cha estatos fo a fte populato ea usg double saplg. Joual of Appled tatstcal cece AHMED M.. KHAN.U. AND TRIPATHI T.P. 995&96. Model based egesso estatos usg ultphase saplg. Algah Joual of tatstcs 5& CHAND L oe ato-tpe estatos based o two o oe auxla vaables. Ph.D. thess subtted to Iowa tate Uvest Aes Iowa. COCHRAN W.G aplg Techques Fst edto 953 ecod edto 963. Joh Wle ad os New Yo. DIANAG.TOMMAIC.AND PREOP.4. Estato fo fte populato ea ude ult-phase saplg. Att della LII Ruoe cetfca I Ba KIREGYERA B. 98. A cha ato-tpe estato fte populato two phase saplg usg two-auxla vaables. Meta