ADAPTIVE CLUSTER SAMPLING USING AUXILIARY VARIABLE

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1 Joural o Mathematcs ad tatstcs 9 (3): 49-55, 03 I: cece Publcatos do:0.3844/jmssp Publshed Ole 9 (3) 03 ( ADAPTIVE CLUTER AMPLIG UIG AUXILIARY VARIABLE papor Chutma Departmet o Mathematcs, Fult o cece, Mahasarakham Uverst, Maha arakham, Thalad Receved , Revsed ; Accepted ABTRACT I ths stud we stud the estmators o the populato mea adaptve cluster samplg b usg the ormato o the aular varable. The estmators ths stud are the classcal rato estmator, the rato estmator usg the populato coecet o varato ad the coecet o kurtoss o the aular varable, the regresso estmator ad the derece estmator. mulatos showed that the derece estmator had the smallest estmated mea square error whe compared to the rato estmators ad the regresso estmator. Kewords: Adaptve Cluster amplg, Aular Varable, Rato Estmator, Regresso Estmator, Derece Estmator cece Publcatos. ITRODUCTIO Adaptve cluster samplg, proposed b Thompso (990), s a ecet method or samplg rare ad hdde clustered populatos. I adaptve cluster samplg, a tal sample o uts s selected b smple radom samplg. I the value o the varable o terest rom a sampled ut satses a pre-speced codto C, that s {, c}, the the ut s eghborhood wll also be added to the sample. I a other uts that are adaptvel added also sats the codto C, the ther eghborhoods are also added to the sample. Ths process s cotued utl o more uts that sats the codto are oud. The set o all uts selected ad all eghborg uts that sats the codto s called a etwork. The adaptve sample uts, whch do ot sats the codto are called edge uts. A etwork ad ts assocated edge uts are called a cluster. I a ut s selected the tal sample ad does ot sats the codto C, the there s ol oe ut the etwork. A eghborhood must be deed such that ut s the eghborhood o ut j the ut j s the eghborhood o ut. I ths stud, a eghborhood o a ut s deed as the our spatall adjet uts, that s to the let, rght, top ad bottom o that ut as show Fg.. Fgure llustrates the eample o a etwork. The ut wth a star s the tal ut selected. The codto 49 to adaptvel added uts s a value greater tha or equal to. Uts that are to the let, rght, top ad bottom o oe aother make up a eghborhood. The uts the gra shadg orm a sgle etwork. The uts bold umbers are edge uts o the etwork. The etwork ad ts edge uts make up a cluster. ometmes other varables are related to the varable o terest. We ca obta addtoal ormato or estmatg the populato mea. Use o a aular varable s a commo method to mprove the precso o estmates o a populato mea. I ths stud, we wll stud the estmator o populato mea adaptve cluster samplg usg a aular varable. ome comparsos are made usg a smulato. Fg.. The eample o etwork where a ut eghborhood s deed as our spatall adjet uts

2 papor Chutma / Joural o Mathematcs ad tatstcs 9 (3): 49-55, 03.. mple Radom amplg Usg Aular Varable Let be the varable o terest deed o the te populato ad the populato cossts o a set o uts {u, u,.u }de b ther labels {,,,}. Wth ut s assocated the varable o terest ad the aular varable. The populato mea o s ad the populato mea o s. Let be the sample mea o the varable o terest ad be the sample mea o the aular varable the smple radom samplg. The rato estmate o the populato mea o R R The appromate ME o the rato estmate o the populato total o ME R R ( R ) ( + ) where, s the sample sze, R, s the populato rato, s the populato vare o the aular varable, s the populato vare o the varable o terest ad s the populato covare betwee the aular varable ad the varable o terest. soda ad Dwved (98) suggested the rato estmator or the populato mea o as: R R _ D D ME( R _ D ) C α( α K) C where, α ad K ρ, C s the C populato coecet o varato o the aular varable, C s the populato coecet o varato o the varable o terest ad ρ s the coecet o correlato betwee the aular varable ad the varable o terest. gh ad Kakra (993) suggested the rato estmator or the populato mea o as: where, + β R + β ( ) R _ K K + β K + β The appromate ME o R _ K ME( R _ K ) C δ( δ K) δ + β ad β () s the populato coecet o kurtoss o the aular varable. Upadhaa ad gh (999) cosdered both coecet o varato ad kurtoss rato estmator as: β R β ( ) R _ UK UK β UK β The appromate ME o R _ UK cece Publcatos D The appromate ME o R _ D 50 ME( R _ UK) C ( K)

3 papor Chutma / Joural o Mathematcs ad tatstcs 9 (3): 49-55, 03 cece Publcatos β β C + β R C + β ( ) R _ UK UK C + β UK C + β The appromate ME o R _ UK ME( R _ UK ) C ( K), C C + β The regresso estmate o the populato mea o lr + β β The appromate ME o lr ME ( lr ) ( ρ ) The derece estmate o the populato mea o D + The appromate ME o D ME ( D ) ( + ).. Adaptve Cluster amplg Let be the varable o terest deed o the te populato ad the populato cossts o a set o uts {u, u, u } de b ther labels {,,,}. Wth 5 ut s assocated the varable o terest. The populato mea o s. Let a be the estmator o the populato total adaptve cluster samplg. Let deote the tal sample sze ad v deote the al sample sze. Let ψ deote the etwork that cludes ut ad m be the umber o uts that etwork. The tal sample o uts s selected b smple radom samplg wthout replemet. The Hase-Hurwtz estmator o the populato mea or the varable o terest ca be wrtte as (Thompso, 990; Thompso ad eber, 996): where, (w ) s the average o the varable o terest the etwork that cludes ut o the tal sample, that ( w ) The vare o V ( ) m j ψ j ( ).3. Rato Estmator Adaptve Cluster amplg Drver ad Chao (007) proposed the rato estmator adaptve cluster samplg as: Ad: R _ where, (w ) s the average o the aular varable the etwork that cludes ut o the tal sample, that

4 papor Chutma / Joural o Mathematcs ad tatstcs 9 (3): 49-55, 03 cece Publcatos m j ψ The appromate ME o R _ Ad: ME R R ( R _ ) ( w w w + w ) ( ) w ( ) w ( ) j w w w w.4. Proposed Estmator Adaptve Cluster The rato estmator or the populato mea o adaptve cluster samplg based o soda ad Dwved (98) as: R w R _ w w where, ad C w s the populato w coecet o w. ME o ths estmator ca be oud b usg Talor seres method deed as: Ad: ( ) h, h(, ) h (, ) + h, + ( ),, w h, R _ w 5 h, w w R _ w w + ( ), ( ), ( ) ( ) Ad: ( ) + ( ) w ME( R _ ) E( R _ ) ( ) w w w w,, + E w + ( ) w V Cov, + V ( ) ( w ) ( w ) w w + ( w ) ρ w www ( Cw ) + C + α C α ρ C C ME( R _ ) w w w w w w w w C + α α w w w ( ) K w α w w

5 papor Chutma / Joural o Mathematcs ad tatstcs 9 (3): 49-55, 03 cece Publcatos K w ρ w The rato estmator or the populato mea o adaptve cluster samplg based o gh ad Kakra (993) as: where, + β R + β w C C w w ( ) R _ + β + β ad β (w ) s the populato coecet o kurtoss o w. The Talor seres method s used or ths estmator the same wa to obta the ME: ME( R _ ) Cw wδw ( δw Kw ) δ w + β The rato estmator or the populato mea o adaptve cluster samplg based o Upadhaa ad gh (999) as: 3 β w β w R _ 3 β w w β w w 3 w ad the Talor seres method s used or ths estmator the same wa to obta the ME: ME( R _ 3 ) Cw ww ( w K w ) β w β w w 53 C + β R C + β w ( ) w R _ 4 4 w Cw + β 4 C + β w ad the Talor seres method s used or ths estmator the same wa to obta the ME: ME( R _ 4 ) Cw ww ( w K w ) C w w Cw + β The regresso estmate o the populato mea o adaptve cluster samplg + β lr _ w β w w w w The appromate ME o lr _ ME ( lr _ ) w ( ρw w ) The derece estmate o the populato mea o adaptve cluster samplg + D _ The appromate ME o D _ Table. Data statstcs varable varable w C w C w.9008 β (w ) β (w ) w:w ρ w:w w,

6 papor Chutma / Joural o Mathematcs ad tatstcs 9 (3): 49-55, 03 Fg.. Y values Fg. 3. X values Table. The estmated ME o the estmators or the populato mea o the varable o terest ME ME( R _ ) ME( R _ ) ME( R _ ) ME( R _ 3 ) ME( R _ 4 ) ME ( lr _ ) ME ( D _ ) cece Publcatos 54

7 papor Chutma / Joural o Mathematcs ad tatstcs 9 (3): 49-55, 03 ME.5. mulato tud ( D _ ) ( w w w + w ) I ths secto, the smulato -values ad -values rom Pocha (008) were studed. The populatos were show Fg. -3 ad the data statstcs o ths populatos were show Table. For eh terato, a tal sample o uts s selected b smple radom samplg wthout replemet. The -values are obtaed or keepg the sample etwork. I eh the sample etwork the -values are obtaed. The codto or added uts the sample s deed b C {: >0}. For eh estmator 5,000 teratos were perormed to obta a cur estmate. Ital R szes were vared 5, 0, 5, 0, 30 ad 40 were used. The estmated mea square error o the estmate mea ME 5,000 ( ) 5,000 where, s the value or the relevat estmator or sample.. COCLUIO 4. REFERECE Drver, A.L. ad C. Chao, 007. Rato estmators adaptve cluster samplg. Evrometrcs, 8: DOI: 0.00/ev.838 Pocha,., 008. Rato estmator usg two aular varables or adaptve cluster samplg. J. Tha tatst. Assoc., 6: gh, H.P. ad M.. Kakra, 993. A moded rato estmator usg kow coecet o kurtoss o a aular charter. soda, B.V.. ad V.K. Dwved, 98. A moded rato estmator usg coecet o varato o aular varable. J. Ida oc. Agrc. tatst., 33: 3-8. Thompso,.K. ad G.A.F. eber, 996. Adaptve amplg. st Ed., Wle, ew York, IB-0: , pp: 65. Thompso,.K., 990. Adaptve cluster samplg. J. Am. tatst. Assoc., 85: DOI: 0.080/ Upadhaa, L.. ad H.P. gh, 999. Use o trasormed aular varable estmatg the te populato mea. Bometr. J., 4: DOI: 0.00/(ICI)5-4036(99909)4:5<67::AID-BIMJ67>3.0.CO;- W Adaptve cluster samplg s a ecet method or samplg rare ad hdde clustered populatos. From the estmated ME o the estmators Table showed that the derece estmator had the smallest estmated mea square error whe compared to the rato estmators ad the regresso estmator. The rato estmator or the populato mea ( R _ ) had the smaller estmated mea square error whe compared to the rato estmators usg ormato o C w ad β (w ). The estmator or the populato mea dd ot use aular varable had the hgher estmated mea square error whe compared to the estmator or the populato mea usg aular varable cece Publcatos 3. ACKOWLEDGEMET Ths research was all supported b Mahasarakham Uverst. We would also lke to prooudl thaks Mr. Pavee Chutma or hs programmg advce. 55