Ratio-Cum-Product Estimator Using Multiple Auxiliary Attributes in Two-Phase Sampling

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1 On Jounal of Sascs, 04, 4, Publshd Onln Jun 04 n Scs. h:// h://dx.do.o/0.436/os ao-um-poduc Esmao Usn Mull Auxla Abus n Two-Phas Samln John Kun u, Lo Odono Damn of Mahmacs, Knaa Unvs, Naob, Kna Emal: ohnunu08@ahoo.com cvd Mach 04; vsd Al 04; accd 8 Al 04 oh 04 b auhos and Scnfc sach Publshn Inc. Ths wo s lcnsd und h av ommons Abuon Innaonal Lcns ( BY. h://cavcommons.o/lcnss/b/4.0/ Absac In hs a, w hav oosd h classs of ao-cum-oduc smaos fo sman oulaon man of sud vaabl fo wo-has samln usn mul-auxla abus fo full nfomaon, aal nfomaon and no nfomaon cass. Th xssons fo man sua os a dvd. An mcal sud s vn o coma h fomanc of h smao wh h xsn smao ha ulzs auxla abu o mull auxla abus. Th ao-cum-oduc smao n wo-has samln fo full nfomaon cas has bn found o b mo ffcn han xsn smaos and also ao-cum-oduc smao n wo-has samln fo boh aal and no nfomaon cas. Fnall, ao-cum-oduc smao n wo-has samln fo aal nfomaon cas has bn found o b mo ffcn han ao-cum-oduc smao n wo-has samln fo no nfomaon cas. Kwods ao-um-poduc Esmao, Mull Auxla Abus, Two-Phas Samln and B-Sal olaon offcn. Inoducon Th us of sulmna nfomaon s a wdl dscussd ssu n samln ho. Auxla vaabls a commonl usd n saml suv accs o oban movd dsns and o achv hh cson n h smas of som oulaon aams such as h man o h vaanc of a sud vaabl. Th conc of ao smaon was noducd n saml suv b ochan []. I s fd whn h sud vaabl s hhl osvl colad wh h auxla vaabl. Muh [] oosd h oduc smao fo navl colad sud vaabl(s and auxla vaabl whch was smla o ao smao. Oln [3] was h fs auho o us nfomaon on mo han on sulmna chaacsc, whch s o- How o c hs a: Kun u, J. and Odono, L. (04 ao-um-poduc Esmao Usn Mull Auxla Abus n Two-Phas Samln. On Jounal of Sascs, 4, h://dx.do.o/0.436/os

2 svl colad wh h vaabl und sud, usn a lna combnaon of ao smao basd on ach auxla vaabl. a [4] susd a mhod of usn mul-auxla nfomaon n saml suv. Usn hs da, Snh [5] oosd a mulvaa xsson of oduc smao wh h sud vaabl was navl colad wh h mul-auxla vaabl. In h sam a, Snh [6] oosd a ao-cum-oduc smao and s mul-vaabl xsson. Snh and Talo [7] oosd a ao-cum-oduc smao fo fn oulaon man n sml andom samln usn coffcn of vaaon and coffcn of uoss whch was mo ffcn han h vous ao-cum oduc smao. Jha, Shama and Gov [8] oosd a nal faml of smaos usn nfomaon on auxla abu. Th usd nown nfomaon of oulaon ooon ossssn an abu (hhl colad wh sud vaabl Y. Th abu a nomall usd whn h auxla vaabls a no avalabl,.. amoun of ml oducd, a acula bd of cow o amoun of ld of wha and a acula va of wha. Bahland Tua [9] oosd ao and oduc xonnal smaos usn auxla abu. ash, Pana, Nmala and Flonns [0] usd h nfomaon on auxla abu n ao smao n sman oulaon man of h vaabl of ns usn nown abus such as coffcn of vaaon, coffcn uoss and on bsal colaon coffcn. Th smao fomd b han h usual saml man and Naand Gua [8] smao. ash, Pana, Nmala and Flonns [0] also usd h auxla abus n ao-oduc xonnal smao follown h wo of Bahland Tua [9], h smao was mo ffcn comad o man un, ao and oduc xonnal smaos as wll as Na and Gua [] smao. Th conc of doubl samln was fs oosd b Nman [] n samln human oulaons whn h man of auxla vaabl(s was unnown. I was la xndd o mulhas b obson [3]. I s advanaous whn h an n cson s subsanal as comad o h ncas n h cos du o collcon of nfomaon on h auxla vaabl fo la samls. In mos suv, h auxla nfomaon s alwas avalabl and v fom of auxla nfomaon should b usd n dvlon samln sas. Samuddnand Hanf [4] noducd h follown aoach of usn auxla vaabl. Full nfomaon cas: Infomaon fo all auxla vaabls s avalabl No nfomaon cas: Infomaon fo all auxla vaabls s no avalabl. 3 Paal nfomaon cas: Infomaon fo som auxla vaabl s avalabl fo all oulaon uns. W hav usd hs sas o dvlo ao-cum-oduc smaos usn mull auxla abus fo full nfomaon, aal and no nfomaon cass. Hanf, Ha and Shahbaz [5] oosd a nal faml of smaos usn mull auxla abu n snl and doubl has samln. Th smaos had a small MSE comad o ha of Jha, Shama and Gov [8]. Th also xndd h wo o ao, oduc and sson smaos whch w nalzaon of Na and Gua [] smao n snl and doubl has samln wh full nfomaon, aal nfomaon and no nfomaon. Th conc of mull auxla abus was oosd b Hanf, Ha and Shahbaz [6], and hn xndd o ao and oduc smaos. In hs a, w hav ncooad h mull auxla abus n aocum-oduc smao n wo-has samln as oosd b Snh [7] and usd sas noducd b Samuddnand Hanf [4] and also ncooa Aoa and Bans [7] aoach n wn down h man suad o.. Plmnas and Noaons.. Noaons onsd a oulaon of N uns. L Y b h vaabl fo whch w wan o sma h oulaon man and P, P,, P a auxla abus. Fo wo-has samln dsn, l n and n ( n < n b saml szs fo fs and scond has scvl. In dfnn h abus w assum coml dchoom so ha h h, f un of oulaon ossss auxla abu = 0, ohws L A N = and = a n = b h oal numb of uns n h oulaon and saml scvl os- = 47

3 A a sssn abu. L P = and = b h cosondn ooon of uns ossssn a scfc N n abus and s h man of h man vaabl a scond has. L and dno h h auxla abu fom fs and scond has samls scvl and dno h vaabl of ns fom scond has. L P and dno h oulaon mans and coffcn of vaaon of h auxla abu scvl and Pb dnos h oulaon b-sal colaon coffcn of Y and P. L b ooon of uns ossssn abu n fs has saml of sz n whl b ooon of uns ossssn abu n scond has saml of sz n. Fuh, l = Y, θ = n N P θ = n N P =, ( θ < θ = (,,, wh, and ( a samln o whch a assumd o b v small. W l = (.0 v =, P P v = =, P P v = + (. P whl v d =, v d = = P P H w shall a ( d v d ( = = + P + P P v o, m of od n as ( (. vd fo =,,, (.3 P S S L = and = a h coffcn of vaaon of sud vaabl and h auxla vaabls Y P scvl. Th b-sal colaon coffcn bwn sud vaabl and auxla abus s vn b S Pb =. Thn fo sml andom samln whou lacmn fo boh fs and scond hass w w SS b usn has ws oaon of xcaons as: ( = θ, ( = θ, ( = θ J Pb ( (, = θ θ ( = θ ( ( ( ( ( (, ; = θ θ = θ θ ( = ( θ θ P, E( ( ( ; PP = θ θ E Y E P E YP E YP E PP E X E PP E If A s a sua max, s nvs can b wn usn ad on max as, A A ( (.4 T = = (.5 Ad A A 48

4 (. = Aoa and La [] (.6 Th follown noaons wll b usd n dvn h man sua os of oosd smaos Dmnan of oulaon colaon max of abus,,,, and. Dmnan of Pb h mno of cosondn o h h lmn of Pb. Dnos h mull coffcn of dmnaon of on,,, and. Dnos h mull coffcn of dmnaon of on,,,, and. Pb Dmnan of oulaon colaon max of abus,,, and. Dmnan of h colaon max of,,,, and. Dmnan of h colaon max of,,,, and.. Dmnan of h mno cosondn o.. (,,,,, and. Dmnan of h mno cosondn o.. ( Pb of h colaon max of Pb of h colaon max,,,,, and. (.7.. Man Un n Two-Phas Samln Th saml man usn sml andom samln whou lacmn s vn b, Whl h vaanc of s vn b, n n = = (.8 θ V = Y (.9.3. ao and Poduc Esmao n Two-Phas Samln Usn On Auxla Abus In od o hav an sma of h oulaon man Y of h sud vaabl, assumn h nowld of h oulaon ooon P, Na and Gua [8] dfnd ao and oduc smaos of oulaon man whn h o nfomaon of oulaon ooon of uns ossssn h sam abu s vaabl. Na and Gua [8] oosd h follown smaos: P P = = P Th MSE of and P u o h fs od of aoxmaon a vn scvl b, ( θ P P P Pb ( θ P P P Pb (.0 (. MSE = Y + (. MSE = Y + + (.3 Pb Pb Th omum valu a = and = fo ao and oduc smao scvl. 49

5 .3. ao and Poduc Esmao Usn Mull Auxla Abus n Two-Phas Samln Th ao and oduc smaos b Hanf, Ha and Shahbaz [5] fo snl has samln usn nfomaon on mull auxla abus a vn scvl b, P β β β P P P = (.3 ( ( ( ( P P P = (.4 Th MSE of h and P u o h fs od of aoxmaon a vn scvl b, 3. Mhodolo ( = θ ( Pb ( = θ ( P Pb MSE Y MSE Y 3.. ao-um-poduc Esmao Usn Mull Auxla Abus fo Full Infomaon as n Two-Phas Samln (.5 (.6 If w sma a sud vaabl whn nfomaon on all auxla vaabls s avalabl fom oulaon, s ulzd n h fom of h mans. B an h advana of ao-cum-oduc chnu fo wo-has samln, a nalzd smao fo sman oulaon man of sud vaabl Y wh h us of mul auxla abus s oosd as: P( 3. β+ β+ β P P P + + ( ( P + P + P = (3. Usn (.0, (. n (3. and non h scond and hh ms fo ach xanson of oduc and af smlfcaon, w w, h+ = Y Y = + β Th man suad o of ao-cum-oduc smao s: (3. P( 3. = P = + P MSE E h+ = Y Y ( ( 3. = P + β = P = + P (3.3 W dffna Euaon (3.3 aall wh sc o ( =,,, and β ( = +, +,, hn ua o zo, usn (.5, (.7 and (.4, w = + β = + P P Usn nomal uaons ha a usd o fnd h omum valus of lfd fom as: and (3.4 (3.5 β (3.3 can b wn n sm- 50

6 Usn (.4 n (3.6, w, MSE E Y Y h+ = ( ( 3. = P + β = P = + P MSE( Y P P Usn h omum valu and O O (3.6 + h= ( 3. = θ P Pb + β Pb = = (3.7 β n (3.4 and (3.5 and (3.7, w, MSE Y P P + h= ( ( 3. = θ + ( ( P Pb + Pb = P P = + Usn (.6 n (3., w, θy ( ( 3. = ( P + = MSE MSE Y ( ( 3. = θ P = θ 3. (. MSE( Y P Pb 3.. ao-um-poduc Esmao Usn Mull Auxla Abus fo Paal Infomaon as n Two-Phas Samln Pb (3.8 (3.9 (3. (3. In hs scon, w oosd a ao-cum-oduc smao usn mull auxla abus fo aal nfomaon cas n wo-has samln usn auxla abus wh s nown and s unnown abus whch a osvl colad wh sud vaabl Y and auxla abus wh nown and + unnown abus whch a navl colad wh sud vaabl (Y. Th oosd ao-cum-oduc smao fo aal nfomaon cas s as follows, P( 3. = β β β P P s P s + + γ+ λ+ s+ s P P + + λ γ h γh γh γ + λh + + h h h + + h+ P + h P h h h + + γ + (3.3 Usn (.0, (. n (3.3 and non h scond and hh ms fo ach xanson of oduc and af smlfcaon, w w, ( ( Y Y Y = + + P( 3. = P = P = + P + = h + = h Y Y Y ( + = + P = + P = + h+ P (3.4 5

7 Man suad o of P( 3. s vn b ( ( MSE E W dffna Euaon (3.5 wh sc o Y Y Y ( ( 3. = P + + = P = P = + P + = h + = h Y Y Y ( = + P = + P = h+ P + ( =,,,, β ( =,,,, (,,, = + +, γ ( = +, +,,, λ ( = +, +,,, γ ( = +, +,, h h and ua o zo and us (.4, (.6 and (.7. Th omum valu s as follows, + = ( β = m m m + m h h (3.5 (3.6 (3.7 + = ( + γ = ( λ = m γ = x m m + ( + m (3.8 (3.9 (3.0 (3. Usn nomal uaons ha a usd o fnd h omum valus of, β,, γ, λ and γ (3.5 can b wn n smlfd fom as MSE ( P( 3. ( x x x s+ = x x β = P = P = + P = EE / + Y Y Y + Y ( x x += h x x += h x x x Yγ Yλ γ P = + P = h+ P P Subsun (.4, (3.4 o (3.9 n (3.0, w + (3. 5

8 O O MSE + m + m ( ( 3. = Y θ + ( θθ ( Pb ( P θ Pb = = m m s h + = + = + + m + ( θθ ( Pb + ( θθ ( = + = + m h + = + m + θ Pb + θθ γ Pb = + h = + m = Y s= ( θ θ Pb Pb = = + + = h + ( Pb + ( γ Pb = + h = + + = h + θ + ( Pb + = = + m m Pb ( θ θ = Y + = θ Pb + = h m m + Pb + + = m = + m m Pb Pb (3.3 (3.4 (3.5 O Usn (.6 n (3.6, w, O MSE ( = Y P 3. ( θ θ + θ m m ( = 3. ( θ θ( Pb + θ( P Pb m MSE Y ( = 3. θ( Pb + θ P ( Pb Pb m MSE Y (3.6 (3.7 ( ao-um-poduc Esmao n Two-Phas Samln (No Infomaon as If w sma a sud vaabl whn nfomaon on all auxla vaabls s unavalabl fom oulaon, s 53

9 ulzd n h fom of h mans. B an h advana of ao-cum-oduc chnu fo wo-has samln, a nalzd smao fo sman oulaon man of sud vaabl Y wh h us of mul auxla vaabls a susd as: P( 3.3 = s γ+ γ+ γ + + ( ( + + (3.9 Usn (.0, (. n (3.9 and non h scond and hh ms fo ach xanson of oduc and af smlfcaon, w w, = + Y Y P( 3.3 Man suad o of P( 3.3 smao s vn b x x + h= x x γ = P = P (3.30 MSE E Y Y x x + h= x x ( ( 3.3 = + γ P = P = P (3.3 W dffna Euaon (3.3 aall wh sc o ( =,,, and β ( = +, +,, hn ua o zo, usn (.5, (.7 and (.4, w : + = ( (3.3 + β = ( Usn nomal uaons ha a usd o fnd h omum valus of smlfd fom as: Usn (.4 n (3.34, w, and x x + h= x x MSE ( P( 3.3 = E + Y Yγ = P = P (3.33 β (3.3 can b wn n ( h= Y θ θ θ P θ θ β P Pb Pb = = + (3.35 = + Subsun uaon (3.3 and (3.33 n (3.35, w O MSE ( + h= = θ Y 3.3 θ + θ θ P Pb + θ θ Pb = = ( θ θ MSE ( ( 3.3 = Y + ( P Pb + θ = = Y ( θ θ + θ (3.36 (3.37 (

10 Usn (.6 n (3.38, w, ( ( = 3.3 θ( P Pb + θ Pb MSE Y ( Bas and onssnc of ao-um-poduc Esmaos Ths ao-cum-oduc smaos usn mull auxla abus n wo-has samln a basd. Howv, hs bass a nlbl fo moda and la samls. I s asl shown ha h ao-cum-oduc smaos a conssn smaos usn mull auxla vaabls snc h a lna combnaons of conssn smaos follows ha h a also conssn. 4. Smulaon, sul and Dscusson In hs scon, w cad ou som daa smulaon xmns o coma h fomanc of ao-cum oduc smao n wo-has samln usn mull auxla abus wh xsn smaos of fn oulaon ha uss on o mull auxla abus naml man un, ao and oduc smao usn on auxla abus and ao and oduc smaos usn wo auxla abus. Th smulad daa fo h mcal sud nclud a sud vaabl and auxla abus ha a nomall dsbud wh h follown vaabls N = 300, n = 45, Man = 45, sandad dvaon = 5 = Pb = Pb Pb = 0.68 = 0.88 = In od o valua h ffcnc an w could achv b usn h oosd smaos, w hav calculad h vaanc of man un and h Man suad o of all smaos w hav consdd. W hav hn calculad cn lav ffcnc of ach smao n laon o vaanc of man un. W hav hn comad h cn lav ffcnc of ach smao, h smao wh h hhs cn lav ffcnc s consdd o b h mos ffcn han h oh smao. Th ffcnc s calculad usn h follown fomula: ( ˆ ˆ MSE ff ( Y = 00 (4.0 MSE Y ˆ Tabl shows h cn lav ffcnc of xsn and oosd smao wh sc o man un smao fo wo-has samln. I s obsvd ha ao and oduc smaos usn on auxla abu a mo ffcn han man un n h wo-has samln. Aan, ao and oduc smao usn mull auxla abus a mo ffcn han man un and ao and oduc smao usn on auxla abu n h wo-has samln. Fnall, ao-cum-oduc smao n h wo-has samln fo full nfomaon cas usn mull auxla abus s h mos ffcn of h fv smaos snc has h hhs cn lav ffcnc. Tabl shows cn lav ffcnc of ao-cum-oduc smaos wh sc o man un smao n wo-has samln. I s obsvd ha h ao-cum-oduc smaos a mo ffcn han man un n h scond has samln. Fnall, Tabl 3 comas h ffcnc of full nfomaon cas and aal cas o no nfomaon cas and full o aal nfomaon cas. I s obsvd ha h full nfomaon cas and aal nfomaon cas a mo ffcn han no nfomaon cas bcaus h hav hh Pcn lav Effcnc han no nfomaon cas. In addon, h full nfomaon cas s mo ffcn han h aal nfomaon cas bcaus has a hh Pcn lav Effcnc han aal nfomaon cas. 5. oncluson ao-cum-oduc smao usn mull auxla abus n full nfomaon cas n wo-has samln s commndd o sma oulaon man as oufoms oh smao n wo-has samln. If som auxla abus a nown, h ao-cum-oduc smao usn mull auxla abus n aal nfo- 55

11 Tabl. lav ffcnc of xsn and oosd smao wh sc o man un smao fo wo-has samln. Esmao lav cn ffcnc wh sc o man un 00 4 P 6 93 P 79 P( 3. 0 Tabl. lav ffcnc of xsn and oosd smaos wh sc o man un smao fo wo-has samln. Esmao lav ffcnc of xsn and oosd smaos wh sc o man un smao fo wo-has samln P( P( 3. P( 3. Tabl 3. omasons of full, aal and no nfomaon cass fo oosd ao-cum-oduc smao usn mull auxla vaabls. Poulaon Pcn lav ffcnc of full and aal o no nfomaon Pcn lav ffcnc of full o aal n fomaon cas Esmao P( 3. P( 3. P( 3.0 P( 3. P( 3.0 lav cn ffcnc maon cas should b usd bu f all h auxla abus a unnown, and ao-cum-oduc smao usn mull auxla abus n no nfomaon cas should b usd o sma fn oulaon man. Ths s cla fom Tabl 3. fncs [] ochan, W.G. (940 Th Esmaon of h Ylds of h al Exmns b Samln fo h ao of Gan o Toal Poduc. Th Jounal of Aculual Scnc, 30, h://dx.do.o/0.07/s [] Muh, M.N. (964 Poduc Mhod of Esmao. Th Indan Jounal of Sascs Ss, 6, [3] Oln, I. (958 Mulvaa ao Esmaon fo Fn Poulaon. Boma, 45, h://dx.do.o/0.093/bom/ [4] a, D. (965 On a Mhod of Usn Mul-Auxla Infomaon n Saml Suvs. Jounals of h Amcan Sascal Assocaon, 60, h://dx.do.o/0.080/ [5] Snh, M.P. (967 Mulvaa Poduc Mhod of Esmaon fo Fn Poulaon. Jounal of h Indan Soc of Aculu Sascs, 3, [6] Snh, M.P. (967 Mulvaa Poduc Mhod of Esmaon fo Fn Poulaon. Jounal of h Indan Soc of Aculu Sascs, 3, [7] Snh, H.P. and Talo,. (003 Us of Known olaon offcn n Esman h Fn Poulaon Man. Sascs n Tanslaon, 6, [8] Na, V.D. and Gua, P.. (996 A No on Esmaon of Man wh Known Poulaon of Auxla haac. Jounal of h Indan Soc of Aculual Sascs, 48, [9] Bahl, S. and Tua,.K. (99 ao and Poduc T Esmao. Infomaon and Omzaon Scnc,, 56

12 h://dx.do.o/0.080/ [0] ash, S., Pana,., Nmala, S. and Flonns, S. (007 ao-poduc T Exonnal Esmao fo Esman Fn Poulaon Man Usn Infomaon on Auxla Abus. nassanc Hh Pss, USA. [] Jha, H.S., Shama, M.K. and Gov, L.K. (006 A Faml of Esmao of Poulaon Man Usn Infomaon on Auxla Abus. Pasan Jounal of Sascs,, [] Nman, J. (938 onbuon o h Tho of Samln Human Poulaons. Jounal of h Amcan Sascal Assocaon, 33, 0-6. h://dx.do.o/0.080/ [3] obson, D.S. (95 Mull Samln of Abus. Jounal of h Amcan Sascal Assocaon, 47, h://dx.do.o/0.080/ [4] Smuddn, M. and Hanf, M. (007 Esmaon of Poulaon Man n Snl and Two Phas Samln wh o whou Addonal Infomaon. Pasan Jounal of Sascs, 3, [5] Hanf, M., Ha, I.U. and Shahbaz, M.Q. (009 On a Nw Faml of Esmao Usn Mull Auxla Abus. Wold Ald Scnc Jounal,, [6] Ha, I.U. (009 A Faml of Esmaos fo Two-Phas Samln Usn Mul-Auxla Abus. Ph.D. Thss, Naonal oll of Busnss Admnsaon & Economcs, Laho. [7] Aoa, S. and Bans, Lal. (989 Nw Mahmacal Sascs. Saa Paashan, Nw Dlh. 57