Raio-Produc T Exonnial Esimaor For Esimaing Fini Poulaion Man Using Informaion On Auxiliar Aribu Rajsh Singh, Pankaj hauhan, and Nirmala Sawan, School of Saisics, DAVV, Indor (M.P., India (rsinghsa@ahoo.com Flornin Smarandach hair of Darmn of Mahmaics, Univrsi of Nw Mxico, Gallu, USA (smarand@unm.du Absrac In racic, h informaion rgarding h oulaion roorion ossssing crain aribu is asil availabl s Jhajj.al. (6. For simaing h oulaion man Y of h sud variabl, following Bahl and Tuja (99, a raio-roduc xonnial simaor has bn roosd b using h known informaion of oulaion roorion ossssing an aribu (highl corrlad wih in siml random samling. Th xrssions for h bias and h man-squard rror (MSE of h simaor and is minimum valu hav bn obaind. Th roosd simaor has an imrovmn ovr man r uni simaor, raio and roduc xonnial simaors as wll as Naik and Gua (996 simaors. Th rsuls hav also bn xndd o h cas of wo has samling. Th rsuls obaind hav bn illusrad numricall b aking som mirical oulaions considrd in h liraur. Kwords: Proorion, bias, man-squard rror, wo has samling.. Inroducion
In surv samling, h us of auxiliar informaion can incras h rcision of an simaor whn sud variabl is highl corrlad wih h auxiliar variabl x. bu in svral racical siuaions, insad of xisnc of auxiliar variabls hr xiss som auxiliar aribus, which ar highl corrlad wih sud variabl, such as (i Amoun of milk roducd and a aricular brd of cow. (ii Yild of wha cro and a aricular vari of wha c. (s Shabbir and Gua(6. In such siuaions, aking h advanag of oin bisrial corrlaion bwn h sud variabl and h auxiliar aribu, h simaors of aramrs of inrs can b consrucd b using rior knowldg of h aramrs of auxiliar aribu. onsidr a saml of siz n drawn b siml random samling wihou rlacmn (SRSWOR from a oulaion of siz N. l i and i dno h obsrvaions on variabl and rscivl for h i h uni ( i =,,..., N. W no ha i =, if i h uni of oulaion osssss aribu and i =, ohrwis. L N A = i and a = i= n i= i dno h oal numbr of unis in h oulaion and saml rscivl ossssing aribu. L A P = and N a = dno h roorion of unis n in h oulaion and saml rscivl ossssing aribu. In ordr o hav an sima of h oulaion man Y of h sud variabl, assuming h knowldg of h oulaion roorion P, Naik and Gua (996 dfind raio and roduc simaors of oulaion whn h rior informaion of oulaion roorion of unis, ossssing h sam aribu is availabl. Naik and Gua (996 roosd following simaors:
P = (. = P (. Th MSE of and u o h firs ordr of aroximaion ar MSE MSE ( f Y [ ( K ] = (.3 ( f Y [ ( K ] = (. S S whr =, =, Y P f =, K n N N = ρb, S = (i Y, N i= N N S = ( i P N, S = ii NPY and i= N i= S ρb = is h oin bisrial corrlaion cofficin. S S Following Bahl and Tuja (99, w roos h following raio and roduc xonnial simaors P 3 = x (. P P = x (.6 P. Bias and MSE of 3 and To obain h bias and MSE of 3 o h firs dgr of aroximaion, w dfin ( Y ( P =, =, hrfor E ( i =. i = (,, Y P E ( = f, f, E( = fρb. E ( = 3
Exrssing (. in rms of s, w hav = P( P P( P x Y( 3 = ( x ( Y (. Exanding h righ hand sid of (. and raining rms u o scond owrs of s, w hav = Y 3 (. Taking xcaions of boh sids of (. and hn subracing Y from boh sids, w g h bias of h simaor 3 u o h firs ordr of aroximaion, as ( = 3 K Y f B (.3 From (., w hav ( Y Y 3 (. Squaring boh sids of (. and hn aking xcaions w g MSE of h simaor 3, u o h firs ordr of aroximaion as = K ( Y f ( MSE 3 (. To obain h bias and MSE of o h firs dgr of aroximaion, w xrss (.6 in rms of s
P( P = Y( x (.6 P( P and following h abov rocdur, w g h bias and MSE of as follows B ( = fy K (.7 MSE( = fy ( K (. 3. Proosd class of simaors I has bn horicall sablishd ha, in gnral, h linar rgrssion simaor is mor fficin han h raio (roduc simaor xc whn h rgrssion lin of on x asss hrough h nighborhood of h origin, in which cas h fficincis of hs simaors ar almos qual. Also in man racical siuaions h rgrssion lin dos no ass hrough h nighborhood of h origin. In hs siuaions, h raio simaor dos no rform as good as h linar rgrssion simaor. Th raio simaor dos no rform wll as h linar rgrssion simaor dos. Following Singh and Esjo (3, w roos following class of raio-roduc xonnial simaors: P P = α x ( αx (3. P P whr α is a ral consan o b drmind such ha h MSE of is minimum. For α =, rducs o h simaor P 3 = x and for α =, i rducs o P P = x. P
Bias and MSE of : Exrssing (3. in rms of s, w hav = Y ( P P α x P P ( ( P( P ( αx P( P ( α x ( αx = Y (3. Exanding h righ hand sid of (3. and raining rms u o scond owrs of s, w hav = Y α α (3.3 Taking xcaions of boh sids of (3.3 and hn subracing Y from boh sids, w g h bias of h simaor u o h firs ordr of aroximaion, as B( = fy ρb α (3. From (3.3, w hav ( Y Y α (3. Squaring boh sids of (3. and hn aking xcaions w g MSE of h simaor, u o h firs ordr of aroximaion as MSE( = fy α α ρ b α (3.6 Minimizaion of (3.6 wih rsc o α ilds oimum valu of as K α = = α (Sa (3.7 6
Subsiuion of (3.7 in (3. ilds h oimum simaor for as ( o (sawih minimum MSE as min.mse( = f Y ( ρ =M( o (3. b which is sam as ha of radiional linar rgrssion simaor.. Efficinc comarisons In his scion, h condiions for which h roosd simaor is br han,,, 3, and hav bn obaind. Th varianc of is givn b var( = f (. Y To comar h fficinc of h roosd simaor wih h xising simaor, from (. and (.3, (., (., (. and (3., w hav var( M( = ρ b. (. ( ρ MSE( M( = b. (.3 ( ρ MSE( M( = b. (. MSE( 3 M( = ρb. (. MSE( M( = ρb. (.6 Using (.-(.6, w conclud ha h roosd simaor ourforms,,, 3, and.. Emirical sud W now comar h rformanc of various simaors considrd hr using h following daa ss: 7
Poulaion. [Sourc: Sukham and Sukham (97,. 6] = numbr of villags in h circls and = A circl consising mor han fiv villags. N = 9, Y =3.36, P =.36, ρ b =.766, =.6, =.9. Poulaion. [Sourc: Mukhoadhaa, (,. ] Y= Houshold siz and = A houshold ha availd an agriculural loan from a bank. N =, Y =9., P =., ρ b = -.37, =.7, =.77. Th rcn rlaiv fficinc (PRE s of h simaors, - and ( o wih rsc o unusual unbiasd simaor hav bn comud and comild in abl.. Tabl.: PRE of various simaors wih rsc o. Esimaor PRE s (., Poulaion I II.63.9.7.9 3 66..7.. (.9 7.6
Tabl. shows ha h roosd simaor undr oimum condiion rforms br han h usual saml man, Naik and Gua (996 simaors ( and and h raio and roduc xonnial simaors ( 3 and. 6. Doubl samling In som racical siuaions whn P is no known ariori, h chniqu of wo-has samling is usd. L dno h roorion of unis ossssing aribu in h firs has saml of siz n ; dno h roorion of unis ossssing aribu in h scond has saml of siz n < n and dno h man of h sud variabl in h scond has saml. Whn P is no known, wo-has raio and roduc xonnial simaor ar givn b ' 6 = x (6. ' ' 7 = x (6. ' To obain h bias and MSE of 6 and 7, w wri such ha and = Y(, P( E ( = E ( = ( ' =, ' = P( ' E =. E ( = f, E ( = f, E ( ' = f, E( ' = f ρb. 9
whr f =. n' N Exrssing (6. in rms of s, w hav 6 = Y( P( ' x P( ' P( P( ' = Y( x (6.3 Exanding h righ hand sid of (6.3 and raining rms u o scond owrs of s, w hav ' ' ' ' 6 = Y (6. Taking xcaions of boh sids of (6. and hn subracing Y from boh sids, w g h bias of h simaor 6 u o h firs ordr of aroximaion, as B = (6. ( f Y ( 6 3 K From (6., w hav (' 6 (6.6 ( Y Y Squaring boh sids of (6.6 and hn aking xcaions w g MSE of h simaor 6, u o h firs ordr of aroximaion as MSE ( 6 = Y f f 3 ( K (6.7 To obain h bias and MSE of 7 o h firs dgr of aroximaion, w xrss (6. in rms of s as
7 = Y( P( x P( P( ' P( ' ' = Y( x (6. Exanding h righ hand sid of (6. and raining rms u o scond owrs of s, w hav ' ' ' ' 7 = Y (6.9 Taking xcaions of boh sids of (6.9 and hn subracing Y from boh sids, w g h bias of h simaor 7 u o h firs ordr of aroximaion, as B ( 7 = f 3Y ( K (6. From (6.9, w hav ( ' 7 (6. ( Y Y Squaring boh sids of (6. and hn aking xcaions w g MSE of h simaor 7, u o h firs ordr of aroximaion as MSE ( 7 = Y f f 3 ( K (6. 7. Proosd class of simaors in doubl samling W roos h following class of simaors in doubl samling ' ' = α x ( α x (7. ' ' whr α is a ral consan o b drmind such ha h MSE of is minimum.
For α =, rducs o h simaor ' 6 = x and for α =, i rducs o ' ' 7 = x. ' Bias and MSE of : Exrssing (7. in rms of s, w hav ( P( ( P( ' ( α x P( P( ' P( ' P = Y( α x P( ' P ' ' = Y( α x ( αx (7. Exanding h righ hand sid of (7. and raining rms u o scond owrs of s, w hav = Y[ ' α α' ' ' ' α ' α ] (7.3 Taking xcaions of boh sids of (7.3 and hn subracing Y from boh sids, w g h bias of h simaor u o h firs ordr of aroximaion, as ( = f 3Y K α (7. B From (7.3, w hav Y Y α α ' (7. (
Squaring boh sids of (7. and hn aking xcaions w g MSE of h simaor, u o h firs ordr of aroximaion as MSE ( = Y f f 3 α α K (7.6 Minimizaion of (7.6 wih rsc o α ilds oimum valu of as K α = = α (Sa (7.7 Subsiuion of (7.7 in (7. ilds h oimum simaor for as ( o (sa wih minimum MSE as min.mse( = Y ( f f ρ =M( o, (sa (7. which is sam as ha of radiional linar rgrssion simaor.. Efficinc comarisons Th MSE of usual wo-has raio and roduc simaor is givn b MSE MSE ( Y [ f f ( K ] 9 3 3 b = (. ( Y [ f f ( K ] 3 = (. From (., (6.7, (6., (., (. and (7. w hav var( M( = f ρ. (.3 MSE( MSE( 3 b M( = f 3 ρb. (. 6 M( = f 3 ρb. (. 7 ( ρ MSE( M( = f. (.6 9 3 b 3
MSE( ( ρ M( = f. (.7 3 b From (.3-(.7, w conclud ha our roosd simaor is br han, 6, 7, 9, and. 9. Emirical sud Th various rsuls obaind in h rvious scion ar now xamind wih h hl of following daa: Poulaion. [Sourc: Sukham and Sukham( 97,. 6] N = 9, n =, n = 3, = 3, =.3, =.333, ρ b =., =.69, =.7. Poulaion. [Sourc: Mukhoadhaa(,. ] N =, n = 3, n = 7, = 7.3, =.9, =.3, ρ b = -.3, =.36, =.37. Tabl 9.: PRE of various simaors (doubl samling wih rsc o. Esimaor PRE s (., Poulaion I II 6.9. 7.9.9 9.6.9 7.6.9 (.3 6.7
Tabl 9. shows ha h roosd simaor undr oimum condiion rforms br han h usual saml man, 6, 7, 9, and. Rfrncs Bahl, S. and Tuja, R.K. (99: Raio and Produc xonnial simaor, Informaion and Oimizaion scincs, Vol.XII, I, 9-63. Jhajj, H. S., Sharma, M. K. and Grovr, L. K. (6: A famil of simaors of oulaion man using informaion on auxiliar aribu. Pak. J. Sais., (, 3-. Mukhoadhaa, P.(: Thor and mhods of surv samling. Prnic Hall of India, Nw Dlhi, India. Naik,V.D. and Gua, P.. (996: A no on simaion of man wih known oulaion roorion of an auxiliar characr. Jour. Ind. Soc. Agr. Sa., (,-. Shabbir,J. and Gua, S.(7 : On simaing h fini oulaion man wih known oulaion roorion of an auxiliar variabl. Pak. J. Sais.,3(,-9. Singh, H.P. and Esjo, M.R. (3: On linar rgrssion and raio-roduc simaion of a fini oulaion man. Th saisician,,, 9-67. Sukham, P.V. and Sukham, B.V. (97: Samling hor of survs wih alicaions. Iowa Sa Univrsi Prss, Ams, U.S.A.