Improved estimation of population variance using information on auxiliary attribute in simple random sampling. Rajesh Singh and Sachin Malik

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1 Imrovd imaio of oulaio variac uig iformaio o auxiliar ariu i iml radom amlig Rajh igh ad achi alik Darm of aiic, Baara Hidu Uivri Varaai-5, Idia (righa@gmail.com, achikurava999@gmail.com) Arac igh ad Kumar () uggd imaor for calculaig oulaio variac uig auxiliar ariu. Thi ar roo a famil of imaor ad o a adaaio of h imaor rd Kadilar ad igi () ad igh al. (7), ad iroduc a w famil of imaor uig auxiliar ariu. Th xrio of h ma quar rror (E) of h adad ad rood famili ar drivd. I i how ha adad imaor ad uggd imaor ar mor ffici ha igh ad Kumar () imaor. Th horical fidig ar uord a umrical xaml. K word: iml radom amlig, auxiliar ariu, oulaio variac, ma quar rror, fficic.. Iroducio I i wll kow ha h auxiliar iformaio i h hor of amlig i ud o icra h fficic of imaor of oulaio aramr. Ou of ma raio, rgrio ad roduc mhod of imaio ar good xaml i hi cox. Thr xi iuaio wh iformaio i availal i h form of ariu which i highl corrlad wih. Takig io coidraio h oi i-rial corrlaio coffici w auxiliar ariu ad ud varial, vral auhor icludig Naik ad Gua (996), Jhajj al. (6), hair ad Gua (7), igh al. (7, 8), igh al. (), Ad- Elfaah al. (), alik ad igh (3a,,c), igh (3), igh ad alik (3) ad harma al. (3) dfid raio imaor of oulaio ma wh h rior iformaio of oulaio roorio of ui, oig h am ariu i availal. I ma iuaio, h rolm of imaig h oulaio variac of ud varial aum imorac. Wh h rior iformaio o aramr of auxiliar varial() i

2 availal, Da ad Triahi (978), Iaki (983), Praad ad igh (99), Kadilar ad igi (6, 7) ad igh al. (7) hav uggd variou imaor of. ad oidr a aml of iz draw RWOR from a oulaio of iz N. L i i do h orvaio o varial ad rcivl for h i h ui (i=,,3...n). I i aumd ha ariu ak ol h wo valu ad accordig a =, if i h ui of h oulaio o ariu =, if ohrwi. Th variac of h uual uiad imaor Ŝ i giv var( Ŝ) λ (.) whr, μ λrq, μ rq r/ q/ μ μ rq N r i Y i P i N q. I hi ar w hav rood a famil of imaor for h oulaio variac wh auxiliar varial i i h form of ariu. For mai rul w cofi ourlv o amlig chm RWOR igorig h fii oulaio corrcio.. Eimaor i liraur I ordr o hav a ima of h ud varial, aumig h kowldg of h oulaio roorio P, igh ad Kumar () rood h followig imaor (.) (.)

3 3 x (.3) Th E xrio of h imaor ad variac of ar giv, rcivl, E λ λ λ (.) λ λ λ V (.5) O diffriaig (.5) wih rc o ad quaig o zro w oai λ (.6) λ x uiuig h oimum valu of i (.5), w g h miimum variac of h imaor, a λ λ mi.v λ (.7) Th E xrio of h imaor 3 i giv λ E 3 λ λ (.8) 3. Th adad imaor Followig Kadilar ad igi (), w roo h followig variac imaor uig kow valu of om oulaio aramr(), K (3.) K (3.)

4 K3 (3.3) K (3.) whr, ad ar uiad imaor of oulaio variac To oai h ia ad E, w wri-, uch ha E E ad λ E, E ad k. λ, E λ, ad rcivl. Th E xrio of ŜK (i,,3, ) o h fir ordr of aroximaio ar rcivl giv i λ ω λ ω λ,(i,,3, ) E( Ki ) i i (3.5) whr, ω,ω, ω3, ω. Followig igh al. (7), w roo imaor a (3.6) whr, ar ihr ral umr or h fucio of h kow aramr of ariu uch a,, ad k. Exrig quaio (3.6) i rm of, w hav

5 A whr,. A U o fir ordr of aroximaio, h E of i giv A E E A A A A A A λ λ E (3.7) Tal 3. r om of h imora imaor of h oulaio variac, which ca oaid uial choic of coa,. Tal 3.: mr of Eimaor Valu of

6 8 9 oivad igh al. (7), w roo aohr imrovd raio- imaor R for h oulaio variac a η ν αη ν αη ν R (3.8) whr η, ν ar ihr ral umr or h fucio of h kow aramr of ariu uch a,, ad k. Exrig quaio (3.8) i rm of, w hav R A η whr, A η -. ν U o fir ordr of aroximaio, h E of R i giv E E A α R E R λ A α λ A αλ (3.9) iimizaio of (3.9) wih rc o α ild i oimum valu a α o A λ λ.

7 uiuig oimum valu of α i (3.9), w g h miimum variac of R. Tal 3. r om of h imora imaor of h oulaio variac, which ca oaid uial choic of coa η, ν. Tal 3.: mr of R Eimaor Valu of α η ν R R R R3 R R5 R6 -. Th uggd cla of imaor W ugg aohr imrovd cla of imaor for oulaio variac a δ μ δ μ δ μ δ μ m m x γ (.)

8 whr δ ad μ ar ihr ral umr or fucio of kow aramr of h auxiliar ariu uch a,, ad k.th calar γ ak valu - ad + for raio ad roduc imaor, rcivl. Exrig quaio (.) i rm of ad raiig rm u o cod dgr of, w hav γ λθ γ θ (.) m m U o fir ordr of aroximaio, h E of h imaor i E whr, m R m R m m R m R m R (.3) γ λ γ θ λ γ θ λ γλ, R λ, R λ γλ, R 3 γ R γ θ γλ λ. R 5 λ γθλ, 3 O ariall diffriaig (.3) wih rc o m i i,, w g oimum valu of m ad m, rcivl a m (o) R R R R R R R RR 5 R 3R. ad m (o) R R R 3 5

9 5. Efficic omario Fir, w comar h fficic of rood imaor udr oimum codiio wih uual imaor: Ŝ E λ m R m R m m R m R m R V 3 5 (5.) Nx, w comar h fficic of rood imaor udr oimum codiio wih h raio imaor, xoial imaor, rgrio imaor ad ohr imaor lid i h ar. From (.) ad (.3), w hav E E( ) From (.5) ad (.3), w hav λ λ λ m R m R m m R m R m R (5.) 3 5 E( ) λ λ λ V From (.8) ad (.3), w hav m R m R m m R m R m R (5.3) 3 5 λ E 3 E( ) λ λ From (3.5) ad (.3), w hav m R m R m m R m R m R (5.) 3 5 E( Ki ) E( ) i i From (3.7) ad (.3), w hav E λ ω λ ω λ m R m R m m R m R m R (5.5) 3 5 E( ) λ λ A A A

10 m R m R m m R m R m R (5.6) 3 5 E R - E( ) λ A α λ A αλ m R m R m m R m R m R (5.7) 3 5 Uig (5.)-(5.7), w coclud ha h rood imaor udr oimum codiio rform r ha h ohr imaor dicud i hi ar. 6. Emirical ud For mirical ud, w u h daa giv i ukham ad ukham (97),. 56. Th varial of h ir ar: Y i umr of villag i h circl, ad rr a circl coiig mor ha fiv villag. Th valu of rquird aramr ar: N = 89, =3, =.7, =., =.6, = , 6.6, 3.996, λ 3.8, λ 6.6, k Tal 6.: PRE of variou imaor Eimaor PRE Eimaor PRE K K 6. 9

11 6. K K.89 R K 9.35 R 6. K R3. K 5.6 R R R R I Tal 6., h rc rlaiv fficici of h rood imaor Ki (i,,3,), i (i,,..., ) ad Ri (i,,...,6) ar lid wh w choo diffr valu of ad i ca of h imaor i ad α, η ad ν i ca of h imaor Ri rcivl. Alo, h PRE of rgrio imaor R i 6.8. Tal 6.: PRE of uggd imaor wih diffr valu of coa Valu of δ ad μ wh ( γ =) δ μ PRE 6.5 N 7.3 N f 6.57 N g=(-f) 7.7 k

12 N 75. N N k 7.93 f I h Tal 6., PRE of h rood imaor wih rc o i calculad for diffr valu of aramr. I i orvd h high PRE 8.57 i oaid for ad. I ha alo orvd ha h uggd cla of imaor udr oimum codiio i mor ffici ha uual uiad imaor, uual rgrio imaor, igh ad Kumar () imaor ad ohr imaor dicud i hi ar. Hc, for rood choic of aramr h rood imaor imaor coidrd i hi ar. Ackowldgm i amog all h Th auhor ar vr idd o h dior i chif Prof. lvi co ad aomou rfr for hir valual uggio ladig o imrovm of h quali of co ad raio of h origial maucri. Rfrc Ad-Elfaah, A.. El-hri, E.A. ohamd,.. Adou, O. F. (): Imrovm i imaig h oulaio ma i iml radom amlig uig iformaio o auxiliar ariu. Al. ah. ad om. doi:.6/j.amc.9... Da A.K. ad Triahi T.P.(978): U of auxiliar iformaio i imaig h fii oulaio variac. akha,, Iaki,. T. (983). Variac imaio uig auxiliar iformaio, Jour. of Amr. ai. Ao.78, 7 3.

13 Jhajj, H.., harma,. K. ad Grovr, L. K. (6) : A famil of imaor of oulaio ma uig iformaio o auxiliar ariu. Pak. Jour. of a., (), 3-5. Kadilar,. ad igi, H. (): Raio imaor i iml radom amlig. Al. ah. ad om. 5, Kadilar,. ad igi, H. (6): Nw raio imaor uig corrlaio coffici. Ira, -. Kadilar,., igi, H. (7): Imrovm i variac imaio i iml radom amlig. omm. i a. Tho. ad h. 36, alik,. ad igh, R. (3a) : A imrovd imaor uig wo auxiliar ariu. Ali. ah. om., 9, alik,. ad igh, R. (3) : A famil of imaor of oulaio ma uig iformaio o oi i-rial ad hi corrlaio coffici. I. Jour. a. Eco. (), alik,. ad igh, R. (3c) : Dual o raio cum roduc imaor of fii oulaio ma uig auxiliar ariu() i raifid radom amlig. WAJ, 8(9), Naik, V.D., Gua, P.. (996): A o o imaio of ma wih kow oulaio roorio of a auxiliar characr. Jour. of h Id. oc. of Agr. a. 8() Praad, B., igh, H. P. (99). om imrovd raio- imaor of fii oulaio variac i aml urv. omm. i a. Tho. ad h. 9:7 39. hair, J., Gua,. (7): O imaig h fii oulaio ma wih kow oulaio roorio of a auxiliar varial. Pak. Jour. of ai. 3 () 9. harma, P., Vrma, H., aaullah, A. ad igh, R. (3): om xoial raio-roduc imaor uig iformaio o auxiliar ariu udr cod ordr aroximaio. I. Jour. a. Eco., (3), igh, R. (3) : O imaio of oulaio ma uig iformaio o auxiliar ariu. PJOR 9(), igh, R., auha, P., awa, N. ad maradach, F. (7): Auxiliar iformaio ad a riori valu i corucio of imrovd imaor. Raiac High r.

14 igh, R. hauha, P. awa, N. ad maradach, F. (8): Raio imaor i iml radom amlig uig iformaio o auxiliar ariu, Pak. Jour. a. Or. Rc. (), igh, R., Kumar,. ad maradach, F. (): Raio imaor i iml radom amlig wh ud varial i a ariu. Worl. A. ci. Jour. (5), igh R. ad Kumar. (): A famil of imaor of oulaio variac uig iformaio o auxiliar ariu. udi i amlig chiqu ad im ri aali. Zi Pulihig, igh, R., alik,. (3) : A famil of imaor of oulaio ma uig iformaio o wo auxiliar ariu. Worl. A. ci. Jour., 3(7), ukham, P.V. ad ukham, B.V. (97) : amlig hor of urv wih alicaio. Iowa a Uivri Pr, Am, U..A.

Ratio-Product Type Exponential Estimator For Estimating Finite Population Mean Using Information On Auxiliary Attribute

Ratio-Product Type Exponential Estimator For Estimating Finite Population Mean Using Information On Auxiliary Attribute 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