Improved Exponential Estimator for Population Variance Using Two Auxiliary Variables

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1 Improvd Epoal Emaor for Populao Varac Ug Two Aular Varabl Rajh gh Dparm of ac,baara Hdu Uvr(U.P., Ida Pakaj Chauha ad rmala awa chool of ac, DAVV, Idor (M.P., Ida Flor maradach Dparm of Mahmac, Uvr of w Mco, Gallup, UA (marad@um.du Abrac I h papr poal rao ad poal produc p maor ug wo aular varabl ar propod for mag ukow populao varac. Problm dd o h ca of wo-pha amplg. Thorcal rul ar uppord b a mprcal ud. K word: Aular formao, poal maor, ma quard rror.. Iroduco I commo pracc o u h aular varabl for mprovg h prco of h ma of a paramr. Ou of ma rao ad produc mhod of mao ar good ampl h co. Wh h corrlao bw h ud vara ad h aular vara pov (hgh rao mhod of mao qu ffcv. O h ohr had, wh h corrlao gav (hgh produc mhod of mao ca b mplod ffcvl. L ad (, do h ud vara ad aular vara akg h valu ad (, rpcvl, o h u U (,,,, whr povl corrlad wh ad gavl corrlad wh. To ma (, aumd

2 ha ( X ad ( Z ar kow. Aum ha populao larg o ha h f populao corrco rm ar gord. Aum ha a mpl radom ampl of draw whou rplacm (RWOR from U. Th uual ubad maor of whr ( (. h ampl ma of. Wh h populao varac rao maor for a ( X kow, Iak (983 propod a k (. whr ( X a ubad maor of. Upo h fr ordr of appromao, h varac of populao corrco (fpc rm ar rpcvl gv b ad ME of k (gorg h f (.3 ( [ ] var ME (. ( [ ] k μ pqr whr δ pqr, p / q / r / μ μ μ μ pqr p q r ( Y ( X ( Z ; p, q, r bg h o-gav gr.

3 Followg Bahl ad Tuja (99, w propo poal rao p ad poal produc p maor for mag populao varac a p (.5 p (.6. Ba ad ME of propod maor To oba h ba ad ME of, w wr (, ( uch ha E( E( E E Afr mplfcao w g h ba ad ME of a ad (, (, ( (. E 3 B (. 8 8 ME (. To oba h ba ad ME of, w wr (, ( uch ha E( E( E( (, ( ( E Afr mplfcao w g h ba ad ME of a 5 B( ( ( (. ME 3. Improvd Emaor Followg Kadlar ad Cg (6 ad gh. al. (7, w propo a mprovd maor for mag populao varac a-

4 α p ( αp (3. whr α a ral coa o b drmd uch ha h ME of mmum. Eprg rm of, w hav ( α α p p (3. Epadg h rgh had d of (3. ad rag rm up o cod powr of, w hav 8 α 8 α 8 α α ( Takg pcao of boh d of (3.3 ad h ubracg h ba of h maor, up o h fr ordr of appromao, a ( α ( α ( α ( ( 8 8 B From (3., w hav ( α α from boh d, w g α ( (3. (3.5 quarg boh h d of (3.5 ad h akg pcao, w g ME of h maor, up o h fr ordr of appromao, a α α ( ( ME ( α ( α α( ( α( (3.6 Mmao of (3.6 wh rpc o α ld opmum valu a { ( 6} ( α α (a (3.7 ubuo of α from (3.7 o (3.6 gv mmum valu of ME of.

5 . Propod maor wo-pha amplg I cra praccal uao wh o kow a pror, h chqu of wopha or doubl amplg ud. Th chm rqur collco of formao o ad h fr pha ampl of ( < ad o for h cod pha ampl of (< from h fr pha ampl. Th maor, ad wo-pha amplg wll ak h followg form, rpcvl ' d p (. ' ' d p (. ' ' ' d k p ( kp (.3 ' ' To oba h ba ad ME of d, d, d, w wr (, (, ' ( ' (, ' ( ' ' whr ' ( ', ' ' ' Alo, E( E(, ', E(' ' ' ' ' (, (' ( E(' ' ' (, E ' ( ' Eprg d, d, ad d rm of ad followg h procdur plad co ad co3 w g h ME of h maor, rpcvl a-

6 ME( d ' ( ( (. ' ( ME( d ' ( ( (.5 ' ( ( d k ' k ' ( ME ' ' ( ( k ( k k(k ( (.6 ' Mmao of (.6 wh rpc o k ld opmum valu a { ( 6} ( k k (a (.7 ubuo of k from (.7 o (.6 gv mmum valu of ME of d. 5. Emprcal ud To llura h prformac of varou maor of Murh(967, p.-6. Th vara ar: : oupu, : umbr of workr, : fd capal, 8, 5,., w codr h daa gv.667, 3. 65,. 866,. 3377,. 8, 3. Th prc rlav ffcc (PRE of varou maor of wh rpc o covoal maor ha b compud ad dplad abl 5..

7 Tabl 5. : PRE of,, ad m. ME ( wh rpc o Emaor PRE(., I abl 5. PRE of varou maor of wo-pha amplg wh rpc o dplad. Tabl 5. : PRE of, d, d ad m.me ( d wh rpc o Emaor PRE (., d 7.76 d d 7.77 ar 6. Cocluo From abl 5. ad 5., w fr ha h propod maor prform br ha covoal maor ad ohr mod maor. Rfrc Iak, C. T. (983: Varac mao ug aular formao. Joural of Amrca acal Aocao. Bahl,. ad Tuja, R.K. (99: Rao ad Produc p poal maor, Iformao ad Opmao cc, Vol.XII, I, Kadlar,C. ad Cg,H. (6 : Improvm mag h populao ma mpl radom amplg. Appld Mahmac Lr 9 (

8 gh, R. Chauha, P. awa,. ad maradach, F. (7: Aular formao ad a pror valu coruco of mprovd maor. Raac hgh pr, UA. Murh, M..(967: amplg Thor ad Mhod. acal Publhg oc, Calcua, Ida.

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