A MODIFIED RATIO-CUM-PRODUCT ESTIMATOR OF FINITE POPULATION MEAN IN STRATIFIED RANDOM SAMPLING
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1 Data cience Journal, Volume 8, 4 eptember 009 MODIFIED IO-CUM-PODUC EIMO OF FINIE POPULION MEN IN IFIED NDOM MPLING ajes ail cool of tudies in tatiics, Viram University, Ujjain-45600, MP, India tailraj@gmailcom BC is paper sugges a ratio-cum-product eimat of finite population mean using a crelation coefficient between udy variate and auiliary variate in ratified random sampling Bias and mean squared epressions of te suggeed eimat are derived and compared wit combined ratio eimat and several oter eimats considered by Kadilar and Cingi (003 n empirical udy is also carried out to eamine te perfmance of te proposed eimat Key wds: Finite population mean, Crelation coefficient, tratified random sampling, Bias, Mean squared err INODUCION uiliary infmation is often used to improve te efficiency of eimats atio, product, and regression metods of eimation are good eamples of tis contet en te crelation coefficient between te udy variate and auiliary variate is positive (ig, ratio type eimats are used On te oter and, if tis crelation is negative, product type eimats are used In te recent pa, ratio-cum-product eimats ave drawn te attention of researcers, see ing and uiz Espejo (003 and ing and ail (005 is encouraged te aut to mae an attempt to udy te beavi of ratio-cum-product eimats isodia and Dwivedi (98 ave used te coefficient of te variation of auiliary variate in conructing a ratio type eimat in simple random sampling Upadyaya and ing (999 used infmation on te coefficient of urtosis and coefficient of variation wereas ing et al (004 used only te coefficient of urtosis f eimating te population mean ing and ail (003 utilized infmation on te crelation coefficient between udy variate and auiliary variate Kadilar and Cingi (003 defined various ratio type eimats in ratified random sampling ail and ing (005 proposed a ratio-cum-product eimat by using a coefficient of variation is led te aut to sugge a modified ratio-cum-product eimat f eimating te population mean using te crelation coefficient in ratified random sampling Let U ( U, U,, U N be a finite population of size N, wic is divided into omogeneous rata of size N (,, sample of size n is drawn from eac ratum using simple random sampling witout replacement Let y be te udy variate taing values y i auiliary variate taing values i Let ( i t observation from t ratum, and similarly, let i be te y and y population mean Y and X of te udy variate and auiliary variate respectively, be te unbiased eimats of te were (N / N is te weigt of t ratum, n y (/ n y is te sample mean of te udy variate y f j j t ratum, and 8
2 Data cience Journal, Volume 8, 4 eptember 009 n t (/ n is te sample mean of te auiliary variate f te ratum j j e combined ratio and product eimats f population mean Y respectively are y / ( P y / X ( e mean squared err (ME epressions of te combined ratio and product eimats up to te fir degree of approimation are ME( (y y, (3 were ME( Y / X, P (y N N n, y (/(N (y j Y, N n j N (/(N j X and (/(N j y (4 N y (y j Y j X j isodia and Dwivedi (98 suggeed a ratio eimat of population mean Y using te coefficient of variation (C of auiliary variate as [ C / C ] y (5 Here, (, y are te sample means f, y ing et al (004 proposed anoter ratio eimat f Y, using te coefficient of urtosis β of auiliary variate as y β / β (6 [ ] Upadyaya and ing (999 suggeed two eimats using infmation on te coefficient of variation coefficient of urtosis β ( f Y as and [ C / β C ] 3 y β, (7 [ β / C β ] 4 y C (8 C and te ing and ail (003 defined a modified ratio eimat of Y using te ρ y, crelation coefficient between udy variate and auiliary variate as X ρ y 5 y (9 ρ y Kadilar and Cingi (003 defined 3, and 4 in a ratified random sampling respectively as 83
3 Data cience Journal, Volume 8, 4 eptember 009 y C / C, (0 y β / β, ( 3 y β C / β C, ( 4 y C β / C β, (3 ail et al (008 ave given 5 in ratified random sampling as 5 y ρ y / ρ y (4 o te fir degree of approimation, te mean squared errs of 3 4 and 5 respectively are were ME( ME( ME( ME( ME( ( ( ( ( ( y y y y y D K U U β C D K y y D ( Y / C, ( Y / y, (5, (6 U U C y, (7 y, (8 K β, U ( Yβ / β C U ( Y C / C β ( Y / ρ y, POPOED IO EIMO ssuming tat te crelation coefficient ρ y between y and in te t ratum is nown f all rata, te proposed ratio-cum-product eimat is ρ y ρ y y α ( α (0 ρ ρ y y (9 84
4 Data cience Journal, Volume 8, 4 eptember 009 ere y [ α5 ( α 6 ] 6 y ρ y / ρ y is proposed by ail et al (008 Here α is a suitably cosen scalar e note tat f α reduces to te eimat 5 reduces to te eimat 5 wile f α 0, it It is to be mentioned tat if te scalar α closes to unity, te ratio eimat 5 is to be used, wereas 6 used wen α is closer to zero o obtain te bias and mean squared err of let y Y( e0 and Y( e, suc tat E(e0 E(e 0 and E(e 0 y, E(e Y X and E(e0e y Y X Epressing (0 in terms of e i { ( α( λe } ( e α( λe Y 0 X were λ X ρ e now assume tat λ e so tat we may epand ( e λ of approimation, te bias and mean squared err of te proposed eimat [( α αλ ], ( as a series in powers of λ e o te fir degree are Bias( ( λ /X, ( y [ ( α ( α ] ME (, (3 Eimat at optimum α e mean squared err of were β y y is minimized f α y y β is 85
5 By te subitution of α in (0 we get te asymptotically optimum eimat (OE f Y as ρ ρ β ρ ρ β y y y y y (opt ubituting te value of α in (3, te minimum mean squared err of ME ( of (opt is β y, wic is equivalent to te variance of te regression eimat in ratified random sampling en α is not nown, ten it is advisable to eimate (opt α from te sample data at and 3 EFFICIENCY COMPION e variance of te unbiased eimat y in ratified random sampling is y (y V (4 From (3, (5, (6, (7, (8, (9, (3, and (4 (i ( ME (y ME if α α B B eiter (5 were y L L B and (ii ( ME ( ME if α α B B eiter (6 (iii ( ME ( ME P if α α B B eiter (7 Data cience Journal, Volume 8, 4 eptember
6 Data cience Journal, Volume 8, 4 eptember 009 (iv ME( ME( eiter if (v ME( ME( eiter D if K α B α B D K α α B D B K D K (8 (9 (vi ME( ME( 3 eiter if (vii ME( ME( 4 eiter U if (viii ME( ME( 5 eiter U if 4 EMPIICL UDY α B α B B α B α U U α α B U B U F empirical udy, we used te data given in Kadilar and Cingi (003 U U (30 (3 (3 able Data tatiics N 854 N 06 N 06 N 3 94 N N 6 n 40 n 9 n 7 n 3 38 n 4 67 n 5 7 n 6 N 5 73 X X 4375 X 74 X X X5 644 X Y 930 Y 536 Y Y Y Y5 967 Y
7 Data cience Journal, Volume 8, 4 eptember 009 β 307 β 5 7 β β β β β β y 9584 C 0 C 0 C 3 C C 5 7 C 6 9 C 385 C y 4 8 C y 5 C y C y C y 5 47 C y 6 34 C y y 645 y 55 y y y y y 706 ρ 0 8 ρ 0 86 ρ ρ ρ ρ ρ χ 0975 ω 0 05 ω 0 05 ω ω ω ω able Mean quared Errs of y P and Eimats Mean quared Errs y P at optimum α able sows tat te suggeed eimat subantial gain tan te usual unbiased eimat eimat P as te lowe mean squared err, ie, it is me efficient (wit y, te combined ratio eimat, te combined product, te eimats proposed by Kadilar and Cingi (003 i (i to 4, and te ail et al (008 eimats 5 and 6 us te proposed eimat is recommended f use in practice 5 CKNOLEDGEMEN e aut is tanful to te referee f is valuable suggeions regarding te improvement of te paper 6 EFEENCE Kadilar, C & Cingi, H (003 atio eimats in ratified random sampling Biometrical Journal 45(:
8 Data cience Journal, Volume 8, 4 eptember 009 ing, HP &ail, (003 Use of nown crelation coefficient in eimating te finite population mean tatiics in ransition 6 (4: ing, HP & uiz, E(003 On linear regression and ratio-product eimation of a finite population mean e tatiician 5, ing, HP, ail,, ail,, & Karan, (004 n Improved Eimat of population Mean Using Power ransfmation Jour Ind oc gril tati 58(, 3-30 ing, HP & ail, (005 Eimation of finite population mean wit nown coefficient of variation of an auiliary caracter tatiica, anna LXV,n3, isodia, BV & Dwivedi, VK (98 modified ratio eimat using coefficient of variation of auiliary variable Jour Ind oc gri tat, 33(, 3-8 ail, ing, VP, & ail, (008 ome ratio eimats of finite population mean in ratified random sampling ubmitted to Jour Ind oc gril tati Upadyaya, LN & ing, HP (999 Use of transfmed auiliary variable in eimating te finite population mean Biometrical Jour 4(5, (rticle iy: eceived 4 February 009, ccepted 8 May 009, vailable online 6 June
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