Eponential atio Tpe Eimators In tratified andom ampling ajes ing, Mukes Kumar,. D. ing, M. K. Caudar Department of tatiics, B.H.U., Varanasi (U.P.-India Corresponding autor Abract Kadilar and Cingi (003 ave introduced a famil of eimators using auiliar information in ratified random sampling. In tis paper, e propose te ratio eimator for te eimation of population mean in te ratified random sampling b using te eimators in Bal and Tuteja (99 and Kadilar and Cingi (003. Obtaining te mean square error (ME equations of te proposed eimators, e find teoretical conditions tat te proposed eimators are more efficient tan te oter eimators. Tese teoretical findings are supported b a numerical eample. Ke ords: tratified random sampling, eponential ratio-tpe eimator, bias, mean squared error.. Introduction et a finite population aving N diinct and identifiable units be divided into rata. et n be te size of te sample dran from t ratum of size N b using simple random sampling itout replacement. et n N n and N.
et and be te response and auiliar variables respectivel, assuming values i and i for te i t unit in te t ratum. et te ratum means be Y N N i i and N N i i respectivel. as A commonl used eimator for Y is te traditional combined ratio eimator defined C ere,.,, (. n n i i and i, i n n N /N and. Te ME of C, to a fir degree of approimation, is given b ME( C γ [ + ] (. ere γ (, Y n N Y is te population ratio, is te population variance of a variate of intere in ratum and is te population variance of auiliar
variate in ratum and of intere in ratum. is te population covariance beteen auiliar variate and variate Auiliar variables are commonl used in surve sampling to improve te precision of eimates. Wenever tere is auiliar information available, te researcers ant to utilize it in te metod of eimation to obtain te mo efficient eimator. In some cases, in addition to mean of auiliar, various parameters related to auiliar variable, suc as andard deviation, coefficient of variation, skeness, kurtosis, etc. ma also be knon (Koncu and Kadilar (009. In recent ears, a number of researc papers on ratio tpe and regression tpe eimators ave appeared, based on different tpes of transformation. ome of te contributions in tis area are due to isodia and Divedi (98, Upadaa and ing (999, ing and Tailor (003, Kadilar and Cingi (003, 00, 006, ing et.al.(00, Kosnevisan et.al. (007, ing et.al. (007 and ing et.al. (008. In tis article, e ud some of tese transformations and propose an improved eimator.. Kadilar and Cingi Eimator Kadilar and Cingi (003 ave suggeed folloing modified eimator KC,ab,ab (. ere, ab ( a + b,, ab ( a + b. and a, b suitabl cosen scalars, tese are eiter functions of te auiliar variable suc as coefficient of variation C, coefficient of kurtosis β ( etc or some oter conants.
Te ME of te eimator ME ( KC W γ KC is given b [ ] + ab ab (. ere a,b Y..a.,ab.( Bal and Tuteja (99 suggeed an eponential ratio tpe eimator BT ep + (.3 Te eimator BT is more efficient tan te usual ratio eimator under certain conditions. In recent ears, man autors suc as ing et. al. (007, ing and Visakarma (007 and Gupta and abbir (007 ave used Bal and Tuteja (99 eimator to propose improved eimators. Folloing Bal and Tuteja (99 and Kadilar and Cingi (003, e ave proposed some eponential ratio tpe eimators in ratified random sampling. 3. Proposed eimators Te Bal and Tuteja (99 eimator in ratified sampling takes te folloing form t ep + (3. Te bias and ME of t, to a fir degree of approimation, are given b Bias(t ME(t 3 ( γ 8 γ [ + ] (3. (3.3
3. isodia- Divedi eimator Wen te population coefficient of variation C is knon, isodia and Divedi (98 suggeed a modified ratio eimator for Y as- D + C + C (3. In ratified random sampling, using tis transformation te eimator t ill take te form t D ep ( + C ( + C ( + C + ( + C D D ep (3.5 D + D ere D ( C + and D ( C. + Te bias and ME of t D, are respectivel given b D Bias (t D D ( γ θ 8 (3.6 D ME (t D θ [ D + ] (3.7 ere D Y D Y ( + C and θ D. D
3. ing-kakran eimator Motivated b isodia and Divedi (98, ing and Kakran (993 suggeed anoter ratiotpe eimator for eimating Y as- K + β + β ( ( (3.8 Using (3.8, te eimator t at (3. ill take te folloing form in ratified random sampling- t K ep ( +β ( ( +β ( ( +β ( + ( +β ( K K ep (3.9 K + K ere, ( + β ( K and K ( + β (. Bias and ME of t K, are respectivel given b ( K Bias (t K K γ θ (3.0 8 ME (t K [ K γ K + ] (3. K Y ere and ( + β ( θ K K.
3.3 Upadaa-ing eimator Upadaa and ing (999 considered bot coefficients of variation and kurtosis in teir ratio tpe eimator as U β β (3. ( + C ( + C We adopt tis modification in te eimator t proposed at (3. (β ( + C t U ep ( β ( + C (3.3 t U at (3.3 can be re-ritten as t U ep U U + U U ere U ( β ( + C U ( β + C. (3. Bias and ME of t U, to fir degree of approimation, are respectivel given b Bias U U 8 (3.5 ( t γ θ U ME ( t U γ + U U (3.6 ere us Y.. β ( ( β ( + C and θ U. β U (.
Upadaa and ing (999 proposed anoter eimator b canging te place of coefficient of kurtosis and coefficient of variation as U C + β( (3.7 C + β ( Incorporating tis modification in te proposed eimator t, e ave- U t U U ep U + U (3.8 ere ( C +β ( and us us (C +β ( Bias and ME of t U, are respectivel given b Bias(t ME(t U U U ( γ θ 8 (3.9 U + [ U γ U ] (3.0 ere U Y. ( C + β (.C. and θu C. U 3.. G.N. ing Eimator Folloing ing (00, using values of and β (, e propose folloing to eimators. tgn GN ep GN + GN GN (3.
ere GN ( + σ and GN ( + σ Te Bias and ME of t GN to a fir degree of approimation, are respectivel given b Bias(t GN γ θ GN ( GN 8 (3. ME(tGN γ GN + [ GN ] (3.3 ere GN Y (, θ GN. GN +σ imilarl, e propose anoter eimator t GN GN ep GN GN + GN (3. ere, ( β ( + σ, GN GN ( β + σ Te Bias and ME of t GN to a fir degree of approimation are respectivel given b Bias(t ME(t GN 3GN GN ( γ θ 8 (3.5 GN + [ GN γ GN ] (3.6 ere, GN Y (, β ( + σ θ GN i. β GN (
. Improved Eimator Motivated b ing et. al. (008, e propose a ne famil of eimators given b- t MK,ab, ab ep,ab +,ab α (. ere a and b are suitabl cosen scalars and α is a conant. Te bias and ME of t MK up to fir order of approimation, are respectivel given b Bias ( t MK γ θ ab ( α α + 8 ab (. ME ( t MK γ α ab α + ab (.3 Te ME(t MK is minimized for te optimal value of α given b- γ α i ab γ i Putting tis value of α in equation (.3, e get te minimum ME of te eimator t MK as- ME(t MK min. i γ ( ρ c (. ere,ρ is combined correlation coefficient in ratified sampling across all rata. It is c γ ρ i calculated as ρ c. γ γ i We note ere tat min ME of t MK is independent of a and b. terefore, e conclude tat it same for an (all values of a and b.
5. Efficienc comparisons Fir e compare te efficienc of te eimator t at (3. it eimator t D. We ave ME(t D < ME(t < γ + D D γ + (5. D γ + D < γ + et A γ and B γ Ten equation (5. can be re-ritten as- D D A +.B < A +.B -A ( D + B/( D - ( D + < 0 (5. From (5., e get to conditions θ D + D < θ + (i Wen ( D - ( D + > 0 B < A /( D + (5.3 (ii Wen ( D ( D + < 0 B > A / ( D + (5. ere A θ and B θ Wen eiter of tese conditions is satisfied, eimator t D ill be more efficient tan te eimator t.
Te same conditions also olds true for te eimators t K, t U, t U, t GN and t GN if e replace D b K, U, U, GN and GN respectivel in conditions (i and (ii. Net e compare te efficiencies of t opt it te oter proposed eimators. ME(t MK min. < ME(t ab γ ρ < γ ab ( c + ab i i (5.5 On putting te value of and rearranging te terms e get ρ c ab γ 0 γ > i i (5.6 Tis is alas true. Hence te eimator t MK under optimum condition ill be more efficient tan oter proposed eimators in all conditions. 6. Data description and results For empirical ud e use te data set earlier used b Kadilar and Cingi (003. Y is apple production amount in 85 villages of turke in 999, and is te numbers of apple trees in 85 villages of turke in 999. Te data are ratified b te region of turke from eac ratum, and villages are selected randoml using te Neman allocation as n N N
Table 6.: Data tatiics N 06 N 06 N 3 9 N 7 N 5 0 N 6 73 n 9 n 7 n 3 38 n 67 n 5 7 n 6 375 7 3 709 7365 5 6 6 98 Y i536 Y Y 3 938 Y 5588 Y 5 967 Y 6 0 β 5.7 β 3. 57 β 3 6. β 97.60 β 7.7 β 5 6 8. 0 C.0 C.0 C 3. C 3.8 C 5.7 C 6.9 C.8 C 5. C 3 3.9 C 5.3 C 5.7 C 6.3 989 576 3 60757 85603 5 503 6 879 65 55 3 9907 863 5 390 6 96 ρ 0.8 ρ 0. 86 ρ 3 0. 90 ρ 0.99 ρ 7 5 0. ρ 6 0. 89 γ 0.0 γ 0. 09 γ 3 0. 06 γ 0.009 γ 5 0. 38 γ 6 0. 006 0.05 0. 05 3 0. 0 0.0 5 0. 057 6 0. 0 N85 n0 β 3. 07 C 3.85 C 5.8 79 706 ρ 0. 9 37600 Y 930 0.07793 D 0.0779 K 0.0778 U.07789 U 0.07786 GN 0.0663 GN 0.07765
Table 6. : Eimators it teir ME values Eimators ME values t t D t K t U t U t GN t GN t MK(opt 35969.59 35969.688 359890.33 359739.875 359830.5 360007.985 36079.9 837.8898 From Table 6., e conclude tat te eimator t MK as te minimum ME and ence it is mo efficient among te discussed eimators. 7. Conclusion In te present paper e ave eamined te properties of eponential ratio tpe eimators in ratified random sampling. We ave derived te ME of te proposed eimators and also tat of some modified eimators and compared teir efficiencies teoreticall and empiricall.
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