J.Nat.Sci.Foudatio Sri Laka 08 46 (: 45-46 DOI: http://d.doi.org/0.408/jsfsr.v46i.8498 RESEARCH ARTICLE Ehacig ratio estimators for estimatig populatio mea usig maimum value of auiliar variable Nasir Abbas, Muhammad Abid,*, Muhammad Tahir, Nasir Abbas Zawar Hussai 4 Departmet of Statistics, Govermet College Uiversit, Faisalabad 8000, Pakista. Departmet of Mathematics, Istitute of Statistics, Zhejiag Uiversit, Hagzhou 007, Chia. Departmet of Statistics, Govermet Postgraduate College, Jhag, Pakista. 4 Departmet of Statistics, Quaid-i-Azam Uiversit, Islamabad 44000, Pakista. Revised: 9 Ma 08; Accepted: 5 Ma 08 Abstract: This article proposes some ewl modified ratio estimators for the estimatio of populatio mea of the stud variable b icorporatig the maimum value of the auiliar variable. The epressios of the bias mea squared error were derived. The coditios i which the suggested estimators have miimum values of mea squared error i compariso to the eistig estimators were also derived. A empirical simulatio stud was also coducted. From the empirical simulatio stud, it was foud that the suggested estimators perform more efficietl as compared to the eistig estimators cosidered i this stud. Kewords: Bias, coefficiet of variatio, maimum value, ratio estimator, simulatio stud. INTRODUCTION I surve research, if iformatio is available i ever uit of populatio this iformatio is correlated with the stud variable, the such iformatio is called auiliar iformatio. B usig this auiliar iformatio oe ca propose umerous tpes of estimators for estimatig the populatio mea b icorporatig product, ratio regressio methods of estimatio. The ratio product methods of estimatio are helpful for estimatig the populatio mea whe the correlatio betwee the stud variable the auiliar variable is positive egative, respectivel (Cochra, 940; Murth, 967. Suppose a fiite populatio V {V,V,...,V N } cosists of N differet idetifiable uits. Let be a measurable variable of iterest with values i beig ascertaied o V i ; i,,..., N resultig i a set of observatios {,,..., N }. The mai purpose of this stud is to N estimate the populatio mea i b drawig N i a simple rom sample (SRS from the populatio. Suppose that the iformatio o auiliar variable Χ, which is correlated with the stud variable, is also available agaist ever uit of the populatio. With this additioal iformatio oe ca develop various tpes of estimators for populatio variace b usig the ratio, regressio, product ratio-cum product methods of estimatio. Therefore, the ratio estimator for estimatig the populatio mea of the stud variable is defied as: R R,...( where R is the estimate of R. The ratio estimator represeted i equatio ( produced efficiet results for estimatig i compariso to the usual sample mea for a high positive correlatio. * Correspodig author (mabid@gcuf.edu.pk; https://orcid.org/0000-000-8996-0 This article is published uder the Creative Commos CC-B-ND Licese (http://creativecommos.org/liceses/b-d/4.0/. This licese permits use, distributio reproductio, commercial o-commercial, provided that the origial work is properl cited is ot chaged i awa.
454 Nasir Abbas et al. Geerall, estimatio of b usig auiliar iformatio is dealt i the cotet of augmetig the covetioal parameters of a auiliar variable through ratio or regressio methods of estimatio to achieve greater efficiec. Mostl, coefficiet of skewess, coefficiet of kurtosis, coefficiet of variatio of the auiliar variable coefficiet of correlatio betwee the stud the auiliar variable are used i liear combiatio with some other covetioal parameters of the auiliar variable to estimate the populatio mea. For istace, refer Sisodia Dwivedi (98, Rao (99, Upadhaa Sigh (999, Sigh et al. (004, Kadilar Cigi (004; 006, a Tia (00, Jeelai et al. (0 the refereces therei. For the ratio estimator for estimatig based o the o-covetioal measures of locatio dispersio, refer the work of Abid et al. (06a; 06b; 06c; 06d. The preset stud was focused o estimatio of the populatio mea b icorporatig iformatio o a maimum value of the auiliar variable with a liear combiatio of the coefficiet of variatio correlatio coefficiet uder ratio tpe structure of estimatio. The mai motivatio behid suggestig the proposed estimators was to use additioal iformatio cotaied i the maimum value of the auiliar to get relativel more precise estimators. METHODOLOG ratio estimators Kadilar Cigi (004 proposed the ratio estimators based o the C β of a auiliar variable showed that their suggested estimators are more efficiet i compariso to the usual ratio estimator of populatio mea. Kadilar Cigi (004 estimators are metioed b: ( + b + ( ( C + b + C + ( ( β ( + β b + + ( ( β C β + C b 4 + + ( ( C β ( C + β b + 5 The values of biases, costats mea squared errors (MSEs of the above-give estimators are: B ( S R R ( R S + S ( ρ MSE( B ( S R R ( R S + S ( ρ MSE( B ( S R R ( R S + S ( ρ MSE( B ( 4 S R4 R4 ( R4 S + S ( ρ MSE( 4 B ( 5 S R5 C ( + ( + β β C ( β + C R5 C + β ( R5 S + S ( ρ MSE( 5 where N is the populatio size, is the sample size, f /N represet the samplig fractio, μ is the populatio mea of a auiliar variable X, μ is the populatio mea of stud variable, S is the populatio stard deviatio of X, S is the populatio stard deviatio of, C is the coefficiets of variatio of X, C is the coefficiets of variatio of ρ is the coefficiet of correlatio betwee X. Kadilar Cigi (006 suggested some ratio estimators b usig the correlatio coefficiet of the auiliar variable. It was proved that the suggested estimators are smaller values of MSEs compared to the other eistig estimators (Cadilar & Cigi, 006. The estimators are represeted below: September 08 Joural of the Natioal Sciece Foudatio of Sri Laka 46(
Ehacig ratio estimators usig maimum value 455 ( ( ρ + b 6 + + ( ( C ρ + ρ b 7 + C + ( ( ρ C ρ + C b 8 + + ( ( β ρ ( β + ρ b 9 + + ( ( ρ β ( ρ + β b 0 + The values of biases, costats MSEs for Kadilar Cigi (006 are represeted as: B ( 6 S R 6 R6 ( R6 S + S ( ρ MSE( 6 B ( 7 S R 7 R7 ( R7 S + S ( ρ MSE( 7 B ( 8 S R 8 R8 ( R8 S + S ( ρ MSE( 8 B ( 9 + ρ C C + ρ ρ ρ + C S β R 9 R 9 β + ρ ( R9 S + S ( ρ MSE( 9 B ( 0 S R 0 R0 ( R0S + S ( ρ MSE( 0 ρ ρ + β a Tia (00 proposed the estimators based o the values of β β of a auiliar variable these estimators are give as: ( ( β ( + β + b + + ( ( β β ( β + β b + where β β are the coefficiet of skewess coefficiet of kurtosis of X, respectivel. The values of biases, costats MSEs for a Tia (00 are as follows: B ( S R R ( RS + S ( ρ MSE( B ( S R R ( RS + S ( ρ MSE( ( + β β ( β + β Subramai Kumarapia (0a; 0b; 0c; 0d itroduced some ew estimators usig the liear combiatio of M d, C, β, β deciles of a auiliar variable for estimatig the populatio mea. Subramai Kumarapia (0a; 0b; 0c; 0d estimators are give as: + ( ( Md ( + M b + d + b( X d C + M 4 C + M d + ( ( d B+ M b 5 B + M d + ( ( d B + M b 6 B + M d ( ( D ( + D + b 7 + Joural of the Natioal Sciece Foudatio of Sri Laka 46( September 08
456 Nasir Abbas et al. ( ( D ( + D + b 8 + ( ( D ( + D + b 9 + ( ( D ( + D + b 0 + 4 4 ( ( D ( + D + b + 5 5 ( ( D ( + D + b + 6 6 ( ( D ( + D + b + 7 7 ( ( D ( + D + b 4 + 8 8 ( ( D ( + D + b 5 + 9 9 + ( ( D ( + D b 6 + 0 0 where M d is media of X D i,i,,,0 are the deciles of X. The values of biases, costats MSEs of Subramai Kumarapia (0a; 0b; 0c; 0d estimators are show below: S B R, R ( ρ MSE R S S + S B 4 R 4, R 4 ( ρ MSE 4 R 4 S S + S B 5 R 5, R 5 ( ρ MSE 5 R 5 S S + d ( + M C d ( C + M B d ( B + M B S 6 R 6, R 6 ( ρ MSE 6 R 6 S S + B S 7 R 7, R 7 ( ρ MSE 7 R 7 S S + B S 8 R 8, R 8 ( ρ MSE 8 R 8 S S + S B 9 R 9, R 9 ( ρ MSE 9 R 9 S S + S B 0 R 0, R 0 ( ρ MSE 0 R 0 S S + S B R, B ( B + M ( + D ( + D ( + D ( + D 4 R 5 ( ρ MSE R S S + S B R, R ( ρ MSE R S S + S B R, R ( ρ MSE R S S + S B 4 R 4, R 4 ( ρ MSE 4 R 4 S S + ( + D ( + D 6 ( + D 7 ( + D 8 d September 08 Joural of the Natioal Sciece Foudatio of Sri Laka 46(
Ehacig ratio estimators usig maimum value 457 B S 5 R 5, R 5 ( ρ MSE 5 R 5 S S + B S 6 R 6, R 6 ( ρ MSE 6 R 6 S S + ( + D 9 ( + D Jeelai et al. (0 developed the ratio estimator based o the value of quartile deviatio β of the auiliar variable for estimatig the populatio mea it is give as: + ( ( β QD β + QD b 7 + where QD is the quartile deviatio of X. The value of bias, costat the MSE are specified as: B ( 7 R7 S R 7 β ( β + QD ( R7S + S ( ρ MSE( 7 All the above estimators utilise auiliar iformatio i oe wa or the other. Good auiliar variables are those which are similar to the stud variable ca be take as good proies of the stud variable. I additio, if correlatio betwee the stud auiliar variable is high (positive or egative, the based o the auiliar iformatio the characteristics of the stud variable (such as mea, variace, coefficiet of variatio, maimum, miimum ca be precisel predicted. If the auiliar variable is available i advace, i the past eperiece or i the pilot stud, its etreme values (miimum maimum are easil available. Whe correlatio betwee the stud variable the auiliar variable is positive, the selectig the larger (smaller value of the auiliar variable will more likel result i the larger (smaller value of stud variable i the sample. 0 Therefore, i ma populatios there eist some large or small values, to estimate the populatio parameters without cosiderig this iformatio is sesitive. As is evidet, i case of presece of a etremel high (low outlier i the distributio of auiliar variable, its mea is larger (smaller. I either case the result will be overestimatio or uderestimatio of the mea. I the et sectio, we ited to propose estimators of mea b givig the same icremet (the maimum of auiliar variable to populatio sample meas of auiliar variable. ratio estimators Geerall whe the correlatio betwee the stud variable the auiliar variable is positive, the selectio of the maimum value of the auiliar variable is highl associated with the selectio of the maimum value of the stud variable. I this sectio, ratio estimators are proposed for the estimatio of fiite populatio mea usig iformatio of the maimum value of the auiliar variable with the liear combiatio of the coefficiet of variatio coefficiet of correlatio of a auiliar variable. The proposed estimators are give b: ( ( M( + b p + ( + M( ( + b ( ( C + M( p C + M ( + b ( ρ M( ρ + M( p + where M ( is the maimum value of a auiliar variable. It is to be oted that i all the proposed estimators the ratio part is augmeted b usig the populatio maimum of the auiliar variable. It is aticipated that usig the maimum of auiliar variable will result i improved estimatio of mea. The values of biases, costats MSEs of the proposed estimators are specified below: B( p S R p ( + ( ρ MSE( p Rp S S Rp ( + M( Joural of the Natioal Sciece Foudatio of Sri Laka 46( September 08
458 Nasir Abbas et al. B( p S R p Rp ( + ( ρ MSE( p Rp S S S B ( p Rp Rp ( + ( ρ MSE( p Rp S S Efficiec comparisos ( C + M( ( ρ + M( I this sectio, we derive the coditios i which the suggested estimators perform more efficietl i compariso to the eistig estimators. The proposed estimator performs more efficietl if ol if; MSE < MSE pk l, f ( R pk S + S( ρ < Rl S S ( + ρ f R pk S < R l S R pk < R l...( where k,, for the proposed estimators l,,...,7 for the eistig estimators. RESULTS AND DISCUSSION Practical stud To evaluate the performace of the proposed estimators agaist their competig estimators which are give i the Methodolog, five atural populatios have bee selected. s I II were obtaied from Sigh Chaudhar (986, populatio III was take from Cochra (940 populatios IV V were obtaied from Murth (967. The characteristics of the populatios cosidered i this stud are preseted i Table. The values of biases costats are give i Tables for eistig proposed estimators, respectivel. The MSE values are represeted i Tables 4 5 for the eistig proposed estimators, respectivel. It was revealed that the values of costats of the proposed estimators, i.e., R pk are smaller as compared to the values of the costats of the eistig estimators i.e., R l which satisfied equatio ( (Table vs Table. Table : Values of populatio characteristics Parameter I II III IV V N 4 4 49 80 80 0 0 0 0 0 856.4 856.4 7.796 58.67 58.67 08.88 99.44 0.4 85.5 6.46 ρ 0.449 0.445 0.9874 0.95 0.94 S 7.4 7.4. 85.659 85.659 C 0.857 0.856 0.96 0.54 0.54 S 50.506 50.5 04.405 79.49 845.60 C 0.7 0.75.0 0.948 0.75 β 0.098.045 5.4.0-0.06 β 0.978.8.55 0.698.050 M d 50 4.500 64 48 757.500 QD 80.5 89.75 8.50 79.75 588.5 M ( 564 64 507 095 485 September 08 Joural of the Natioal Sciece Foudatio of Sri Laka 46(
Ehacig ratio estimators usig maimum value 459 From Tables, it was oted that the suggested estimators have miimum values of biases compared to the eistig estimators. As epected, MSE values of all the proposed estimators were smaller i compariso to the eistig estimators for all the populatios cosidered i this stud, which revealed the supremac of the proposed estimators (Table 4 vs Table 5. It was also observed that the suggested estimators based o the liear combiatio of correlatio coefficiet the maimum value of auiliar variable, i.e.,, have the miimum MSE value as compared to the other proposed estimators, i.e., (Table 5. Hece, we ca sa that the suggested estimators perform more efficietl compared to the eistig estimators based o the five real datasets cosidered i this stud. Table : The values of costats biases for eistig estimators Estimator I II Costat III IV V I II Bias III IV V 4.00 4.94.9 8.77 4.60 9.54 0.00.875 74.8 09.5 4.086 4.78.7 8.6 4.598 9.09 9.97.800 7.67 09.7 4.098 4.7.80 8. 4.60 9.45 9.898.56 7.98 09.5.960 4.79.7 8.090 4.650 8.59 9.90.860 7.8.86 4 5 4.097 4.64.8 8.0 4.60 9.4 9.865.50 7.9 09.5 4.09 4.85.7 8.9 4.597 9.5 9.957.80 7.7 09.4 6 4.088 4.8.8 8.5 4.596 9.099 9.94.80 7.65 09.7 7 4.069 4.58.7 8. 4.598 9.05 9.84.799 7.57 09.6 8 4.0 4.85.7 8.094 4.66 8.76 9.960.86 7..46 9 4.096 4.44.79 8.8 4.60 9.5 9.77.509 7.90 09.5 0 4.08 4.69. 8.094 4.597 9.069 9.885.7 7.4 09. 4.098 4.75. 8.4 4.60 9.45 9.94.709 74.7 09.5.86.5046 0.765.966.75.0.40.476 75.76 9.5.05.04 0.768.747.47.96.64.489 7.0 0.47 4.64.676 0.97.99.804.04.886.8 89. 40.69 5 0.489.550.05 0.4-0.478 0.0.56.085 57.48.86 6.068.9 0.96 4.8.464 5.4 5.88.664 6.75 6.08 7.998.0 0.89 4.88.66 4.89 4.987.08 08.08 55.88 8.70.84 0.860.605.007.975 4.58.868 97.95 46.786 9.5.757 0.88.686.874.490 4..705 85.6 4.77 0.86.505 0.765.966.75.00.40.4756 75.764 9.54.964.09 0.75 0.6.6.00.70.90 59.604 5.567.865.846 0.650 8.047.985.894.8484.0650 4.68 0.85.57.700 0.5 6.588.765.47.567 0.760.964 6.7 4.8.496 0.66 5..49 0.960. 0.89 4.67 0.567 5.08.08 0.09.755.4 0.6686 0.577 0.07 7.46 6.55 6.948.8.06.5.07 4.79 5.60.85 79.4 48.85 7 Joural of the Natioal Sciece Foudatio of Sri Laka 46( September 08
460 Nasir Abbas et al. Table : The values of costats biases for proposed estimators Costat Bias Estimator I II III IV V I II III IV V.08.08 0.09.755.4 0.669 0.57 0. 7.46 6.55 p 0.864 0.8 0..600 0.898 0.406 0.67 0. 6.858 4.76 p 0.585 0.58 0.06.497.07 0.86 0.5 0.07 6.47 5.960 p Table 4: MSE values of the eistig estimators for all selected populatios Estimator MSE I II III IV V 667.45 747.65 5.4 96660.70 58994.0 669.64 77. 50.84 9066.50 588.50 6666.4 748.6 465.5 94.50 58058.0 646.6 776.04 509.5 9808.0 5949.80 4 5 666. 79.75 466.0 900.40 58079.0 669.85 799.5 50. 9087.0 58046.80 6 666.87 787.08 50. 90560.0 5807.70 7 6554.4 794.9 50.66 9008.0 589.40 8 68.65 740.4 509.59 988.80 59760.90 9 6657.9 79.66 464.7 98.60 5806.0 0 6600.54 76.98 490.45 98450.90 58097.60 6665.98 76.6 490. 960.70 58055.0 489.70 785.70 04.80 40.80 79.80 0800.4 7.47 06.55 99044.90 7.80 4 440.8 99.76 0.78 48450.4 59.5 5 8945.9 89.07 40.50 8486.7 0545.47 6.5 90.4 9. 68. 645.8 7 05.0 4.8 7. 58044. 0046.7 8 6. 604.0 55.0 580.0 56874.4 9 8.0 40.5 7.4 4694.7 5888.6 0 489.7 785.7 04.8 40.8 79.8 06.6 090. 8. 9476.5 987. 0455.5 0454.6 5. 9869.8 0048.6 9987.7 0.8 07.7 9586.0 9794.6 4 9656. 990.7 59.5 950.7 696.6 5 9406.6 96. 0.4 5945.5 4866.6 6 875.4 76.0 80.8 464.6 67595. 7 September 08 Joural of the Natioal Sciece Foudatio of Sri Laka 46(
Ehacig ratio estimators usig maimum value 46 Joural of the Natioal Sciece Foudatio of Sri Laka 46( September 08 Estimator MSE I II III IV V p 9406.6 96. 0.4 5945.5 4866.6 p 98.8 985.9 0.7 564.7 609. p 899.4 9000.9 0.0 547. 4586.9 Table 5: MSE values of the suggested estimators for all selected populatios Simulatio stud We also coducted a simulatio stud betwee the proposed estimators the eistig estimators b usig the real dataset of populatio IV. We used the simulatio stud to fid the MSE of the eistig estimators the suggested estimators. The algorithm is desiged i R laguage. The simulatios were coducted accordig to the followig steps: i. Select samples of size from the real dataset of populatio IV usig simple rom samplig without replacemet. ii. Use the data from samples i (i to fid the value of. So, we obtai 5000 values of from 5000 samples for 0. 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.9709 0.9675 0.9655 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.988 0.9864 0.9846 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.97 0.9677 0.9656 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.9865 0.9846 0.988 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.97 0.9678 0.9657 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.9869 0.985 0.98 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.975 0.968 0.966 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.980 0.9777 0.9758 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.9680 0.9646 0.964 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.984 0.980 0.9784 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.970 0.9676 0.9656 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.984 0.98 0.980 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.9679 0.9644 0.96 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.9849 0.988 0.980 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.959 0.9555 0.95 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.987 0.985 0.985 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.975 0.968 0.9660 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.9907 0.989 0.987 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.9594 0.9557 0.95 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.999 0.99 0.9896 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.97 0.9678 0.9657 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.994 0.997 0.990 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.976 0.969 0.967 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.9959 0.9945 0.998 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.980 0.9809 0.9790 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From.0000 0.9988 0.997 The compariso betwee the eistig proposed estimators are doe o the basis of relative efficiec (RE. The RE of suggested estimators with respect to eistig estimators are foud b usig the epressio give below:, where,,,,7,,. Table 6: Relative efficiec of suggested estimators with respect to eistig estimators 0.97 0.9677 0.9656 0.9865 0.9846 0.988 0.97 0.9678 0.9657 0.9869 0.985 0.98 0.9680 0.9646 0.964 0.984 0.980 0.9784 0.970 0.9676 0.9656 0.984 0.98 0.980 0.9679 0.9644 0.96 0.9849 0.988 0.980 0.959 0.9555 0.95 0.987 0.985 0.985 0.9594 0.9557 0.95 0.999 0.99 0.9896 0.97 0.9678 0.9657 0.994 0.997 0.990 0.976 0.969 0.967 0.9959 0.9945 0.998 0.987 0.985 0.985 estimators are performed more capable based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of compared to the eistig estimators used i this stud. maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From 0.987 0.985 0.985 Table 6: Relative efficiec of suggested estimators with respect to eistig estimators
46 Nasir Abbas et al. iii. For 0, the MSE of are computed b the formulas give below: (...( (...(4 where is the populatio mea of the stud variable. The compariso betwee the eistig proposed estimators were doe o the basis of relative efficiec (RE. The RE of the suggested estimators with respect to eistig estimators were foud b usig the epressio give below:, where l,,,...,7 k,,. The values of RE are reported i Table 6. From Table 6, it ca be cocluded that all the suggested estimators perform better based o the values of RE i compariso to the eistig estimators cosidered i this stud also these results are i accordace with the results of the practical stud. The preset stud proposes some ew ratio estimators b itegratig the iformatio o the maimum value of a auiliar variable. The compariso betwee proposed estimators eistig estimators are doe o the basis of mea square error relative efficiec. From the empirical stud, it was revealed that the suggested estimators have smaller mea square error values i compariso to the eistig estimators, which idicate the supremac of the proposed estimators compared to the eistig estimators cosidered i this stud. It is oted that the performace of the proposed estimator is relativel higher as compared to the other proposed estimators. The results of simulatio stud also support the theoretical fidigs of the stud. Therefore, based o the fidigs obtaied from this stud, we strogl recommed the use of proposed estimators for the practical applicatios agaist the eistig estimators. REFERENCES Abid M., Abbas N., Nazir Z.A. & Li Z. (06a. Ehacig the mea ratio estimators for estimatig populatio mea usig o-covetioal locatio parameters. Revista Colombiaa de Estadistica 9(: 6 79. DOI: https://doi.org/0.5446/rce.v9.559 Abid M., Abbas N. & Riaz M. (06b. Improved modified ratio estimators of populatio mea based o deciles. Chiag Mai Joural of Sciece 4(:. Abid M., Abbas N., Sherwai R.A.K. & Nazir Z.A. (06c. Improved ratio estimators for the populatio mea usig o-covetioal measures of dispersio. Pakista Joural of Statistics Operatio Research (: 5 67. DOI: https://doi.org/0.887/pjsor.vi.8 Abid M., Sherwai R.A.K., Abbas N. & Nawaz T. (06d. Some improved modified ratio estimators based o decile mea of a auiliar variable. Pakista Joural of Statistics Operatio Research (4: 787 797. DOI: https://doi.org/0.887/pjsor.vi4.408 Cochra W.G. (940. Samplig Techiques, rd editio. Wile Easter Limited, Idia. Ferrell E.B. (95. Cotrol charts usig midrages medias. Idustrial Qualit Cotrol 9: 0 4. Hettmasperger T.P. & McKea J.W. (998. Robust Noparametric Statistical Methods. Joh Wile Sos, New ork, USA. Jeelai M.I., Maqbool S. & Mir S.A. (0. Modified ratio estimators of populatio mea usig liear combiatio of coefficiet of skewess quartile deviatio. Iteratioal Joural of Moder Mathematical Scieces 6: 74 8. Kadilar C. & Cigi H. (004. Ratio estimators i simple rom samplig. Applied Mathematics Computatio 5: 89 90. Kadilar C. & Cigi H. (006. A improvemet i estimatig the populatio mea b usig the correlatio coefficiet. Hacettepe Joural of Mathematics Statistics 5: 0 09. Murth M.N. (967. Samplig Theor Methods. Statistical Publishig Societ, Calcutta, Idia. Rao T.J. (99. O certai methods of improvig ratio regressio estimators. Commuicatios i Statistics: Theor Methods 0: 5 40. DOI: https://doi.org/0.080/060990880705 Sigh D. & Chaudhar F.S. (986. Theor Aalsis of Sample Surve Desigs. New Age Iteratioal Publisher, Idia. Sigh G.N. (00. O the improvemet of product method of estimatio i sample surves. Joural of the Idia Societ of Agricultural Statistics 56: 67 75. Sigh H.P. & Tailor R. (00. Use of kow correlatio coefficiet i estimatig the fiite populatio meas. September 08 Joural of the Natioal Sciece Foudatio of Sri Laka 46(