Optimal Search of Developed Class of Modified Ratio Estimators for Estimation of Population Variance

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1 American Journal of Operational Research 05 5(4): 8-95 DOI: 0.593/j.ajor Optimal earch of Developed Class of Modified Ratio Estimators for Estimation of Population Variance Alok Kumar hukla heela Misra.. Mishra 3* ubhash Kumar Yadav 3 Department of tatistics D.A-V. College Kanpur U.P. India Department of tatistics University of Luckno Luckno U.P. India 3 Department of Mathematics and tatistics (A Centre of Ecellence) Dr. RML Avadh University Faizabad U.P. India Abstract This manuscript deals ith the estimation of population variance using auiliary variable through optimally searching developed class of modified ratio type estimators of population mean of the study variable. We have proposed a generalized class of estimators of population variance of the main variable under study. The proposed estimator is the generalization of the ubramani and Kumarapandiyan (05) estimator ith a characterizing scalar constant α hich is to be obtained. The epressions for the bias and mean squared error of the proposed class of estimators have been obtained up to the first order of approimation. The optimum value of the characterizing scalar has been obtained by minimizing the mean squared error of the proposed class of estimators. The minimum value of the mean squared error has also been obtained for this optimum value of the characterizing scalar. At last an empirical study is also carried out through hich e infer that the proposed estimator has minimum mean square error among the class of estimators of population variance as an optimal search condition. Keyords Auiliary variable Parameter Estimator Bias Mean quared Error Efficiency. Introduction It has been seen in practice that the variance is one of the important measure of dispersion. For eample e can see ho the variation in production variation in sales variation in blood pressure etc affects the human beings. It plays a very important role in the field of Medical sciences Biological sciences Agriculture Industry etc. o it is of great significance in above fields to estimate the variation. To estimate any parameter of the variable under study the most appropriate estimator for the parameter is the corresponding statistic and in case of variance estimation the most suitable estimator is the sample variance of the study variable. Being unbiased estimator of the population parameter the sample variance should be preferred for the estimation but it has a big amount of variation among its values. Therefore e seek for such estimator hich may be biased but it should have minimum mean squared error. Auiliary variable hich is generally highly positively or negatively correlated ith the main variable under study fulfill the need in searching such estimator and its use lift up the efficiency of the estimator. When the main variable and the auiliary variable are positively correlated ratio type estimators are used for the * Corresponding author: sant_003@yahoo.co.in (.. Mishra) Published online at Copyright 05 cientific & Academic Publishing. All Rights Reserved estimation of population parameters under considerations. While the Product type estimators are used hen main and auiliary variables are negatively correlated to each other and the line of regression Y on X passes through origin and in other case regression estimators are used for the estimation of population parameters. In the present manuscript e have considered only positively correlated case using ratio type estimators of population variance. Let us consider that the finite population under investigation consist of N distinct and identifiable units and let ( i yi) i... n be a bivariate sample of size n taken from (X Y) using a simple random sampling ithout replacement (RWOR) scheme. Let X and Y respectively be the population means of the auiliary and the study variables and let and y be the corresponding sample means. and y are the unbiased estimators of the population means X and Y respectively. In the present manuscript e are searching for such an estimator of population variance hich should have minimum mean squared error among the class of such estimators. The folloing notations early used by ubramani and Kumarapandiyan (05) have been used in this manuscript as: N - ize of the population n - ize of the sample Y - tudy variable X - Auiliary variable

2 American Journal of Operational Research 05 5(4): M d - Median of the auiliary variable ρ - Correlation coefficient beteen X and Y Y X - Population means y - ample means y s y C y - Population variances s - ample variances C - Coefficient of variations ( f ) γ n n f N µ λ rs rs r s µ µ µ / / 0 0 N r s rs ( yi Y)( i X) N i µ 03 ( ) keness of the auiliary variable 3 µ 0 µ 04 ( ) Kurtosis of the auiliary variable µ 0 Q - First quartile of the auiliary variable Q3 - Third quartile of the auiliary variable Qr - Inter quartile range of the auiliary variable Qd - emi quartile range of the auiliary variable Qa - emi quartile average of the auiliary variable Di - i th Decile of the auiliary variable B(.) - Bias of the estimator V (.) - Variance of the estimator ME(.) - Mean squared error of the estimator ME( t ) PRE( t ) e e t p *00 - Percentage relative ME( t p ) efficiency of the estimator t p over t e.. Revie of Literature of Variance Estimators The simplest and the most appropriate estimator of population variance is the sample variance given by: ˆ s y 0 (.) It is unbiased and its variance up to the first degree of approimation is: V( ˆ ) y ) (.) 4 0 γ λ40 Isaki (983) proposed the folloing ratio estimator of population variance using auiliary information as: ˆ here n sy n ( yi y) i (.3) s R sy N N ( i ) i X X X N Xi N i Y Y i n i n i n ( i ) n i s N i N n i y y n. i The Bias and Mean quare Error (ME) of the estimator in (.3) up to the first order of approimation are respectively given by B ( ˆ ) γ [ ( λ )] (.4) R y 04 4 ( λ40 ) + ( λ04 ) ME( tr) γ ( λ ) (.5) Various authors in the literature have proposed so many estimators of population variance latest being ingh and olanki (03) ubramani and Kumarapandiyan (0a b c 03) tailor and hrama (0) and Yadav and Kadilar (03a b) Yadav et al (05a b) utilizing different parameters of auiliary variable. Table- early given by ubramani and Kumarpandiyan (05) represents different estimators of population variance ith their Bias and ME. Where R i i...5 and C + i ( ) 3 ( ) 4 ρ 5 6 Md 7 Q 8 Q3 9 Qr 0 Qd Qa D 3 D 4 D3 5 D4 6 D5 7 D6 8 D7 9 D8 0 D9 D0 ( ) C ( ) 3 4 C ( ) C ( ) 5 C

3 84 Alok Kumar hukla et al.: Optimal earch of Developed Class of Modified Ratio Estimators for Estimation of Population Variance C ρ ρ C C C M 30 3 d M d C ( ) 33 ( ) ( ) ( ) C ( ) 3 ( ) ρ ( ) ρ ( ) ( ) 38 M d M 39 d ( ) ( ) 4 43 ( ) M ρ ( ) 40 4 ρ ( ) ρ d ( ) ρ M Table. Bias and ME of Various Estimators 49 d 50 5 d. ( ) M d Estimator Bias ME M ( ) 44 M d ( ) 48 M d ˆ + C sy s + C Kadilar and Cingi (006b) ˆ + ( ) sy s + ( ) Upadhyaya and ingh (999) ˆ + ( ) 3 sy s + ( ) ˆ + ρ + ρ 4 sy s ˆ + 5 sy s + ˆ + Md 6 sy s + Md ubramani and Kumarpandiyan (0a) ˆ + Q 7 sy s + Q ubramani and Kumarpandiyan (0b) ˆ + Q3 8 sy s + Q3 ubramani and Kumarpandiyan (0b) ˆ + Qr 9 sy s + Qr ubramani and Kumarpandiyan (0b) ( λ04 ) γ y R ( λ ) 04 y λ γ R 3 04 y 3 λ γ R 4 04 y 4 λ γ R 5 04 y 5 λ γ R 6 04 y 6 λ γ R 7 04 y 7 λ γ R 8 04 y 8 λ γ R 9 04 y 9 λ γ R γ γ γ γ γ γ γ γ γ ( ) 4 λ40 R λ04 R λ 4 λ40 R λ04 R λ 4 λ40 R3 λ04 R3 λ 4 λ40 R4 λ04 R4 λ 4 λ40 R5 λ04 R5 λ 4 λ40 R6 λ04 R6 λ 4 λ40 R7 λ04 R7 λ 4 λ40 R8 λ04 R8 λ 4 λ40 R9 λ04 R9 λ

4 American Journal of Operational Research 05 5(4): ˆ + Qd 0 sy s + Qd ubramani and Kumarpandiyan (0b) ˆ + Qa sy s + Qa ubramani and Kumarpandiyan (0b) ˆ + D sy s + D ubramani and Kumarpandiyan (0c) ˆ + D 3 sy s + D ubramani and Kumarpandiyan (0c) ˆ + D3 4 sy s + D3 ubramani and Kumarpandiyan (0c) ˆ + D4 5 sy s + D4 ubramani and Kumarpandiyan (0c) ˆ + D5 6 sy s + D5 ubramani and Kumarpandiyan (0c) ˆ + D6 7 sy s + D6 ubramani and Kumarpandiyan (0c) ˆ + D7 8 sy s + D7 ubramani and Kumarpandiyan (0c) ˆ + D8 9 sy s + D8 ubramani and Kumarpandiyan (0c) ˆ + D9 0 sy s + D9 ubramani and Kumarpandiyan (0c) 0 04 y 0 λ γ R 04 y λ γ R ) 04 y λ γ R ) ( ) 3 04 y 3 λ γ R 4 04 y 4 λ γ R ) 5 04 y 5 λ γ R 6 04 y 6 λ γ R ) 7 04 y 7 λ γ R 8 04 y 8 λ γ R 9 04 y 9 λ γ R ) ) 0 04 y 0 λ γ R ) γ γ γ γ γ γ γ γ γ γ γ 4 λ40 R0 λ04 R0 λ 4 λ40 R λ04 R λ 4 λ40 R λ04 R λ 4 λ40 R3 λ04 R3 λ 4 λ40 R4 λ04 R4 λ 4 λ40 R5 λ04 R5 λ 4 λ40 R6 λ04 R6 λ 4 λ40 R7 λ04 R7 λ 4 λ40 R8 λ04 R8 λ 4 λ40 R9 λ04 R9 λ 4 λ40 R0 λ04 R0 λ

5 86 Alok Kumar hukla et al.: Optimal earch of Developed Class of Modified Ratio Estimators for Estimation of Population Variance ˆ + D0 sy s + D0 ubramani and Kumarpandiyan (0c) ˆ ( ) + C sy s ( ) + C Kadilar and Cingi (006b) ˆ C + ( ) 3 sy sc + ( ) Kadilar and Cingi (006b) ˆ ( ) + C 4 sy s ( ) + C ˆ C + ( ) 5 sy sc + ( ) ˆ ρ + C 6 sy sρ + C ˆ C 7 sy sc + ρ + ρ ˆ + C 8 sy s + C ˆ C + 9 sy sc + ˆ Md + C 30 sy sm d + C ˆ C + Md 3 sy sc + Md ubramani and Kumarpandiyan (03) ˆ ( ) + ( ) 3 sy s ( ) + ( ) ˆ ( ) + ( ) 33 sy s ( ) + ( ) 04 y λ γ R ) 04 y λ γ R 3 04 y 3 λ γ R 4 04 y 4 λ γ R ( ) 5 04 y 5 λ γ R 6 04 y 6 λ γ R 7 04 y 7 λ γ R 8 04 y 8 λ γ R 9 04 y 9 λ γ R y 30 λ γ R 3 04 y 3 λ γ R ) ) 3 04 y 3 λ γ R y 33 λ γ R ) γ γ γ γ γ γ γ γ γ γ γ γ γ 4 λ40 R λ04 R λ 4 λ40 R λ04 R λ 4 λ40 R3 λ04 R3 λ 4 λ40 R4 λ04 R4 λ 4 λ40 R5 λ04 R5 λ 4 λ40 R6 λ04 R6 λ 4 λ40 R7 λ04 R7 λ 4 λ40 R8 λ04 R8 λ 4 λ40 R9 λ04 R9 λ 4 λ40 R30 λ04 R30 λ 4 λ40 R3 λ04 R3 λ 4 λ40 R3 λ04 R3 λ 4 λ40 R33 λ04 R33 λ

6 American Journal of Operational Research 05 5(4): ˆ ρ+ ( ) 34 sy sρ+ ( ) ˆ ( ) 35 sy s ( ) + ρ + ρ ˆ + ( ) 36 sy s + ( ) ˆ ( ) + 37 sy s ( ) + ˆ Md + ( ) 38 sy sm d + ( ) ˆ ( ) + Md 39 sy s ( ) + Md ˆ ρ+ ( ) 40 sy sρ+ ( ) ˆ ( ) 4 sy s ( ) + ρ + ρ ˆ + ( ) 4 sy s + ( ) ˆ ( ) + 43 sy s ( ) + ˆ Md + ( ) 44 sy sm d + ( ) ˆ ( ) + Md 45 sy s ( ) + Md ˆ 46 sy s + ρ + ρ y 34 λ γ R y 35 λ γ R y 36 λ γ R ) y 37 λ γ R y 38 λ γ R y 39 λ γ R y 40 λ γ R 4 04 y 4 λ γ R ) ) ) ) 4 04 y 4 λ γ R y 43 λ γ R y 44 λ γ R ) ) y 45 λ γ R y 46 λ γ R ) γ γ γ γ γ γ γ γ γ γ γ γ γ 4 λ40 R34 λ04 R34 λ 4 λ40 R35 λ04 R35 λ 4 λ40 R36 λ04 R36 λ 4 λ40 R37 λ04 R37 λ 4 λ40 R38 λ04 R38 λ 4 λ40 R39 λ04 R39 λ 4 λ40 R40 λ04 R40 λ 4 λ40 R4 λ04 R4 λ 4 λ40 R4 λ04 R4 λ 4 λ40 R43 λ04 R43 λ 4 λ40 R44 λ04 R44 λ 4 λ40 R45 λ04 R45 λ 4 λ40 R46 λ04 R46 λ

7 88 Alok Kumar hukla et al.: Optimal earch of Developed Class of Modified Ratio Estimators for Estimation of Population Variance ˆ ρ + 47 sy sρ + ˆ Md 48 sy sm d + ρ + ρ ˆ ρ + Md 49 sy sρ + Md ˆ Md + 50 sy sm d + ˆ + Md 5 sy s + Md y 47 λ γ R y 48 λ γ R y 49 λ γ R y 50 λ γ R 5 04 y 5 λ γ R ) ) ) ( ) γ γ γ γ γ 4 λ40 R47 λ04 R47 λ 4 λ40 R48 λ04 R48 λ 4 λ40 R49 λ04 R49 λ 4 λ40 R50 λ04 R50 λ 4 λ40 R5 λ04 R5 λ Thus the bias and ME of 5 estimators mentioned above in the table may be ritten as i y i i [ ] B ( ˆ ) γ R R ( λ ) i ˆ ( λ40 ) + R ( 04 ) ( ) i λ ME i γ Ri ( λ ) i...5 (.6) Upadhyaya and ingh (00) proposed the folloing estimator of population variance utilizing the population mean of the auiliary variable as ˆ X 5 sy (.7) The bias and the mean squared error of estimator up to the first order of approimation respectively are B ( ˆ 5) γ y C λc 4 ME( ˆ 5) γ y ( λ40 ) + C λc µ here λ µ µ (.8) (.9) 0 0 ubramani and Kumarpandiyan (05) using population mean of the auiliary variable proposed the folloing modified estimators for population variance as ˆ X + i + i Pi sy i...5 (.0) The bias and the mean squared error of the estimator in (.0) up to the first order of approimations respectively are B ( ˆ Pi ) γ y θpic θpiλc (.) Where 4 ˆ ( λ40 ) + θ ( ) PiC ME Pi γ θpiλc i...5 (.) X θ Pi X + i i Proposed Class of Estimators Motivated by Yadav et al (05b) e proposed the folloing class of estimators using a characterizing scalar α as ˆ X ( ) i i s + α α + α (3.) + i i...5 Where α is a characterizing scalar obtained by minimizing the mean squared error of the proposed estimator ˆ α i. To study the large sample properties of the proposed class of estimators e define sy y ( + e0 ) and X( + e ) such that Ee ( 0) Ee ( ) 0 and Ee ( 0 ) γ( λ40 )

8 American Journal of Operational Research 05 5(4): Ee ( ) γ C Eee ( 0 ) γλc. Using above approimations the proposed estimator may be ritten as ˆ αi y( + 0) α + ( α) e X + i X ( + e ) + i y ( 0 ) ( ) X + e i + α + α X + Xe + i y ( + e0 ) α + ( α) X e + X + i + + y ( e0 ) α ( α ) + θ pie y 0 α α θpi ( + e )[ + ( )( + e ) ] α + ( α)( θpie y ( + e0 ) + θpie...) y + e0 θpie + θpie θpiee 0 pie pie piee 0 + αθ αθ + αθ Through the order of approimations retaining the terms up to the first order e have + e 0 θ ˆ pie + θpie θpiee 0 αi y + αθ pie αθ pie + αθ piee 0 ubtracting y on both sides of (3.) e have e0 θ ˆ pie + θpie θpiee 0 αi y y + αθ pie αθ pie + αθ piee 0 (3.) (3.3) Taking epectation on both the sides of equation (3.3) e get Ee ( 0) θpiee ( ) + θpi ( e) ˆ E ( αi y ) y θ pi ( ee 0) + αθ pi ( e) αθ pi ( e ) + αθ pi ( ee 0 ) Putting the values of different epectations in above equation e get the bias of the proposed class of estimators as B ( ˆ ) αi γ y θpic θpiλc αθ + αθ λ pic pi C (3.4) quaring on both sides of (3.3) simplifying and retaining the terms up to the first order of approimation e have e0 + θpie + αθpie ˆ 4 [ αi y ] y θ piee 0+ αθ piee 0 αθ pie (3.5) Taking epectations on both the sides of equation (3.5) e get the mean squared error of the proposed class of estimators up to the first order of approimation as Ee ( 0) + θ piee ( ) ˆ 4 ME ( αi ) y + αθpiee ( ) θpieee ( 0 ) (3.6) + αθ pieee ( 0) αθ piee ( )] Putting the values of different epectations in above equations and simplifying e have ( λ ) + θ C 40 pi ˆ 4 ME( αi ) γ y + αθpic θpiλc (3.7) + αθ piλc αθ pic The optimum value of the characterizing scalar α is obtained by minimizing ME in (3.7) using the method of maima-minima as αθ pic + θ piλc θ pic 0 pic pi C θ pic θ θ λ µ α (3.8) Where µ θpic θpiλc and θpic The minimum value of bias of the proposed class of estimators is obtained by putting optimum value of α in (3.4) as ˆ µ B ( αi ) γ ( θpic θpiλc ) (3.9) Minimum value of the ME of the proposed class of estimators is obtained by putting the optimum value of α in (3.7) and thus the minimum ME is given as γ{( λ40 ) + θpic ˆ 4 ME( αi) µ θpiλc} (3.0)

9 90 Alok Kumar hukla et al.: Optimal earch of Developed Class of Modified Ratio Estimators for Estimation of Population Variance 4. Efficiency of the Proposed Class of Estimators From (.) and (3.0) e have: if V ( t ) ME ( ˆ ) 0 min αi 4µ γθ { pic θpiλc} > 0 µ γθ { pic θpiλc} < (4.) From (.5) and (3.0) e have: If ME( t ) ME ( ˆ ) R min αi { } γ λ λ 04 4 µ > γθ { pic θpiλc} + { θ pic θpiλ C } µ γ < { ( λ04 ) ( λ ) } From (.6) and (3.0) e have: If ME( ˆ ) ME ( ˆ α ) i min i { Ri ( 04 ) Ri( ) } γ λ λ 4 > 0 µ γθ { pic θpiλc} + { θ θ λ } µ γ < { } 0 (4.) pic pi C Ri λ04 R i λ From (.9) and (3.0) e have: If ( i...5) (4.3) ME( ˆ ˆ 5) MEmin ( αi ) { C λc} γ 4 > 0 µ γθ { pic θpiλc} + { θ pic θpiλ C} µ γ < { C λc} (4.4) From (.) and (3.0) e have: ˆ ˆ 4 µ ME( Pi ) MEmin ( α i ) > 0 if 5. Numerical tudy µ > (4.5) To judge the performances of the proposed class of estimators along ith the eisting modified ratio type estimators of population variance given in this manuscript e have considered to natural populations the population- and Population-. Both the populations have been taken from the book by ingh and Chaudhary (986) at page 77. The population parameters of above populations are given belo as: Population-: ingh and Chaudhary (986) N 34 n 0 Y X λ y C λ λ ( ) λ M d 5 Q Q Q r 6.05 Q 6.05 Q 7.45 D 7.03 D 7.68 d a D D 4.94 D 5 5 D 6.7 D D D D Population-: ingh and Chaudhary (986) N 34 n 0 Y X λ y C λ λ ( ).758 λ M d 4.5 Q 9.95 Q Q r Q Q D 6.06 d a D 8.30 D D 4. D D 6.0 D D D D The folloing table represents the Bias and the Mean quared Error (ME) of proposed class of estimators ith the eisting modified ratio type estimators of population variance mentioned in the manuscript as

10 American Journal of Operational Research 05 5(4): Table. Bias and ME of Various Estimators for Population- Estimator Bias (.) ME (.) Estimator Bias (.) ME (.) Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP

11 9 Alok Kumar hukla et al.: Optimal earch of Developed Class of Modified Ratio Estimators for Estimation of Population Variance Ŝ Ŝ Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ ˆP Ŝ Ŝ α ˆR

12 American Journal of Operational Research 05 5(4): Table 3. Bias and ME of Various Estimators for Population- Estimator Bias (.) ME (.) Estimator Bias (.) ME (.) Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP

13 94 Alok Kumar hukla et al.: Optimal earch of Developed Class of Modified Ratio Estimators for Estimation of Population Variance Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ ˆR Ŝ ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP ˆP Ŝ α

14 American Journal of Operational Research 05 5(4): Note: ince the estimator ˆP5 of the class of estimators proposed by ubramani and Kumarapandiyan (05) has minimum ME among its class and the rest estimators mentioned in above table ecept the proposed estimator. Therefore e have computed only Ŝ α 5 estimator of proposed class of estimators for both the populations corresponding to ˆP5 estimator. 6. Conclusions In the present paper e develop a generalized improved class of estimators for population variance using population mean of the auiliary variable by applying the scalar constant to ubramani and Kumarapandiyan (05) estimator of population variance. The bias and the mean squared error of the proposed general class of estimators have been obtained up to the first order of approimation. The optimum value of the characterizing scalar has been obtained up to the first order of approimation and for this optimum value of the scalar the minimum mean squared error of the proposed class of estimators have been obtained. From table- and table-3 e observe that the estimator Ŝ α 5 of the proposed class of estimators has the minimum mean squared error as compared to other mentioned eisting estimators of population variance in this manuscript. Therefore it is strongly recommended that the proposed generalized class of estimators should be preferably used for the estimation of population variance using auiliary variable under simple random sampling scheme. REFERENCE [] IAKI C.T Variance estimation using auiliary information. Journal of the American tatistical Association 78: 7-3. [] INGH D. and CHAUDHARY F Theory and analysis of sample survey designs. Ne Age International Publisher. [3] INGH H.P. and OLANKI R A ne procedure for variance estimation in simple random sampling using auiliary information. tatistical Papers [4] UBRAMANI J. and KUMARAPANDIYAN G. 0a. Variance estimation using median of the auiliary variable. International Journal of Probability and tatistics Vol. (3) [5] UBRAMANI J. and KUMARAPANDIYAN G. 0b. Variance estimation using quartiles and their functions of an auiliary variable International Journal of tatistics and Applications 0 Vol. (5) [6] UBRAMANI J. and KUMARAPANDIYAN G. 0c. Estimation of variance using deciles of an auiliary variable. Proceedings of International Conference on Frontiers of tatistics and Its Applications Bonfring Publisher [7] UBRAMANI J. and KUMARAPANDIYAN G. 03. Estimation of variance using knon coefficient of variation and median of an auiliary variable. Journal of Modern Applied tatistical Methods Vol. () [8] UBRAMANI J. and KUMARAPANDIYAN G. 05. A Class of Modified Ratio Estimators for Estimation of Population Variance. JAMI 9-4. [9] TAILOR R. and HARMA B. 0. Modified estimators of population variance in presence of auiliary information. tatistics in Transition-Ne series 3() [0] UPADHYAYA L.N. and INGH H.P. 00. Estimation of population standard deviation Using auiliary information. American Journal of Mathematics and Management ciences (3-4) [] YADAV.K. and KADILAR C. 03a. A class of ratio-cum-dual to ratio estimator of Population variance. Journal of Reliability and tatistical tudies 6() [] YADAV.K. and KADILAR C. 03b. Improved Eponential type ratio estimator of Population variance. Colombian Journal of tatistics 36() [3] YADAV.K. KADILAR C. HABBIR J. and GUPTA. (05a). Improved Family of Estimators of Population Variance in imple Random ampling Journal of tatistical Theory and Practice [4] YADAV.K. MIHRA.. HUKLA A.K. and TIWARI V. (05b). Improvement of Estimator for Population Variance using Correlation Coefficient and Quartiles of The Auiliary Variable Journal of tatistics Applications and Probability