International Journal of Computational Intellience Research ISSN 0973-873 Volume 3, Number (07), pp. 3-33 Research India Publications http://www.ripublication.com A Generalized Class of Double Samplin Estimators Usin Auxiliary Information on an Attribute and an Auxiliary Variable Shashi Bhushan, Praveen Kumar Misra and Sachin Kumar adav Department of Mathematics and Statistics, Dr. Shakuntala Misra National Rehabilitation University, India. Department of Statistics, University of Lucknow, India Abstract In the present study, we have proposed a eneralized class of double samplin estimators usin auxiliary information in both the form variable and attribute. The bias and mean square error of proposed eneralized class is obtained has been shown that it attains minimum value under some optimum conditions. It has been shown that many of the well-known double samplin estimators are the member of the proposed class. Also the Bias and MSE table of some proposed estimators is iven in the last. Keywords: Double samplin, auxiliary information, bias, mean square error. INTRODUCTION Consider the followin notations = Study variable X = Auxiliary variable = Auxiliary attribute N = Size of population N i = Population mean of study variable N i
4 Shashi Bhushan, Praveen Kumar Misra and Sachin Kumar adav X N Xi = Population mean of auxiliary variable N i N P i = Population mean of auxiliary attribute N i S N ( i ) N i S X X S N X ( i ) N i N ( i P) N i = Population variance of study variable =Population variance of auxiliary variable =Population variance of auxiliary attribute If the information about the auxiliary variable and attribute is not known then in double samplin scheme these auxiliary characteristics are replaced by the correspondin sample values comes from the lare preliminary simple random sample of size n drawn without replacement from a population of size N in the first phase. Also the characteristic of interest and the auxiliary characteristic X and are observed on the second phase sample of size n drawn from the first phase sample by simple random sample without replacement. Let x n i X n i n i, p n i be the sample mean of auxiliary variable and attribute respectively based on the first phase sample of size n. Also denote the followin sample information as n y i = Sample mean of study variable, n x Xi = Sample mean of auxiliary n i n i variable n p = Sample mean of auxiliary attribute n i i With this available information we propose a class of estimators for mean value of study variable as y( x, p) (.) Where ( x, p) is the function of x and p such that (i) ( x, p)
A Generalized Class of Double Samplin Estimators Usin Auxiliary Information 5 (ii) The function ( x, p ) is continuous and bounded in the closed interval R of real line. (iii) The first and second order partial derivatives of the function ( x, p) are exist and are continuous and bounded in R. It is to be mentioned that proposed class of estimators is very lare. Consider the followin member of this class. Followin Olkin(958) () x p y ( ) x p. Followin Sinh (967) () x p y ( ) x p 3. Followin Shukla (966) and john (969) (3) x( ) p y x( ) p 4. Followin Sahai et al (980) (4) x( ) p y x( ) p 5. Followin Mohanty and Pattanaik (984) (5) x p y x p 6. Followin Mohanty and Pattanaik (984) (6) x p y ( ) x p 7. Followin Tuteja and Bahl (99) (7) x p y x p 8. Followin Tuteja and Bahl (99) (8) x p y ( ) x p
6 Shashi Bhushan, Praveen Kumar Misra and Sachin Kumar adav 9. Followin Naik and Gupta (996) and Abu Dayyeh (003) proposed Ratio type estimator by us[0] (9) x p y x p 0. Followin Sahai and Ray (980) (0) x p y x p. Followin Walsh (970) x () y. x x x p p p ( ) ( ). Followin Srivastava (97) () x p yexplo lo x p 3. Followin Srivastava (97) (3) x p yexp x p 4. Followin Srivastava (97) x p exp lo ( ) exp lo x p (4) y 5. Proposed Estimator by us in our paper (5) x x p p yexp exp xx p p p (.) Many such estimators utilizin the information in both the forms auxiliary variable and attribute can be constructed as the member of this class. BIAS AND MSE OF THE PROPOSED GENERALIZED CLASS OF DOUBLE SAMPLING ESTIMATORS To obtain the bias of the proposed class of estimators, we further assume that the third order partial derivatives of ( x, p ) exist and are bounded and continues. Then takin Taylor s expansion about the point ( x, p ) up to third order terms, we et
A Generalized Class of Double Samplin Estimators Usin Auxiliary Information 7 ( x, p) ( x x) x( x, p) ( p p) p( x, p)! ( x x) xx ( x, p) ( x x)( p p) xp( x, p) y (.)! ( p p) pp( x, p) 3 ( x x) xxx ( x, p ) 3( x x) ( p p) xxp ( x, p ) 3 3! 3( x x)( p p) xpp( x, p ) ( p p) ppp( x, p ) where x x ( x x), p p ( p p),0 and a( x, p), ab( x, p) and (, ) abc x p are the first order, second order and third order partial derivatives of the function ( x, p) at the point ( x, p ) about a, a and b and a, b and c respectively in eneral terms. Now let us take y e 0, x X e, p P e, x X e, pp e ( ) ( ) ( ) ( ) ( ) 0 (.) with E e0 E e E e E e E e And the results iven in Sukhatme and Sukhathhme (997) E( e0 ) fns, E( e ) fnsx, E( e ) fnsp, E( e ) fn SX, E( e ) fn SP E( e0e) fnx S SX, E( e0e ) fnx S SX, E( e0e) fnps SP E( e0e ) fnps SP, E( ee ) fnxpsx SP, E( e e) fnxpsx SP E( ee ) fnxpsx SP, E( e e) fnxpsx SP where f n (.3) n N Substitutin the values from (.) in (.) and nelectin the terms of ei ' s( i 0,, ) havin powers reater than two, we et e e x( x, p) e e p( x, p)! e e xx ( x, p) ( ee ee e e e e ) xp ( x, p) (.4)! e e pp ( x, p) e0 e0 e e x( x, p) e0 e e p( x, p)!
8 Shashi Bhushan, Praveen Kumar Misra and Sachin Kumar adav On takin expectation and substitutin the results from (.3), we et ( f f E ) S ( x, p) S S ( x, p) n n X xx XP X P xp SP pp ( x, p) fn fn X S SX x ( x, p) PS SP p( x, p) Showin that is a biased estimator of and its bias is iven by Bias( ) E( ) f Bias S x p S S x p S x p f S S ( x, p) S S ( x, p) nn ( ) X xx (, ) XP X P xp (, ) P pp(, ) nn X X x P P p n n where f f f nn n n The mean square error (MSE) of is iven by usin (.4) MSE( ) E( ) (, ) (, ) x p E e0 e e x p e e x p (To the first order of approximation) E e e x( x, p) E e e p( x, p) E( e0 ) E( ee ee e e e e ) x( x, p) p( x, p) E e0e e0e x( x, p) E e0e e0e p( x, p ) Substitutin the results from (.3), we et S (, ) (, ) X x x p SP p x p ( fns fnn MSE ) XP X P x (, ) p(, ) S S x p x p fnn X S SX x ( x, p) PS SP p( x, p ) where n nn B( say) (.5) f S f M (Say) (.6) M S ( x, p) S ( x, p) S S ( x, p) ( x, p) X x P p XP X P x p S S ( x, p) S S ( x, p) X X x P P p
A Generalized Class of Double Samplin Estimators Usin Auxiliary Information 9 This will be minimum when ( ) S ( x, p) x p X P XP ( XP ) SX ( ) S ( x, p) P X XP ( XP ) SP (.7) (.8) And its minimum value under the optimum values of the characteristic scalars is iven by min MSE( ) ( f f R ) S (.9) n nn. XP BIAS AND MSE OF THE PROPOSED MEMBER OF THE CLASS The bias and MSE of some proposed members of the class defined in equation (.) are iven in the followin table Table 3.: Derivatives of ( x, p) of proposed members of the eneralized class Estimator (, ) x x p (, ) p x p (, ) xx x p (, ) pp x p (, ) xp x p () ( ) x p x ( ) p 0 () x ( ) p 0 0 0 (3) ( ) K K K ( ) K ( ) K (4) ( ) K K 0 0 0 (5) ( ) x p ( ) x ( )( ) p ( ) xp (6) ( ) x p x ( ) p ( ) xp (7) x ( ) p ( ) x ( ) p ( ) xp
30 Shashi Bhushan, Praveen Kumar Misra and Sachin Kumar adav (8) x ( ) p ( ) x ( ) p ( ) xp (9) x p ( ) x ( ) p xp (0) x ( ) p x ( ) p xp () ( ) x ( ) p ( ) x ( ) p ( )( ) xp () x p ( ) x ( ) p xp (3) x p x p xp (4) x p x p 0 (5) x p ( ) 4x ( ) 4 p 4xp Estimator Bias () () Table 3.: Bias of the proposed members of the eneralized class fnn CX ( ) CP X C CX ( ) PC C P fnn X C CX ( ) PC CP (3) fnn ( ) ( ) ( ) X P XP X P X X P P (4) S S S S C S K C S K K f K nn S S ( ) S S X X P P (5) ( ) CX ( )( ) CP ( ) XPCXCP nn XC C X f ( ) CC P P
A Generalized Class of Double Samplin Estimators Usin Auxiliary Information 3 (6) fnn CX ( ) CP ( ) XPCXCP X C CX ( ) PC C P (7) fnn ( ) CX ( ) CP ( ) XPCXCP X C CX ( ) PC CP (8) fnn ( ) CX ( ) CP ( ) XPCXCP X C CX ( ) PC C P (9) f nn ( ) CX ( ) CP XPCXCP X C CX PC CP (0) fnn ( ) CX ( ) CP XPCXCP X C CX PC CP () ( ) CX ( ) CP ( )( ) XPCXC P fnn ( ) X C CX ( ) PC CP () fnn ( ) CX ( ) CP XPCXCP X C CX PC CP (3) fnn CX CP XPCXCP X C CX PC CP (4) f nn CX CP X C CX PC CP (5) CX CP CX CP XCXC PCPC XPCXC P fnn 4 4 8 8 4 Table 3.3: MSE of the proposed members of the eneralized class Estimator MSE () () (3) (4) CX ( ) CP ( ) XPCXCP n nn X C CX ( ) PC CP CX ( ) CP ( ) XPCXCP n nn X C CX ( ) PC CP S ( ) S ( ) S S X P XP X P fnc fnn K X C SX K ( ) PC SPK S ( ) S ( ) S S X P XP X P f ns fnn K X S SX ( ) PS SP K
3 Shashi Bhushan, Praveen Kumar Misra and Sachin Kumar adav (5) (6) (7) (8) (9) (0) () () (3) (4) (5) CX ( ) CP ( ) XPCXCP n nn X C CX ( ) PC CP CX ( ) CP ( ) XPCXCP n nn X C CX ( ) PC CP CX ( ) CP ( ) XPCXCP n nn X C CX ( ) PC CP CX ( ) CP ( ) XPCXCP n nn X C CX ( ) PC CP C C C C X P XP X P n nn X C CX PC CP C C C C X P XP X P n nn X C CX PC CP X P XP X P n nn ( ) X C CX ( ) PC CP ( ) C ( ) C ( )( ) C C C C C C X P XP X P n nn X C CX PC CP C C C C X P XP X P n nn X C CX PC CP C C C C X P XP X P n nn X C CX PC CP C C C C C C C C 4 4 X P XP X P n nn X X P P CONCLUSION The above theoretical study establishes the importance of the proposed eneralized class of double samplin estimators due to its wider applicability and emphasizes its superiority over the proposed eneralized class of estimators in the sense of unbiasedness. Also it attains minimum value of MSE under the optimum values of the characterizin parameters. min MSE( ) f ( R ) S. n. XP
A Generalized Class of Double Samplin Estimators Usin Auxiliary Information 33 ACKNOWLEDGEMENT The financial aid rendered by UGC is ratefully acknowleded. REFERENCES [] Bhushan, S. (03). Improved Samplin Strateies in Finite Population. Scholars Press, Germany. [] Bhushan S. (0). Some Efficient Samplin Strateies based on Ratio Type Estimator, Electronic Journal of Applied Statistical Analysis, 5(), 74 88. [3] Bhushan S. and Katara, S. (00). On Classes of Unbiased Samplin Strateies, Journal of Reliability and Statistical Studies, 3(), 93-0. [4] Bhushan, S. and Kumar S. (06). Recent advances in Applied Statistics and its applications. LAP Publishin. [5] Bhushan S. and Pandey A. (00). Modified Samplin Strateies usin Correlation Coefficient for Estimatin Population Mean, Journal of Statistical Research of Iran, 7(), - 3. [6] Bhushan S., Sinh, R. K. and Katara, S. (009). Improved Estimation under Midzuno Lahiri Sen-type Samplin Scheme, Journal of Reliability and Statistical Studies, (), 59 66. [7] Bhushan S., Masaldan R. N. and Gupta P. K. (0). Improved Samplin Strateies based on Modified Ratio Estimator, International Journal of Aricultural and Statistical Sciences, 7(), 63-75. [8] Bahli, S. And Tuteja, R.K. (99): Ratio and Product type exponential estimator, Information and Optimization sciences, Vol.XII, I, 59-63. [9] Gray H. L. and Schucany W.R. (97). The Generalized Jack-knife Statistic, Marcel Dekker, New ork. [0] Naik, V.D. and Gupta, P.C. (996): A note on estimation of mean with known population proportion of an auxiliary character. Jour. Ind. Soc. Ar. Stat., 48(), 5-58. [] Sukhatme, P.V. andsukhatme, B.V. (970): Samplin theory of surveys with applications. Iowa State University Press, Ames, U.S.A. [] Srivastava, S. K. (967). An estimator usin auxiliary information. Calcutta Statistical Association. Bulletin, 6, -3. [3] Cochran, W. G. (977), Samplin Techniques, 3rd ed. New ork: John Wiley and Sons.
34 Shashi Bhushan, Praveen Kumar Misra and Sachin Kumar adav