International Journal of Computational and Applied Mathematics. ISSN 89-966 Volume, Number (07), pp. -0 Research India ublications http://www.ripublication.com On The Class of Double Sampling Exponential Ratio Type Estimator Using Auxiliary Information on an Attribute and an Auxiliary Variable Shashi Bhushan, raveen 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 this paper, we have proposed a class of double sampling exponential ratio type estimator using the auxiliary information in both the form attribute and variable. The proposed class of estimator utilizes the auxiliary information on means of the auxiliary variable and attribute available in the first phase sample. The bias and mean square error (MSE) of the proposed class of estimator is obtained. In the conclusion, it has been shown that the proposed class of double sampling estimator is better than the most commonly used double sampling estimators discussed in the literature. An empirical study is included for illustration. Keywords: Auxiliary information, Double Sampling Exponential type Estimator, Bias, Mean Square Error, and Multiple Correlation Coefficient INTRODUCTION It is well known to all that the auxiliary information is used to improve the precision of the estimator. So in the case when auxiliary information is not known, double sampling strategy helps in improving the precision of the estimator. Many biased double sampling ratio type, double sampling ratio type and the biased double sampling estimator obtained through parametric combination of ratio type and the usual unbiased estimators are available for estimating the population mean. The use of an auxiliary variable and an attribute to improve the efficiency of the population mean has been discussed recently by us in among others.
Shashi Bhushan, raveen Kumar Misra and Sachin Kumar adav Consider the following notations = Study variable = Auxiliary Variable Ф = Auxiliary Attribute N= Size of the population n = Size of the sample N i N i = opulation mean of study variable N i N i = opulation mean of auxiliary variable N i i N = opulation mean of auxiliary attribute S N ( i ) N i = opulation variance of study variable S N ( i ) N i = opulation variance of auxiliary variable N ( i ) N i S = opulation variance of auxiliary attribute (.) If the information about the auxiliary variable and attribute is not known then in double sampling scheme these auxiliary characteristics are replaced by the corresponding sample values comes from the large 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 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 n i = First phase sample mean of auxiliary variable p y x n i n i = First phase sample mean of auxiliary attribute n i n i = Second phase sub-sample mean value of study variable n i n i = Second phase sub-sample mean value of auxiliary variable
On the Class of Double Sampling Exponential Ratio Type Estimator 3 n i p = Second phase sub-sample value of auxiliary attribute (.) n i Consider the following usual unbiased and biased double sampling ratio and exponential ratio type estimator of population mean developed in the past (i) (ii) General estimator of population mean in case of SRSWOR y = Sample mean with MSE f C (.3) ( ) n Double sampling ratio estimator using auxiliary variable x y x (iii) (iv) (v) with MSE( ) f C ( ) n f C nn CyC (.) Naik and Gupta (996) double sampling ratio estimator using auxiliary attribute p 3 y p with MSE( 3 ) f C ( ) n f C nn C yc (.5) Bahl and Tuteja(99) double sampling exponential ratio estimator using auxiliary variable yexp x x x x MSE( ) f C ( ) n f C nn CyC with (.6) Following Bahl and Tuteja (99), double sampling exponential estimator given by Nirmala Sawan (00) using auxiliary attribute 5 yexp p p p p MSE( 5 ) f C ( ) n f C nn C yc with where f n n N and fnn fn fn n n (.7)
Shashi Bhushan, raveen Kumar Misra and Sachin Kumar adav ROOSED CLASS OF DOUBLE SAMLING EONENTIAL RATIO TE ESTIMATORS Following Bahl and Tuteja (99) and Singh (967) we propose a class of Double sampling Ratio type estimators assuming that the auxiliary population means and auxiliary population proportions are known x x p p p yexp exp xx p p (.) It may be noted that the estimators given from (.3), (.6) and (.7) are the special cases of the proposed study using no auxiliary variable, using one auxiliary variable and using one auxiliary attribute respectively. In order to obtain the Bias and Mean square error (MSE), let us denote y e ( 0), x e ( ), p ( e ), x ( e ), p( e ) with E e0 E e E e E e E e ( ) ( ) ( ) ( ) ( ) 0 (.) And the results given in Sukhatme and Sukhatme (997) E e f C, E e fnc, E e fn C, E e fnc, E e 0 n, Ee e f C C, E e e f C C 0 n 0 n E e e f C C, 0 n f C n Ee e f C C, Eee Ee e E ee f C C (.3) E e e f C C 0 n n n Substituting the values from (.) in (.) and on solving, we get p ( e e ) ( e e ) ( e e ee ) ( e e ) e0 8 ( e e ) ( e e e e ) e0 ( e e ) e0 ( e e ) 8 ( ee e e ee e e ) Taking expectation and substituting the values from (.3), we get (.) Bias p E p C C C C fn fn fn fn fn fn fn fn 8 8 C C C C C C fn fn fn fn fn fn
On the Class of Double Sampling Exponential Ratio Type Estimator 5 where C C C C C C ( f ) 8 8 n fn CC C C = f nn A (Say) (.5) C C C C C C C C C C A 8 8 For mean square of error (MSE), taking (.) up to the first order of approximation ( e e ) ( e e ) p e0 (.6) Squaring and taking expectation both the sides, we get p p MSE E where C fnc ( fn fn) C C ( fn fn) C C ( fn fn) C C C ( fn fn) C C C C = f C f C C C C n nn fnc fnn B (Say) (.7) C C C C B C C C C The minimum value of MSE is obtained if optimum values of and are ( ) C opt( ) ( ) C (.8) ( ) C opt( ) ( ) C (.9) and the minimum mean square error under the optimizing values of the characteristic scalars is given by p n nn. min MSE( ) ( f f R ) C M (Say) (.0)
6 Shashi Bhushan, raveen Kumar Misra and Sachin Kumar adav CLASS OF ESTIMATORS BASED ON THE ESTIMATED VALUES OF THE CHARACTERIZING SCALARS The optimum values of the characterizing scalars are rarely known in practice hence they may be estimated by estimators based on the sample values. The optimizing values of the characterizing scalars can be written as ( ) C opt( ) ( ) C ( ) C opt( ) ( ) C where S S S S S S S S S S S S S S N a b c abc ( i ) ( i ) ( i ) N i 0 00 00 00 00 0 000 0 0 00 00 0 We may take and as the unbiased estimator of and such that 0 00 0 0 00 00 0 x y 0 00 0 0 00 00 0 p y (3.) (3.) where 0 m 0, 0 m0, 0 m0, 00 m00 and 00 m00 are the estimators of the population parameters 0, 0, 0, 00 and 00 respectively such that m y y x x p n a b c abc i i i n i Let us take e e, e 0 0 3, e e 00 00 6 00 00 7 0 0 0 0 5 with E( e ) 0, i 3,,5,6,7 i Then under the estimated values of characterizing scalars and, the proposed class of ratio estimators becomes x x p p s yexp exp xx p p (3.3) utting the values from (.) and simplifying we get ( ) ( ) e e e e s e O e 0 ( )
On the Class of Double Sampling Exponential Ratio Type Estimator 7 To the first order of approximation e e e e s e0 ( ) ( ) For MSE squaring and taking expectation both the sides, we get s s MSE E C fnc ( fn fn) C C ( fn fn) C C ( fn fn) C C C ( fn fn) C C C C = f C f C C C C n nn ( f f R ) C (On substituting the values from (3.)) n nn. min MSE( p ) (3.) This is the same as that of minimum value of mean square error of proposed class of estimator. COMARISON WITH THE AVAILABLE ESTIMATORS Consider the following estimator of the study variable (i) Sample Mean in case of SRSWOR y Vs. S From (.3) and (3.) (ii) nn. MSE( ) M f R C 0 (.) Double sampling ratio estimator using auxiliary variable Vs. s From (.) and (3.) nn. MSE( ) M f C C C R C fnn C C C C C R. C f ( C C ) ( R ) C 0 nn. As R. (.)
8 Shashi Bhushan, raveen Kumar Misra and Sachin Kumar adav (iii) Double sampling ratio estimator using auxiliary attribute 3 Vs. s From (.) and (3.) 3 nn. MSE( ) M f C C C R C fnn C C C C C R. C f ( C C ) ( R ) C 0 nn. (.3) (iv) As R. Bahl and Tuteja (99) double sampling exponential ratio estimator using auxiliary attribute Vs. s From (.5) and (3.) C MSE( ) M fnn C C R. C C fnn C C C C R. C C fnn ( C ) ( R. ) C 0 (.) (v) As R. Following Bahl and Tuteja (99), roposed double sampling exponential ratio estimator using auxiliary attribute by Nirmal Sawan (00) 5 Vs. s From (.6) and (3.) Cp MSE( 5 ) M fnn C C y R. C C fnn C C C C R. C C fnn ( C ) ( R. ) C 0 (.5) As R.
On the Class of Double Sampling Exponential Ratio Type Estimator 9 EMIRICAL STUD opulation : [Source: William G. Cochran (977), age-3] = Food Cost, = Family Income, = Family of size more than 3 = 7.0, = 7.55, = 0.5, = 0.388, = -0.53, C = 0.6, C = 0.369, C = 0.985, R = 0.9879, n = 6, n =, N = 33. = 0.5, opulation II: [Source: Advance Data from Vital and Health Statistics, Number 37, October 7, 00(CDC)] = Height of the people, = Weight of the people, = Sex of the people = 0.8, = 39.63, = 0.50, = 0.973, = 0.07, = 0.073, C = 0.965, C = 0.8337, R = 0.9673, n =, n =, N = 36. C =.0, Table 5.: RE of various estimators with respect to sample mean ESTIMATOR opulation I RE opulation II 00.00 00.00 0.33 50.5 3 7. 5. 03.3 3.89 5 7.70 8. s 5.5 8.09 CONCLUSION The comparative study of the proposed class of double sampling exponential ratio type estimators establishes its superiority in the sense of minimum mean square of error over the sample mean, double sampling ratio estimator, and double sampling exponential ratio estimator using auxiliary variable and attribute at one time under the estimated values of optimizing scalars.
0 Shashi Bhushan, raveen Kumar Misra and Sachin Kumar adav ACKNOWLEDGEMENT The financial aid rendered by UGC is gratefully acknowledged. REFERENCES [] Bhushan, S. (03). Improved Sampling Strategies in Finite opulation. Scholars ress, Germany. [] Bhushan S. (0). Some Efficient Sampling Strategies based on Ratio Type Estimator, Electronic Journal of Applied Statistical Analysis, 5(), 7 88. [3] Bhushan S. and Katara, S. (00). On Classes of Unbiased Sampling Strategies, Journal of Reliability and Statistical Studies, 3(), 93-0. [] Bhushan, S. and Kumar S. (06). Recent advances in Applied Statistics and its applications. LA ublishing. [5] Bhushan S. and andey A. (00). Modified Sampling Strategies using Correlation Coefficient for Estimating opulation Mean, Journal of Statistical Research of Iran, 7(), - 3. [6] Bhushan S., Singh, R. K. and Katara, S. (009). Improved Estimation under Midzuno Lahiri Sen-type Sampling Scheme, Journal of Reliability and Statistical Studies, (), 59 66. [7] Bhushan S., Masaldan R. N. and Gupta. K. (0). Improved Sampling Strategies based on Modified Ratio Estimator, International Journal of Agricultural and Statistical Sciences, 7(), 63-75. [8] Bahli, S. And Tuteja, R.K. (99): Ratio and roduct type exponential estimator, Information and Optimization sciences, Vol.II, 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,.C. (996): A note on estimation of mean with known population proportion of an auxiliary character. Jour. Ind. Soc. Agr. Stat., 8(), 5-58. [] Sukhatme,.V. andsukhatme, B.V. (970): Sampling theory of surveys with applications. Iowa State University ress, Ames, U.S.A. [] Srivastava, S. K. (967). An estimator using auxiliary information. Calcutta Statistical Association. Bulletin, 6, -3. [3] Cochran, W. G. (977), Sampling Techniques, 3rd ed. New ork: John Wiley and Sons.