International Journal of Statistics and Systems ISSN 0973-675 Volume, Number (07), pp. 5-3 Research India Publications http://www.ripublication.com On The Class of Double Sampling Ratio Estimators Using 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 this paper, we have proposed a class of ratio estimators using the double sampling estimators as the auxiliary information on an attribute and an auxiliary variable. The bias and mean square error (MSE) of the proposed class are obtained. Further, it has been shown that the proposed class has minimum MSE under the optimum values of the characterizing scalar. Also, it has been shown that under the estimated values of characterizing scalar it attains its minimum MSE. An empirical study is included as illustration. Keywords: Double Sampling, Ratio Estimator, Mean Square error, Bias. INTRODUCTION In sampling, prior information about the study variable or information on the some variable closely related with the study variable is called auxiliary information. If it is available completely then we can improve the precision of our estimates. But absence of prior information is not a serious problem in sampling. In such cases where prior information is not available we reside on double sampling technique. 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.
6 Shashi Bhushan, Praveen 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 = Population mean of study variable N i N i = Population mean of auxiliary variable N i i P N = Population mean of auxiliary attribute S N ( i ) N i = Population variance of study variable S N ( i ) N i = Population variance of auxiliary variable N ( i P) N i S = Population 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 Ratio Estimators Using Auxiliary Information 7 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) (iii) (iv) (v) 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 with MSE( ) f C ( ) n f C nn CyC (.4) Naik and Gupta (996) double sampling ratio estimator using auxiliary attribute p 3 y p with MSE( 3 ) f C ( ) n f C nn P PC yc P (.5) Bahl and Tuteja (99) double sampling exponential ratio estimator using auxiliary variable 4 yexp x x x x MSE( 4 ) f C ( ) n f C nn CyC 4 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 P PC yc P 4 with where f n n N and fnn fn fn n n (.7)
8 Shashi Bhushan, Praveen Kumar Misra and Sachin Kumar adav PROPOSED CLASS OF DOUBLE SAMPLING RATIO ESTIMATORS Following the estimator proposed by us in our previous paper, we proposed the following class of double sampling ratio estimators x p = y x p (.) It may be noted that the estimators given from (.3), (.4) and (.5) 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= P e, x= e, p= P e with Ee Ee Ee Ee E e 0 0 (.) And the results given in Sukhatme and Sukhtame (997) E e f C, E e fnc, E e fn C, E e fncp, E e 0 n, E e e f C C 0 n, E e e f C C 0 n P P E e e f C C 0 n E e e f C C 0 n P P f C n P, Eee Ee e E ee f C C (.3) E e e f C C n P P n P P Substituting the values from (.) in (.), we get e e e e e e e e e e e e e ee ee e e e e 0 0 0 0 0 e e e e e e e e Taking Expectation and substituting the results from (.3), we get Bias E C C PC CP fn fn ( ) ( ) PC CP C CP where fnn ( fn fn) n n ( ) ( ) A C C C C C C C C P P P P P (.4) f A ( say) (.5) nn
On The Class of Double Sampling Ratio Estimators Using Auxiliary Information 9 Up to first order of approximation (.) can be written as e e e e e 0 On squaring and taking expectation both the sides, we get MSE E f C f C C C C C C C C n nn P P P P P n nn f C f B (Say) (.6) Where B C C C C C C C C P P P P P The minimum value of MSE is obtained if the optimum value of and are ( ) C opt( ) P P ( P ) C (.7) ( ) C opt( ) P P ( P ) CP (.8) And the minimum value of MSE under the optimum values of characterizing scalars is n nn. P min MSE ( f f R ) C M (Say) (.9) 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( ) P P ( P ) C ( ) C opt( ) where P P ( P ) CP S SP SPSP S S S P P SP S S SP P S S S P P N a b c abc ( i ) ( i ) ( i P) N i 0 00 0 0 00 00 0 0 00 0 0 00 00 0 P We may take and as the unbiased estimator of and such that (3.)
0 Shashi Bhushan, Praveen Kumar Misra and Sachin Kumar adav 0 00 0 0 00 00 0 x y 0 00 0 0 00 00 0 p y (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, e 0 0 3 00 00 7 0 0 4 0 0 5 e with E( e ) 0, i 3,4,5,6,7 i 00 00 6 Then under the estimated values of characterizing scalars and, the proposed class of ratio estimators becomes x p y x p Substituting the values from (.) and simplifying, we get e e e e e O( e ) 0 To the first order of approximation it can be written as e0 e e e e On squaring and taking expectation both the sides, we get MSE E f C f C C C C C C C C n nn P P P P P ( f f R ) C M (On substituting the values from (3.)) n nn. P min MSE (3.3) (3.4) This is same as minimum value of the mean square error of the proposed class of estimator. COMPARATIVE STUD Consider the following estimator (i) Sample Mean (SRSWOR) Ŷ y vs. From (.3) and (3.4)
On The Class of Double Sampling Ratio Estimators Using Auxiliary Information nn. P MSE( ) M f R C 0 (4.) (ii) Double sampling ratio estimator using auxiliary variable i.e. vs. From (.4) and (3.4), we have nn. P MSE( ) M f C C C R C fnn C C C C C R. PC f ( C C ) ( R ) C 0 nn. P As R. P (4.) (iii) Double sampling ratio estimator using auxiliary attribute i.e. 3 vs. From (.5) and (3.4), we have 3 nn P P P. P MSE( ) M f C C C R C fnn CP PC CP PC PC R. PC f ( C C ) ( R ) C 0 nn P P. P P (4.3) (iv) As R. P P Bahl and Tuteja (99) double sampling exponential ratio estimator using auxiliary variable i.e. 4 vs. From (.6) and (3.4), we have C MSE( 4 ) M fnn C C R. PC 4 C fnn C C C C R. PC 4 C fnn ( C ) ( R. P ) C 0 (4.4) (v) As R. P Following Bahl and Tuteja (99), Nirmala Sawan s (00) Proposed double sampling exponential ratio estimator using auxiliary attribute 5 vs. From (.7) and (3.4), we have
Shashi Bhushan, Praveen Kumar Misra and Sachin Kumar adav Cp MSE( 5 ) M fnn CP PC y R. PC 4 CP fnn PC CP PC PC R. PC 4 CP fnn ( PC ) ( R. P P ) C 0 (4.5) As R. P P EMPIRICAL STUD Population : [Source: William G. Cochran (977), Page-34] = Food Cost, = Family Income, = Family of size more than 3 = 7.40, = 7.55, P = 0.5, P = 0.388, P = -0.53, C = 0.46, C = 0.369, C P = 0.985, R = 0.49879, n = 5, n = 5, N = 33. P = 0.5, Population II: [Source: Advance Data from Vital and Health Statistics, Number 347, October 7, 004(CDC)] = Height of the people, = Weight of the people, = Sex of the people = 40.8, = 39.63, P = 0.50, = 0.973, P = 0.07, P = 0.073, C = 0.9654, C = 0.48337, R = 0.94673, n = 8, n = 5, N = 36. P C P =.04, Table 5.: PRE of the various estimators with respect to the sample mean ESTIMATOR 3 4 5 PRE Population I Population II 00.00 00.00 03.5 47.6.5 4.49 04.65 307.63 64.65 6.0.44 38.34
On The Class of Double Sampling Ratio Estimators Using Auxiliary Information 3 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. ACKNOWLEDGEMENT The financial aid rendered by UGC is gratefully acknowledged. REFERENCES [] Bhushan, S. (03). Improved Sampling Strategies in Finite Population. Scholars Press, Germany. [] Bhushan S. (0). Some Efficient Sampling Strategies 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 Sampling Strategies, 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 Publishing. [5] Bhushan S. and Pandey A. (00). Modified Sampling Strategies using Correlation Coefficient for Estimating Population 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 P. 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 Product 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, P.C. (996): A note on estimation of mean with known population proportion of an auxiliary character. Jour. Ind. Soc. Agr. Stat., 48(), 5-58. [] Sukhatme, P.V. andsukhatme, B.V. (970): Sampling theory of surveys with applications. Iowa State University Press, 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.
4 Shashi Bhushan, Praveen Kumar Misra and Sachin Kumar adav