ESTIMATION OF FINITE POPULATION MEAN USING KNOWN CORRELATION COEFFICIENT BETWEEN AUXILIARY CHARACTERS

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1 STATISTICA, anno LXV, n. 4, 005 ESTIMATION OF FINITE POPULATION MEAN USING KNOWN CORRELATION COEFFICIENT BETWEEN AUXILIAR CHARACTERS. INTRODUCTION Let U { U, U,..., U N } be a finite population of N units. Suppose two auxiliary variables X X are observed on Ui ( i,,..., N ), where X is positively X is negatively correlated with the study variable. A simple rom sample without replacement (SRSWOR) of size n with n < N, is drawn from the population U to estimate N N yi / N, the population mean of, when the i i population means X xi / N X x i / N of X X are respectively, known. For estimating, Singh (967) suggested a ratio-cum-product estimator N i X x y x X () where n y y / n, x x / n x x / n. i i n i i Wide applicability of the estimator has led many authors to suggest unbiased versions of with their properties, for instance, see Sahoo Swain (980), Biradar Singh (99-93) Tracy et al. (998). Sahai Sahai (985) Singh (987 b) have mentioned that the past association with experimental material might provide a close guess for the correlation coefficient between study variate auxiliary character X i.e. can be guessed quite accurately. Recently, Singh Tailor (003) have utilized the information on suggested a modified ratio estimator for with its properties. Further Singh n i i

2 408 Singh (984) advocated that the correlation coefficient xx between auxiliary variates X X may be known in many practical situations hence utilizing the known value of xx suggested a class of estimators for population variance y of with its properties. This led authors to suggest modified ratiocum-product estimator using xx with its properties. A jackknife version of the suggested estimator is also given its properties are studied. An empirical study is carried out in support of the proposed estimator.. SUGGESTED RATIO-CUM-PRODUCT ESTIMATOR Assuming that the correlation coefficient xx between auxiliary characters X X is known, we define a ratio-cum-product estimator of as X x. () y x xx X xx To the first degree of approximation, the bias mean square error (MSE) of the proposed estimator are respectively, given by B( ) [ C ( K ) C ( K K )] (3) x x MSE( ) [ C C ( K ) C { ( K K )}], (4) where y x x K ( C / C ), y x K ( C / C ), y x K ( C / C ), i Xi /( Xi x ) x, i (,) ; n N, C y S y /, C S / X,( i,) ; S /( S S ),( i,) ; i i i y x y x i i i N y i j S ( y ) /( N ), N xi ij i j S ( x X ) /( N ),( i,)

3 Estimation of finite population mean using known correlation coefficient between auxiliary characters 409 N S ( y )( x X )/( N ),( i,). i i ij i j When no auxiliary information is used the estimator reduces to the conventional unbiased estimator y. If the information only on auxiliary variate X is used, then the estimator tends to the usual ratio estimator yr y( X / x ). On the other h if the information is available on auxiliary variate X only, reduces to the usual product estimator yp y( x / X ). It is well known that sample mean y is an unbiased estimator of its variance under SRSWOR sampling scheme is given by V ( y) C y. (5) To the first degree of approximation, the biases MSEs of are respectively given by y R, y P R x B( y ) C ( K ), (6) P x B( y ) C K, (7) B( ) [ C ( K ) C ( K K )], (8) x y y x R y x MSE( y ) [ C C ( K )], (9) P y x MSE( y ) [ C C ( K )], (0) MSE( ) [ C C ( K ) C { ( K K )}]. () y x y y x 3. EFFICIENC COMPARISIONS It follows from (4), (5), (9), (0) () that (i) MSE( y ) V( y ) if R K ()

4 40 (ii) MSE( y ) V ( y) if P K (3) (iii) MSE( ) V ( y) if x y y x [ C ( K ) C { ( K K )}] 0 which is always true if K K K (iv) MSE( ) V( y) if x y [ C ( K ) C { ( K K )}] 0 (4) which always holds if K K K (5) (v) MSE( ) MSE( y R ) if K K xx (6) (vi) MSE( ) MSE( y P ) if K K xx, (7) where K ( C / C ). (vii) MSE( ) MSE( y R ) if K C x K K C x [( ){ ( )} { ( )} ] 0 which is always true if K ( ) (viii) MSE( ) MSE( y P ) if K K K K C x K C x [ { ( )} ( ){( ) } ] 0 (8)

5 Estimation of finite population mean using known correlation coefficient between auxiliary characters 4 which always holds if K K K ( ) (9) (ix) MSE( ) MSE( ) if x x [ C ( ){ K ( )} C { K ( ) which is always true if ( )( K }] 0 ( ) K xx ( ) ( ) K K. (0) ( ) Now combining (), (6) (0) we get that the proposed estimator is more efficient than y, y R Singh s (967) estimator i.e. MSE( ) MSE( ) MSE( y ) V( y) if R ( ) K xx ( ) ( ) K K K xx. () ( ) Further combining (0), (7) (3) we obtained that the suggested estimator is more efficient than y, y P Singh s (967) estimator i.e. MSE( ) MSE( ) MSE( y ) V( y) if P K ( ) K K xx ( ) ( ) K. ( ) () It is to be noted that the suggested estimator is biased. In some applications, bias is a major disadvantage. Keeping this in view, we have discussed the unbiasedness of the proposed estimator, using the technique suggested by Quenouille (956) known as Jack-knife technique, proposed a family of almost unbiased estimators with its properties.

6 4 4. FAMIL OF UNBIASED ESTIMATORS OF POPULATION MEAN USING JACKKNIFE TECHNIQUE Let a simple rom sample of size n = gm drawn without replacement split at rom into g sub-samples, each of size m. Then we define the Jack-knife ratio-cum-product estimator for population mean as where (3) g X x j J y j g j x j xx X xx j sample means group th y ( n y m y )/( n m) y j j ij i ij x ( n x m x )/( n m), i, ; are the based on a sample of (n-m) units obtained by omitting the x ( i,; j,,..., g ) are the sample means based on ij the j sub samples of size m = n/g. The bias of, to terms of order n, can be easily obtained as J th j ( N n m) B( J ) [ C x ( K ) C x ( K Kx )] x. (4) N( n m) From (3) (4) we have B( ) ( N n)( n m) B( ) n( N n m) J (5) ( N n)( n m) ( ) ( ) n( N n m) or B B J ( N n)( n m) B( ) B( ) 0 n( N n m) or J or ( ) ( ) 0 (6) B B J for any scalar, we have ( N n)( n m). (7) n( N n m) From (6), we have

7 Estimation of finite population mean using known correlation coefficient between auxiliary characters 43 ( ) ( E E ) 0 J or or ( ) ( E y E y ) 0 J E[ y{ ( ) }]. J Hence, the general family of almost unbiased ratio-cum-product estimators of as [ y{ ( )} ] (8) u J see Singh (987 a). Remark 4.. For 0, u yields the usual unbiased estimator y while ( ), gives an almost unbiased estimator for as u ( N n m) X x g y N x xx X xx ( N n)( g ) g y X x j j Ng j x j xx X xx which is Jack-knifed version of the proposed estimator. Many other almost unbiased estimator from (8) can be generated by putting suitable values of. (9) 5. SEARCH OF AN OPTIMUM ESTIMATOR IN FAMIL u AT (8) The family of almost unbiased estimator u at (8) can be expressed as y y, (30) u where by y [( ) y y ] y. The variance of J u is given V( ) V( y ) V( y ) Cov( y, y ) (3) u which is minimized for

8 44 Cov( y, y )/ V ( y ). (3) Substitution of (3) in (3) yields minimum variance of u as { Cov( y, y )} min. V( u ) V( y) V( y ) 0 V( y)( ), (33) where 0 is the correlation coefficient between y y. From (33) it is immediate that min. V( ) V( y). u To obtain the explicit expression of the variance of, we write the following results to terms of order n, as J J u MSE( ) Cov(, ) MSE( ) (34) Cov( y, ) Cov( y, ) [ C C C C C ] (35) J y y x y x where MSE( ) is given by (4). Now using the results from (4), (5) (35) into (3) we get the variance of to the terms of order n as u V( ) [ C ( ) ( C C C C ) u y x x C yc x C yc x ( )( )] (36) which is minimized for ( C yc x C yc x ) opt ( )( C x C x C x C x ). (37) Substitution of opt in u yields the optimum estimator u( opt ) (say). Thus the resulting minimum variance of u is given by

9 Estimation of finite population mean using known correlation coefficient between auxiliary characters 45 V C V ( C x C x ) u y u( opt ) ( C x C x C x C x ) min. ( ) ( ) From (4), () (38) we have ( C x C ) x V y V u C y ( C x C x x ) x C x C x ( ) min. ( ) 0 MSE( ) min. V( ) u ( ) Cx Cx xx C x C x C ycx C ycx ( C x C x C x C x ) Thus from (39) (40) we have the following inequalities: u. (38) (39) 0. (40) min. V( ) V ( y) (4) min. V( ) MSE( ) (4) u which follows that u with opt is more efficient than y. When u does not coincide with V( ) V( y) if opt then from (5) (36) we note that either or 0 opt opt 0 (43) It is observed from () (36) that MSE( u ) MSE( ) if B ( B AC ) B ( B AC ) ( ) A ( ) A, (44) x x A ( C C C C ),

10 46 y x y x B ( C C C C ), x y y x C [ C ( K ) C { ( K K )}]. We also note from (4) (36) that the estimator u is better than ( or ) if u either or opt ( ) ( ) opt ( ) ( ) (45) The optimum value opt of can be obtained quite accurately through past data or experience. 6. EMPIRICAL STUD To observe the relative performance of different estimators of, we consider a natural population data set given in Steel Torrie (960, p.8). The population description is given below: y : Log of leaf burn in sec. x : Potassiam percentage x : Clorine percentage. The required population values are: , C , X , y C 0.95, X , C , x x 0.794, N=30, , n=6, , g=. The percentage relative efficiencies (PREs) of various estimators of with respect to y have been computed presented in Table. Estimator y y R TABLE Percent relative efficiencies of different estimators of with respect to y yp u u ( ) with PRE (, y) opt.975

11 Estimation of finite population mean using known correlation coefficient between auxiliary characters 47 with Table clearly indicates that the suggested estimators ( or u ) = opt, are more efficient than usual unbiased estimator y, ratio estimator y R, product estimator y P, Singh s (967) ratio-cum-product estimator with considerable gain in efficiency. u 7. CONCLUDING REMARKS Usually information regarding correlation coefficient xx between the two auxiliary variates X X is known or can made known to the experimenter through past studies or with the familiarity of experimental material. When x x is known an improved version u of Singh s (967) estimator is suggested with its properties. Using Jack-knife technique envisaged by Quenouille (956), a family of unbiased estimators u is also proposed. A large number of unbiased estimators can be generated from u. Asymptotically optimum estimator (AOE) in the family of estimators u is identified with its variance formula. It is shown that the suggested family of estimators u is more efficient than y at optimum conditions. Empirical study also suggests that the suggested estimators ( or u ) u with opt are better than y, y R, y P Singh s (967) estimator. Thus we conclude that the proposed estimators ( or u ) are to be preferred in practice. u School of Studies in Statistics Vikram University, Ujjain, India Department of Educational Surveys Data Processing New Delhi, India HOUSILA P. SINGH RAJESH TAILOR ACKNOWLEDGEMENTS Authors are thankful to the referee for his valuable suggestions regarding improvement of the paper.

12 48 REFERENCES R.S. BIRADAR, H.P. SINGH (99-93), Almost unbiased ratio-cum-product estimators, Aligarh Journal of Statistics,, pp M.H. QUENOUILLE (956), Notes on bias in estimation, Biometrika, 43, pp A. SAHAI, A. SAHAI (985), On efficient use of auxiliary information, Journal of Statistical Planning Inference,, pp L.N. SAHOO, A.K.P.C. SWAIN (980), Unbiased ratio-cum-product estimator, Sankhya, C, 4, pp H.P. SINGH (987 a), Class of almost unbiased ratio product-type estimators for finite population mean applying Quenouilles method, Journal of Indian Society of Agriculture Statistics, 39, pp H.P. SINGH (987 b), On the estimation of population mean when the correlation coefficient is known in two phase sampling, Assam Statistical Review,, pp. 7-. H.P. SINGH, R. TAILOR (003), Use of known correlation coefficient in estimating the finite population mean, Statistics in Transition, 6, pp M.P. SINGH (967), Ratio-cum-product method of estimation, Metrika,, pp R.K. SINGH, G. SINGH (984), A class of estimators for population variance using information on two auxiliary variates, Aligarh Journal of Statistics, (3-4), pp D.S. TRAC, H.P. SINGH, R. SINGH (998), A class of almost unbiased estimators for finite population mean using two auxiliary variables, Biometrical Journal, 40, pp R.G.D. STEEL, J.H. TORRIE (960), Principles Procedures of Statistics, Mc Graw Hill Book Co. RIASSUNTO Stima della media di un popolazione finita con coefficiente di correlazione tra caratteri ausiliari noto Il contributo propone uno stimatore ratio-cum-product modificato della media di una popolazione finita di una variabile oggetto di studio sfrutto il coefficiente di correlazione noto tra due caratteri ausiliari X e X. Si ottiene uno stimatore ratio-cumproduct quasi corretto attraverso la tecnica Jacknife del tipo previsto da Quenille (956). In seguito vengono esaminati con un esempio numerico i meriti dello stimatore proposto. SUMMAR Estimation of finite population mean using known correlation coefficient between auxiliary characters This paper proposes a modified ratio-cum-product estimator of finite population mean of the study variate using known correlation coefficient between two auxiliary characters X X. An almost unbiased ratio-cum-product estimator has also been obtained by using Jackknife technique envisaged by Quenouille (956). The merits of the proposed estimator are examined through a numerical illustration.