An Unbiased Estimator For Population Mean Using An Attribute And An Auxiliary Variable
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1 Advance in Dynamical Sytem and Application. ISS , Volume, umber, (07 pp Reearch India Publication An Unbiaed Etimator For Population Mean Uing An Attribute And An Auxiliary Variable Shahi Bhuhan, Praveen Kumar Mira and Sachin adav Department of Mathematic and Statitic, Dr. Shakuntala Mira ational Rehabilitation Univerity, India. Department of Statitic, Lucknow Univerity, Lucknow (U.P., India. Abtract In thi paper, we have conidered a cla of exponential type ratio etimator uing the auxiliary information in both the form attribute and variable. Here wepropoe an improved cla of etimator uing Jack-Knife technique.it ha been hown that the propoed Jack-Knife etimator i unbiaed and ha leer minimum mean quare error under the optimum value of characterizing parameter a compared to ome commonly ued etimator available in the literature. An empirical tudy i included a an illutration. Keyword: Exponential Type Ratio Etimator, Jack-Knife technique, Unbiaedne and Mean Square Error.. ITRODUCTIO In many ituation auxiliary information i ued to improve the preciion of theetimator. Thi auxiliary information may be in the form of a variable or an attribute or both. For example - Height i different for male and female which how that ex i a helpful attribute while dealing with height and alo height i related with the weight which i a variable; amount of milk produced by cow depend on the breed a well a on the diet; yield of wheat crop depend on the variety of wheat and manure a well etc.in uch ituation, we can take the advantage of the available information on variable and attribute to increae the efficiency of the etimator. Conider the following notation = Study variable, X = Auxiliary variable, = Auxiliary attribute, = Size of population
2 30 Shahi Bhuhan, Praveen Kumar Miraand Sachin adav i i = Population mean of tudy variable, auxiliary variable i i Xi i X = Population mean of P = Population mean of auxiliary attribute, S ( = Population variance of tudy variable S X X X ( i i =Population variance of auxiliary i variable, SP ( i P =Population variance of auxiliary attribute i Auming that a imple random ample of ize n i drawn from the population on the tudy variable and auxiliary character X and. Denote the following ample information a y variable n Xi n i n i n i i = Sample mean of tudy n i x = Sample mean of auxiliary variable, p = Sample mean of auxiliary n i attribute n =Sample variance of the tudy variable, X ( X i X = Sample n ( i n i n i variance of the auxiliary variable n ( i p =Sample variance of the auxiliary n i attribute (. ow, conider ome etimator of population mean developed in the pat uing the auxiliary information (i General etimator of mean in cae of SRSWOR (ii Sample mean ywith MSE( fn C (. Ratio etimator uing auxiliary variable X y With MSE( fn ( C CX X CyCX (.3 x (iii (iv aik and Gupta(996 Ratio etimator uing auxiliary attribute P 3 y With MSE( 3 fn ( C CP PC ycp (.4 p Bahl and Tuteja(99Exponential Ratio type etimator uing auxiliary variable
3 An Unbiaed Etimator For Population Mean Uing An Attribute. 3 (v (vi (vii (viii (ix 4 yexp X x X x With MSE( 4 fn ( C CX X CyCX (.5 4 Following Bahl and Tuteja(99, Exponential Ratio etimator given by irmalasawan (00 uing auxiliary attribute 5 yexp P p P p With MSE( 5 fn ( C CP PC ycp (.6 4 Product etimator uing auxiliary variable x 6 y With MSE( 6 fn ( C CX X CyCX (.7 X aik and Gupta (996 Product etimator uing auxiliary attribute p 7 y With MSE( 7 fn ( C CP PC ycp (.8 P Bahl and Tuteja (99 Exponential product type etimator uing auxiliary variable 8 yexp x X x X With MSE( 8 fn ( C CX X CyCX (.9 4 Following Bahl and Tuteja (99, Exponential product etimator given by irmala Sawan (00 uing auxiliary attribute 9 yexp p P P p With MSE( 9 fn ( C CP PC ycp (.0 4 Following Bahl and Tuteja (99 and Singh (967 we propoe a cla of etimator auming that the auxiliary population mean and auxiliary population proportion are known X x P p p yexp exp X x P p (. it may be noted that the etimator given in (., (.5 and (.6 are the pecial cae of the propoed tudy uing no auxiliary information, one auxiliary variable and one auxiliary attribute repectively. In order to obtain the Bia and Mean quare error (MSE, let u denote y e ( 0, x X e (, p P( e And the reult given in Sukhatme and Sukhatme (99 E( e0 fnc, E( e fncx, E( e fncp, E( e0e fnx C CX E( e0 E( e E( e 0 With (. E( e0e fnpc CP, E( ee fnxpcx CP (.3
4 3 Shahi Bhuhan, Praveen Kumar Miraand Sachin adav Subtituting the value from (.in (., we get e e e e e e e e 0 ee e e 0 p e (.4 Taking expectation both the ide,we get E( e E( e E( e E( e Ee ( E( p E( e ( E( e E e e 0 E( e ( e0 E e e Subtituting the value from(. and(.3, we get (.5 Bia( E( p p CX CP CX CP XCX C PCPC XPCX C P f n = f n A (Say (.6 C C C C C C C C C C Where A X P X P X X P P XP X P Squaring (.4up to the firt order ofapproximation e e 0 e Taking expectation, we get (.7 E( e E( e E( ee MSE( p E( e0 E( e0e E( e0e 4 4 Subtituting the value from (.3, we get CX CP XPCX C P fn C XCX C PCPC 4 4 fn B (.8 Where C C C C B C C C C C 4 4 X P XP X P X X P P The minimum value of MSE i obtained if optimum value of and are ( C opt( X P PX ( XP CX ( C P X XP, opt( ( XP CP (.9
5 An Unbiaed Etimator For Population Mean Uing An Attribute. 33 And the minimum mean quare error under the optimiing value of the characteritic calar i given by p n. XP min MSE( f ( R C (.0. THE PROPOSED JACK-KIFE ESTIMATOR Conider a random ample of ize n = m drawn from the finite population of ize by SRSWOR and plit it into two random ub-ample of ize m each. Define the following etimator ( ( ( ( X x P p y exp exp ( ( X x P p, ( ( ( ( X x P p y exp exp ( ( X x P p (3 X x P p yexp exp X x P p (. ( ( ( ( where y, y, x, x and p ( (, p are the repective ample mean of the tudy variable, auxiliary variable and auxiliary attribute baed on ample and each of ize m and yxand, p are the repective mean of the tudy variable, auxiliary variable and auxiliary attribute baed on the entire ample. From (.6, we have Bia ( ( m Let u define ( (3 f A, Bia( f ma, Bia( f n A K ( ( (Say (. (.3 Thi i an alternative etimator of population mean. So Bia( f ma K (Say (.4 K fn ( m Define the ratio K K f ( m Uing K, we propoe the Jack-knife etimator m (3 K K (.5 (.6 (3 E( KE( Taking expectation both the ide, we get E ( K Subtituting the value from (. and (.4, ( m ( m E( ( m ( m (.7
6 34 Shahi Bhuhan, Praveen Kumar Miraand Sachin adav Showing that the propoed Jack-Knife etimator i an unbiaed etimator of population mean. Mean quare error of the propoed etimator i defined a MSE( E (3 K (3 (3 E E( K E( KE( ( (.8From K ( K (3 (3 (.8, we have E ( MSE ( m B (.9 ( ( Conider E( E ( ( ( ( 4 E E E ( i ( i From (.8 E( MSE( B ; i, m Taking ( i ( i ( i ( i ( i ( i 0 ( ( ( y ( e, x X ( e, p P( e ; i, With E( e i E( e i E( e i 0; i, Uing (. we can write 0 (.0 (. (. ( ( ( ( i i i i ( i ( i e e e e e e0 ( i ( i ( i ( i ( i ( i ( i ( i e e e0 e e0 e e 8 4 (.3 To the firt order of approximation ( ( ( ( ( ( ( e e ( e e E( ( E e0 e0 ( ( ( ( ( ( ( ( ( ( E( e0 e0 E( e0 e E( e0 e E( e0 e E( e0 e ( ( ( ( ( ( ( ( E( e e E( e e E( e e E( e e Uing the reult given in Sukhatme and Sukhatme (997 ( ( ( ( ( ( ( ( E( e0 e0 C, E( e e CX, E( e e CP, E( e0 e X C CX ( ( E( e0 e X C CX, ( ( E( e0 e PC CP, ( ( E( e0 e PC CP
7 An Unbiaed Etimator For Population Mean Uing An Attribute. 35 E e e, ( ( E( e e XPCX CP ( ( ( XPCX CP ( ( We get E( ( B (.4 Putting the value from (. and (.4 in (.0, we get E( MSE( B B 4 m B m (.5 ow conider ( ( (3 (3 E( ( E( (3 ( (3 ( E E (.6 To the firt order of term ( i ( i (3 ( i e e ( i e e E( ( E e0 e0 ; i, ( i ( i ( i ( i ( i E( e0e0 E( e0e E( e0e E( ee 0 E( ee0 ( i ( i ( i ( i E( ee E( ee E( ee E( ee ; i, Subtituting the following reult given by Sukhatme and Sukhatme (997 E( e e f C ( i 0 0 n E( e e f C C ( i 0 ( i, E( e e f C n X X E( e e f C C We get n XP X P ( i n X, ( i 0, E( e e f C ( i n P E( e e f C C n P P,, E( e e f C C ( i 0 ( i 0 E( e e f C C n P P n X X, E( e e f C C ( i n XP X P (3 ( i E( ( B ; i, m (.7 Putting the value from (.7 in (.6, we get (3 E( ( B B m m m B (.8
8 36 Shahi Bhuhan, Praveen Kumar Miraand Sachin adav Subtituting the value from (.9, (.5, and (.8 in (.8, we get MSE ( K K B fn B (.9 ( m K The minimum value of (.9 i obtained when ( C ( C opt(, opt( X P PX P X XP ( XP CX ( XP CP The minimum value of mean quare error, under the optimum value of the characteritic calar i given by min MSE ( R C n. XP ˆ min MSE min MSE( M ( ay (.0 p Thi i ame a that of minimum mean quare error of the family of etimator P 3. COMPARISO WITH THE AVAILABLE ESTIMATORS Conider the following etimator of the tudy variable (i Sample mean (SRSWOR y v. ( MSE M R C From (. and (.0 we have ( 0 (3. (ii Ratio etimator uing auxiliary variable. XP v. ( From (.3 and (.0, we have MSE( M 0 Since R. ( 3.. XP X (iii Ratio etimator uing auxiliary attribute 3 v. ( (iv (v From (.4 and (.0, we have MSE( 3 M 0A R 4. (3.3. XP P Bahl and Tuteja(99Exponential Ratio etimator uing auxiliary variable v. ( From (.5 and (.0 we have MSE( 4 M 0 A R. (3.4. XP X Following Bahl and Tuteja (99Exponential Ratio etimator uing auxiliary variable, propoed etimator of irmalasawan (00 uing auxiliary attribute 5 v. ( From (.6 and (.0, we have MSE 5 ( M 0A R. (3.5. XP P
9 An Unbiaed Etimator For Population Mean Uing An Attribute EMPIRICAL STUD Population : [Source: William G. Cochran (977, Sampling Technique, 3 rd Edition ew ork: John Wiley And Son,Page-34] = Food Cot, X = Family Income,ɸ= Family of ize more than , X 7.55, P 0.5, C 0.46, C 0.369, C 0.985, X P 0.5, 0.388, 0.53, n, n 6, 33 X P XP Population : [Source: Advance Data from Vital and Health Statitic, umber 347, October 7, 004(CDC] = Height of the people, X = Weight of the People, ɸ = Sex of the people n X P C C C 36,, 40.8, 39.63, 0.50, , X , P , , , R X P XP. XP Table 4.: PRE of variou etimator with repect to ample mean Etimator Population I Population II PRE COCLUSIO The comparative tudy of the propoed Jack-Knife etimator etablihe it uperiority in the ene of unbiaed and minimum mean quare of error over ample mean, ratio etimator and exponential ratio etimator uing auxiliary variable and auxiliary attribute under the optimum condition.
10 38 Shahi Bhuhan, Praveen Kumar Miraand Sachin adav REFERECES [] Abu Dayyeh W. A., AhmedM.S., AhmedR.A. and Muttlak H. A. (003 :Some etimator of a finite population mean uing auxiliary information, Applied Mathematic and Computation, 39, [] Bahl S. and Tuteja R. K. (99: Ratio and Product type exponential etimator, Information and Optimization cience, Vol.XII, I, [3] Bhuhan, S. (03. Improved Sampling Strategie in Finite Population. Scholar Pre, Germany. [4] Bhuhan S. (0. Some Efficient Sampling Strategie baed on Ratio Type Etimator, Electronic Journal of Applied Statitical Analyi, 5(, [5] Bhuhan S., Gupta R. and Pandey S. K. (05. Some log-type clae of etimator uing auxiliary information, International Journal of Agricultural and Statitical Science, (, [6] Bhuhan S. and Katara, S. (00. On Clae of Unbiaed Sampling Strategie, Journal of Reliability and Statitical Studie, 3(, [7] Bhuhan, S. and Kumar S. (06. Recent advance in Applied Statitic and it application. LAP Publihing. [8] Bhuhan S., Pandey A. and Singh R.K. (009 Improved Clae of Regreion Type Etimator ; International Journal of Agricultural and Statitical Science (ISS: , 5(, [9] Bhuhan S. and Pandey A. (00. Modified Sampling Strategie uing Correlation Coefficient for Etimating Population Mean, Journal of Statitical Reearch of Iran, 7(, - 3. [0] Bhuhan S., Singh, R. K. and Katara, S. (009. Improved Etimation under Midzuno Lahiri Sen-type Sampling Scheme, Journal of Reliability and Statitical Studie, (, [] Bhuhan S., Maaldan R.. and Gupta P. K. (0. Improved Sampling Strategie baed on Modified Ratio Etimator, International Journal of Agricultural and Statitical Science, 7(, [] Gray H. L. and Schucany W. R. (97: The Generalized Jack-knife Statitic, Marcel Dekker, ew ork.
11 An Unbiaed Etimator For Population Mean Uing An Attribute. 39 [3] aik V. D. and Gupta P. C. (996: A note on etimation of mean with known population proportion of an auxiliary character. Jour. Ind. Soc. Agr. Stat., 48(, [4] Quenouille M. H. (956: ote on bia in etimation, Biometrika, 43, p [5] S.K.Srivatava (967: An etimator uing auxiliary information in ample urvey, Calcutta Statitical Aociation Bulletin, 6, -3. [6] Sukhatme P. V. and Sukhatme B. V. (970: Sampling theory of urvey with application. Iowa State Univerity Pre, Ame, U.S.A.
12 40 Shahi Bhuhan, Praveen Kumar Miraand Sachin adav
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