A Generalized Class of Unbiased Estimators for Population Mean Using Auxiliary Information on an Attribute and an Auxiliary Variable
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1 Iteratioal Joural of Computatioal ad Applied Mathematics. ISSN Volume, Number 07, pp. -8 Research Idia ublicatios A Geeralized Class of Ubiased Estimators for opulatio Mea Usi Auxiliary Iformatio o a Attribute ad a Auxiliary Variable Shashi Bhusha, ravee umar Misra ad Sachi umar adav Departmet of Mathematics ad Statistics, Dr. Shakutala Misra Natioal Rehabilitatio Uiversity, Idia. Departmet of Statistics, Uiversity of Luckow, Idia. Abstract I this paper, we have proposed a eeralized class of estimators usi auxiliary iformatio i both the form attribute ad variable. We also propose a class of ubiased estimators usi the Jack-kife techique. The bias, mea square error MSE ad miimum MSE uder certai optimum coditios for the proposed class are obtaied. Fially, some cocludi remarks are made clearly demostrati that some importat classes of estimators are special cases of the proposed study. eywords: opulatio mea, auxiliary iformatio, Bias, Mea square error, Jack-kife. INTRODUCTION We all kow that i may situatios auxiliary iformatio is helpful i icreasi the efficiecy of the estimator i sampli. This auxiliary iformatio may be i the form of a variable or a attribute which is kow i advace or collected while the survey is bei coducted without icreasi the cost of the survey or at a little cost to improve the efficiecy of the estimators. For example Heiht is closely related with Weiht ad it is also differet for male ad female showi that sex is a helpful attribute while deali with heiht, Amout of milk produced by the cow depeds o the breed as well as o the diet, yield of the crop depeds o the variety of the seed ad maure used as well etc. i such situatio we ca take the advataes of auxiliary iformatio available i the form of variable ad attribute to improve the precisio of the estimator. May statisticia have utilizes the auxiliary iformatio either i the form of variable or i the form of attribute at a time. Here we have cosidered both type of iformatio i our study.
2 Shashi Bhusha, ravee umar Misra ad Sachi umar adav Cosider the followi otatios = Study variable = Auxiliary variable = Auxiliary attribute N = Size of populatio N i = opulatio mea of study variable N i N i = opulatio mea of auxiliary variable N i N i = opulatio mea of auxiliary attribute N i S N i N i S S N i N i N i N i = opulatio variace of study variable = opulatio variace of auxiliary variable = opulatio variace of auxiliary attribute Assumi that a simple radom sample of size is draw from the populatio o the study variable ad auxiliary characters ad. Deote the followi sample iformatio as y i = Sample mea of study variable i x i = Sample mea of auxiliary variable i p = Sample mea of auxiliary attribute i i With this available iformatio we propose a class of estimators for mea value of study variable as y x, p.
3 A Geeralized Class of Ubiased Estimators for opulatio Mea 3 Where x, p is the fuctio of x ad p such that i, ii The fuctio x, p is cotiuous ad bouded i the closed iterval R of real lie. iii The first ad secod order partial derivatives of the fuctio x, p are exist ad are cotiuous ad bouded i R. It is to be metioed that proposed class of estimators is very lare. Cosider the followi member of this class. Followi Olki 958 y x p. Followi Sih 967 x p y 3. Followi Shukla 966 ad joh y x p 4. Followi Sahai et al x p y, where, where 5. Followi Mohaty ad attaaik y x p 6. Followi Mohaty ad attaaik x p y 7. Followi Tuteja ad Bahl 99 7 x p y
4 4 Shashi Bhusha, ravee umar Misra ad Sachi umar adav 8. Followi Tuteja ad Bahl 99 8 y x p 9. Followi Naik ad Gupta 996 ad Abu Dayyeh x p y 0. Followi Sahai ad Ray x p y. Followi Walsh 970 y x x p. Followi Srivastava 97 x p yexplo lo 3. Followi Srivastava 97 3 x p yexp 4. Followi Srivastava 97 4 y. x p exp lo exp lo 5. roposed Estimator by us i our paper 5 x p yexp exp x p p. May such estimators utilizi the iformatio i both the forms auxiliary variable ad attribute ca be costructed as the member of this class. BIAS AND MSE OF THE ROOSED GENERALISED CLASS To obtai the bias of the proposed class of estimators, we further assume that the third order partial derivatives of x, p exist ad are bouded ad cotiues. The taki Taylor s expasio about the poit, up to third order terms, we et
5 A Geeralized Class of Ubiased Estimators for opulatio Mea 5, x x, p p,! x xx, x p xp, y! p pp, 3 x xxx, 3 x p xxp, 3 3! 3 x p xpp, p ppp,. where x, p,0 ad,,, ad, abc are the first order, secod order ad third order partial derivatives of the fuctio x, p at the poit, about a, a ad b ad a, b ad c respectively i eeral terms. a ab Now let us take, y e0, x e p e With the assumptio that E e E e E e 0. 0 Ad the results ive by Sukhatme ad Sukhatme 997 E e0 fs, E e fs, E e fs, E e0e f S S E e0e fs S, E ee fs S where f N.3 Substituti the values from. i. ad electi the terms of ei ' s i 0,, havi powers reater tha two, we et,, e x e p! e xx, ee xp, e pp,.4! e0 e0e x, e0e p,!
6 6 Shashi Bhusha, ravee umar Misra ad Sachi umar adav Taki expectatio ad substituti the results from.3, we et f E S S S S xx, xp, pp, f S S, S S, x p Showi that is a biased estimator of ad its bias is ive by Bias E f Bias S S S S xx, xp, pp, f S S, S S, x p B say.5 The mea square error MSE of is ive by usi.4 MSE E,, x p E e0 e e To the first order of approximatio E e0 E e x, E e p, E ee x, p, E e0e x, E e0e p, Substituti the results from.3, we et MSE f where S S x, S p, S Sx, p, S S x, S S p, fm Say.6 M S, S, S S,, x p x p S S, S S, S x p
7 A Geeralized Class of Ubiased Estimators for opulatio Mea 7 This will be miimum whe S, x p S S, S.7.8 Ad its miimum value uder the optimum values of the characteristic scalars is ive by mi MSE f R S.9. ROOSED GENERALISED CLASS OF UNBIASED JAC-NIFE ESTIMATORS Cosider a simple radom sample of size = m draw from the fiite populatio of size N by SRSWOR ad split it ito two sub-samples of size m each. Let us defie the followi estimators y x, p, y x, p, 3 y x, p 3. where y, y, x, x ad p, p are the respective sample meas of the study variable, auxiliary variable ad auxiliary attribute based o sample ad each of size m ad yx, ad p are the respective mea of the study variable, auxiliary variable ad auxiliary attribute based o the etire sample. From the equatio.4 fm Bias S xx, S Sxp, S pp, f S S, S S, m x p fm Bias S xx, S Sxp, S pp, f S S, S S, m x p f Bias S S S S 3 m xx, xp, pp, f S S, S S, m x p say 3.
8 8 Shashi Bhusha, ravee umar Misra ad Sachi umar adav Let us defie 3.3 This is a alterative estimator of populatio mea, so f Bias S S S S m xx, xp, pp, f S S, S S, m x p say 3.4 Defie the ratio m f m f m N N m N m m N Usi we proposed the followi eeralized class of ubiased Jack-ife estimators as 3 Taki expectatio both the sides, we et E E 3 E Substituti the values from equatio 3. ad 3.4 we et 3.5 fm fm S xx, S Sxp, S pp, E fm fm S S x, S S p, Substituti the value of, we et E 3.6 Showi that, the proposed eeralized class of estimators is ubiased. Mea square error of the proposed class is defied as MSE E E 3
9 A Geeralized Class of Ubiased Estimators for opulatio Mea E E E From equatio.6, we have E MSE f M 3.8 m ad from equatio 3.3 E E E E E E 4 aai from equatio.6 E MSE i i 3.9 Taki f M ; i, 3.0 m i i i y e, x e, p e i i 0 with the assumptio that i i i 0 i E e E e E e 0; i, 3. Usi.4 we ca write i i i,, i i i e xx e e xp e x, e p, i e pp, i i i i i e0 e0 e x, e0 e p,! So, to the first order of approximatio e x, e p, e 0 E E e x, e p, e 0
10 0 Shashi Bhusha, ravee umar Misra ad Sachi umar adav E e e x, E e e x, p, E e e p, x, E e e p, E e e0 x, E e e0 p, E e 0 e0 E e0 e x, E e0 e p, Substituti the followi results ive by the Sukhatme ad Sukhatme 997 E e0 e0 S. E e e S, E e e S N N N E e e N, E e0 e S S N, E e0 e S S N 0 S S E e0 e S S N, E e e S S N, E e e S S N We et, E M N 3. utti the values from equatio 3.0 ad 3. i equatio 3.9, we et E fmm M fmm N Now cosider 3 3 E E E E 3 3 Usi., to the first order of terms we ca write e, e, e E E i x p 0 3 i ;, i i i e x, e p, e 0 3.4
11 A Geeralized Class of Ubiased Estimators for opulatio Mea i i i E ee x, E ee x, p, E ee p, x, i E ee p, E e e, E e e, E e e, E e e, E e e i 0 0 i i i i 0 x 0 p 0 x 0 p Substituti the followi results ive by the Sukhatme ad Sukhatme 997 E e e f S, E e e i f S, E e e i f S 0 0 m m m E e e f S S, E e i e f S S 0 m 0 m E e e f S S 0 m E e e f S S, i E e e f S S, E e i e f S S ad f 0 m m m N m m We et S x, S Sx, p, 3 i E, f S m p S S x, S S p, S f M 3.5 m Substituti the results from 3.5 to 3.4, we et 3 E fmm fmm m f M 3.6 Now substituti the results from 3.8, 3.3 ad 3.6 i equatio 3.7, we et MSE f M f M f M m m m f m M f S x, S Sx, p, S p, m N S S x, S S p, S m M S x, S Sx, p, S S p, N S S x, S S p, 3.7
12 Shashi Bhusha, ravee umar Misra ad Sachi umar adav Which is the same as that of.6 but proposed class of Jack-ife estimator has a advatae over the proposed class of estimators i the sese of ubiasedess. Also the optimum values of the parameters are the same as ive i.7 ad.8 ad thereby ivi the same miimum MSE as that of.9 i.e mi MSE f R S. mi MSE 3.8 BIAS AND MSE OF THE ROOSED MEMBERS OF THE CLASS The bias ad MSE of some proposed members of the class defied i equatio. are ive i the followi table Table 4.: Derivatives of x, p of proposed members of the eeralized class Estimator x,, xx,,, p pp xp
13 A Geeralized Class of Ubiased Estimators for opulatio Mea Table 4.: Bias of the proposed members of the eeralized class Estimator Bias f C C C C C C f C C C C 3 f S S S S C S C S 4 f S S S S 5 f C C C C C C CC 6 C C C C C C f CC
14 4 Shashi Bhusha, ravee umar Misra ad Sachi umar adav 7 f CC C C CC C C 8 f CC C C CC C C 9 0 f f C C C C C C CC C C C C C C CC C C CC f C C C C f C C C C C C CC 3 f C C CC C C C C 4 f C C C C C C 5 C C C C C C C C f CC 4
15 A Geeralized Class of Ubiased Estimators for opulatio Mea 5 Table 4.3: MSE of the proposed members of the eeralized class Estimator MSE f f f f f f f f C C C CC C C C C C C C CC C C C C C S S S S C S C S S S S S S S S S S C C C CC C C C C C C C CC C C C C C C C CC C C C C C C C CC C C C C
16 6 Shashi Bhusha, ravee umar Misra ad Sachi umar adav f f f f f f f C C C CC C C C C C C C CC C C C C C C C CC C C C C C C C CC C C C C C C C CC C C C C C C C C C C C C C C C C C 4 4 CC CC C CONCLUSION The above theoretical study establishes the importace of the proposed eeralized class of jack-kife estimator due to its wider applicability ad emphasize its superiority over the proposed eeralized class of estimators i the sese of ubiasedess. Also it attais miimum value of MSE uder the optimum values of the characterizi parameters. mi MSE f R S..
17 A Geeralized Class of Ubiased Estimators for opulatio Mea 7 ACNOWLEDGEMENT The fiacial aid redered by UGC is ratefully ackowleded. REFERENCES [] Bhusha, S. 03. Improved Sampli Strateies i Fiite opulatio. Scholars ress, Germay. [] Bhusha S. 0. Some Efficiet Sampli Strateies based o Ratio Type Estimator, Electroic Joural of Applied Statistical Aalysis, 5, [3] Bhusha S. ad atara, S. 00. O Classes of Ubiased Sampli Strateies, Joural of Reliability ad Statistical Studies, 3, [4] Bhusha, S. ad umar S. 06. Recet advaces i Applied Statistics ad its applicatios. LA ublishi. [5] Bhusha S. ad adey A. 00. Modified Sampli Strateies usi Correlatio Coefficiet for Estimati opulatio Mea, Joural of Statistical Research of Ira, 7, - 3. [6] Bhusha S., Sih, R.. ad atara, S Improved Estimatio uder Midzuo Lahiri Se-type Sampli Scheme, Joural of Reliability ad Statistical Studies,, [7] Bhusha S., Masalda R. N. ad Gupta.. 0. Improved Sampli Strateies based o Modified Ratio Estimator, Iteratioal Joural of Aricultural ad Statistical Scieces, 7, [8] Bahli, S. Ad Tuteja, R.. 99: Ratio ad roduct type expoetial estimator, Iformatio ad Optimizatio scieces, Vol.II, I, [9] Gray H. L. ad Schucay W.R. 97. The Geeralized Jack-kife Statistic, Marcel Dekker, New ork. [0] Naik, V.D. ad Gupta,.C. 996: A ote o estimatio of mea with kow populatio proportio of a auxiliary character. Jour. Id. Soc. Ar. Stat., 48, [] Sukhatme,.V. adsukhatme, B.V. 970: Sampli theory of surveys with applicatios. Iowa State Uiversity ress, Ames, U.S.A. [] Srivastava, S A estimator usi auxiliary iformatio. Calcutta Statistical Associatio. Bulleti, 6, -3. [3] Cochra, W. G. 977, Sampli Techiques, 3rd ed. New ork: Joh Wiley ad Sos.
18 8 Shashi Bhusha, ravee umar Misra ad Sachi umar adav
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