Improvement in Estimating the Population Mean Using Exponential Estimator in Simple Random Sampling

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1 Bulletn of Statstcs & Economcs Autumn 009; Volume 3; Number A09; Bull. Stat. Econ. ISSN ; Copyrght 009 by BSE CESER Improvement n Estmatng the Populaton Mean Usng Eponental Estmator n Smple Random Samplng Rajesh Sngh 1 Pankaj Chauhan Nrmala Sawan and Florentn Smarandache 3 1 Department of StatstcsBanaras Hndu Unversty(U.P.) Inda (E-mal: rsnghstat@yahoo.com) School of Statstcs DAVV Indore (M.P.) Inda 3 Department of Mathematcs Unversty of New Meco Gallup USA (E-mal: smarand@unm.edu) ABSTRACT Ths study proposes some eponental rato-type estmators for estmatng the populaton mean of the varable under study usng known values of certan populaton parameter(s). Under smple random samplng wthout replacement (SRSWOR) scheme mean square error (MSE) equatons of all proposed estmators are obtaned and compared wth each other. The theoretcal results are supported by a numercal llustraton. Keywords: Eponental estmator aulary varable smple random samplng effcency. Mathematcs Subject Classfcaton Number: 6D05 Journal of Economc Lterature (JEL) Classfcaton Number: C83 1. INTRODUCTION Consder a fnte populaton U U 1 U... UN of N untes. Let y and stand for the varable under study and aulary varable respectvely. Let ( y ) 1... n denote the values of the unts ncluded n a sample s n of sze n drawn by smple random samplng wthout replacement (SRSWOR). The aulary nformaton has been used n mprovng the precson of the estmate of a parameter (see Cochran (1977) Sukhatme and Sukhatme (1970) and the reference cted there n). Out of many methods rato and product methods of estmaton are good llustratons n ths contet. In order to have a survey estmate of the populaton mean Y of the study character y assumng the knowledge of the populaton mean X of the aulary character the well-known rato estmator s X t r y (1.1) Bahl and Tuteja (1991) suggested an eponental rato type estmator as X t 1 yep (1.) X Several authors have used pror value of certan populaton parameter(s) to fnd more precse estmates. Ssodya and Dwved (1981) Sen (1978) and Upadhyaya and Sngh (1984) used the known coeffcent of varaton (CV) of the aulary character for estmatng populaton mean of a study character n rato method of estmaton. The use of pror value of coeffcent of kurtoss n estmatng

2 14 Bulletn of Statstcs & Economcs the populaton varance of study character y was frst made by Sngh et.al.(1973). Later used by Sngh and Kakaran (1993) n the estmaton of populaton mean of study character. Sngh and Talor (003) proposed a modfed rato estmator by usng the known value of correlaton coeffcent. Kadlar and Cng (006(a)) and Khoshnevsan et.al.(007) have suggested modfed rato estmators by usng dfferent pars of known value of populaton parameter(s). In ths paper under SRSWOR we have suggested mproved eponental rato-type estmators for estmatng populaton mean usng some known value of populaton parameter(s).. THE SUGGESTED ESTIMATOR Followng Kadlar and Cng (006(a)) and Khoshnevsanet.al. (007) we defne modfed eponental estmator for estmatng Y as (ax b) (a b) t yep (.1) (ax b) (a b) where a( 0) b are ether real numbers or the functons of the known parameters of the aulary varable such as coeffcent of varaton ( C ) coeffcent of kurtoss ( ()) and correlaton coeffcent (). To obtan the bas and MSE of t we wrte y Y(1 e0) X(1 e1) such that E(e0) E(e1) 0 and E(e0) f1c y E(e1 ) f1c E(e0e1) f1c yc where N n Sy S f 1 Cy C. nn Y X Epressng t n terms of e s we have ax ax(1 e 1) t Y(1 e0)ep ax b ax(1 e1) 1 t Y(1 e0 )ep e1(1 e1) (.) ax where. (ax b) Epandng the rght hand sde of (.) and retanng terms up to second power of e s we have t Y(1 e e e e e ) (.3) Takng epectatons of both sdes of (.3) and then subtractng Y from both sdes we get the bas of the estmator t up to the frst order of appromaton as B(t) f1y( Cy CyC ) (.4) From (.3) we have

3 Autumn 009; Bull. Stat. Econ.; Volume 3; Number A09 15 ( 0 1 t Y) Y(e e ) (.5) Squarng both sdes of (.5) and then takng epectaton we get MSE of the estmator t up to the frst order of appromaton as MSE(t) f1y (Cy C CyCc ) (.6) 3. SOME MEMBERS OF THE SUGGESTED ESTIMATOR t The followng scheme presents some estmators of the populaton mean whch can be obtaned by sutable choce of constants a and b. Estmator a Values of b t 0 y The mean per unt estmator X t 1 yep X Bahl and Tuteja (1991) estmator X t yep X () X t3 yep X Cc () X 1 t 4 yep X ()(X ) t5 yep ()(X ) C C (X ) t6 yep C (X ) () t 7 C (X ) y ep C (X ) (X ) t8 yep (X ) C ()(X ) t9 yep ()(X ) t 10 (X ) y ep (X ) () 1 () C C C () C () C () In addton to above estmators a large number of estmators can also be generated from the proposed estmator t at (.1) just by puttng dfferent values of a and b.

4 16 Bulletn of Statstcs & Economcs It s observed that the epressons of the frst order of appromaton of bas and MSE of the gven member of the famly can be obtaned by mere substtutng the values of a and b n (.4) and (.6) respectvely. 4. MODIFIED ESTIMATORS Followng Kadlar and Cng (006(b)) we propose modfed estmators combnng estmator t 1 and t ( ) as follows t t (1 ) t ( ) (4.1) 1 where s a real constant to be determned such that the MSE of t s mnmum and t ( ) are estmators lsted n secton 3. Followng the procedure of secton () we get the MSE of t to the frst order of appromaton as where MSE (t ) f1y Cy C CyC (4.) X X () X 3 X C X 4 X ) CX 6 C X () ()X 5 ()X C ) CX C X 7 X 8 X C ()X () 9 X 10. () Mnmzaton of (4.) wth respect to yelds ts optmum value as (K ) opt (say) (4.3) (1 ) Cy where K. C Substtuton of (4.3) n (4.10) gves optmum estmator t o (say) wth mnmum MSE as mn MSE(t ) f1y Cy(1 ) MSE(t ) o (4.4) The mn MSE(t ) at (4.4) s same as that of the appromate varance of the usual lnear regresson estmator.

5 Autumn 009; Bull. Stat. Econ.; Volume 3; Number A EFFICIENCY COMPARISON It s well known that under SRSWOR the varance of the sample mean s 1Y Cy Var(y) f (5.1) we frst compare the MSE of the proposed estmators gven n (.6) wth the varance of the sample mean we have the followng condton: K (5.) When ths condton s satsfed proposed estmators are more effcent than the sample mean. Net we compare the MSE of proposed estmators t ( ) n (4.4) wth the MSE of estmators lsted n secton 3. We obtan the followng condton y ( C C ) (5.3) We can nfer that all proposed estmators t ( ) are more effcent than estmators proposed n secton 3 n all condtons because the condton gven n (5.1) s always satsfed. 6. NUMERICAL ILLUSTRATION To llustrate the performance of varous estmators of Y we consder the data gven n Murthy (1967 pg-6). The varates are: y : Output : number of workers X = Y = C y = C = = () = We have computed the percent relatve effcency (PRE) of dfferent estmators of Y wth respect to usual estmator y and compled n table 6.1: TABLE 6.1: PRE OF DIFFERENT ESTIMATORS OF Y WITH RESPECT TO y Estmator PRE y 100 t t t t t t t t t t to

6 18 Bulletn of Statstcs & Economcs 7. CONCLUSION We have developed some eponental rato type estmators usng some known value of the populaton parameter(s) lsted n secton 3. We have also suggested modfed estmators t ( ). From table 6.1 we conclude that the proposed estmators are better than Bahl and Tuteja (1991) estmator t 1. Also the modfed estmator t ( ) under optmum condton performs better than the estmators proposed and lsted n secton 3 and than the Bahl and Tuteja (1991) estmator t 1. The choce of the estmator manly depends upon the avalablty of nformaton about known values of the parameter(s) ( C () etc.). REFERENCES Bahl S. and Tuteja R.K. (1991): Rato and Product type eponental estmator Informaton and Optmzaton scences Vol.XII I Cochran W.G.(1977): Samplng technques. Thrd U.S. edton. Wley eastern lmted 35. Kadlar C. and Cng H. (006(a)): A new rato estmator usng correlaton coeffcent. Inter Stat Kadlar C. and Cng H. (006(b)): Improvement n estmatng the populaton mean n smple random samplng. Appled Mathematcs Letters Khoshnevsan M. Sngh R. Chauhan P. Sawan N. and Smarandache F. (007): A general famly of estmators for estmatng populaton mean usng known value of some populaton parameter(s). Far East Journal of Theoretcal Statstcs () Murthy M.N.(1967): Samplng Theory and Methods Statstcal Publshng Socety Calcutta. Sen A.R.(1978) : Estmaton of the populaton mean when the coeffcent of varaton s known. Comm. Stat.-Theory Methods A Sngh H.P. and Kakran M.S. (1993): A modfed rato estmator usng coeffcent of varaton of aulary character. Unpublshed. Sngh H.P. and Talor R. (003): Use of known correlaton coeffcent n estmatng the fnte populaton mean. Statstcs n Transton Sngh J. Pandey B.N. and Hrano K. (1973): On the utlzaton of a known coeffcent of kurtoss n the estmaton procedure of varance. Ann. Inst. Stat. Math Ssodya B.V.S. and Dwved V.K. (1981): A modfed rato estmator usng coeffcent of varaton of aulary varable. Jour. Ind. Soc. Agr. Stat Sukhatme P.V. and Sukhatme B.V.(1970) : Samplng theory of surveys wth applcatons. Iowa State Unversty Press Ames U.S.A. Upadhyaya L.N. and Sngh H.P. (1984): On the estmaton of the populaton mean wth known coeffcent of varaton. Bometrcal Journal