Almost Unbiased Estimator for Estimating Population Mean Using Known Value of Some Population Parameter(s)

Transcription

1 Almos Unbiased Esimaor for Esimaing Populaion Mean Using Known Value of Some Populaion Parameers Rajesh Singh Deparmen of Saisics, Banaras Hindu Universi U.P., India Mukesh Kumar Deparmen of Saisics, Banaras Hindu Universi U.P., India Florenin Smarandache Deparmen of Mahemaics, Universi of New Meico, Gallup, USA Absrac In his paper we have proposed an almos unbiased using known value of some populaion parameers. Various eising s are shown paricular members of he proposed. Under simple random sampling wihou replacemen SRSWOR scheme he epressions for bias and mean suare error MSE are derived. The sud is eended o he wo phase sampling. Empirical sud is carried ou o demonsrae he superiori of he proposed. Ke words: Auiliar informaion, bias, mean suare error, unbiased, wo phase sampling. 1. Inroducion Consider a finie populaion U = U 1, U, U N of N unis. Le and sand for he variable under sud and auiliar variable respecivel. Le i, i, i=1,,.., n denoe he values of he unis included in a sample s n of size n drawn b simple random sampling wihou replacemen SRSWOR. The auiliar informaion has been used in improving he precision of he esimae of a parameer See Cochran 1977, Sukhame e. al and he referenc es cied here in. Ou of man mehods, raio and produc mehods of esimaion are good illusraions in his cone. In order o have a surve esimae of he populaion mean characer, assuming he knowledge of he populaion mean characer, he well-known raio is of he sud of he auiliar Bahl and Tueja 1991 suggesed an eponenial raio pe as re ep Pak.j.sa.oper.res. Vol.IV No. 008 pp63-76

2 Rajesh Singh, Mukesh Kumar, Florenin Smarandache Several auhors have used prior value of cerain populaion parameers o find more precise esimaes. Sisodia and Dwivedi 1981, Sen 1978 and Upadhaa and Singh 1984 used he known coefficien of variaion CV of he auiliar characer for esimaing populaion mean of a sud characer in raio mehod of esimaion. The use of prior value of coefficien of kurosis in esimaing he populaion variance of sud characer was firs made b Singh e. al Laer used b Singh and Kakaran 1993 in he esimaion of populaion mean of sud characer. Singh and Tailor 003 proposed a modified raio b using he known value of correlaion coefficien. Kadilar and Cingi 006, Khosnevisan e. al. 007, Singh e. al. 007 Singh and Kumar 009 and Singh e. al. 009 have suggesed modified raio s b using differen pairs of known value of populaion parameers. In his paper under SRSWOR, we have proposed almos unbiased for esimaing Y.. Almos unbiased raio pe Suppose, Such ha denoes he se of all possible raio pe s for esimaing he populaion mean. B definiion he se is a linear varie, if.1 for, R. i=0, 1, denoes he saisical consans and R denoes he se of real numbers. To obain he bias and MSE of Y1, 1 such ha e 0, e 1 E e 0 =E e 1 =0. Ee0 f 1 C, e1 f 1 C 1 1 f 1, n N S C Y, S C, S 1 N 1 N i1, we wrie E, Ee e f C C. Y. i i S 1 N 1, C K, C N i Y i1 S, S S S 1 N 1, N i i1 64 Pak.j.sa.oper.res. Vol.IV No. 008 pp63-76

3 Almos Unbiased Esimaor for Esimaing Populaion Mean using Known Value of Some Populaion Parameers Epressing in erms of e s, we have.3. Epanding he righ hand side of.3 and reaining erms up o second order of e s, we have.4..5 Taking epecaions of boh side of.4 and hen subracing from boh side, we ge he bias of he, up o he firs order of approimaion as From.4, we have Suaring boh sides of.9 and hen aking epecaions, we ge MSE of he, up o he firs order of approimaion, as This is minimum when Puing his value of in.10, we ge he minimum MSE of as min..1 which is same as ha of radiional linear regression from.5 and.11, we have.13 From. and.13, we have onl wo euaions in hree unknowns. I is no possible o find he uniue values for In order o ge uniue values for we shall impose he linear resricion.14 Euaions.,.11 and.14 can be wrien as in he mari form as =.15 Pak.j.sa.oper.res. Vol.IV No. 008 pp

4 Rajesh Singh, Mukesh Kumar, Florenin Smarandache Using.15,we ge uniue values of i=0,1, as Use of hese i=0,1, remove he bias up o erms of order n -1 a Produc pe s Suppose,, such ha,, Q denoes he se of all possible produc pe s for esimaing he populaion mean varie if. B definiion, he se Q is linear 3.1 for, 3. Epressing. i=0,1, denoes he saisical consans. in erms of e s, we have 3.3 Epanding he righ hand side of 3.3 and reaining erms up o second power of e s, we have Taking epecaions of boh sides of 3.4 and hen subracing from boh sides, we ge he bias of he, up o he firs order of approimaion as Pak.j.sa.oper.res. Vol.IV No. 008 pp63-76

5 Almos Unbiased Esimaor for Esimaing Populaion Mean using Known Value of Some Populaion Parameers Bias epression for he s and is given b From 3.4, we have 3.9 Suaring boh he sides of 3.9 and hen aking epecaions, we ge MSE of he, up o he firs order of approimaion, as which is minimum for Puing his value of =-k in 3.10,we ge he minimum MSE of as min. 3.1 which is same as ha of radiional linear regression. From 3.5 and 3.11, we have 3.13 From 3. and 3.13, we have onl wo euaions in hree unknowns. I is no possible o find he uniue values for i s, i=0,1,. In order o ge uniue values of i s, we shall impose he linear resricion Euaions 3.,3.13 and 3.14 can be wrien in he mari form as 3.14 = 3.15 Solving 3.15, we ge he uniue values of i s i=0,1, as Pak.j.sa.oper.res. Vol.IV No. 008 pp

6 Rajesh Singh, Mukesh Kumar, Florenin Smarandache 3.17 Use of hese i s i=0,1, remove he bias up o erms of order on -1 a 3.1. In Appendi A we have lised some of he imporan known s of he populaion mean, which can be obained b suiable choice of consans, and a and b. 4. Proposed s in wo phase sampling When is unknown, i is someimes esimaed from a preliminar large sample of size n on which onl he characerisic is measured for deails see Singh e. al Then a second phase sample of size n n < n is drawn on which boh and characerisics are measured. Le denoe he sample mean of based on firs phase sample of size n, and be he sample means of and respecivel based on second phase of size n. In wo phase sampling he will ake he following form 4.1 for, 4. and To obain he bias and MSE of Y1, 1 e 0, e 1 such ha E e 0 = E e 1 = Ee 1 =0., we wrie Ee0 f 1 C, Ee1 f 1 C, e' 1 f C E e 0e f C C E 1 1 E e0e' 1 fcc f 1 1 n 1 N Ee, 1e' 1 f C 1 f ' n 1 N Following he procedure menioned in secion and 3, we ge bias and MSE of wd as Pak.j.sa.oper.res. Vol.IV No. 008 pp63-76

7 Almos Unbiased Esimaor for Esimaing Populaion Mean using Known Value of Some Populaion Parameers =. 4.4 MSE is minimum, when 4.5 Puing his value of in 4.4, we ge he minimum MSE of as min. 4.6 This is same as ha of radiional wo phase linear regression. The bias epression for he s and is respecivel given b From 4. and 4.5, we have onl wo euaions in hree unknowns. I is no possible o find he uniue values for w id s i=0,1,. In order o ge uniue values of, we shall impose linear resricion 4.9 Euaions 4., 4.5 and 4.9 can be wrien in mari form, as = 4.10 Solving 4.10, we ge he uniue values of as Use of hese w id s i=0,1, will remove he bias up o erms of order On -1 a 4.1. Pak.j.sa.oper.res. Vol.IV No. 008 pp

8 Rajesh Singh, Mukesh Kumar, Florenin Smarandache The wrien in 3.1, in wo phase sampling, will ake following form 4.13 For, i=0,1, denoes he saisical consans. The s and are and Following he procedure of secion 4, we ge he uniue values of id s i=0,1, as The minimum MSE of is given b. 5. Empirical sud For empirical sud we use he daa ses earlier used b Kadilar and Cingi 006 populaion 1 and Khosnevisan e. al. 007 populaion o verif he heoreical resuls. 70 Pak.j.sa.oper.res. Vol.IV No. 008 pp63-76

9 Almos Unbiased Esimaor for Esimaing Populaion Mean using Known Value of Some Populaion Parameers Daa saisics Populaion N n Populaion Populaion Table 5.1: Values of i s and i s Populaion 1 Populaion The percen relaive efficiencies PRE of various s of and presened in Table 5. below. are compued Table 5.: PRE of differen s of Esimaor PRE Pop I Esimaor PRE Pop II w op op Pak.j.sa.oper.res. Vol.IV No. 008 pp

10 Rajesh Singh, Mukesh Kumar, Florenin Smarandache In order o see he performance of he suggesed s in wo phase sampling we use he daa se of Murh 1967 Populaion III and Seel and Torrie 1960 Populaion IV. Populaion N n n Populaion Populaion Table 5.3 : The values of and Populaion III Populaion IV The percen relaive efficiencies of various s of are compued and presened in Table 5.4 below. in wo phase sampling Table 5.4 : PRE of differen s of in wo phase sampling Esimaor PRE Populaion I PRE Populaion II op Conclusion From heoreical discussion and empirical sud we conclude ha he proposed s under opimum condiions perform beer han oher s considered in he aricle. The relaive efficiencies of various s are lised in Table 5. and Pak.j.sa.oper.res. Vol.IV No. 008 pp63-76

11 Almos Unbiased Esimaor for Esimaing Populaion Mean using Known Value of Some Populaion Parameers Appendi A Table A.1: Some members of he proposed famil of s - a b Raio Esimaor corresponding o i=0,1, Produc Esimaor corresponding o i=0,1, The mean per uni The mean per uni 1 o The usual raio The usual produc C C Sisodia and Dwivedi Singh e. al. 004 C C Upadhaa and Singh 1999 C C Pande and Dube Singh e. al. 004 C 4 C Upadhaa and Singh C C Upadhaa and Singh 1999 C 5 C Upadhaa and Singh Singh and Tailor ep Bahl and Tueja 1991 Singh and Tailor ep Bahl and Tueja 1991 Pak.j.sa.oper.res. Vol.IV No. 008 pp

12 Rajesh Singh, Mukesh Kumar, Florenin Smarandache Pak.j.sa.oper.res. Vol.IV No. 008 pp a b Raio Esimaor corresponding o i=0,1, Produc Esimaor corresponding o i=0,1, ep 8 Singh e. al. 007 ep 8 Singh e. al C ep Singh e.al. 007 C ep 9 Singh e.al ep 10 Singh e. al.007 ep 10 Singh e. al C ep 11 Singh e. al. 007 C ep 11 Singh e. al ep 1 C C Singh e. al. 007 ep 1 C C Singh e. al ep 13 C C Singh e. al. 007 ep 13 C C Singh e. al C ep 14 Singh e. al. 007 C ep 14 Singh e. al ep 15 Singh e. al. 007 ep 15 Singh e. al ep 16 Singh e. al. 007 ep 16 Singh e. al. 007

13 Almos Unbiased Esimaor for Esimaing Populaion Mean using Known Value of Some Populaion Parameers In addiion o above s a large number of s can also be generaed from he proposed s jus b puing differen values of consans, and a and b. Acknowledgemens The second auhor Mukesh Kumar is hankful o UGC, New Delhi, India, for providing financial assisance. The auhors would like o hank he referee for his consrucive suggesions on an earlier draf of he paper. References 1. Bahl, S. and Tueja, R.K. 1991: Raio and produc pe eponenial. Infrm. and Opim. Sci., II, I, Kadilar, C. and Cingi, H. 006: New raio s using correlaion coefficien. InerSa, Khoshnevisan,M., Singh, R., Chauhan, P., Sawan, N. and Smarandache, F.007: A general famil of s for esimaing populaion means using known value of some populaion parameers. Far eas journal of saisics,, Pande, B.N. and Dube, V. 1988: Modified produc esim aor using coefficien of variaion of auiliar variae. Assam Saisical Rev., Sen, A.R. 1978: Esimaion of he populaion mean when he coefficien of variaion is known. Commun. Sa. Theo. Meh. A 7, Singh, H.P. and Kakran, M.S. 1993: A modified raio using coefficien of variaion of auiliar characer. Unpublished. 7. Singh, H.P. and Tailor, R. 003: Use of known correlaion coefficien in esimaing he finie populaion mean. Saisics in Transiion, 6, 4, Singh, H.P., Tailor, R, Tailor, R. and Kakran, M.S. 004: An improved of populaion mean using power ransformaion. Jor. Ind. Soc. Agri. Sais. 58, Singh, J., Pande, B.N. and Hirano, K. 1973: On he uilizaion of a known coefficien of kurosis in he esimaion procedure of variance. Ann. Ins. Sa. Mah., 5, Singh, R. and Kumar, M. 009 : A noe on ransformaions on auiliar variable in surve sampling. o appear in MASA 11. Singh, R., Cauhan, P., Sawan, N. and Smarandache, F. 007: Auiliar informaion and a prior values in consrucion of improved s. Renaissance High press, USA. Pak.j.sa.oper.res. Vol.IV No. 008 pp

14 Rajesh Singh, Mukesh Kumar, Florenin Smarandache 1. Singh, R., Singh, J. and Smarandache, F. 009: Sudies in saisical inference, sampling echniues and demograph. ProQues Informaion & Learning, USA. 13. Sisodia, B. V. S. and Dwivedi, V.K. 1981: A modified raio using coefficien of variaion of auiliar variable. Jour. Ind. Soc. Agri. Sa., 33, Sukhame, P.V., Sukhame, B.V., Sukhame, S. and Ashok, C. 1984: Sampling heor of surves wih applicaions. Iowa Sae Universi Press, USA. 15. Upadhaa, L.N. and Singh, H.P. 1984: On he esimaion of he populaion mean wih known coefficien of variaion. Biom. Jour., 68, Pak.j.sa.oper.res. Vol.IV No. 008 pp63-76