Some Ratio and Product Estimators Using Known Value of Population Parameters

Transcription

1 Rajes Sing Deparmen of Maemaics, SRM Universi Deli NCR, Sonepa- 9, India Sacin Malik Deparmen of Saisics, Banaras Hindu Universi Varanasi-, India Florenin Smarandace Deparmen of Maemaics, Universi of New Meico Gallup, NM 87, USA Some Raio and Produc Esimaors Using Known Value of Populaion Parameers Publised in: Sacin Malik, Neeraj Kumar, Florenin Smarandace (Ediors USES OF SAMPLING TECHNIQUES & INVENTORY CONTROL WITH CAPACITY CONSTRAINTS Pons Ediions, Brussels, Belgium, ISBN pp. 9-9

2 Absrac In e presen aricle, we proposed a famil of esimaors for esimaing populaion means using known value of some populaion parameers. Kosnevisan e al. [] proposed a general famil of esimaors for esimaing populaion means using known value of some populaion parameer(s wic afer some subsiuions led o some raio and produc esimaors iniiall proposed b Sisodia and Dwivedi [], Sing and Tailor [], Pande and Dube [], Adewara e al. [], adav and Kadilar []. Te presen famil of esimaors provides us significan improvemen over previous families in eor. An empirical sud is carried ou o judge e meri of e proposed esimaor. Kewords: Raio Esimaor, Produc Esimaor, Populaion Parameer, Efficienc, Mean Square Error.. Inroducion Te problem of esimaing e populaion mean in e presence of an auiliar variable as been widel discussed in finie populaion sampling lieraure. Raio, produc and difference meods of esimaion are good eamples in is cone. Raio meod of esimaion is quie effecive wen ere is ig posiive correlaion beween sud and auiliar variables. On e oer and, if correlaion is negaive (ig, e produc meod of esimaion can be emploed efficienl. 9

3 Uses of Sampling Tecniques & Invenor Conrol wi Capaci Consrains In recen ears, a number of researc papers on raio-pe, eponenial raio-pe and regression-pe esimaors ave appeared, based on differen pes of ransformaions. Some imporan conribuions in is area are due o Sing and Tailor [], Sabbir and Gupa [7,8], Kadilar and Cingi [9,], Kosnevisan e. al.(7. Kosnevisan e al. [] defined eir famil of esimaors as a b [ α(a b ( α(a b g ] were a(, b are eier real numbers or e funcions of e known parameers of e auiliar variable suc as sandard deviaion ( σ, Coefficien of Variaion ( C, Skewness ( β (, Kurosis ( β ( and Correlaion Coefficien (ρ. (i. Wen α=, a==b, g=, we ave e mean per uni esimaor, wi MSE( N n (. ( Y C (ii. Wen α=, a=, b=, g=, we ave e usual raio esimaor, ( wi N n MSE( ( Y (C C ρρ C (. (iii. Wen α=, a=, b=, g=-, we ave e usual produc esimaor, ( wi MSE( N n ( Y (C C ρρ C (. (iv. Wen α=, a=, b= C, g=, we ave Sisodia and Dwivedi [] raio esimaor, C ( wi C N n MSE( ( Y (C ( C ( ρρ C (. C C (v. Wen α=, a=, b= C, g=-

4 Some Raio and Produc Esimaors Using Known Value of Populaion Parameers C we ave Pande and Dube [] produc esimaor, ( wi C MSE( N n ( Y (C ( C C ( C ρρ C (. ρ (vi. Wen α=, a=, b=, g=, we ave Sing and Talor [] raio esimaor, ( ρ wi N n MSE( ( Y (C ( C ( ρρ C (. ρ ρ (vii. Wen α=, a=, b= ρ, g=-, we ave Sing and Talor [] produc esimaor, ρ ( wi ρ MSE( N n ( Y (C ( C ( ρρ C (.7 ρ ρ Tere are oer raio and produc esimaors from ese families a are no inferred ere bu is paper will be limied o ose ones a made use of Coefficien of Variaion ( C and Correlaion Coefficien ( ρ since e conclusion obained ere can also be inferred on all oers a made use of oer populaion parameers suc as e sandard deviaion ( σ, Skewness ( β ( and Kurosis ( ( in e same famil. β. On e Modified Raio and Produc Esimaors. Adoping Adewara (, Adewara e al. ( proposed e following esimaors as (, (. (, (. C (, (. C

5 Uses of Sampling Tecniques & Invenor Conrol wi Capaci Consrains C (, (. C ρ ( and (. ρ ρ (, (. ρ Were and are e sample means of e auiliar variables and variable of ineres e o be drawn wi e relaionsips (i Srivenkaaramana and Srina []. f ( f and (ii. Y f ( f. Te Mean Square Errors of ese esimaors i, i =,,, are as follows: n (i. MSE( ( MSE( (.7 N n n (ii. MSE( ( MSE( N n (.8 n (iii. MSE( ( MSE( (.9 N n n (iv. MSE( ( MSE( N n (. n (v. MSE( ( MSE( (. N n n (vi. MSE( ( MSE( (. N n Following Adewara e al [], Yadav and Kadilar [] proposed some improved raio and produc esimaors for esimaing e populaion mean of e sud variable as follows k (, (. η η k (, (.

6 Some Raio and Produc Esimaors Using Known Value of Populaion Parameers η η C k (, (. C C k (, (. C ρ k ( ρ η (.7 η ρ k (, (.8 ρ Te mean square error of ese esimaors η i, i=,,, are as follows MSE(η MSE(η MSE(η MSE(η MSE(η MSE(η Y Y Y Y Y Y k λc k k λc k k λc k k λc k λc k k λc k k λc k k ν λc ν k k λc k k λc k ν λc ν k k λc k k λc k k ν λc ν k k λc k k λc k ν λc ν k k λc k (.9 (. (. (. (. (. Were, λ N - n, n,c N - n S Y,C S,C S, ν Y C, ν C S andρ S S And λc λc λc,k C λ C λ λc C λ C λ λc k, λν C ν λc λνλc ν C λ ν C λ λc k,k ν Cλ νc λ λc λν C ν λc λν λc ν C λ ν C λ λc k,andk ν Cλ ν C λ λc

7 Uses of Sampling Tecniques & Invenor Conrol wi Capaci Consrains. Te Proposed famil of esimaors Following Malik Sing [], we define e following class of esimaors for populaion mean Y as M α ψ δ ω μ ω μ m m ep ψ δ ω μ ω μ β (. Were m and mare suiabl cosen consans. ψ, δ, ω, and μ are eier real numbers or funcion of known parameers of e auiliar variable. Te scalar α and β akes values + and - for raio and produc pe esimaors respecivel. To obain e MSE, le us define Y, e e suc a E(e i, i=, and E(e λc, E(e λc, E(ee λρc C epressing equaion (. in erms of e s and reaining onl erms up o second degree of e s, we ave M m Y e m e ψ δ ψ e δ α ωe ep ω μ - ωe β m Y e m e R e -α epβre R e R e α α e αe my β ee α ee βe αβ e β e 8 β e e m e α e β e (.

8 Some Raio and Produc Esimaors Using Known Value of Populaion Parameers were, ψ, R ψ δ R ω ω μ Subracing Y from bo e sides of (., we ave Y m Y- e L e L e L e e m e L e Y M (. were, L L L L βr αr αα R αβ RR β R α R β R α R β R 8 β R Squaring bo sides of (. and neglecing erms of e s aving power greaer an wo, we ave MSE Y m T m T m m T m T m T M (. were, T Y λ C L λc T λ C T YL λc LλC λρc C T Y LλC LλρC C T YL λc L λρc C L λc L λρc C minimizaion of (. wi respec o m and m ields opimum values as T T TT, m TT T m T T T T T T T. Empirical Sud: Populaion I: Kadilar and Cingi [9] N =, n =, ρ.8, C., C., Y.9 and 7.7

9 Uses of Sampling Tecniques & Invenor Conrol wi Capaci Consrains Populaion II: Maddala [] N =, n =, ρ.8, C.78, C.98, Y 7.7 and 7.. Resuls: Table.: Sowing e esimaes obained for bo e Kosnevisan e al. [] esimaors and Adewara e al. [] esimaors Esimaor Populaion I ( ρ Populaion II ( ρ

10 Some Raio and Produc Esimaors Using Known Value of Populaion Parameers Table.: Sowing e esimaes obained for Yadav and Kadilar [] esimaors Esimaor Populaion I ( ρ Populaion II ( ρ η.7 - η -.7 η 8. - η -.7 η η -.7 Table.: MSE of suggesed esimaors wi differen values of consans MSE m m α β ψ δ ω μ esimaor PopI PopII C C ρ ρ m - - η.7 - m η -.7 7

11 Uses of Sampling Tecniques & Invenor Conrol wi Capaci Consrains m C - - η 8. - m - C - - η -.7 m ρ - - η m - ρ - - η -.7 m m 7. - M m m M Since convenionall, for raio esimaors o old, ρ and also for produc esimaors o old, ρ. Terefore wo daa ses are used in is paper, one o deermine e efficienc of e modified raio esimaors and e oer o deermine a of e produc esimaors as saed below.. Conclusion In is paper, we ave proposed a new famil of esimaor for esimaing unknown populaion mean of sud variable using auiliar variable. Epressions for e MSE of e esimaor are derived up o firs order of approimaion. Te proposed famil of esimaor is compared wi e several eising esimaors in lieraure. From able., we observe a e new famil of esimaors performs beer an e oer esimaors considered in is paper for bo of e daa ses. References [] Kosnevisan, M., Sing, R., Cauan, P., Sawan, N. and Smarandace, F.(7: A general famil of esimaors for esimaing populaion mean using known value of some populaion parameer(s. Far Eas Jour. of eor. sais. (, 8 9. [] Sisodia, B.V.S. and Dwivedi, V.K.(98: A modified raio esimaor using coefficien of variaion of auiliar variable. Journal Ind. Soc. Agril. Sais.,(, 8. [] Sing, H.P. and Tailor, R. ( : Use of known correlaion coefficien in esimaing e finie populaion mean. Sais. in Trans. (.. 8

12 Some Raio and Produc Esimaors Using Known Value of Populaion Parameers [] Pande, B.N. and Dube, Vas (988: Modified produc esimaor using coefficien of variaion of auiliar variae, Assam Saisical Rev., (, [] A.A. Adewara, R. Sing, M. Kumar( : Efficienc of some modified raio and produc esimaors using known value of some populaion parameers, Inernaional Journal of Applied Science and Tecnolog ( [] Yadav, S.K. and Kadilar, C. (: Improved class of raio and produc esimaors. Applied maemaics and compuaion. [7] Sabbir, J., Gupa, S. (. : Improved raio esimaors in sraified sampling. Amer. J. Maema. Manag. Sci. :9 [8] Sabbir, J. and Gupa, S. ( : A new esimaor of populaion mean in sraified sampling, Commun. Sa. Teo. Me. : 9 [9] Kadilar, C. and Cingi, H. ( : Raio Esimaors in Simple Random Sampling. Appl. Mae. and Compu., 89-9 [] Kadilar, C. and Cingi, H. (.: Raio esimaors for e populaion variance in sample and sraified random sampling. Appl. Mae. and Compu. 7, 7 9. [] Adewara, A.A. (. : Effecs of improving bo e auiliar and variable of ineres in raio and produc esimaors. Proc. Pakisan Acad. Sci.(: [] Srivenkaaramana, T. and Srina, K.P. (97 : Raio and Produc meods of esimaion in sample surves wen e wo variables are moderael correlaed. Vignana Barai : 8 [] Maddala, G.S. (977: Economerics. McGraw Hills Pub. Co. New York. [] Malik and Sing (: Esimaion of populaion mean using informaion on auiliar aribue in wo-pase sampling. Applied Maemaics and compuaion, ( - 9