Chapter-2: A Generalized Ratio and Product Type Estimator for the Population Mean in Stratified Random Sampling CHAPTER-2

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1 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling CHAPTER- A GEERALIZED RATIO AD PRODUCT TYPE ETIMATOR FOR THE POPULATIO MEA I TRATIFIED RADOM AMPLIG. Introduction Te importance of ratification in surve sampling is well nown for te case of eterogeneous population. ometimes, it becomes necessar to ratif te population to get te valid, accurate and efficient results regarding te parameter. To increase te efficienc of te eimators, te population is ratified in suc a wa tat te units belonging to same rata are as muc omogeneous as possible wit respect to te variable under ud. If te diribution of te ud variable is completel nown ten te boundar points of te rata are determined on te basis of diribution of variable so tat te variance of te eimator becomes minimum for a given allocation. Oterwise, te nowledge about te diribution of auiliar variable, igl correlated wit ud variable, ma be used for te same purpose. 9

2 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling In suc situation, cuomar separate ratio eimator and combined ratio eimator ave been defined for eimating te parameter depending upon te information available on population mean of auiliar variable in eac ratum. Te separate ratio eimator of population mean Y of ud variable is given as w R R R were R is te ratio eimate of te population mean Y for te t ratum and w is ratum weigt. Te combined ratio eimator is given b w RC w were, denote te sample means of variables, respectivel in t ratum and is te nown population mean of auiliar variable. Following isodia and Dwivedi (98), ing and Karan (993), Upadaa and ing (999), Kadilar and Cingi (003) ave defined four 30

3 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling eimators of population mean in ratified random sampling. Te ave sown tat eimators defined b tem are more efficient tan te traditional combined ratio eimator under some conditions. In 989, Prasad as defined improved ratio-tpe eimator of population mean and Kadilar and Cingi (005) ave defined similar eimator in ratified random sampling as p κ RC (..) were κ is an conant. Te ave sown tat eimator p defined in (..) is more efficient tan te usual combined ratio eimator RC. In te present capter, we propose a generalized ratio and product tpe eimator of population mean under ratified random sampling using nown information on parameters of auiliar variable. Te eimators defined b Kadilar and Cingi (003) are particular cases of proposed eimator. Te epressions for mean square error and its minimum value ave been obtained. Te comparison of proposed eimator wit te eimator defined b Kadilar and Cingi (005) ave been made teoreticall. Te results ave also been illurated numericall as well as grapicall. 3

4 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling. ampling Design and Terminolog Let population of units be divided into rata and simple random sample of size n is drawn from t ratum of size wit. Let Y i and i denote te values of variable under ud and auiliar variable on te t i unit of te population in te small letters denote te values in te sample. Defining t ratum respectivel. Te corresponding n i, n i n i, n i Y Yi, i i, i w, Y wy, w, w, 3

5 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling RC, R Y, Rˆ, w, ρ, γ ( n ), n Y Y ( ) i ( ), i i, i Y Y ( )( ) i i i.3 Te Proposed Eimator Under te sampling design defined in te section (.), wen information on two oter population parameters θ and φ in addition to population mean 33

6 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling of te auiliar variable are nown in advance, we propose a generalized ratio and product tpe eimator of population mean as α (.3.) Here w ( + φ ) θ, w( + φ) θ, w and α is an arbitrar conant. Using te nown information on coefficient of variation ( C ), population mean ( ) ( ) and coefficient of urtosis ( ) β of auiliar variable in te ratum, Kadilar and Cingi (003) ave defined te following ratio-tpe eimators t ( ) D ( + ) w C ( + ) w C 34

7 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling ( ) K ( + β ( )) w ( + β ( )) w ( 3 ) U ( β ( ) + ) w C ( β ( ) + ) w C ( 4 ) U ( + β ( )) w C ( + β ( )) w C Te eimators D, K, U and U defined above are particular members of te proposed eimator defined in (.3.), wic can be obtained, respectivel, from b taing ( ) α, θ, φ C ( ) α, θ, φ β ( ) 35

8 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling ( ) α θ β ( ) 3,, φ C, and ( 4 ) α, θ C, φ β ( ) Defining ε η Y uc tat E ( ε) E( η) 0 E Y ( ε ) γ w, ( η ) E w, γ θ E( εη) γ wθ Y To obtain te bias and mean square error (ME) of proposed eimator, we epand in terms of ε and η as 36

9 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling ( ) ˆ α α Y + ε+ αη+ αεη+ η +... (.3.) Taing epectation on bot sides of (.3.), we ave ( ) ( ˆ α α E ) Y + E( ε) + α E( η) + α E( εη) + E( η ) +... ubituting te values of epectations and retaining terms up to fir order of approimation, we ave ( ) E( ˆ α α α ) Y Y w w + γ θ + γ θ Y ( ) Bias( ˆ α α α ) Y w + w γ θ γ θ Y (.3.3) From (.3.3), we see tat te bias of ˆ is of fir order. o its contribution to te ME of ˆ will be of second order. 37

10 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling From (.3.) and using te values of epectations, te ME of proposed eimator ˆ, up to fir order of approimation, is obtained as ( ˆ ) ( ˆ ) ME E Y Y E ε α η + + αεη ( ε ) α ( η ) α ( εη) + + Y E E E + + γ Y Y w α θ α θ ( + R α θ + αr θ ) γ w (.3.4) were R Y w Y ( θ + φ ) Differentiating epression (.3.4) w.r.t. α wile eeping θ fied and equating to zero, we get 38

11 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling α Rγwθ + R 0 γwθ After solving, we get ( α) opt R γ wθ γ wθ (.3.5) ubituting te optimum value of α obtained in (.3.5) in (.3.4), te minimum ME of is Min. ME ( ˆ ) γ w γ w θ γ w θ (.3.6) Te results are summarized in te following teorem: Teorem.3.: Up to fir order of approimation, te bias of is ( ) Bias( ˆ α α α ) Y w + w γ θ γ θ Y 39

12 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling 40 Teorem.3.: Up to fir order of approimation, te ME of is ( ) ME ˆ ( ) + + R R w θ α θ α γ wic is minimised for ( ) opt w w R θ γ θ γ α and its minimum value is ( ) w w w Min ME ˆ. θ γ θ γ γ pecial Case: For θ β (regression coefficient of on in te t ratum) and for an value of φ, te minimum ME of reduces to

13 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling ( ˆ ) w ( ) Min. ME ρ γ (.3.7) were ρ is te correlation coefficient between and in te t ratum..4 Comparison For comparing te proposed eimator wit te eimator p given b Kadilar and Cingi (005), we fir obtain te epression for ME of teir eimator up to fir order of approimation. Appling te same procedure as adopted b tem, under te sampling design given in section (.), we get ( p) γ ( + ) Min. ME w R R ME( RC ) (.4.) wic sows tat optimum eimator of p is not efficient tan combined ratio eimator RC, up to fir order of approimation. 4

14 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling Using te epression (.3.6) and (.4.), we obtain ( ) ( ˆ p ) γ ( + ) Min. ME Min. ME w R R γ w + γ wθ γ wθ γ w R R + ( ) γ wθ γ wθ (.4.) > 0 if R β < (.4.3) From (.4.3) we see tat proposed eimator is alwas better tan Kadilar and Cingi (005) eimator for an value of θ wen regression coefficient in eac ratum is less tan te alf of te ratio R. In spite of tis we consider following special cases b taing te different values of θ. 4

15 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling pecial Case I: For θ, te epression (.4.) reduces to Min. ME p Rγ w γ w ( ) Min. ME( ˆ ) (.4.4) 0 pecial Case II: For θ, te epression (.4.) reduces to. ( ). ( ˆ ) Min ME p Min ME γ w R (.4.5) 0 From (.4.4) and (.4.5), we see tat te epressions on te rigt and side are alwas positive unless regression coefficient in eac ratum is equal to conant ratio R, wic sows te superiorit of proposed eimator ˆ over Kadilar and Cingi (005) eimator p. 43

16 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling.5 umerical Eample To ave an idea about te efficienc of proposed eimator ˆ, we illurate te results numericall b taing data on apple production amount (as variable of intere) and number of apple trees (as auiliar variable) in 854 villages of Ture collected in 999 (ource : Initute of tatiics, Republic of Ture) and used b Kadilar and Cingi (005). Te data ave been ratified b regions of Ture and from eac ratum, random sample (villages) is drawn. Te sample sizes are cosen b using te Neman allocation, n n Te values of population parameters obtained are given in table.5. and efficiencies of te eimators are given in table

17 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling Table.5.: Values of Population Parameters n 40 n 9 n 7 n 38 n 67 n 7 n Y 930 Y 536 Y Y 9384 Y 5588 Y 967 Y C 3.85 C.0 C.0 C 3. C C 5.7 C 6.9 β 3.07 β 5.7 β β β β β R ρ 0. 8 ρ 0.86 ρ 0.90 ρ ρ 0.7 ρ κ * γ 0. 0 γ γ 0.06 γ γ 0.38 γ ν w w w 0. 0 w w w

18 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling Table.5.: Te efficiencies of eimators up to fir order of approimation. Eimator ME Efficienc U U K D p (for θ ) (for θ )

19 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling Te comparison of proposed eimator wit te oter eimator as also been sown grapicall b taing some particular value of θ and φ. Taing θ and φ β ( ) ( ) ( D ) ( K ) ( U ) ( U ) ( p ) ME α Figure.5.: Mean quare Errors (ME) of different eimators wit respect to te values of α. 47

20 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling From te grap, we can find te range of variation of α in wic proposed eimator will be efficient tan te eimators proposed b Kadilar and Cingi (003, 005) under te given conditions. From table.5., we see tat proposed eimator is almo si times efficient tan te eimator U and almo two times efficient tan U, K and D defined b Kadilar and Cingi (003) wen θ is taen as regression coefficient of on in te t ratum. It is also almo two times efficient tan te eimator p defined b Kadilar and Cingi (005). From te grap, we also find tat te proposed eimator is efficient tan te eimators defined b Kadilar and Cingi (003, 005) for some range of variation of α wic can be easil obtained from te pa eperience or from te pilot sample surve. Hence we conclude tat te optimum eimator of te proposed eimator can be conructed aving iger efficienc tan te eiing ones..6 ummar In surve sampling, using information on auiliar variable, igl correlated wit ud variable, various ratio and product tpe eimators for eimating te population mean of ud variable under ratified random sampling ave been defined b large number of autors. We ave proposed a 48

21 Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling generalized ratio and product tpe eimator of population mean under ratified random sampling using nown information on parameters of auiliar variable. Te eimators defined b Kadilar and Cingi (003) are particular members of te proposed eimator. Te epressions for te bias and mean square error ave been obtained. We ave also derived te epression for mean square error of te proposed eimator for θ β (regression coefficient of on in te t ratum) as a special case. Te comparison of te proposed eimator wit te eiing ones is made wit respect to teir mean square errors. It as been sown tat te proposed eimator is more efficient tan te Kadilar and Cingi (005) eimator. Te efficiencies of te proposed eimator and tat of te eiing ones ave been sown numericall and grapicall. *** ** * 49