A GENERAL FAMILY OF ESTIMATORS FOR ESTIMATING POPULATION MEAN USING KNOWN VALUE OF SOME POPULATION PARAMETER(S) Dr. M. Khoshnevisan, GBS, Griffith University, Australia ([email protected]) Dr. Rajesh Singh, Pankaj Chauhan and Nirmala Sawan School of Statistics, DAVV, Indore (M. P.), India Dr. Florentin Smarandache University of New Mexico, USA ([email protected]) Abstract A general family of estimators for estimating the population mean of the variable under study, which make use of known value of certain population parameter(s), is proposed. Under Simple Random Sampling Without Replacement (SRSWOR) scheme, the expressions of bias and mean- squared error (MSE) up to first order of approximation are derived. Some well known estimators have been shown as particular member of this family. An empirical study is carried out to illustrate the performance of the constructed estimator over others. Keywords: Auxiliary information, general family of estimators, bias, mean-squared error, population parameter(s). 1. Introduction 1 Let y and x be the real valued functions defined on a finite population ( ) U = U ,U ,.....,U and Yand X be the population means of the study character y and 1 2 N auxiliary character x respectively. Consider a simple random sample of size n drawn without replacement from population U. In order to have a survey estimate of the population mean Y of the study character y, assuming the knowledge of population meanX of the auxiliary character x, the well-known ratio estimator is X t = y (1.1) 1 x Product method of estimation is well-known technique for estimating the populations mean of a study character when population mean of an auxiliary character is known and it is negatively correlated with study character. The conventional product estimator for Y is defined as x t = y (1.2) 2 X Several authors have used prior value of certain population parameters (s) to find more precise estimates. Searls (1964) used Coefficient of Variation (CV) of study character at estimation stage. In practice this CV is seldom known. Motivated by Searls (1964) work, Sisodiya and Dwivedi (1981) used the known CV of the auxiliary character for estimating population mean of a study character in ratio method of estimation. The use of prior value of Coefficient of Kurtosis in estimating the population variance of study character y was first made by Singh et.al.(1973). Later, used by Sen (1978), Upadhyaya and Singh (1984) and Searls and Interpanich (1990) in the estimation of population mean of study character. Recently Singh and Tailor (2003) proposed a modified ratio estimator by using the known value of correlation coefficient. 2 In this paper, under SRSWOR, we have suggested a general family of estimators for estimating the population mean Y. The expressions of bias and MSE, up to the first order of approximation, have been obtained, which will enable us to obtain the said expressions for any member of this family. Some well known estimators have been shown as particular member of this family. 2. The suggested family of estimators Following Walsh (1970), Reddy (1973) and Srivastava (1967), we define a family of estimators Y as g ⎡ aX+b ⎤ t = y (2.1) ⎢ ( ) ( )( )⎥ ⎣α ax+b + 1−α aX+b ⎦ where a(≠0), b are either real numbers or the functions of the known parameters of the auxiliary ( ) variable x such as standard deviation (σ ), Coefficients of Variation (C ), Skewness (β x ), x X 1 Kurtosis (β (x)) and correlation coefficient (ρ). 2 To obtain the bias and MSE of t, we write ( ) ( ) y = Y1+e ,x = X1+e 0 1 such that E (e )=E (e )=0, 0 1 and E(e2)=f C2,E(e2)=f C2,E(e e )=f ρC C , 0 1 y 1 1 x 0 1 1 y x where N−n S2 S2 f = , C2 = y , C2 = x . 1 nN y Y2 x X2 3 Expressing t in terms of e’s, we have t = Y(1+e )(1+αλe )−g (2.2) 0 1 aX where λ = . (2.3) aX+b We assume that αλe <1 so that (1+αλe )−g is expandable. 1 1 Expanding the right hand side of (2.2) and retaining terms up to the second powers of e’s, we have ⎡ g(g+1) ⎤ t ≈ Y 1+e −αλge + α2λ2e2 −αλge e (2.4) ⎢⎣ 0 1 2 1 0 1⎥⎦ Taking expectation of both sides in (2.4) and then subtracting Y from both sides, we get the bias of the estimator t, up to the first order of approximation, as ⎛1 1 ⎞ ⎡g(g+1) ⎤ B(t) ≈⎜ − ⎟Y α2λ2C2 −αλgρC C (2.5) ⎝n N⎠ ⎢⎣ 2 x y x⎥⎦ From (2.4), we have ( ) [ ] t−Y ≅ Y e −αλge (2.6) 0 1 Squaring both sides of (2.6) and then taking expectations, we get the MSE of the estimator t, up to the first order of approximation, as ⎛1 1 ⎞ [ ] MSE(t) ≈⎜ − ⎟Y2 C2 +α2λ2g2C2 −2αλgρC C (2.7) ⎝n N⎠ y x y x Minimization of (2.7) with respect to α yields its optimum value as K α = =α (say) (2.8) λg opt where 4 C K =ρ y . C x Substitution of (2.8) in (2.7) yields the minimum value of MSE (t) as min.MSE(t) =f Y2C2(1−ρ2) = MSE(t) (2.9) 1 y 0 The min. MSE (t) at (2.9) is same as that of the approximate variance of the usual linear regression estimator. 3. Some members of the proposed family of the estimators’ t The following scheme presents some of the important known estimators of the population mean which can be obtained by suitable choice of constants α, a and b: Estimator Values of α a b g 1. t = y 0 0 0 0 0 The mean per unit estimator ⎛X⎞ 1 1 0 1 2. t = y⎜ ⎟ 1 ⎜⎝ x ⎟⎠ The usual ratio estimator ⎛ x ⎞ 1 1 0 -1 3. t = y⎜ ⎟ 2 ⎝X⎠ The usual product estimator ⎛X+C ⎞ 1 1 Cx 1 4.t = y⎜ x ⎟ 3 ⎜ x+C ⎟ ⎝ ⎠ x Sisodia and Dwivedi 5 (1981) estimator ⎛ x+C ⎞ 1 1 Cx -1 5.t = y⎜ x ⎟ 4 ⎜X+C ⎟ ⎝ ⎠ x Pandey and Dubey (1988) estimator ⎡β (x)x+C ⎤ 1 β(x) Cx -1 6.t = y 2 x 2 ⎢ ⎥ 5 β (x)X+C ⎣ ⎦ 2 x Upadhyaya and Singh (1999) estimator ⎡C x+β (x)⎤ 1 Cx β(x) -1 7.t = y x 2 2 ⎢ ⎥ 6 C X+β (x) ⎣ ⎦ x 2 Upadhyaya, Singh (1999) estimator ⎡x+σ ⎤ 1 1 σ -1 8. t = y x x ⎢ ⎥ 7 X+σ ⎣ ⎦ x G.N.Singh (2003) estimator ⎡β (x)x+σ ⎤ 1 β(x) σ -1 9.t = y 1 x 1 x ⎢ ⎥ 8 β (x)X+σ ⎣ ⎦ 1 x G.N.Singh (2003) estimator ⎡β (x)x+σ ⎤ 1 β(x) σ -1 10. t = y 2 x 2 x ⎢ ⎥ 9 β (x)X+σ ⎣ ⎦ 2 x G.N.Singh (2003) estimator ⎡X+ρ⎤ 1 1 ρ 1 11.t = y ⎢ ⎥ 10 ⎣x+ρ⎦ Singh, Tailor (2003) 6 estimator ⎡x+ρ⎤ 1 1 ρ -1 12.t = y ⎢ ⎥ 11 ⎣X+ρ⎦ Singh, Tailor (2003) estimator ⎡X+β (x)⎤ 1 1 β(x) 1 13.t = y 2 2 ⎢ ⎥ 12 x+β (x) ⎣ ⎦ 2 Singh et.al. (2004) estimator ⎡x+β (x)⎤ 1 1 β(x) -1 14.t = y 2 2 ⎢ ⎥ 13 X+β (x) ⎣ ⎦ 2 Singh et.al. (2004) estimator In addition to these estimators a large number of estimators can also be generated from the proposed family of estimators t at (2.1) just by putting values of α,g, a, and b. It is observed that the expression of the first order approximation of bias and MSE/Variance of the given member of the family can be obtained by mere substituting the values of α,g, a and b in (2.5) and (2.7) respectively. 4. Efficiency Comparisons Up to the first order of approximation, the variance/MSE expressions of various estimators are: V(t ) =f Y2C2 (4.1) 0 1 y [ ] MSE(t ) = f Y2 C2 +C2 −2ρC C (4.2) 1 1 y x y x [ ] MSE(t ) = f Y2 C2 +C2 +2ρC C (4.3) 2 1 y x y x [ ] MSE(t ) = f Y2 C2 +θ2C2 −2θ ρC C (4.4) 3 1 y 1 x 1 y x 7 [ ] MSE(t ) = f Y2 C2 +θ2C2 +2θ ρC C (4.5) 4 1 y 1 x 1 y x [ ] MSE(t ) =f Y2 C2 +θ2C2 +2θ ρC C (4.6) 5 1 y 2 x 2 y x [ ] MSE(t ) = f Y2 C2 +θ2C2 +2θ ρC C (4.7) 6 1 y 3 x 3 y x [ ] MSE(t ) = f Y2 C2 +θ2C2 +2θ ρC C (4.8) 7 1 y 4 x 4 y x [ ] MSE(t ) =f Y2 C2 +θ2C2 +2θ ρC C (4.9) 8 1 y 5 x 5 y x [ ] MSE(t ) =f Y2 C2 +θ2C2 +2θ ρC C (4.10) 9 1 y 6 x 6 y x [ ] MSE(t ) =f Y2 C2 +θ2C2 −2θ ρC C (4.11) 10 1 y 7 x 7 y x [ ] MSE(t ) =f Y2 C2 +θ2C2 +2θ ρC C (4.12) 11 1 y 7 x 7 y x [ ] MSE(t ) =f Y2 C2 +θ2C2 −2θ ρC C (4.13) 12 1 y 8 x 8 y x [ ] MSE(t ) =f Y2 C2 +θ2C2 +2θ ρC C (4.14) 13 1 y 8 x 8 y x where X β (x)X C X θ = ,θ = 2 ,θ = x , 1 X+C 2 β (x)+C 3 C X+C x 2 x x x X β (x)X β (x)X θ = ,θ = 1 ,θ = 2 , 4 X+σ 5 β (x)+σ 6 β (x)X+σ x 1 x 2 x X X θ = ,θ = . 7 X+ρ 8 X+β (x) 2 To compare the efficiency of the proposed estimator t with the existing estimators t -t , using 0 13 (2.9) and (4.1)-(4.14), we can, after some algebra, obtain V(t )−MSE(t) =C2ρ2 >0 (4.15) 0 0 y MSE(t )−MSE(t) = (C −ρC )2 > 0 (4.16) 1 0 x y 8 MSE(t )−MSE(t) =(C +ρC )2 > 0 (4.17) 2 0 x y MSE(t )−MSE(t) =(θ C −ρC )2 > 0 (4.18) 3 0 1 x y MSE(t )−MSE(t) =(θ C +ρC )2 > 0 (4.19) 4 0 1 x y MSE(t )−MSE(t) = (θ C +ρC )2 > 0 (4.20) 5 0 2 x y MSE(t )−MSE(t) =(θ C +ρC )2 >0 (4.21) 6 0 3 x y MSE(t )−MSE(t) =(θ C +ρC )2 >0 (4.22) 7 0 4 x y MSE(t )−MSE(t) = (θ C +ρC )2 >0 (4.23) 8 0 5 x y MSE(t )−MSE(t) =(θ C +ρC )2 >0 (4.24) 9 0 6 x y MSE(t )−MSE(t) = (θ C −ρC )2 >0 (4.25) 10 0 7 x y MSE(t )−MSE(t) =(θ C +ρC )2 >0 (4.26) 11 0 7 x y MSE(t )−MSE(t) = (θ C −ρC )2 >0 (4.27) 12 0 8 x y MSE(t )−MSE(t) =(θ C +ρC )2 >0 (4.28) 13 0 8 x y Thus from (4.15) to (4.28), it follows that the proposed family of estimators ‘t’ is more efficient than other existing estimators t to t . Hence, we conclude that the proposed family of 0 13 estimators ‘t’ is the best (in the sense of having minimum MSE). 5. Numerical illustrations We consider the data used by Pandey and Dubey (1988) to demonstrate what we have discussed earlier. The population constants are as follows: 9 N=20,n=8,Y =19.55,X =18.8,C2 =0.1555,C2 =0.1262,ρ = −0.9199,β (x) =0.5473, x y yx 1 β (x) =3.0613, θ =0.7172. 2 4 We have computed the percent relative efficiency (PRE) of different estimators of Y with respect to usual unbiased estimator y and compiled in table 5.1. Table 5.1: Percent relative efficiency of different estimators of Ywith respect to y Estimator PRE y 100 t1 23.39 t2 526.45 t3 23.91 t 550.05 4 t 534.49 5 t 582.17 6 t 591.37 7 t 436.19 8 t 633.64 9 t 22.17 10 t 465.25 11 t 27.21 12 t 644.17 13 t 650.26 (opt) 10