A GENERAL FAMILY OF ESTIMATORS FOR ESTIMATING POPULATION MEAN USING KNOWN VALUE OF SOME POPULATION PARAMETER(S) Dr. M. Khoshnevisan, GBS, Griffith Universit, Australia (m.khoshnevisan@griffith.edu.au) Dr. Rajesh Singh, Pankaj Chauhan and Nirmala Sawan School of Statistics, DAVV, Indore (M. P.), India Dr. Florentin Smarandache Universit of New Meico, USA (smarandache@unm.edu) Abstract A general famil of s for estimating the population mean of the variable under stud, which make use of known value of certain population parameter(s), is proposed. Under Simple Random Sampling Without Replacement (SRSWOR) scheme, the epressions of bias and meansquared error (MSE) up to first order of approimation are derived. Some well known s have been shown as particular member of this famil. An empirical stud is carried out to illustrate the performance of the constructed over others. Kewords: Auiliar information, general famil of s, bias, mean-squared error, population parameter(s).. Introduction
Let and be the real valued functions defined on a finite population ( U, U,..., ) U = and Y and X be the population means of the stud character and U N auiliar character respectivel. Consider a simple random sample of size n drawn without replacement from population U. In order to have a surve estimate of the population mean Y of the stud character, assuming the knowledge of population mean X of the auiliar character, the well-known ratio is X t = (.) Product method of estimation is well-known technique for estimating the populations mean of a stud character when population mean of an auiliar character is known and it is negativel correlated with stud character. The conventional product for Y is defined as t = (.) X Several authors have used prior value of certain population parameters (s) to find more precise estimates. Searls (964) used Coefficient of Variation (CV) of stud character at estimation stage. In practice this CV is seldom known. Motivated b Searls (964) work, Sisodia and Dwivedi (98) used the known CV of the auiliar character for estimating population mean of a stud character in ratio method of estimation. The use of prior value of Coefficient of Kurtosis in estimating the population variance of stud character was first made b Singh et.al.(973). Later, used b Sen (978), Upadhaa and Singh (984) and Searls and Interpanich (990) in the estimation of population mean of stud character. Recentl Singh and Tailor (003) proposed a modified ratio b using the known value of correlation coefficient.
In this paper, under SRSWOR, we have suggested a general famil of s for estimating the population mean Y. The epressions of bias and MSE, up to the first order of approimation, have been obtained, which will enable us to obtain the said epressions for an member of this famil. Some well known s have been shown as particular member of this famil.. The suggested famil of s Following Walsh (970), Redd (973) and Srivastava (967), we define a famil of s Y as g ax + b t ( a b) ( )( ax b) (.) α + + α + where a( 0), b are either real numbers or the functions of the known parameters of the auiliar variable such as standard deviation ( Kurtosis ( β ( ) ) and correlation coefficient (ρ). such that and To obtain the bias and MSE of t, we write = Y( ), = X( + ) + e 0 E (e 0 )=E (e )=0, e σ ), Coefficients of Variation (C X ), Skewness ( ( ) β ), E (e0 ) = fc, E (e ) = fc, E(e0e) = f C, where f N n S S =, C =, C =. nn Y X 3
Epressing t in terms of e s, we have ( + e )( + αλ ) g t = Y (.) 0 e where ax λ =. (.3) ax + b We assume that e < e αλ so that ( ) g + αλ is epandable. Epanding the right hand side of (.) and retaining terms up to the second powers of e s, we have g(g + ) t Y + e0 αλge + α λ e αλge0e (.4) Taking epectation of both sides in (.4) and then subtracting Y from both sides, we get the bias of the t, up to the first order of approimation, as g(g + ) B (t) Y α λ C αλg C n N (.5) From (.4), we have ( t Y) [ αλ Y e 0 ge (.6) Squaring both sides of (.6) and then taking epectations, we get the MSE of the t, up to the first order of approimation, as ) n N [ C + α λ g C αλg C Y (.7) Minimization of (.7) with respect to α ields its optimum value as K α = = α opt (sa) (.8) λg where 4
C K = ρ. C Substitution of (.8) in (.7) ields the minimum value of MSE (t) as min.) = f = (.9) Y C ( ρ ) ) 0 The min. MSE (t) at (.9) is same as that of the approimate variance of the usual linear regression. 3. Some members of the proposed famil of the s t The following scheme presents some of the important known s of the population mean which can be obtained b suitable choice of constants α, a and b:. t 0 = Estimator The mean per unit X. t = The usual ratio 3. t = X The usual product X + C 4. t = 3 + C Sisodia and Dwivedi Values of α a b g 0 0 0 0 0 0 - C 5
(98) + C 5. t = 4 X + C Pande and Dube (988) C - 6. t 5 β β ( ) + C ( ) X + C Upadhaa and Singh (999) C + β () 7. t 6 C X + β () Upadhaa, Singh (999) + σ 8. t 7 X + σ G.N.Singh (003) β () + σ 9. t 8 β()x + σ G.N.Singh (003) β () + σ 0. t 9 β ()X + σ G.N.Singh (003) X + ρ. t 0 + ρ Singh, Tailor (003) ( ) β C - C ( ) ( ) ( ) β β β - σ - σ - σ - ρ 6
+ ρ. t X + ρ Singh, Tailor (003) X + β () 3. t + β () Singh et.al. (004) + β () 4. t3 X + β () Singh et.al. (004) ρ - ( ) ( ) β β - In addition to these s a large number of s can also be generated from the proposed famil of s t at (.) just b putting values of α,g, a, and b. It is observed that the epression of the first order approimation of bias and MSE/Variance of the given member of the famil can be obtained b mere substituting the values of α,g, a and b in (.5) and (.7) respectivel. 4. Efficienc Comparisons Up to the first order of approimation, the variance/mse epressions of various s are: V (t = (4.) 0 C [ C + C ) fy C = (4.) [ C + C + C = (4.3) [ C + θ C θ 3 C = (4.4) 7
[ C + θ C + θ 4 C = (4.5) [ C + θ C + θ 5 C = (4.6) [ C + θ C + θ 6 3 3 C = (4.7) [ C + θ C + θ 7 4 4 C = (4.8) [ C + θ C + θ 8 5 5 C = (4.9) [ C + θ C + θ 9 6 6 C = (4.0) [ C + θ C θ 0 7 7 C = (4.) [ C + θ C + θ ) fy 7 7 C = (4.) [ C + θ C θ 8 8 C = (4.3) [ C + θ C + θ 3) fy 8 8 C = (4.4) where X β ()X C X θ =, θ =, θ 3 =, X + C β () + C C X + C X β()x β ()X θ 4 =, θ 5 =, θ 6 =, X + σ β () + σ β ()X + σ θ X = X + ρ 7, X θ 8 =. X + β () To compare the efficienc of the proposed t with the eisting s t 0 -t 3, using (.9) and (4.)-(4.4), we can, after some algebra, obtain V(t ) ) = C ρ 0 (4.5) 0 0 ) ) = (C (4.6) 0 8
) ) = (C + (4.7) 0 ) ) = ( θ C (4.8) 3 0 ) ) = ( θ C + (4.9) 4 0 ) ) = ( θ C + (4.0) 5 0 ) ) = ( θ C + (4.) 6 0 3 ) ) = ( θ C + (4.) 7 0 4 ) ) = ( θ C + (4.3) 8 0 5 ) ) = ( θ C + (4.4) 9 0 6 ) ) = ( θ C (4.5) 0 0 7 ) ) = ( θ C + (4.6) 0 7 ) ) = ( θ C (4.7) 0 8 ) ) = ( θ C + (4.8) 3 0 8 Thus from (4.5) to (4.8), it follows that the proposed famil of s t is more efficient than other eisting s t 0 to t 3. Hence, we conclude that the proposed famil of s t is the best (in the sense of having minimum MSE). 5. Numerical illustrations We consider the data used b Pande and Dube (988) to demonstrate what we have discussed earlier. The population constants are as follows: 9
N=0,n=8, Y = 9. 55, X = 8. 8, C = 0. 555, C = 0. 6, ρ 0. 999 =, β () = 0. 5473, β () = 3.063, θ = 0. 4 77. We have computed the percent relative efficienc (PRE) of different s of Y with respect to usual unbiased and compiled in table 5.. Table 5.: Percent relative efficienc of different s of Y with respect to Estimator PRE 00 t 3.39 t 56.45 t 3 3.9 t 4 550.05 t 5 534.49 t 6 58.7 t 7 59.37 t 8 436.9 t 9 633.64 t 0.7 t 465.5 t 7. t 3 644.7 t (opt) 650.6 0
where t (opt) is the value of t at (.) and replacing α b α (opt) given in (.8) and the resulting MSE given b (.9). Conclusion From table 5., we observe that the proposed general famil of s is preferable over all the considered s under optimum condition. The choice of the mainl depends upon the availabilit of information about the known parameters of the auiliar variable such as standard deviation ( and correlation coefficient (ρ). σ ), Coefficients of Variation (C X ), Skewness ( ( ) β ), Kurtosis ( β ( ) ) References Pande, B.N. and Dube, Vas (988): Modified product using coefficient of variation of auiliar variate, Assam Statistical Rev., (), 64-66. Redd, V.N. (973): On ratio and product methods of estimation. Sankha, B, 35(3), 307-36. Singh, G.N. (003): On the improvement of product method of estimation in sample surves. Jour. Ind. Soc. Agri. Statistics, 56(3), 67-75. Singh H.P. And Tailor, R. (003): Use of known correlation coefficient in estimating the finite population mean. Statistics in Transition, 6,4,555-560. Singh H.P.,Tailor, R. and Kakaran, M.S. (004): An of Population mean using power transformation. J.I.S.A.S., 58(), 3-30. Singh, J. Pande, B.N. and Hirano, K. (973): On the utilization of a known coefficient of kurtosis in the estimation procedure of variance. Ann. Inst. Stat. Math., 5, 5-55. Sisodia, B.V.S. And Dwivedi, V.K. (98): A modified ratio using coefficient of variation of auiliar variable. Journ. Ind. Soc. Agril. Statist., 33,, 3-8. Searls, D.T. (964): The utilization of known coefficient of variation in the estimation procedure. Journal of American Statistical Association, 59, 5-6.
Searls, D.T. and Intarapanich, P. (990): A note on an for the variance that utilizes the kurtosis. The American Statistician, 44, 4, 95-96. Sen, A.R. (978): Estimation of the population mean when the coefficient of variation is known. Commun. Statist., Theor Meth. A (7), 657-67. Srivastava, S.K. (967): An using auiliar information. Calcutta Statist. Assoc. Bull., 6,-3. Upadhaa, L.N. and Singh, H.P. (999): Use of transformed auiliar variable in estimating the finite population mean. Biometrical Journal, 4, 5, 67-636. Walsh, J.E. (970): Generalization of ratio for population total. Sankha, A, 3, 99-06.