Variance Using Multivariate Auxiliary Information ... Arcos and Rueda 1] suggested an almost unbiased estimator .... the minimum variance of d(Reg)k.
Information and Management Sciences Volume 11, Number 1, pp.49-54, 2000
Some Competitive Estimators of Finite Population Variance Using Multivariate Auxiliary Information M. S. Ahmed M. S. Raman Jahangirnagar University Jahangirnagar University Bangladesh Bangladesh M. I. Hossain Jahangirnagar University Bangladesh
Abstract Two estimators for a nite population variance have been proposed using multivariate auxiliary information. Both the estimators are superior to Arcos and Rueda [1] as well as Isaki [2]. Further one of them is unbiased up to rst order approximation. Computationally these estimators need only the additional information about the correlation coecients among the variables.
Keywords: Finite Population, Multivariate Auxiliary Information.
1. Introduction Isaki [2] proposed the ratio and regression estimators for the estimation of population variance by using multivariate auxiliary information and compared their properties in di erent sampling designs. Arcos and Rueda [1] suggested an almost unbiased estimator with same variance but their claim was not true about the almost unbiasedness due to improper calculation. Here, we have derived the corrected bias that estimator in matrix form. Further, we have suggested an estimator, which is superior to Arcos and Rueda [1] with respect to bias. The properties of these estimators are studied and the comparison is given for a multivariate normal population.
2. Estimators and Their Properties Let U = (U1 ; U2 ; : : : ; UN ) be a nite population. A random sample of size n is drawn and selected, (yi ; xji; j = 1; 2; : : : ; k) i = 1; 2; : : : ; n are observed, and assume that Received May 1999; Revised July 1999 and Accepted September 1999.
50
Information and Management Sciences, Vol. 11, No. 1, March, 2000
Sx2 (j = 1; 2; : : : ; k), the nite population variance of the variable xji(j = 1; 2; : : : ; k) are j
known. P We are desired to estimate Sy2 = N (N ; 1);1 y2 with y2 = N ;1 Ni=1 (yi ; Y )2 where Y is the population mean.
S2
(y; y) 40 (xj ; xj ) De ne Rj = S 2y ; 2 (y) = 40 2 (y; y ) ; 2 (xj ) = 2 (x ; x ) ; xj
20 j
20
j 22(xj ; xj ) 20(xj ; xj )02(xj ; xj )
yx = (y;22x ()y; xj()y; x ) and x x = 20 j 02 j P where rs(z; t) = N ;1 Ni=1 (zi ; Z )r (ti ; T )s , r; s = 0; 1; 2; 3 and 4 0
j
j j0
0
0
(2.1)
Arcos and Rueda [1] proposed the estimator
k " S 2 # j Y xj 2 d1k = sy 2x s j j =1
(2.2)
where 1 ; : : : ; k 2 R, s2y and s2x ; (j = 1; 2; : : : ; k) are usual estimators of Sy2 and Sx2 ; j = 1; 2; : : : ; k respectively. They also discussed its properties under multivariate normal distribution. We propose the following two new estimators j
j
k " Y 2 d2k = sy 2 j =1 j Sxj
where 1 ; : : : ; k 2 R, and
2
#
Sx2
j
+ (1 ; j )s2x
(2.3)
j
S2 d3k = s2y 40 + j s2x x j =1 k X
3 j
(2.4)
5
j
P
where 1 ; : : : ; k 2 R, 0 = 1 ; kj=1 j 2 2 s2 ;S 2 Let 0 = s S;2S and j = S 2 , assume that jj j 1 and j j j j 1 (See, Reddy [3]) y
y
y
xj
xj
xj
De ne = ( 1 2 ; : : : ; k )0 ; = ( 1 2 ; : : : ; k )0 ; = (1 2 ; : : : ; k )0 ; = (1 2; : : : ; k )0; v = dig(12 22 ; : : : ; k2 ); E = (1 1; : : : ; 1)0 ; E [0 ] = V0 ; E [0 ] = 0 = ( 0j )k1; E [0 ] = = (jj )kk ; E [v] = V = diag(jj )kk
(2.5)
From the Taylor's series rst order expansions (see, Appendix), we have Bias(d1k ) = Sy2 [ 0 ; 0 0 + 12 ( 0 V + 0 V E )]
(2.6)
0
Some Competitive Estimators of Finite Population Variance Using Multivariate Auxiliary Information
Bias(d2k ) = Sy2 [ 0 ; 0 0 ] Bias(d3k ) = Sy2 [;0 0 + 0 V E ] MSE(d1k ) = Sy4 [V0 + 0 ; 2 0 0 ] MSE(d2k ) = Sy4 [V0 + 0 ; 2 0 0 ] MSE(d3k ) = Sy4 [V0 + 0 ; 20 0 ]
51
(2.7) (2.8) (2.9) (2.10) (2.11)
The optimum values of ; and , which respectively minimizes mean square errors of d1k , d2k and d3k are all the same and given by
opt = opt = opt = ;1 0 = ! = (!j )k1 ; say:
(2.12)
The respective minimum mean square error of d1k , d2k and d3k also keeps the same and is given by
M0 (d1k ) = M0 (d2k ) = M0(d3k ) = Sy4[V0 ; 0 ;1 0 ]M0; say: For optimum and , the biases of d1k , d2k and d3k are respectively given by Bias(d1k ) = 21 Sy2 ( 00 ;1 V ;1 0 + 00 ;1 V E ) Bias(d2k ) = 0 Bias(d3k ) = Sy2 [;0 0 + 0 V E ]
(2.13) (2.14) (2.15) (2.16)
Up to the rst order approximation, it is evident that d2k is an unbiased estimator. For optimum , Arcos and Rueda [1] showed that Bias (d1k ) = 0, which is incorrect and its bias is given by (2.14). On the other hand, Isaki [2] suggested the regression estimator
d(Reg)k = s2y + Let Aj = Bj Rj;1 and A = (Aj )k1
k X j =1
Bj (Sx2 ; s2x ) j
j
Hence, Var (d(Reg)k ) = Sy2 [V0 + A0 A ; 2A0 0 ]
(2.17)
(2.18)
The optimum value of A which minimizes Var (d(Reg)k ) is given by
Aopt = ;1 0 (= !) i.e. B(opt)j = A(opt)j Rj = !j Rj
(2.19)
and the minimum value of Var (d(Reg)k ) is given by
Vmin = Sy4 [V0 ; 0;1 0]
(2.20)
52
Information and Management Sciences, Vol. 11, No. 1, March, 2000
Therefore, the minimum mean square errors of d1k , d2k and d3k are the same as that of the minimum variance of d(Reg)k . For simple random sampling without replacement design, we have
V0 = H [ 2 (y) ; 1]; jj = H [ 2 (xj ) ; 1]; jj = H [2 (xj ; xj ) ; 1]; 0j = H [2(y; xj ) ; 1]; 0
0
where H = (n;1 ; N ;1 ) and for large N , H = n;1
3. Comparison of these Estimators under Multivariate Normal Model Let random vector (y; x1 ; x2 ; : : : ; xk ) possess the same moments as a 1 (k + 1) multivariate normal variable up to fourth order and assume also that x x = yx = , 8j , j 0 and ;k;1 < < 1, From Isaki [2], we have j j0
2 B(opt)j = 1 + (k R;j1)2 = !j Rj ; j = 1; 2; 3 : : : ; k " # 4 2Sy4 k Vmin = n ; 1 1 ; 1 + (k ; 1)2
j
(3.1) (3.2)
From (2.18) and (3.1), we have 2 !j = 1 + (k; 1)2 = !0(; k) = !0 (say); j = 1; 2; : : : ; k; V0 = n ;2 1 4 2 and 00 ;1 0 = (n ; 1)f12+k(k ; 1)2 g = (2nk!;0 1) Then the estimators d1k and d2k can be de ned as k " Sx2 #!0 Y 2 d1 k = s y 2 j =1 sx # 2 k " Y S x d2k = s2y 2 2 j =1 !0 Sx + (1 ; !0 )sx 3 2 k Sx2 X 2 4 and d3k = sy 1 ; k!0 + !0 s2 5 j =1 x j
(3.3)
(3.4)
j
j
j
(3.5)
j
j
(3.6)
j
From the above expression, it is observed that if is known (In many practical situations is available), so is !0, and thus the estimator d2k and d3k are well-de ned by (3.5) and (3.6) respectively. Furthermore, estimators d2k and d3k both are very easy to compute and their precision increase omitting O(n; 21 ) terms, and it attain the minimum mean
Some Competitive Estimators of Finite Population Variance Using Multivariate Auxiliary Information
square error.
"
4 2S 4 M0 = Vmin = n ;y1 1 ; 1 + (kk; 1)2
#
53
(3.7)
2 !j = 1 + (k; 1)2 = !0 (say) ; j = 1; 2; : : : ; k
It is observed that !0 0, since 2 0 4 2 V0 = n ;2 1 ; 00 ;1 0 = (n ; 1)f12+k(k ; 1)2 g = (2nk!;01) Now, the biases of d1k and d3k are respectively given as
k! 2
Bias(d1k ) = (n ;0 1)y (!0 + 1)
(3.8)
Bias(d3k ) = (n ;0 1)y (1 ; 2 )
(3.9)
k! 2
It is obvious that jBias(d3k )j < jBias(d1k )j, since !0 is always nonnegative. Hence, the estimator d3k dominates d1k in the sense of biasedness.
4. Conclusion The estimator d2k dominates d1k in the sense of biasedness up to the rst order approximation. When the multivariate normal model is assumed, d3k dominates d1k in biasedness de ned by (3.8) and (3.9).
Acknowledgements The authors are grateful to the referee for helpful comments and some corrections which lead to substantial improvements of the paper.
Appendix From (2.2)-(2.5), we have
k " Sx2 #aj k Y Y j 2 2 d1k = sy = Sy (1 + 0 ) (1 + j ); j 2x s j j =1 j =1 1 d1k ; Sy2 = Sy2[0 ; 0 0 ; 0 + 2 ( 0 v + 0 vE ) + 0 0 + ] # k " k Y Y Sx2j 2 2 d2k = sy = Sy (1 + 0 ) f j + (1 ; j )(1 + j )g;1 2 2
S + (1 ;
) s j xj j =1 j xj j =1
(A.1) (A.2) (A.3)
54
Information and Management Sciences, Vol. 11, No. 1, March, 2000
d2k ; Sy2 = Sy2[0 ; 0 0 + 0 0 ; 0 ] assume that j j j j 1 8 9 2 3 2 k k < = X X S x d3k = s2y 40 + j s2 5 = Sy2(1 + 0 ) :0 + j (1 + j );1 ; x j =1 j =1 2 2 0 0 0 0 d3k ; Sy = Sy [0 ; 0 + + vE ; ]
(A.4)
Taking the expectation of (A.2) and (A.4) up to the rst order, we have Bias(d1k ) = Sy2 [ 0 ; 0 0 + 21 ( 0 V + 0 V E )] Bias(d2k ) = Sy2 [ 0 ; 0 0 ] Bias(d3k ) = Sy2 [;0 0 + 0 V E ]
(A.5) (A.6) (A.6)
j
(A.4) (A.3)
j
Again, squaring both sides of (A.2) and (A.4) and taking the expectation up to rst order expression, we have MSE(d1k ) = Sy4 [V0 + 0 ; 2 0 0 ] MSE(d2k ) = Sy4 [V0 + 0 ; 2 0 0 ] MSE(d3k ) = Sy4 [V0 + 0 ; 20 0 ]
(A.7) (A.8) (A.7)
References [1] Arcos, A. and Rueda, M., Variance estimation using auxiliary information: An almost unbiased multivariate ratio estimator, Metrika, Vol.45, pp.171-178, 1997. [2] Isaki, C. T., Variance estimation using auxiliary information, Journal of the American Statistical Association, Vol.78, No.381, pp.117-123, 1983. [3] Reddy, V. N., On a transformed ratio method of estimation, Sankhya, Vol.C36, pp.59-70, 1974.
