AN EFFICIENT CLASS OF CHAIN ESTIMATORS OF POPULATION VARIANCE UNDER SUB-SAMPLING SCHEME

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1 J. Japan Statist. Soc. Vol. 35 No AN EFFICIENT CLASS OF CHAIN ESTIMATORS OF POPULATION VARIANCE UNDER SUB-SAMPLING SCHEME H. S. Jhajj*, M. K. Shara* and Lovleen Kuar Grover** For estiating the population variance Sy of study variable y, a class of chain estiators of Sy has been proposed in the presence of two auxiliary variables x and z by using known inforation on population ean and variance of the second auxiliary variable z. In this proposed class, the second auxiliary variable z is directly highly correlated with the first auxiliary variable x, whereas the variable z is correlated with the variable y due to only the high correlation between the variables y and x. Another generalized class of estiators of Sy has also been considered by using the sae available inforation of auxiliary variable z when both the auxiliary variables x and z are directly highly correlated with the study variable y. The asyptotic expressions for the ean square errors and their optiu values have been obtained. A coparison between the two proposed classes of estiators of Sy has been ade epirically. Key words and phrases: Auxiliary variable, chain estiator, consistent estiator, double sapling technique, ean square error, optiu estiator, study variable. 1. Introduction In anufacturing industries and pharaceutical laboratories soeties researchers are interested in the variation of their products. To easure the variations within the values of study variable y, the proble of estiating the population variance of Sy variable y also received a considerable attention of the statistician in the survey sapling. Liu (1974 gave a general class of quadratic estiators for variance and obtained a class of unbiased estiators under certain conditions. Das and Tripathi (1978 defined six estiators of population variance Sy using known inforation on paraeters of auxiliary variable. Using prior inforation on paraeters of auxiliary variable/variables, Srivastava and Jhajj (1980, 1995, Isaki (1983, Singh and Kataria (1990, Prasad and Singh (1990, 199, Ahed et al. (000 have defined estiators or classes of estiators of Sy. In a situation when prior inforation on paraeters of auxiliary variables is not available, using double sapling technique, Singh and Singh (001 defined a ratio-type estiator of Sy. Ahed et al. (003 gave soe chain ratio-type as well as chain product-type estiators of Sy, under two-phase sapling schee. Al-Jararha and Ahed (00 defined two classes of estiators of Sy by using prior inforation on paraeter of one of the two auxiliary variables under double Received April 8, 004. Revised July 8, 004. Accepted February 16, 005. *Departent of Statistics, Punjabi University, Patiala , India. **Departent of Matheatics, Guru Nanak Dev University, Aritsar , India. Eail:

2 74 H. S. JHAJJ ET AL. sapling schee. When the population ean X and population variance Sx of auxiliary variable x (highly correlated with study variable y are known, Srivastava and Jhajj (1980 defined a class of estiators of Sy as ( ˆV g = g s y, x X, s x (1.1 Sx where g(,, is paraetric function satisfying certain regularity conditions; x, s x and s y are saple ean of x and saple variances of x and y respectively for the saple of size n. If X and S x are unknown then following Srivastava and Jhajj (1987, under double sapling technique, one can define a general class of estiators of population variance Sy as ˆV gd = g d (s y, x x, s x (1. s x where g d (,, is a paraetric function such that g d (Sy, 1, 1 = Sy and satisfies certain regularity conditions; x and s x are the saple ean and saple variance of variable x for the preliinary large saple of size n ; x, s x and s y are saple ean of x and saple variances of x and y respectively for the sub saple of size n(n <n under the double sapling technique. In such situation, soeties the inforation on population ean Z and population variance Sz of another auxiliary variable z, highly correlated with study variable y, is available in advance. Following Srivastava and Jhajj (1980, one can generalize the class of estiators defined in (1. as ( ˆV Hd = H s y, x x, s x s, z Z, s z (1.3 x Sz where z and s z are the saple ean and saple variance of variable z in the second phase saple of double sapling. In the class ˆV Hd, both the auxiliary variables x and z are considered to be highly correlated with the study variable y. But soeties in a trivariate distribution consisting study variable y and two auxiliary variables x and z, in which x is highly correlated with both variables y and z; whereas the variables y and z have no direct correlation with each other but they are just correlated with each other due to only their correlation with variable x, such as (i In any agricultural experient, both the yield of crop (say y and the labour deployed (say z are highly correlated with the area under crop (say x. Whereas the yield of crop (y and the labour deployed (z are correlated with each other due to only their correlation with the area under crop (x. (ii In any repetitive survey, the values of a variable of interest corresponding to both the last to last year (say z and the current year (say y are highly

3 EFFICIENT CLASS OF CHAIN ESTIMATORS OF POPULATION VARIANCE 75 correlated with the values of the sae variable corresponding to the last year (say x. Whereas the values corresponding to the last to last year (z and the values corresponding to the current year (y are correlated with each other due to only their correlation with the values of the sae variable corresponding to the last year (x. In such situations, we propose a class of chain estiators of Sy. The word chain estiator eans that first iprove the estiators x and s x of X and S x respectively by using known values of population ean Z and variance Sz of variable z. In turn, these iproved estiators are used for the estiation of Sy which leads to the creation of chain estiators. The asyptotic expressions for the ean squared errors and their iniu values are obtained for the proposed class of chain estiators and the generalized class ˆV Hd. It has been shown that the optiu estiator of the class of chain estiators has sipler for as copared to that of the class ˆV Hd. A coparison aong the different classes of estiators of Sy with respect to their ean squared error is also ade epirically. This coparison shows that the proposed class of chain estiators is ore efficient than the generalized class ˆV Hd and hence recoended in practical applications for estiating Sy.. Notations and expectations Fro the population of size N, select a first phase siple rando saple of size n and observe both the variables x and z for the selected units. A second phase siple rando saple of size n (n <n is selected fro the first phase saple and variables x, y and z are easured on these selected units. Let the values of variables x, y and z be denoted by X j, Y j and Z j respectively on the j-th unit of the population; j = 1,,..., N and the corresponding sall letters x j, y j and z j denote the saple values. We write Ȳ = 1 N 1 N 1 N Y j, X = X j, Z = Z j N N N j=1 j=1 j=1 Sy = 1 N (Y j N 1 Ȳ, Sx = 1 N (X j N 1 X, j=1 j=1 N (Z j Z j=1 µ rst = 1 N (Y j N Ȳ r (X j X s (Z j Z t µ rst, λ rst = j=1 S z = 1 N 1 Obviously µ r/ 00 µs/ 00 µt/ 00 C 0 = S y, C 1 = S, C = Ȳ x X S z Z ρ yx = ρ 01 = λ 110, ρ yz = ρ 0 = λ 101, ρ xz = ρ 1 = λ 011..

4 76 H. S. JHAJJ ET AL. Let z and s z denote the saple ean and saple variance of variable z for the first phase saple of size n. In this paper, all the sapling variances have been defined either with divisor n 1orn 1 depending on first phase saple or second phase saple respectively. Letting ω = s y Sy, u 1 = x x, v 1 = z Z, v 1 = z Z, u = s x s, v = s z x Sz, v = s z Sz. For the sake of siplicity, assue that N is large enough as copared to n and n so that all finite population correction (fpc ters are ignored. For the given double sapling technique when both the saples drawn are siple rando saples (without replaceent, we have the following expectations: E(ω =E(u 1 =E(v 1 =E(v 1=E(u =E(v =E(v =1 E(u 1 1(v 1 1 = E(u 1 1(v 1 = E(u 1(v 1 1 = E(u 1(v 1 = 0 E(v 1 1 = 1 n C, E(v 1 1 = 1 n C and up to the ters of order n 1,wehave E(ω 1 = 1 n (λ 400 1, E(u 1 1 = n 1 n C1 E(u 1 = n 1 n (λ 040 1, E(v 1 = 1 n (λ E(v 1 = 1 n (λ 004 1, E(ω 1(u 1 1 = n 1 n λ 10 C 1 E(ω 1(v 1 1 = 1 n λ 01C, E(ω 1(v 1 1 = 1 n λ 01C E(ω 1(u 1 = n 1 n (λ 0 1, E(ω 1(v 1 = 1 n (λ 0 1 E(ω 1(v 1 = 1 n (λ 0 1, E(u 1 1(v 1 1 = n 1 n λ 011 C 1 C E(v 1 1(v 1 = 1 n λ 003C, E(v 1 1(v 1 = 1 n λ 003C E(u 1 1(u 1 = n 1 n λ 030 C 1, E(u 1 1(v 1 = n 1 n λ 01 C 1 E(v 1 1(u 1 = n 1 n λ 01 C, E(u 1(v 1 = n 1 n (λ 0 1.

5 EFFICIENT CLASS OF CHAIN ESTIMATORS OF POPULATION VARIANCE Proposed class of chain estiators of Sy Suppose that ean X and variance Sx of first auxiliary variable x are unknown but ean Z and variance Sz of second auxiliary variable z are known in advance. It is assued that the variable z is highly correlated with variable x whereas the correlation between the variables y and z exists due to only the high correlation of variable x with the variables y and z. In such situations, we propose a class of chain estiators of Sy as ( ˆV Td = T s y, x x, s x (3.1 s, z x Z, s z Sz = T (s y,u 1,u,v 1,v where T (,,,, is a paraetric function of s y, u 1, u, v 1 and v such that (3. T (Sy, 1, 1, 1, 1 = Sy, for all Sy. Whatever saple is chosen, let the point (s y,u 1,u,v 1,v assue values in a bounded, closed convex subset R of the five diensional real space containing the point (Sy, 1, 1, 1, 1. The function T (,,,, is continuous and bounded having continuous and bounded first and second order partial derivatives in R. Since there are only a finite nuber of saples therefore under the above conditions, the expectation and the ean square error of the estiators of the class ˆV Td exist. On using second order Taylor s series expansion of T (s y,u 1,u,v 1,v about the point (Sy, 1, 1, 1, 1, the ean square error (MSEof ˆV Td, up to the ters of order n 1,is (3.3 MSE( ˆV Td = 1 n S4 y(λ n C T 4 +(λ 004 1T 5 +C S yλ 01 T 4 +Sy(λ 0 1T 5 +C λ 003 T 4 T 5 } + n 1 n C1T +(λ 040 1T3 +C 1 Syλ 10 T +Sy(λ 0 1T 3 +C 1 λ 030 T T 3 } where T i ; i =, 3, 4, 5 denote the first order partial derivatives of T (s y,u 1,u, v 1,v with respect to u 1, u, v 1 and v at the point (S y, 1, 1, 1, 1 respectively. The MSE of ˆV Td as given in (3.3 is iniized for T = S ( y λ030 (λ 0 1 λ 10 (λ (3.4 C 1 λ 040 λ 030 ( 1 T 3 = Sy λ030 λ 10 λ 0 +1 (3.5 λ 040 λ T 4 = S ( y λ003 (λ 0 1 λ 01 (λ (3.6 C λ 004 λ 003 ( 1 T 5 = Sy λ003 λ 01 λ 0 +1 (3.7 λ 004 λ 003 1

6 78 H. S. JHAJJ ET AL. and iniu ean square error of ˆV Td, up to the ters of order n 1,is (3.8 (3.9 Min.MSE( ˆV Td [ 1 = Sy 4 n (λ n λ 01 + (λ 0 λ 01 λ } λ 004 λ n 1 n λ 10 + (λ 0 λ 10 λ }] λ 040 λ = Min.MSE( ˆV gd 1 n S4 y λ 01 + (λ 0 λ 01 λ λ 004 λ } where Min.MSE( ˆV gd is the iniu asyptotic ean square error of the estiators of the class ˆV gd, up to the ters of order n 1, and is given by [ ( Min.MSE( ˆV 1 1 gd =Sy 4 n (λ n 1 (3.10 n λ 10 + (λ 0 λ 10 λ }] λ 040 λ Rewriting(3.9, we have (3.11 Min.MSE( ˆV gd Min.MSE( ˆV Td = 1 n S4 y λ 01 + (λ 0 λ 01 λ } λ 004 λ In (3.11, the right hand side is the su of two non-negative quantities, since λ 004 λ always. Thus we found that Min.MSE( ˆV Td is always saller than Min.MSE( ˆV gd. 4. Coparison of the class ˆV Td with the class ˆV Hd To copare the generalized class ˆV Hd = H(s y,u 1,u,v 1,v with the proposed class of chain estiators ˆV Td = T (s y,u 1,u,v 1,v, we require the ean square error of ˆV Hd. Proceedingin the sae way as in Section 3, the asyptotic ean square error of ˆV Hd (up to the ters of order n 1 is (4.1 MSE( ˆV Hd = 1 n (λ n [C H 4 +(λ 004 1H 5 +C S yλ 01 H 4 +Sy(λ 0 1H 5 +C λ 003 H 4 H 5 ] + n 1 n [C1H +(λ 040 1H3 +C 1 Syλ 10 H +S y(λ 0 1H 3 +C 1 λ 030 H H 3 +C 1 C ρ 1 H H 4 +C 1 λ 01 H H 5 +C λ 01 H 3 H 4 +(λ 0 1H 3 H 5 ]

7 EFFICIENT CLASS OF CHAIN ESTIMATORS OF POPULATION VARIANCE 79 where H i ; i =, 3, 4, 5 denote the first order partial derivatives of the function H(s y,u 1,u,v 1,v with respect to u 1, u, v 1 and v at the point (S y, 1, 1, 1, 1 respectively. The MSE( ˆV Hd is iniized for (4. (4.3 (4.4 (4.5 H = S } y ρ1 (λ 01 ρ 1 λ 10 C 1 ρ + λ ( H 5 (λ 01 kρ 1 (H 3 + θs C yh 5 λ 030 L } 3ρ 1 1 ρ 1 L 3 (λ H 3 = Sy ρ 01 ρ 1 λ 10 L 1 H 5 (L 4 + L 3 k 1 L L 3 H 4 = S y C H 5 = ( λ01 ρ 1 λ 10 ρ 1 ρ 1 1 [ L3 H 3 C ρ + H 5 k L 3θ(1 + S }] y 1 ρ 1 ( ( }+λ 01 λ 10 λ 01 ρ 1 k θ λ 030 L 3 ρ 1 ρ k L 3 θ ρ 1 +θ(λ 0 1 (λ (λ λ 01 k(λ 003 ρ 1 λ 01 where L 1 = λ 0 λ 10 λ 030 1, L = λ 040 λ 030 1, L 3 = λ 01 ρ 1 λ 030, L 4 = λ 0 λ 01 λ = n n n, k = λ 003 ρ 1 λ 01 ρ, θ = L 4 L 3 k 1 L L 3 ρ 1 and the iniu ean square error of ˆV Hd, up to the ters of order n 1,is given by (4.6 Min.MSE( ˆV Hd 1 n n 1 n ( λ 10 + L 1 = 1 n (λ L ( } L 1 λ 01 ρ 1λ 10 L 1L 3 } n L L (1 ρ 1 L 3 ( } ( L 1 ρ 1 L 3 L5 λ 10 ( L9 ( ( L 4 L 3 + L L (1 ρ 1 1 ρ 1 L 3 λ 003 ρ 1 λ 01 }[L ( 1 ρ 1 λ 003L L 7 } ( L10 L 3 } ( L 11 L 6 ( L8 λ 10 Lλ01 ( λ 003L L 7 ]

8 80 H. S. JHAJJ ET AL. ( } = Min.MSE( ˆV gd L Sy 4 1 λ 01 ρ 1λ 10 L 1L 3 } n L L (1 ρ 1 L 3 ( } ( L 1 ρ 1 L 3 L5 ( λ 10 L9 λ 003L L 7 ( ( 1 + L } ( 4 1 ρ 1 L 3 λ 003 ρ 1 λ 01 L10 ( } n L L (1 ρ 1 L 3 }[L 1 ρ 1 L 3 ( L 11 L 6 where ( L8 λ 10 Lλ01 ( λ 003L L 7 L 5 = λ 01 λ 030 L 4, L 6 = λ 01L + L 4, L 7 = ρ 1 λ 01 L + L 3 L 4, L 8 = ρ 1 L λ 030 L 3, L 9 = L (λ 0 1, L 10 = L (λ 0 1, L 11 = L (λ The condition, under which the optiu estiator of the class ˆV Td is ore efficient than that of the class ˆV Hd, is obtained by using the expressions (3.9 and (4.6 in (4.7 Min.MSE( ˆV Td < Min.MSE( ˆV Hd. Fro (4.7, we are not able to get a concrete atheatical result about the efficiency of the class of chain estiators ˆV Td over the generalized class ˆV Hd. To have an idea about the efficiency of one estiator over the other, we have considered the following specific cases: Case 1. When (y, x, z assued to follow trivariate noral distribution then all odd ordered oents are vanish to zero. In this case the expressions (3.8 and (4.6 respectively reduce to (4.8 and (4.9 Min.MSE( ˆV Td Min.MSE( ˆV Hd Using (4.8 and (4.9, we have = 1 n (λ n 1 (λ0 1 n λ (λ 0 1 n λ = 1 n (λ n 1 (λ0 1 n λ (λ 0 1(λ (λ } 0 1(λ 0 1 n (λ (λ 040 1(λ (λ 0 1 ] }. (4.10 Min.MSE( ˆV Hd Min.MSE( ˆV Td = 1 (λ 0 1 n (λ (λ 0 1(λ (λ } 0 1(λ 0 1 }. n (λ (λ 040 1(λ (λ 0 1

9 EFFICIENT CLASS OF CHAIN ESTIMATORS OF POPULATION VARIANCE 81 Case. When n = n, we see that the two proposed classes ˆV Td and ˆV Hd of estiators of Sy coincide with each other, that is, ˆV Td = ˆV Hd = ˆV (say which is defined as ˆV ˆV (s y, z Z, s z = ˆV (s S y,ν z 1,ν. Therefore, up to the ters of order n 1, the iniu ean square error of ˆV is given by (4.11 Min.MSE( ˆV = 1 n [ (λ Using (3.8 and (4.11, we note that λ 01 + (λ 0 λ 01 λ λ 004 λ }]. (4.1 Min.MSE( ˆV Min.MSE( ˆV Td = n 1 n [ (λ 10 λ 01+ (λ0 λ 10 λ λ 040 λ (λ 0 λ 01 λ }] λ 004 λ For trivariate noral distribution (4.1 reduces to (4.13 Min.MSE( ˆV Min.MSE( ˆV Td = n 1 (λ 0 1 n λ (λ 0 1 }. λ We see that even in the specific cases considered above we could not find a concrete atheatical result showing the efficiency of one over the other. But it sees that ˆV Td ust be better than ˆV Hd because ˆV Td akes full use of inforation on n observations of first phase saple whereas ˆV Hd waste the inforation on n n observations of the saple. Reark 4.1. It should be noted that the efficient use of the estiators of the two proposed classes ˆV Hd and ˆV Td presues that the optiu values of H i and T i ; i =, 3, 4, 5 are known. But these optiu values are functions of unknown population paraeters. Srivastava and Jhajj (1983 have shown that the estiators of the class with estiated values of optiu paraeters obtained by their consistent estiators, attain the sae iniu ean square error of estiators of the class based on optiu values, up to the first order of approxiation. Although by using the sae approach of Srivastava and Jhajj (1983, we can construct the estiators of the classes ˆV Hd and ˆV Td. But the optiu estiator of the class ˆV Hd is very uch coplicated as copared to that of ˆV Td. So due to such type of coplexities, we should prefer the proposed class ˆV Td of

10 8 H. S. JHAJJ ET AL. the chain estiators as copared to the general class ˆV Hd, in practice. Reark 4.. The proposed classes ˆV Td and ˆV Hd of estiators of Sy are very large. Any paraetric function T (s y,u 1,u,v 1,v (or H(s y,u 1,u,v 1,v satisfying certain regularity conditions and T (Sy, 1, 1, 1, 1 = Sy (or H(Sy, 1, 1, 1, 1 = Sy for all Sy, can generate an estiator of the class ˆV Td (or ˆV Hd. For exaple, we have the following functions which generate soe of the siple estiators of these classes ˆV Td and ˆV Hd : 1. T(s y,u 1,u,v 1,v =s yu α 1 u β v γ 1 v δ and H(s y,u 1,u,v 1,v =s yu α 1 u β vγ 1 vδ. T(s y,u 1,u,v 1,v =s y[α(u β(u 1 + γ(v δ(v 1] and H(s y,u 1,u,v 1,v =s y[α(u β(u 1 + γ(v δ(v 1] 3. T(s y,u 1,u,v 1,v =s y[a 1 u α 1 u β + a v γ 1 v δ ]; a 1 + a = 1 and H(s y,u 1,u,v 1,v =s y[a 1 u α 1 u β + a v γ 1 vδ ]; a 1 + a =1 4. T(s y,u 1,u,v 1,v = s4 y + s yα(u β(u 1} s y + γ(v δ(v 1 and H(s y,u 1,u,v 1,v = s4 y + s yα(u β(u 1} s y + γ(v δ(v 1 5. T(s y,u 1,u,v 1,v = s ye α(u 1 1+β(u 1 1+γ(v δ(v 1 and H(s s ye α(u 1 1+β(u 1 y,u 1,u,v 1,v = 1+γ(v δ(v 1. Here the optiu values of α, β, γ and δ in these estiators are so deterined that they satisfy the respective noral equations and the resulting estiators should have the sae iniu asyptotic ean square errors as given in (3.8 and (4.6 respectively. 5. Coparison of the proposed class ˆV Td with the existing ones On using the inforation on variances alone for the two auxiliary variables x and z, Al-Jararha and Ahed (00 defined a class of chain estiators of Sy as: ˆbg = f 1 (s y, s x s, s z (5.1 x Sz or ˆbg = f 1 (s y,u,ν. Up to the ters of order n 1, the iniu ean square error of ˆb g is given by (5. Min.MSE(ˆb g = 1 n (λ n 1 (λ0 1 n λ (λ 0 1 n λ

11 EFFICIENT CLASS OF CHAIN ESTIMATORS OF POPULATION VARIANCE 83 In the present paper, using inforation on variances of the two auxiliary variables x and z along with their eans, we have proposed the two classes ˆV Hd and ˆV Td of estiators of S y. Obviously our proposed class of chain estiators ˆV Td is a generalization of the class ˆb g. On using (3.8 and (5., we have (5.3 Min.MSE(ˆb g Min.MSE( ˆV Td = n 1 λ030 (λ 0 1 λ 10 (λ 040 1} n (λ 040 1(λ 040 λ λ 003 (λ 0 1 λ 01 (λ 004 1} n (λ 004 1(λ 004 λ Fro (5.3, we found that the proposed class of chain estiators of Sy i.e. ˆV Td is always ore efficient than the existing class ˆb g. The sign of equality will hold in (5.3 if (y, x, z follows trivariate noral distribution, that is, the two classes ˆV Td and ˆb g becoe equally efficient in trivariate noral distribution. So it is interesting to note that if (y, x, z has trivariate noral distribution then the available inforation regarding eans of auxiliary variables becoes useless for the estiation of population variance Sy. In the sae paper, Al-Jararha and Ahed also defined another wider class of estiators of Sy using the sae inforation on variances alone for the two auxiliary variables x and z as (5.4 ˆbh = f (s y, s x s x, s z s z, s z Sz. Up to the ters of order n 1, the iniu ean square error of ˆb h is given by (5.5 Min.MSE(ˆb h Sy 4 = 1 n (λ n 1 (λ0 1 n λ (λ 0 1 n λ n 1 (λ0 1(λ 0 1 (λ 0 1(λ 040 1} n (λ 040 1(λ 040 1(λ (λ 0 1 }. Using (3.8 and (5.5, we have (5.6 Min.MSE(ˆb h Min.MSE( ˆV Td Sy 4 = 1 (λ 0 1λ 003 (λ 004 1λ 01 } n (λ 004 1(λ 004 λ 003 ( n 1 n

12 84 H. S. JHAJJ ET AL. (λ 004 1λ 10 (λ λ 030 (λ 0 1} (λ 0 1(λ 0 1 (λ 0 1(λ 040 1} (λ 040 1λ 10 (λ 0 1 λ 030 (λ 0 1 } +λ 030 (λ 0 1(λ 0 1λ 10 (λ 0 1 λ 030 (λ 0 1} ( λ040 λ (λ (λ (λ 0 1 }. Fro (5.6, no concrete decision regarding the efficiency of one over the other can be drawn. 6. Nuerical illustration Since in the Sections 4 and 5, we could not obtained any concrete theoretical conditions under which proposed class ˆV Td is better than ˆV Hd and ˆb h. So to have a rough idea about the efficiencies of the proposed class of chain estiators ˆV Td and the generalized class of estiators ˆV Hd, we take the two epirical populations considered in the literature. The source of population; nature of the variables y, x and z; population size N and various possible population correlation coefficients are given in Table 1. The values of requisite population paraeters for the two Table 1. Description of populations. S.No. Source y x z N ρ yx ρ yz ρ xz 1. Murthy (1967 Output Fixed No. of Page 88 Capital Workers Factories Murthy (1967 Area Area Cultivated Page 399 under under Area in Villages 1 34 Wheat Wheat 1961 in 1964 in 1963 Table. Paraeters of two populations. Paraeters Values of paraeters Population No.1 Population No. λ λ λ λ λ λ λ λ λ λ λ λ

13 EFFICIENT CLASS OF CHAIN ESTIMATORS OF POPULATION VARIANCE 85 Table 3. Percentage efficiency of optiu estiators of S y. For population No. 1 Saple Efficiency of Efficiencies of optiu estiators belonging to the class sizes estiator n n s y ˆV gd ˆbg ˆbh ˆVHd ˆVTd For population No. Saple Efficiency of Efficiencies of optiu estiators belonging to the class sizes estiator n n s y ˆV gd ˆbg ˆbh ˆVHd ˆVTd populations are given in Table. Table 3 gives the efficiencies of the optiu estiators of the classes ˆV gd, ˆb g, ˆb h, ˆV Hd and ˆV Td relative to the estiator s y. Table 3 shows that the proposed class of chain estiators ˆV Td is always ore efficient than the others in every case for both the populations considered. 7. Conclusions Fro the Sections 4 and 5, we see that it is very cubersoe to obtain the optiu estiator of the generalized class ˆV Hd as well as its optiu ean square error, whereas it is very siple in the case of the proposed class of chain estiators ˆV Td. Also fro Table 3, we see that for both the epirical populations, the optiu estiator of the proposed class ˆV Td is always ore efficient than that of all the classes ˆV gd, ˆb g, ˆb h and ˆV Hd. So we conclude that the use of chain estiators belonging to class ˆV Td should be preferred over the estiators belonging to the generalized class ˆV Hd when the variable z is highly correlated with variable x instead of variable y, whereas the correlation between the variables y and z exists due to only the high correlation between the variables y and x. Acknowledgeents The authors thank the referees for their coents which iproved the quality of this paper very uch.

14 86 H. S. JHAJJ ET AL. References Ahed, M. S., Raan, M. S. and Hossain, M. I. (000. Soe copetitive estiators of finite population variance using ultivariate auxiliary inforation, Inforation and Manageent Sciences, 11(1, Ahed, M. S., Walid, A. D. and Ahed, A. O. H. (003. Soe estiators for finite population variance under two-phase sapling, Statistics in Transition, 6(1, Al-Jararha, J. and Ahed, M. S. (00. The class of chain estiators for a finite population variance using double sapling, Inforation and Manageent Sciences, 13(, Das, A. K. and Tripathi, T. P. (1978. Use of auxiliary inforation in estiating the finite population variance, Sankhya C, 40, Isaki, C. T. (1983. Variance estiation using auxiliary inforation, J. Aer. Stat. Assoc., 78, Liu, T. P. (1974. A general unbiased estiator for the variance of a finite population, Sankhya C, 36(1, 3 3. Murthy, M. N. (1967. Sapling Theory and Methods, Statistical Publishing Soc., Calcutta, India. Prasad, B. and Singh, H. P. (1990. Soe iproved ratio type estiators of finite population variance in saple surveys, Coun. Statist. - Theory and Methods, 19(3, Prasad, B. and Singh, H. P. (199. Unbiased estiators of finite population variance using auxiliary inforation in saple surveys, Coun. Statist. - Theory and Methods, 1(5, Singh, H. P. and Singh, R. (001. Iproved ratio type estiator for variance using auxiliary inforation, J. Ind. Soc. Agri. Statist., 54(3, Singh, S. and Kataria, P. (1990. An estiator of finite population variance, J. Ind. Soc. Agri. Statist., 4(, Srivastava, S. K. and Jhajj, H. S. (1980. A class of estiators using auxiliary inforation for estiating finite population variance, Sankhya C, 4(1, Srivastava, S. K. and Jhajj, H. S. (1983. A class of estiators of the population ean using ulti-auxiliary inforation, Cal. Stat. Assoc. Bull., 3, Srivastava, S. K. and Jhajj, H. S. (1987. Iproved estiation in two phase and successive sapling, J. Ind. Stat. Assoc., 5, Srivastava, S. K. and Jhajj, H. S. (1995. Classes of estiators of finite population ean and variance using auxiliary infoation, J. Ind. Soc. Agri. Statist., 47(,

Keywords: Estimator, Bias, Mean-squared error, normality, generalized Pareto distribution

Keywords: Estimator, Bias, Mean-squared error, normality, generalized Pareto distribution Testing approxiate norality of an estiator using the estiated MSE and bias with an application to the shape paraeter of the generalized Pareto distribution J. Martin van Zyl Abstract In this work the norality