Mixture Ratio Estimators Using Multi Auxiliary Variables and Attributes for Two Phase Sampling

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http://dx.doi.org/10.4236/ojs.2014.49073

Mixture Ratio Estimators Using

Multi-Auxiliary Variables and Attributes for

Two-Phase Sampling

Paul Mwangi Waweru, John Kung’u, James Kahiri

Department of Statistics and Actuarial Science, Kenyatta University, Nairobi, Kenya Email: [email protected], [email protected]

Received 25 July 2014; revised 27 August 2014; accepted 12 September 2014

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Abstract

In this paper, we have proposed three classes of mixture ratio estimators for estimating popula-tion mean by using informapopula-tion on auxiliary variables and attributes simultaneously in two-phase sampling under full, partial and no information cases and analyzed the properties of the estima-tors. A simulated study was carried out to compare the performance of the proposed estimators with the existing estimators of finite population mean. It has been found that the mixture ratio es-timator in full information case using multiple auxiliary variables and attributes is more efficient than mean per unit, ratio estimator using one auxiliary variable and one attribute, ratio estimator using multiple auxiliary variable and multiple auxiliary attributes and mixture ratio estimators in both partial and no information case in two-phase sampling. A mixture ratio estimator in partial information case is more efficient than mixture ratio estimators in no information case.

Keywords

Ratio Estimator, Multiple Auxiliary Variables, Multiple Auxiliary Attributes, Two-Phase Sampling, Bi-Serial Correlation Coefficient

1. Introduction

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information in selecting the sample with varying probabilities. If regression line is still linear but does not pass through the origin the regression estimator is used. Watson [4] used the regression estimator of leaf area on leaf weight to estimate the average area of the leaves on a plant.

Olkin [5] was the first person to use information on more than one auxiliary variable, which was positively correlated with the variable under study, using a linear combination of ratio estimator based on each auxiliary variable. Raj [6] suggested a method of using multi-auxiliary information in sample survey. Singh [7] proposed a ratio-cum-product estimator and its multi-variable expression.

The concept of double sampling was first proposed by Neyman [1] in sampling human populations when the mean of auxiliary variable was unknown. It was later extended to multi-phase by Robson [8]. Ahmad [9] pro-posed a generalized multivariate ratio and regression estimators for multi-phase sampling while Zahoor, Muh-hamad and Munir [10] suggested a generalized regression-cum-ratio estimator for two-phase sampling using multiple auxiliary variables.

Jhajj, Sharma and Grover [11] proposed a family of estimators using information on auxiliary attribute. They used known information of population proportion possessing an attribute (highly correlated with study variable

Y). The attribute are normally used when the auxiliary variables are not available e.g. amount of milk produced and a particular breed of cow or amount of yield of wheat and a particular variety of wheat. Jhajj, Sharma and Grover [11] used the information on auxiliary attributes in ratio estimator in estimating population mean of the variable of interest using known attributes such as coefficient of variation, coefficient kurtosis and point bi-serial correlation coefficient. The estimator performed better than the usual sample mean and Naik and Gupta [12] es-timator. Jhajj, Sharma and Grover [11] also used the auxiliary attribute in regression, product and ratio type ex-ponential estimator following the work of Bahl and Tuteja [13].

Hanif, Haq and Shahbaz [14] proposed a general family of estimators using multiple auxiliary attribute in sin-gle- and double-phase sampling. The estimator had a smaller MSE compared to that of Jhajj, Sharma and Grover [11]. They also extended their work to ratio estimator which was generalization of Naik and Gupta [12] estima-tor in single- and double-phase samplings with full information, partial information and no information. Kung’u and Odongo [15] and [16] proposed ratio-cum-product estimators using multiple auxiliary attributes in single- and two-phase sampling. Moeen, Shahbaz and Hanif [17] proposed a class of mixture ratio and regression esti-mators for single-phase sampling for estimating population mean by using information on auxiliary variables and attributes simultaneously.

In this paper, we will extend the mixture ratio estimator proposed by Moeen, Shahbaz and Hanif [17] in sin-gle-phase sampling to two-phase sampling under full, partial and no information case strategies introduced by Samiuddin and Hanif [18] and also incorporate Arora and Bansi [19] approach in writing down the mean squared error.

2. Preliminaries

2.1. Notation and Assumption

Consider a population of N units. Let Y be the variable for which we want to estimate the population mean and

1, 2, , k

X X  X are k auxiliary variables. For two-phase sampling design let n1 and n2

(

n2<n1

)

are sample

sizes for first and second phase respectively. xj_{( )}1 and xj( )2 denote the

th

j auxiliary variables form first and second phase samples respectively and y2 denote the variable of interest from second phase. Xj and Cxj

denote the population means and coefficient of variation of jth auxiliary variables respectively and ρyx_j de-notes the population correlation coefficient of Y and Xj.

Further, let ₁ ₂

(

₁ ₂

)

1 2

1 1 1 1

n N n N

θ =_ − _ θ =_ − _ θ <θ

   

( ) ( ) ( ) ( )

( ) ( )

(

)

2 1

2

2 1

2

,

and 1, 2, ,

j

y j j x

j x

j

y Y e x X e

x X e j k

= + = +

= + =  (1.0)

where ey₂, ex_j_{( )}₁ and ex_j_{( )}₂ are sampling error and are very small. We assume that

( )

y2

( )

xj( )1

( )

xj( )2 0.

(3)

Consider a sample of size n drawn by simple random sampling without replacement from a population of size

N. Let yj and τj denotes the observations on variable y and r respectively for the th

j unit where 1, 2, ,

j= +k k+  q.

In defining the attributes we assume complete dichotomy so that,

1, if unit of population possess auxiliary attribute,

0, otherwise.

th th

ij

i j

τ _{= }

 (1.2)

Let

1

N

j ij

j

A τ

=

∑

and

1

n

j ij i

a τ

=

∑

be the total number of units in the population and sample respectively

pos-sessing attribute τj. Let j j

A P

N

= and p_j_{( )}₂ aj n

= be the corresponding proportion of units possessing a specific

attributes τj and y is the mean of the main variable at second phase. Let pj( )1 and pj( )2 denote the

th

j

auxiliary attribute form first and second phase samples respectively and y2 denote the variable of interest from

second phase. The mean of main variable of interest at second phase will be denoted by y2. Also let us define

( )1 ( )1 , ( )2 ( )2

j j j j j j

e_τ = p −P e_τ = p −P (1.3)

where ey₂, eτ_j_{( )}₁ and eτ_j_{( )}₂ are sampling error and are very small. We assume that

( )

y2

( )

j( )1

( )

j( )2 0.

E e =E e_τ =E e_τ =

Let j ( )2 j j

x v

X

= , then _j 1 ( )2j 1 ( )2j j x( )2j

j j j

e

x x X

v

X X X

−

− = − = = . Therefore _j 1 x( )2j

j

e v

X

= + . Similarly, _j 1 ( )2j j

e w

P

τ

= + .

Also ( )

( )

2

1

j j

j

x z

x

= , then

( ) ( ) ( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( ) ( )

2 1 2 1 2 1 2 1 1 1

1

2 1

1 1

1 1 .

1

j j j j j j j j j

j j

x x x x x x x x x j j

j

x

x j j j

j j

j

e e e e e e e e e

x x

v

x x e X e X X

X

−

− − − −  

−

 

− = = = = = +

 

+   _ _

 + 

 

 

We shall take vi−1 to term of order 1

n as

( )2 ( )1

1 x j xi j j

j

e e

v

X

−

− = hence,

( )2 ( )1 ( )1 ( )2

1 x j x j 1 x j x j . j

j j

e e e e

z

X X

− −

= + = −

Similarly,

( )1 ( )2

1 j j .

j

e e

u

P

τ − τ

= − (1.4)

The coefficient of variation and correlation coefficient are given by

2 1 1

1 1

2 2 2

1 1

, and

, , , ,

, _i _i .

y x x yx

y x yx

y x

yz y y

yz Pb Pb

y z y y

S S S S

C C C

S S

Y X P

S S S

S S S S S S

τ

τ τ

ρ

ρ ρ ρ

= = = =

= = =

(1.5)

(4)

( )

(

( ) ( )

)

(

)

(

( ) ( )

)

(

)

( )

(

)

(

( )

)

( )

(

( ) ( )

)

(

)

(

)

(

_{( )}

(

_{( )} _{( )}

)

(

)

( )

2 2 1 2 1 2

2 2 2 2 2 2

2 1 2 2 1 2

2

2 2

2 ₂ ₂ ₂ ₂ ₂ ₂

2 2 1 2 1

2 2

1 2 1 2

, , ,

, ,

j j

j j j j

j j j

j j j j j

j

y y j x x j x

y x j y x yx y j y Pb

y x x j y x yx x x x j x

E e Y C E e e P C E e e X C

E e e YX C C E e e YP C C

E e e e YX C C E e e e X C

E e e

τ

τ τ τ

τ τ

τ

θ θ θ θ θ

θ ρ θ ρ

θ θ ρ θ θ

= − = − − = −

= =

− = − − = −

( ) ( )

(

)

(

)

(

)

(

_{( )} _{( )}

)

(

)

( ) ( )

(

)

(

)

(

_{( )}

(

_{( )} _{( )}

)

(

)

( ) ( )

(

)

(

( ) ( )

)

(

)

(

)

(

)

( ) ( )

(

)

(

( ) ( )

)

(

)

(

)

1 2 2 2

2 2 2 1 2

1 2 1 2 1 2 1 2

2 2

1 2 2

2 1 2

2 1

,

, ,

,

;

j i j i j

j j i j

i j i j j j

i j j j

j i j i

i i j j

j x x i j x x x x

i j Pb y x x j y x yx

i j

x x x x i

e P C E e e X X C C i j

E e e P P C C i j E e e e YX C C

E e e e e P P C C i j

E e e e e X

τ τ τ

τ τ τ τ τ τ

τ τ τ τ τ τ τ τ

θ θ θ ρ

θ ρ θ θ ρ

θ θ ρ θ θ

− = − = ≠

= ≠ − = −

− − = − ≠

− − = −

(

)

( )

(

( ) ( )

)

(

)

(

)

(

)

( )

(

( ) ( )

)

(

)

(

)

(

)

2 1 2 2 1 2

1 2

;

j i j i

j i i

j i j i

i i i

j x x x x

x x x i j x x x x

i j

X C C i j

E e e e X X C C i j

E e_τ e_τ e_τ P P C C_τ _τ _{τ τ} i j

ρ

θ θ ρ

≠

− = − ≠

(1.6)

( )

1 1 T

ij

Adj A

A C

A A

− ₌ ₌

(1.7)

(

2

)

[ ]

.

1 Arora and Lai 19

q

q q

yx

y x x

R

R = −ρ  (1.8)

The following notations will be used in deriving the mean square errors of proposed estimators.

q

yx

R

 : Determinant of population correlation matrix of variables y x, , 1 x2, , xq−1 and xq.

i _q

yx _yx

R



: Determinant of ith minor of

p

yx

R

 corresponding to the

th

i element of ρyx_i.

2

r

yx

ρ : Denotes the multiple coefficient of determination of y on x1, x2, , xr−1 and xr.

2

q

yx

ρ

 : Denotes the multiple coefficient of determination of y on y x, , 1 x2, , xq−1 and xq.

r

x

R



: Determinant of population correlation matrix of variables x1, x2, , xr−1 and xr.

q

x

R

 : Determinant of population correlation matrix of variables , x1 x2, , xq−1 and xq.

i r

y x

R



: Determinant of the correlation matrix of yi, , x1 x2, , xr−1 and xr.

.

i q

y x

R

 : Determinant of the correlation matrix of 1 2 1

, , , , and

i q q

y x x  x₋ x .

. .

i j r

y y x

R



: Determinant of the minor corresponding to

i j

y y

ρ of the correlation matrix of

(

)

1 2 1

, , , , , and

i j r r

y y x x  x− x i≠ j .

. .

i j q

y y x

R



: Determinant of the minor corresponding to ρy y_i _j of the correlation matrix of

(

)

1 2 1

, , , , , and .

i j q q

y y x x  x₋ x i≠ j (1.9) 2.2. Mean per Unit in Two-Phase Sampling

The sample mean y2 using simple random sampling without replacement in two-phase sampling is given by is

given by,

2

2 1 2

1 n i i

y y

n =

(5)

while its variance is given,

( )

2 2

Var y =θY C_y. (2.1) 2.3. Ratio Estimator Using Auxiliary Variable in Two-Phase Sampling

Let

2

2 1 2

1 n i i

x x

n =

=

∑

be the sample mean of the auxiliary variable in two-phase sampling. The ratio estimator when information on one auxiliary variables is available for population (full information case) is:

1

RV 2

2

.

X

t y

x

α

 

= _ _

  (2.2)

The mean square error of tRV can be written as:

( )

2

(

2₂ 2 2

)

RV 2 1 1

MSE t =θY Cy +α Cx −2αC Cx yρyx (2.3)

where 1

y yx

x

C C

ρ

α = and ρyx are the optimum value and the correlation coefficient respectively.

2.4. Ratio Estimator Using Multiple Auxiliary Variables in Two-Phase Sampling

The ratio estimator by Haq [9] when information on k auxiliary variables is available for population (full in-formation case) is:

( ) ( ) ( )

1 2

RMV 2

2 1 2 2 2

.

k

X

X X

t y

x x x

β β β

            =

     

      (2.4)

The optimum values of unknown constants are,

( )

1

1 j k .

j k

yx

j y yx

j

x x

R C

C R

β +

= −  

(2.5)

The mean square error of tRMV can be written as:

(

)

2 2

(

2

)

RMV 2

MSE t =θY Cy 1−ρyx_k .

 (2.6)

2.5. Ratio Estimator Using Auxiliary Attribute in Two-Phase Sampling

In order to have an estimate of the population mean Y of the study variable y, assuming the knowledge of the population proportion P, Naik and Gupta [12] defined ratio estimator of population mean when the prior infor-mation of population proportion of units possessing the same attribute is variable. Naik and Gupta [12] proposed the following estimator:

1

RA 2

2

.

P

t y

p

α

 

= _ _

  (2.7)

The MSE of tRA up to the first order of approximation are given respectively by,

( )

2

(

2 2 2

)

RA 2 1 1

MSE 2

j

y P P y Pb

t =θY C +α C − αC C ρ _τ (2.8)

where 1

y yx

x

C

ρ

α = and ρ_Pbτ are the optimum value and the bi-serial correlation coefficient respectively.

2.6. Ratio Estimator Using Multiple Auxiliary Attributes in Two-Phase Sampling

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auxiliary attributes is given by,

( ) ( ) ( )

1 2

MRA 2

2 1 2 2 2

.

q

P

P P

t y

p p p

γ γ γ

            = _ _{ } _ _ _

      (2.9)

The optimum values of unknown constants are,

( )

1

1 .

j q

y y j y j

R C

C R

τ _τ

τ τ

γ +

= − 



(2.10)

The MSE of the tMRA up to the first order of approximation is given by,

(

)

2 2₂

(

2

)

MRA 2

MSE 1 .

q

y Pb

t =θY C −ρ _τ

 (2.11)

In general these estimators have a bias of order 1

n. Since the standard error of the estimates is of order

1

n,

the quantity bias .s e is of order 1

n and becomes negligible as n becomes large.

3. Methodology

3.1. Mixture Ratio Estimators Using Multi-Auxiliary Variable and Attributes for Two-Phase Sampling (Full Information Case)

If we estimate a study variable when information on all auxiliary variables is available from population, it is uti-lized in the form of their means. By taking the advantage of mixture ratio estimators technique for two-phase sampling, a generalized estimator for estimating population mean of study variable Y with the use of multi auxiliary variables and attributes are suggested as:

( )

( ) ( ) ( ) ( ) ( ) ( )

1 2 1 2

1 2

2 MRVAF 3.0

2 1 2 2 2 2 1 2 2 2

.

k k k t

X P P P

X X

t y

x x x p p p

α α α β+ β+ β

+ +

                        = _ _{ } _ _ _{ } _ _ _ _ _

            (3.0)

Using (1.0), (1.3) and (1.4) in (3.0) and ignoring the second and higher terms for each expansion of product and after simplification, we write,

( ) 2 ( )2 ( )2

MRVAF 3.0

1 1

.

j j

k _x h k t

y i j

j j j k j

e e

t e Y Y Y

X P

τ

α + = β

= = +

= + −

∑

−

∑

(3.1)

The mean squared error of t_{MRVAF 3.0}_{( )} is given by,

( )

(

)

(

( )

)

2 ( )2 ( )2

2 2

MRVAF 3.0 MRVAF 3.0

1 1

MSE j j .

k _x h k t

y j j

j j j k j

e e

t E t Y E e Y Y

X P

τ

α + = β

= = +

 

 

= − = _ − − _



∑



(3.2)

We differentiate the Equation (3.2) partially with respect to αj

(

i=1, 2,,k

)

and βj

(

i= +k 1,k+2,,t

)

then equate to zero, using (1.6), (1.7) and (1.9), we get

( )

1

1 j t

j

j t

yx

j y x

x x

R C

C R

α +

= − (3.3)

( )

1

1 j t .

j t

y

j y

j

R C

C R

τ _τ

τ τ

β _{= −} +



(3.4)

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( )

(

)

( )2 ( )2

2 2

MRVAF 3.0

1 1

MSE j j .

k _x h k t

y y i j

j

j j k j

e e

t E e e Y Y

P X

τ

α + = β

= = +       = _ − − _  _ _ 

∑

 (3.5)

Taking expectation and using (1.6) in (3.5), we get,

( )

(

)

2 2

2 MRVAF 3.0

1 1

MSE .

j j j j

k k h t

y i j y x yx j j y Pb j j j k j

Y Y

t Y C X C C P C C

X P τ

θ α ρ + = β ρ

= = +

 

= _ − − _



∑

 (3.6)

Substituting the optimum (3.3) and (3.4) in (3.6), we get,

( )

(

)

2 2

( )

2 MRVAF 3.0

1 1

MSE 1 1 .

j j

j j j j

j _j j _j

yx y

k k h t

x

j y j y

y y x yx y Pb

j x _x j

R R

C C

t Y C C C C C

C _R C _R

τ _τ τ

τ _τ

θ ρ + = ρ

= =     = _ + − + − _    

∑

(3.7)

Or

( )

(

)

2 2

( )

2 MRVAF 3.0

1 1

MSE 1 1 1 .

i j j j j j j j y yx

k _x k h t

j j

y yx Pb

j j k

x

R R

t Y C

R R

τ τ

τ

θ ρ + = ρ

= = +       =  + − + −        

∑

(3.8)

Or ( )

(

)

( ) ( ) , 2 2 2 MRVAF 3.0 ,

MSE t.

t

y x y

x

R

t Y C

R τ τ θ =   (3.9)

Using (1.8), we get,

( )

(

MRVA 3.0

)

2 2 2

(

2.( ),

)

MSE 1 .

t

y y x

t =θY C −ρ _τ

  (3.10)

3.2. Mixture Ratio Estimator Using Multi-Auxiliary Variable and Attributes in Two-Phase Sampling (Partial Information Case)

In this case suppose we have no information on all s and t auxiliary variables but only for r and g auxiliary va-riables from population. Considering mixture ratio technique of estimating technique, the population mean of study variable Y can be estimated for two-phase sampling using multi-auxiliary variables and attributes as:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 2 2 1 2

1

1 1 ₁ 1 2 ₂ 1 1 1 1 2 1

2 MRVAP 3.1

2 1 2 1 2 2 2 2 2 2 2 1 2 2 2

1 1 ₁

1

2 2 1

s s r r k

k

r _r s s k

r r s s k

k _k

k k

x _X x _X x _X x x x

t y

x x x x x x x x x

p _P

p p

α β α β α β α α α

γ + + + + + + + + ₊ + +                                     =                                                 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2 2 1 2

1 2 ₂ 1 1 1 1 2 1

2

2 2 2 2 2 2 1 2 2 2

.

k k k h h h h q

k _k h _h h h t

k k h h h h t

p _P p _P p p p

p p p p p p p

λ+ γ+ λ+ γ λ γ+ γ+ γ

+ ₊ + + + + + +                                 _ _ _ _ _ _ _{ }   _{ } _ _ _ _ _ (3.11) Using (1.0), (1.3) and (1.4) in (3.1) and ignoring the second and higher terms for each expansion of product and after simplification, we write,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

1 2 2 1 2 1 2

2 1 2

2 MRVAP 3.1

1 1 1 1

1 1

.

j

j j j j j j

j j

j j j

r _x _x r _x s r k _x _x k t h

j i

j j j j j j i i

h q q k t h

j j

j j j h j

e e e e e e e

t y Y Y Y Y

X X X P

e e e

Y Y

P P

τ τ

τ τ τ

α β α γ

λ γ + = + = = = = = + = + = = = + − − − = + − + + − − +

∑

(3.12)

(8)

( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

1 2 2 1 2 1 2

2

2 1 2

MRVAP 3.1

1 2 1

1 1 1 1

2

1 1

.

j

j j j j j j

j j

j j j

r _x _x r _x s r k _x _x k t h

y j i

j j j j j j i i

h q q k t h

j j

j j j h j

t

e e e e e e e

E E e Y Y Y Y

X X X P

e e e

Y Y

P P

τ τ

τ τ τ

α β α γ

λ γ + = + = = = = = + = + = = = + − − −   = + − + +   −   − +  

∑

(3.13)

We differentiate the Equation (3.13) with respect to αi

(

i=1, 2,,r

)

, βi

(

i=1, 2,,r

)

,

(

1, 2, ,

)

i i r r k

α = + +  , γi

(

i= +k 1,k+2,,h

)

, λi

(

i= +k 1,k+2,,h

)

, γi

(

i= +h 1,h+2,,q

)

and equate to zero and use (1.6), (1.7) and (1.9). The optimum values are as follows,

( )

(

)

( )

(

)

( )

1 1 1 1

1 1, 2, , ;

1 1

j j

m w

j m j

w

j _p j

w

j m w

j r j w j w yx yx

j y x x

j

x x yx

x

y _x y

j y

j

yx

j y x

j x x j y j y R R C j r

C R _R

R R

C

j k k h

C R R

R C j r C R C R C R

τ τ _τ

τ τ τ

τ τ α γ β λ + + + +     = − _ − _ =         = − _ − _ = + +     = − = = −   

( )

1 1

1, 2, , ;

1 1, 2, , ;

1 1, 2, , .

j w ji s j s j w j p yx

j y x

j

x x

j y

x

j k k h

R C

j r r k

C R

C

R j h h q

R C τ _τ ψ _τ τ α γ + +   _{= +} ₊     = − = + + = − = + +    (3.14)

Using normal equation that are used to find the optimum values given (3.13) we can write

( )

(

)

( ) ( ) ( ) ( ) ( )

1 2 2 1 2

2

1 2 2 1 2

MRVAP 3.1

1 2 1

1 1 1

MSE

.

j

j j j j

j j

j

j j j j

r _x _x r _x s r k _x _x

y y j

j j j j j j

h q m

k t h k t h

i j j

i i j j j h j

t

e e e e e

E E e e Y Y Y

X X X

e e e e e

Y Y Y

P P P

τ τ τ τ τ

α β α

γ λ γ

+ = = = = + = + = + = = = = +   − −   = _ + − +    − −    + − +  

∑

(3.15)

Taking expectation and using (1.6) in (3.15), we get,

( )

(

)

(

)

(

)

(

)

(

)

3.1

2 2 2

2 2

2 1 2 2 1 2

1 1 1

2 2 2

1 2 2 1 2

1 1 1

+

j j j j j j j j j

j j j j j j j j

MRVAP

r r r s k

j j j

y y x yx y x yx y x yx

j j j

j j j r

t k q t k h q

j y Pb j y Pb j y

i k j i k j i h j

MSE t

Y Y Y

Y C X C C X C C X C C

X X X

Y Y Y

P C C P C C P C C

P τ P τ P τ

θ θ θ α ρ θ β ρ θ θ α ρ

θ θ γ ρ θ λ ρ θ θ γ

+ = = = = + + = + = = + = + = + = + − − + − − − + −

∑

. j Pb ρ (3.16)

(9)

( )

(

)

(

) ( )

( )

(

)

( )

(

)

( )

MRVAP 3.1 .

1 . 1 .

2 2

2 1 2 2

1 1

. .

1 1 .

1 2 1 2

1

MSE

1 1

j j j

q w w

j j

q w w

j

i j

q q w

i

q q w

yx

r _{y x} yx r yx

j y x j y x

y yx yx

j _x j

y

r s k yx _{y x} _{y x} y

j j y

yx

j r _x _x

t

R R R

Y C

R R R

R

R R

R R R

τ τ

τ τ _τ

τ

θ θ θ ρ θ ρ

θ θ ρ θ θ

+ + = = + = ₊ ₊ = +       = _ + − − _ − _ − −          + − − + − − _ −  

∑

   

( )

(

)

( )

1 .

1 . 1

2 1 2

1 1

1 1 .

j

j j

q w

j j j

w q

t k h

Pb j k

y q

t k h y _y

j y j

Pb Pb

j k j h

R R

τ

τ _τ _τ

τ τ

ρ

θ ρ θ θ γ ρ

+ = = + + = ₊ ₊ = + = +       − − + − − _  

∑



∑

   (3.17) Or ( )

(

)

(

) ( )

( )

(

)

( )

(

)

( )

.

1 1 .

2 2

2 1 2 1

MRVAP 3.1

1 1

. .

1 1

1 2 1 2

1 1

MSE 1 1

1 1 j j q w j j q w j i q q i j q q yx

r _{y x} r yx

j j y x

y yx yx

j _x j _x

y

r s k yx _{y x} t k h _y

j j

yx Pb

j r _x j k

R R

t Y C

R R R R R R τ _τ τ

θ θ θ ρ θ ρ

θ θ ρ θ θ ρ

+ + = = + = ₊ + = ₊ = + = +   = _ + − − − −   + − − + − −

∑

 

( )

1 .

(

)

( )

1 .

1 1 2

1 1

1 1 .

j

j _q

w

j j j

w q

y q

t k h y _y

j y j

Pb Pb

j k _x j h

R R

τ

τ _τ _τ

τ

θ + = + ρ θ θ + γ ρ

= + = +   − − + − − _  

∑



∑

  (3.18) Or ( )

(

)

(

)

( )

MRVAP 3.1 . . 2 2 2 1

1 1 1

. .

1

1 1

MSE

1 1 1 1

j j

i

q q q

j j j

q q q

j _i

q w

i j i

q w

yx y

r yx _x r s k _{y x} t k h _y

j j j

y yx yx Pb

j _x j r _x j k

yx

q _j _y r _j yx _{y x} _j

Pb yx

j h j _x

t

R R

R Y C

R R R

R R

τ _τ

τ

θ θ ρ ρ ρ

γ ρ θ ρ

+ = + = = = + = + = + =         =  −  + − + − + −       _ _ _    + − _+ + − + −  

∑

    . 1 . j w j w

t k h yx y x

Pb j k R_x

ρ + = = +                 _ _  

∑

(3.19) Or ( )

(

)

(

)

( ) ( )

( )

( ) ( ) ( ) ( )

( )

( ) , , , , , , , . . 2 1 2 2

MRVAP 3.1 ,

1

1 1

MSE 1

1 1 1

j j q j j q q q j

i _r _w

j j

w w w

y x

q x

j

y x y x

j

x x

r _j yx t k h _j yx x

yx Pb

j j k

x x x

R

t Y C R

R R

R R R

τ _τ

τ τ

τ

τ τ τ

θ θ

ρ

θ _ρ _ρ

(10)

( )

(

)

(

)

( )

( ) ( )

. , . ,

2 2

2 1 1

MRVAP 3.1

, ,

MSE q w .

q w

y x y x

y

x x

R R

t Y C

R R

τ τ

θ θ θ

 

 

= _ − + _

 

 

(3.21)

Using (1.8) in (3.21), we get,

( )

(

MRVAP 3.1

)

2 2

(

2 1

)

(

2( ),

)

1

(

2( ),

)

MSE 1 1 .

q w

y y x y x

t =Y C θ θ− −ρ _τ +θ −ρ _τ

    (3.22)

Simplifying (3.22) we get,

( )

(

MRVAP 3.1

)

2 2

(

2

(

2( ),

)

1

(

2( ), 2( ),

)

MSE 1 .

q q w

y y x y x y x

t =Y C θ −ρ _τ +θ ρ _τ −ρ _τ

      (3.23)

3.3. Mixture Ratio Estimator Using Multi-Auxiliary Variable and Attributes in Two-Phase Sampling (No Information Case)

If we estimate a study variable when information on all auxiliary variables is unavailable from population, it is utilized in the form of their means. By taking the advantage of mixture ratio technique for two-phase sampling, a generalized estimator for estimating population mean of study variable Y with the use of multi auxiliary va-riables and attributes are suggested as:

( ) ( )

( )

( ) ( )

1 2 1 2

1 1 1 2 1 1 1 1 2 1

2 MRVAN 3.2

1

2 1 2 2 2 2 2 2 2

.

s k k t

k k k t

k

k k t

x x x p p p

t y

x x x p p p

α α α γ+ γ+ γ

+ +

                        = _ _{ } _ _ _{ } _ _ _ _ _

            (3.24)

Using (1.0), (1.3) and (1.4) in (3.24) and ignoring the second and higher terms for each expansion of product and after simplification, we write,

( ) 2 ( )1 ( )2 ( )1 ( )2 MRVAN 3.2

1 1

.

j j j j

i

k _x _x k h t _x j

j j j j

e e e e

t y Y Y

X P

τ τ

α + = γ

= =

− −

= +

∑

+

∑

(3.25)

Mean squared error of tMRVAN 3.2_{( )} estimator is given by

( )

(

)

( )1 ( )2 ( )1 ( )2

2

MRVAN 3.2

1 1

MSE _j j j j j .

k _x _x k h t

y j

j j j j

e e e e

t E e Y Y

X P

τ τ

α + = γ

= =

− −

 

 

= _ + + _



∑

 (3.26)

We differentiate the Equation (3.27) partially with respect to αi

(

i=1, 2,,k

)

and βi

(

i= +k 1,k+2,,t

)

then equate to zero, using (1.6), (1.7) and (1.9), we get

( )

1

= 1 j k

j _j

yx

j y x

j

x _x

R C

C _R

α +

− (3.27)

( )

1

= 1 i j .

j _j

y j y j

R C

C _R

τ τ

τ _τ

β +

− (3.28)

Using normal equation that are used to find the optimum values given (3.26) we can write,

( )

(

)

( )1 ( )2 ( )1 ( )2

2 2

MRVAN 3.2

1 1

MSE j j j j .

k _x _x k h t _x _x

y y j j

j i j k j

e e e e

t E e e Y Y

X P

α + = γ

= = +

  − − 

  

= _ + + _

 _ _



∑



(3.29)

Taking expectation and using (1.6) in (3.29), we get

( )

(

)

2 2

(

)

(

)

2 1 2 1 2

MRVAN 3.2

1 1

MSE .

j j j j

k k h t

y j y x yx j y Pb

j j k

t Y C θ θ θ α C C ρ θ θ β C Cτ ρ

+ =

= = +

 

= _ + − + − _



∑

 (3.30)

(11)

( )

(

)

(

) ( )

( )

(

)

( )

. .

2 2

2 2 2 1 2 1

MRVAN 3.2

1 _, 1 _,

MSE 1 1 .

j j

q q

j j

q q

yx y

k k h t

y x y x

j j

y yx Pb

j _x j _x

R R

t Y C

R R

τ

τ τ

θ θ θ θ ρ θ θ + = ρ

= =

 

 

= _ + − − + − − _

 

 

∑

 

(3.31)

Or

( )

(

)

(

)

( )

( ) ( )

. . . .

2 2

2 2 1 1 1

MRVAN 3.2 . .

1 _.

MSE 1 1 .

i j q

y x

q _{y x}

j

y y x

j _x

R

t Y C

R

τ _τ

τ τ

θ θ θ ρ θ θ

=

   

   

=  −  + − + + 

   

 _ _ 

 

∑

 

(3.32)

Or

( )

(

)

(

)

( ) ( )

( )

_{( )} ₍ ₎

2 1

2 2

. . 1

MRVAN 3.2 . _. _. . .

1 .

MSE 1 _i _j .

j j

q _q

q

t j

y x y x _{y x} y x

i x

t Y C R R

R τ τ τ τ

τ

θ θ

ρ θ

=

 ₋ _ _

 

= _ + − + _

 _ _

 

∑

  _{ }

 

(3.33)

Or

( )

(

)

(

)

( )

. ,

2 2

2 1 1

MRVAN 3.2

,

MSE q .

q

y x y

x

R

t Y C

R

τ

θ θ θ

 

 

= _ − + _

 

 



(3.34)

Using (1.8) in (3.34), we get,

( )

(

)

2 2

(

)

(

2_{( )}

)

2 1 1

MRVAN 3.2 ,

MSE t =Y Cy θ θ− 1−ρy xτ +θ .

  (3.35) Simplifying (3.35), we get,

( )

(

MRVAN 3.2

)

2 2

(

2

(

2( ),

)

1 2( ),

)

MSE t =Y Cy θ 1−ρy xτ +θ ρy xτ .

    (3.36)

3.4. Bias and Consistency of Mixture Ratio Estimators

These mixture ratio estimators using multiple auxiliary variables and attributes in two-phase sampling are biased. However, these biases are negligible for large samples that is

(

n≥30

)

. It’s easily shown that the mixture ratio estimators are consistent estimators using multiple auxiliary variables since they are linear combinations of con-sistent estimators it follows that they are also concon-sistent.

4. Simulation, Result and Discussion

We carried out data simulation experiments to compare the performance of mixture ratio estimators using mul-tiple auxiliary variables and attributes in two-phase sampling with ratio estimator using one auxiliary variable and one auxiliary attribute or ratio estimator using multiple auxiliary variable or multiple auxiliary attributes in two-phase sampling estimators for finite population.

All the results were obtained after carrying out two hundred simulations and taking their average. 1) Study variable N=800, n=96, n2 =34, mean=50, standard deviation=6.

2) For ratio estimator the auxiliary variable is positively correlated with the study variable and the line passes through the origin.

2

500, 96, 34, mean 29, standard deviation 6.712.

N= n= n = = =

2

500, 96, 34, mean 38, standard deviation 8.612.

N = n= n = = =

Correlation coefficients

1 0.6820, 0.5788.2

yx yx

ρ = ρ =

3) For ratio estimator the auxiliary attributes is positively correlated with the study variable and the line passes through the origin.

2

500, 96, 34, mean 0.5612.

(12)

500, 96, mean 0.4826.

N= n= =

Correlation coefficients

2 0.7031

yτ

ρ = ,

1 0.6614.

yτ

ρ =

In order to evaluate the efficiency gain we could achieve by using the proposed estimators, we have calcu-lated the variance of mean per unit and the mean squared error of all estimators we have considered. We have then calculated percent relative efficiency of each estimator in relation to variance of mean per unit. We have then compared the percent relative efficiency of each estimator, the estimator with the highest percent relative efficiency is considered to be the more efficient than the other estimators. The percent relative efficiency is cal-culated using the following formulae.

( )

_Var

( )

_{( )}

ˆ ˆ

eff 100

ˆ MSE

y Y

Y

[image:12.595.159.441.481.615.2]

= ∗ (4.0)

[image:12.595.88.538.651.720.2]

Table 1 shows percent relative efficiency of proposed estimator with respect to mean per unit estimator for single-phase sampling. It is very clear from Table 1 that our proposed mixture ratio estimator using multiple auxiliary variables and multiple auxiliary attributes simultaneously is the most efficient compared to ratio esti-mator using one auxiliary variable and one auxiliary attribute or ratio estiesti-mator using multiple auxiliary variable or multiple auxiliary attributes in two-phase sampling.

Table 2 compares the efficiency of full information case and partial case to no information case and full to partial information case of proposed mixture ratio estimators. It is observed that the full information case and partial information case are more efficient than no information case because they have higher percent relative efficiency than no information case. In addition, the full information case is more efficient than the partial in-formation case because it has a higher percent relative efficiency than partial inin-formation case.

5. Conclusion

The proposed mixture ratio estimator under full information case is recommended for estimating the finite pop-ulation mean since it is the most efficient estimator compared to mean per unit, ratio estimator using one aux-iliary variable, ratio estimator using one auxaux-iliary attribute, ratio estimator using multiple auxaux-iliary variable and ratio estimator using multiple auxiliary attributes in two-phase sampling. In case some auxiliary variables or attributes are unknown, we recommend mixture ratio estimator under partial information case since it is more

Table 1.Relative efficiency of existing and proposed estimator with respect to mean per unit estimator for two-phase sampling.

Estimator Relative percent efficiency with respect to mean per unit in two-phase sampling

( )

var y 100

RV

t 146

RA

t 185

MRV

t 212

MRA

t 233

( )

MRVAF 3.0

t (proposed) 285

Table 2. Comparisons of full, partial and no information cases for proposed ratio-cum-product estimator using multiple auxiliary variables.

Population Percent relative efficiency of

full and partial to no information Percent relative efficiency of full to partial in formation case Estimator tMRVAN 3.2( ) tMRVAP 3.1( ) tMRVAF 3.0( ) tMRVAP 3.1( ) tMRVAF 3.0( )

Relative percent

(13)

efficient than the mixture ratio estimator under no information case and if all are unknown, we recommend the mixture ratio estimator under no information case to estimate finite population mean.

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[2] Cochran, W.G. (1940) The Estimation of the Yields of the Cereal Experiments by Sampling for the Ratio of Grain to Total Produce. Journal of Agricultural Science, 30, 262-275. http://dx.doi.org/10.1017/S0021859600048012

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[5] Olikin, I. (1958) Multivariate Ratio Estimation for Finite Population. Biometrika, 45, 154-165.

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