View of SIMULATION STUDY OF SYNTHETIC ESTIMATORS FOR DOMAIN MEAN USING AUXILIARY CHARACTER IN THE PRESENCE OF NON-RESPONSE

(1)

1

SIMULATION STUDY OF SYNTHETIC ESTIMATORS FOR DOMAIN MEAN USING AUXILIARY CHARACTER IN THE PRESENCE OF NON-RESPONSE

Haresh R. Chaudhary

Assistant Professor , Mathematics, R.R. Mehta Science College Palunpur 1. INTRODUCTION

Example surveys would advantageous in discovering estimate to those number total, number mean, populace extent and so forth throughout this way, observing and stock arrangement of all instrumentation may be enha. The vitality of dependable estimate for populace need been accentuated for the improvemen of little zone of the number. In any case over late years, the interest for gauge of parameter for subpopulations (domains) is growing, those parameters from claiming little geological territories for example, such that county, block, region and so on would developing for An quicker speed, because of low extent about units having a place of the investigation character done little area, it is challenging to get the dependable assess of the populace parameter. Done such instances On those estimates would got after that it will bring been unsuitability extensive standard slip.

However, for addition test span done little area, those regulate estimator gives better effect over manufactured estimator. Those estimator coating a few little regions under the supposition about little range Hosting same trademark as vast zone termed Similarly as engineered estimator.

A few manufactured estimators bring been examined by Gonzalez (1973), Gonzalez Also Wakesberg (1973), Schaible et al. (1977) What's more Rao (2003). For those estimation about populace intend of the examine character a few reproduction investigations identified with engineered estimator to little range bring been recommended Eventually Tom's perusing Tikkiwal (1993), Tikkiwal What's more Ghiya (2000), Rai What's more Pandey (2013) Furthermore Singh and seth (2014,2015) and so on.

On there is forgetting about perceptions because of non reaction in the population, that point this will Additionally make non reaction On little zone. Because of non response, the manufactured estimator utilized to the parameters for domain/(small area) In light of the reaction units will need an expand in the standard slip and likewise those

legitimacy of the estimator may be suspected because of those nature from claiming non reaction and only the populace.

Hansen What's more Hurwitz (1946) have suggester a system for sub- sampling from non respondent Furthermore suggested a estimator to number imply utilizing the ponder character in the. Space Toward utilizing accessible data starting with the units sub testing units starting with non- responding and only the test in the area.

Notwithstanding we must Think as of those issue from claiming estimator from claiming parameters from claiming Web-domain in the vicinity from claiming non response, we recommend those engineered estimator Eventually Tom's perusing utilizing Hansen What's more Hurwitz (1946) of the space mean utilizing assistant character in the vicinity non reaction. In this context, basic synthetic, proportion engineered Furthermore summed up engineered estimator to those estimation of the populace mean in the vicinity of non reaction on the contemplate character need been recommend Also their properties would contemplated.

In this paper we recommended An summed up engineered estimator to Web- domain intend utilizing assistant character in the vicinity for non-response.

A similar ponder of the recommended estimator will be constructed for its part.

For those purpose, those information [for MU284 populace provided for in addendum b for Sarndal et al. (1984)]

need been viewed as for those investigation What's more on the groundwork of mimicked relative standard lapse (SRSE) those prevalence of the suggested estimator to space imply will be demonstrated with the proportion engineered estimator for area imply.

1.1 Formulation of the Problem and Notations

Suppose that a finite population U (1,2,3……….N) is divided in to L non- overlapping small domains Ua of size Na

(2)

2

such that

N N

L

a





1

,(a=1,2,3……L). Now we are interested in the estimation of parameter

Y

a of

a

^th domain of size

N

_a. The study character and auxiliary character are denoted by y and x. The population mean of whole population and the population mean of

a

^thdomain (a=1,2,3,……L) for the auxiliary character known, For defining synthetic estimator for the

a

^thdomain. We draw a sample of size n from the population N and we observe that from the selected sample only n1 units respond and n2 units do not

non-respond. We take a sub sample of r (n2/k, k>1) from non-responding n2 units, using Hansen and Hurwitz (1946) technique n1 and r units on y and x character, which is given by

n y n n

y y n

1 2 2 1

1







(2.1)

n x n n

x x n

1 2 2 1 1







(2.2)

Where,

( y

₁

, y

¹₂

)

and (

x

1,

x

¹2)denote the sample based mean of

n

₁and

r

units on y and x character.

2 1

2 ( )

) 1 (

1 X X

S N

N

i i

X 

 





, ²

1

2 ( )

) 1 (

1 Y Y

S N

N

i i

Y 

 





, ₍ ₎₍ ₎

) 1 (

1

X X Y N Y

S _i

N

i i

XY  

 





, Y C_Y S^Y ,

X C

_X 

S

^X

and

X Y

C

_XY

 S

^XY .

Where,

1

W

a _: a a

N N

₁

Response rate for all domain a and

a a

a

N

W

₂

: N

² Non-response rate for all

domain a. ² ²

2 1 2

2

( )

) 1 (

1

²

Y N Y

S

N

i i

Y



  



, 2 ²

2 1 2

2

( )

) 1 (

1

²

X N X

S

N

i i

X



  



,

) )(

) ( 1 (

1

2 2

2 1 2

2

Y Y X N X

S

_i

N

i i

XY

 

  



,

Y C

₂_Y 

S

²^Y

X C

₂_X 

S

²^X and

Y X

C

₂_XY

 S

²^XY _{. (2.3)}

1.2 Various Estimators for Domain Mean Without Non Response (i) Ratio Synthetic Estimator

a a

RS

X

x

T

_,

 y

Rao (2003) (3.1)

) (

)

(

RS_,a a a

X

a

C

²_X

C

_XY

X Y Nn

n Y N

X X T Y

Bias     

(3.2)

2

,

) ( )

(

RSa

X

a

Y

a

X T Y

MSE  

 ³ ⁴  ² ⁽ ^)}

{

a X² Y² XY a _X² _XY

a

X C C C Y C C

X X Y X Y Nn

n

N     



(3.3)

Under the assumption

a

X

X Y

Y 

, MSE (TRS,a) has been reduced which is given as follows:



X Y XY



a a

RS

Y C C C

Nn n T N

MSE

( _, )  ² ²  ² 2 (3.4)

(Ii). Generalized Synthetic Estimator

(3)

3



 



 



 

a a

GS

X

y x

T

_, _, Tikkiwal and Ghiya (2000) (3.5)

a XY X

a a

GS

C C Y

Nn n N X

Y X T

Bias

 



 



  

 



 



  ]

2 ) 1 1 (

[ )

( _, _, ^

 

²



 (3.6)

2 2 2 2

2 2

,

, ) [1 {(2 ) 4 }]

( X Y XY a

a a

GS

C C C Y

Nn n N X

Y X T

MSE

      



 



 

  



2 ] ) 1 1 (

[

2 ² 



 



  

 



 



  _X _XY

a

C C

Nn n N X

Y X

Y

^

  

(3.7)

Under assumption,





 





X X Y

Y

a

, MSE (T^GS,β,a) has reduces which is given as follows:

GS a a

 C

X

C

Y

C

XY



Nn n Y N

T

MSE

( _,__, ) ² 



² ²  ²2



(3.8) 1.3 Proposed Generalized Synthetic

Estimator for Domain Mean in the Presence of Non-Response (

T

_GS^ _,_,_a) Now we propose a generalized synthetic estimator for domain mean in the presence of non response using auxiliary character which is given as follows:



 









 

 



a GS a

X y x

T

_, _, , where β is constant

(4.1)

Particular cases of

T

_GS^ _,_,_a are as follows:

(i)

T

_GS^ _,₀_,_a 

y

^_, _{if β=0} (4.2)

(ii) GS a

X

^a

x T y

_



₁_, 

, , if β=-1 (4.3)

(iii)

a GS a

X y x T

 

 _,₁_,  , if β=1

(4.4)

1.4 Bias and Mean Square Error of Different Estimator for Domain Mean In The Presence Of Non-Response

For large sample approximation, we assume that which are given as follows:

) 1 (  

₀



 Y

y

,

x

^

 X ( 1  

₁

)

, such that

E

(



₀)0,

E ( 

₁

)  0

,



_i

 1 ; i  0 , 1

and we have

2 2 2

2 0

) 1 ( ) ) (

( _Y

C

_hY

n W C g

Nn n

E N



 



 , ₁² ( ) ² ( 1) ² ₂²

)

( _X

C

_X

n W C g

Nn n

E N



 



 and

XY

C

n W C g

Nn n

E

₀ ₁

( N ) ( 1 )

² ₂

)

( 

 

 



, ²

,

h

g  n

g≥1 and W2=N2/N (4.5)

(i) Bias and Mean Square error of

( T

_GS^ _,_₁_,_a

)

a a

GS

X

x

T y

_





 _, ₁_,



(4.6)

(4)

4

)}

) ( 1 ) (

( { )

( )

(

GS_, ₁_,a a a a _X² _XY

W

₂

C

₂²_X

C

₂_XY

n C g

C X X Y Nn

n Y N

X X T Y

Bias  



 





 (4.7)



X Y XY



a a

GS

C C C

Nn n X N

X Y Y

X X T Y

MSE

( ) ( ) { ² ² 4

2 2

, 1

,   



 







 





⁴



^}

) 1 (

2 2

2 2 2 2

XY Y

X

C C

n C W

g

  

 a



X XY



a

C C

Nn n X N

X Y

Y  

 2 {

²

)}

) ( 1 (

2 2 2 2

XY

X

C

n C W

g  



(4.8)

Under the assumption

a

X

X Y

Y 

,

MSE ( T

_GS^ _,_₁_,_a

)

has been reduced which is given as follows:



²



⁽ ¹⁾



²



^}

{

) ² ² ² ² ₂² ₂² ₂

, 1

, a a Y X XY Y X XY

GS

C C C

n W C g

C Nn C

n Y N

MSET

  





 

 



(4.9)

(ii) Bias and Mean Square Error of(

T

_GS^ _,__,_a)



 









 

 



a a

GS

X

y x

T

_, _, (4.10)



 



  

 



 



 



XY X

a a

GS

C C

Nn n N X

Y X T

Bias

^

  

 2

,

, 2

) 1 1 (

[ )

(

X a

XY

C Y

n C W

g  



 



  

  ]

2 ) 1 ) (

1 (

2 2

2

  

(4.11)

} 4 )

2 {(

1 [ )

( ² ² ²

2 2

,

, X Y XY

a a

GS

C C C

Nn n N X

Y X T

MSE   



      



 



 





² ²



²

2

2 2 2

2

( 1)

4 ( 1) ] 2

( 1)

[1 2

a a

Y XY X

a

X XY

g W X

C C C Y Y Y

n X

N n

C C

Nn



  

 

       

 

   

   

2 }]

) {(

) 1 (

2 2

2

XY

X

C

n C W

g









_



(4.12)

For obtaining the value of β, we have differentiate with respect to beta and equating to zero, but equation will not in closed form so it will be difficult to solve it, for value of β which

MSE ( T

_GS^_,__,_a

)

_min. So we will go to further for obtain β.

For the purpose of optimum

min ,

,

)

( T

_GS^ _a

MSE

_ for optimum value of β, we will conduct Simulation study for β which minimize

MSE ( T

_GS^ _,__,_a

)

.

(5)

5 Under assumption,





 





X X Y

Y

a

, MSE(

T

GS^ _,_,a) has reduces which is given as follows:



²



⁽ ¹⁾ ^{ ² ^}]

[ )

( GS_, _,a ²a ² _X² _Y² _XY ² ²

C

₂²_X

C

₂²_Y

C

₂_XY

n

W C g

C Nn C

n Y N

T

MSE

^ _  



 



 



 



(4.13) 2. SIMULATION STUDY

To the reenactment purpose, we bring recognized the information provided for Toward Sarndal et al. ((1992) addendum B). Those populace for sweden regions need aid ordered under eight geological locales (1, 2, 3, 4, 5, 6, 7 Also 8). We think about main five geological areas (2, 3, 4, 5 Furthermore 6) for their particular sizes would (48, 32, 38, 56 Furthermore 41).

We look at those execution for proportion manufactured estimator (TRS,a) Also summed up engineered estimator (TGS,β,a) As far as recreated relative standard lapse (SRSE).

The ponder variable y and the assistant variable x are provided for as takes after:. X: downright amount of metropolitan workers in 1984.

Y: land of land qualities as stated by 1984 appraisal (in millions from claiming kronor).

Currently for the reason for reenactment study we need taken 5000 free test through straightforward arbitrary examining without reinstatement each need extent 43 (20 percentage) starting with the number for size 215, their relative execution of the estimators under consideration, SRSE to each Web-domain would provided for as:.

100 ) ( )

( _, _,  ^, ^, 

a GS a GS a

Y T SMSE T

SRSE _ ^

(5.1)

where,







 ⁵⁰⁰⁰

1

2 , , ,

, ( )

5000 ) 1 (

s

a a s a GS

GS T Y

T

SMSE _ _

(5.2) where,

T

_GS^s _a

,, denote, particular synthetic estimator for s^th sample and

T

GS_,_,a

denote, particular synthetic estimator for domain a=1,2,3,4,5.

Further we must estimate worth of the estimator in the vicinity of non reaction through reenactment examine we need taken 5000 free example through straightforward arbitrary examining without supplanting every have span 43 from the populace from claiming measure 215, we Think as of (70%) are reaction example units Also (30%) need aid non reaction example units utilizing henson Furthermore Hurwitz technique, their relative execution of the estimators under consideration, SRSE to each area which are provided for as takes after:.

100 ) ( )

( _,_,  ^, ^, 



a a GS GS a

Y T SMSE T

SRSE _ ^

(5.3)

where,







  ⁵⁰⁰⁰ 

1

2 , , ,

, ( )

5000 ) 1 (

s

a a s a GS

GS T Y

T

SMSE _ _

(5.4) where, a

s

T

GS^ _,_, denote, particular synthetic estimator for s^th sample and

 a

T

GS_,_, _denote, particular synthetic estimator for domain a=1,2,3,4,5.

The entire Population of size 215 is comprised of five domains. The values of the parameters are given as follows.

The parameters values of the population are given as follows:

N=215,

Y

=2758.851,

X

=1626.549, 2

S

y=12509178,

S

_x²=14334438,

S

xy=12514241,

 XY

=0.935,

S

₂²_y =701859.4, 2

S

2x=288684.9,

S

_2xy² =280682.4. Table 5.1 The Parameter Values of Domains are given as Follows

Domain Values Domain

1 2 3 4 5

Na ₄₈ ₃₂ ₃₈ ₅₆ ₄₁

Ya 2971.1 2498.8 2915.5 3046.5 2175.3

Xa 1658.7 1317.0 1937.7 1950.4 1099.8

2

Sy _11119986 _4164522 _9575690 _27860176 ₂₈₆₀₂₄

2

Sx _4601899 _1989655 _15986129 _38786393 _1020434

(6)

6

Sxy _6920432 _2682303 _11697914 _31770622 _1671325

) (a

YX _0.967 _0.932 _0.946 _0.966 _0.978

The absolute differences for the synthetic assumption taken of ratio synthetic estimator for domain mean (TRS,a) and generalized synthetic estimator for domain mean (TGS,β,a) are given as follows:

Table 5.2

Absolute Difference Under Synthetic Assumption of Ratio Synthetic Estimator for Domain Mean Without Non Response (TRS,A)

Domain Ya

Y

Xa

X Absolute Difference

a

a X

X Y Y 

1 1.063433 0.981592 0.081841

2 1.102465 1.231812 0.129347

3 0.947246 0.835534 0.111713

4 0.905348 0.805079 0.100269

5 1.274951 1.494460 0.219509

Table 5.3

Absolute Difference Under Synthetic Assumption of Generalized Synthetic Estimator for Domain without Non Response (TGS,Β,A)

Domain

Ya

Y _^





 X

Xa Absolute Difference ^



 



 X

X Y

Y a

a

1 1.063433 0.997122 0.066311 2 1.102465 1.042579 0.059886 3 0.947246 1.020861 0.073614 4 0.915348 1.006703 0.091355 5 1.274951 1.080995 0.193956

From the above Table (5.2) and Table (5.3), we observe that absolute difference of synthetic assumption of generalized synthetic estimator (

T

_GS_,_,_a) less than ratio synthetic estimator for domain mean (

T

_GS_,₁_,_a) for all domains (1, 2, 3, 4 and 5).

The absolute differences for the synthetic assumption taken of ratio synthetic estimator for domain mean (

GS a

T

^ _,₁_, ) and generalized synthetic estimator for domain mean (

T

_GS_,_,_a) in the presence of non-response are given as follows:

Table 5.4

Absolute Difference under Synthetic Assumption of Ratio Synthetic Estimator for Domain Mean in the Presence of Non Response (

T

_GS^ _,₁_,_a)

Domain

Ya

Y^

Xa

X^ Absolute Difference

a

a X X

Y Y^  ^

1 1.065833 0.983592 0.082241

2 1.107140 1.237045 0.129905

3 0.944158 0.828565 0.115593

4 0.929658 1.039220 0.109562

5 1.226064 1.465306 0.239242

(7)

7 Table 5.5

Absolute Difference under Synthetic Assumption of Generalized Synthetic Estimator for Domain Mean in the Presence of Non Response (

T

_GS^ _,_,_a)

Domain

Ya

Y^ _^



 



 X

Xa

Absolute Difference





 



 _



X X Y

Y a

a

1 1.066433 0.997319 0.068596

2 1.108141 1.043463 0.064678

3 0.944158 0.983132 0.078974

4 0.909657 1.039220 0.129562

5 1.280651 1.080795 0.199856

From the above Table (5.4) and Table (5.5), we observe that absolute difference of synthetic assumption of generalized synthetic estimator for domain mean in the presence of non response (

T

_GS^ _,_,_a) less

than ratio synthetic estimator for domain mean (

T

_GS^ _,₁_,_a) for all domain (1, 2, 3, 4

and 5).

Table 5.6

Simulated Relative Standard Error of ratio synthetic estimator for domain mean in the presence of non response (

T

_GS^ _,₁_,_a) and generalized synthetic estimator for domain mean in the presence of non response (

T

_GS^ _,_,_a) at optimum value of β for different domains (1, 2, 3, 4 and 5) are given as:

Estimator Domain

1 2 3 4 5

GS a

T

^ _,₁_, 46.174 (36.987) 39.151

(35.290) 62.821

(50.498) 61.391

(46.940) 38.954 (34.637)

GS a

T

^ _,_, 23.114 (20.056) β =-0.1

23.326 (23.152) β =-0.2

23.085 (19.283) β = -0.3

26.522 (22.153) β =-0.2

36.984 (33.967) β =-0.3

The figure in parenthesis () shows Simulated Relative Standard Error without non response.

From the above Table (5.6), we observe that Simulated Relative Standard Error (SRSE) of generalized synthetic estimator for domain mean in the presence of non response (

T

_GS^ _,_,_a) is less than the SRSE of ratio synthetic estimator for domain mean in the presence of non response (

T

_GS^ _,₁_,_a) for all domains (1, 2, 3, 4 and 5). It is also to be noted here that SRSE of generalized synthetic estimator for domain mean without non response (T^GS,β,a) is less than the SRSE of ratio synthetic estimator for domain mean without non response(TGS,- 1,a) for all domains (1,2,3,4 and 5).

3. CONCLUSION

It is momentous will stress here that the supreme Contrast under those manufactured supposition for summed up

engineered estimator for Web-domain imply.

(

T

_GS^ _,_,_a) is Less the outright Contrast under engineered suspicion for those proportion manufactured estimator to area imply. (

T

_GS^ _,₁_,_a) for all domains. So we conclude that the generalized synthetic estimator for domain mean (

GS a

T

^ _,_, ) is more superior than, the ratio synthetic estimator for domain mean (

GS a

T

^ _,₁_, ). We also observe that the SRSE of generalized synthetic estimator for domain mean in the presence of non- response (

T

_GS^ _,_,_a) is less than the SRSE of ratio synthetic estimator for domain mean in the presence of non-response (

T

_GS^ _,₁_,_a) for all domains (1, 2, 3, 4 and 5).

The absolute difference under synthetic assumption and SRSE of the generalized synthetic estimator for

(8)

8 domain mean in the presence of non- response (

T

_GS^ _,_,_a) is less than the ratio synthetic estimator for domain mean in the presence of non-response (

T

_GS^ _,₁_,_a) for all domains (1, 2, 3, 4 and 5).

Hence, we prefer the generalized synthetic estimator for domain mean using auxiliary character in the presence of non-response (

T

_GS^ _,_,_a) in comparison to ratio synthetic estimator for domain mean using auxiliary character in the presence of non response (

T

_GS^ _,₁_,_a).

REFERENCES

1. Hansen, M.H. and Hurwitz, W.N. (1946).

The problem of non-response in Sample Surveys. J. Amer. Stat. Assoc, 41,517-529.

2. Gonzalez, M.E. (1973): Use and evaluation of Synthetic estimators, Proceedings of the social Statistics, Amer. Stat. Assoc, 33-36.

3. Gonzalez, M.E., and Wakesberg, J. (1973), Estimates of the error of Synthetic Estimates, Paper Presented at the First Meeting of the International Association of Survey Statisticians, Vienna, Austria.

4. Schaible et al. (1977): An Empirical comparison of the simple inflation, Synthetic and Composite Estimators for Small Area Statistics. Proc of the Amer.

Stat. Assoc. Social Statics Section, 1017- 1021.

5. Tikkiwal, B. D. (1993): Modeling through Survey data for Small Domains.

Proceedings International Scientific Conference on Small Area Statistics and

Survey Design held in September, 1992 at Warsaw, Poland, 37-75.

6. Cochran, W. G. (1966): Sampling Techniques 3^rd Edition. John Wiley and Sons.

7. Sarndal, C.E. Swensson, B. and Wretman, J.H. (1992): Model Assisted Survey Sampling, Springer-Verlog, New York.

8. Rao. J.N.K. (2003): Small Area Estimation, Wiley Inter-Science, John Wiley and Sons, New Jersey.

9. Tikkiwal, G. C. and Ghiya. A (2000): A generalized Class of Synthetic estimators with application to Crop acreage Estimation for Small domains. Stat. in Tran.- New Series.

10. Pandey, K, K. (2010). Titled “Aspects of small area estimation using auxiliary information.” Book published by VDM Verlog Dr. Muller GmbH & Co. KG, Germany. (ISBN: 978-3-639-31569-1).

11. Aditya, K., Sud, U. C. and Ghrade, Y.

(2014). Estimation of domain mean using two stage sampling with sub sampling of non-respondents. J. Ind. Soc. Agri. Statist.

68, 1, 39-54.

12. Rai, P. K. and Pandey, K. K. (2013):

Synthetic estimators using auxiliary information in Small Domains. Biometrical Journal.42, 7, 65-87.

13. Chandra, H, Sud, U.C. and Ghrade, Y.

(2014) Small area estimation using estimated level auxiliary data. Comm.

Statist-Simulation and Computation. DOI 10.1080103610918.2013.810255.

14. Singh, V.K. and Seth, S.K. (2014): On the Use of One-Parameter Family of Synthetic Estimators for Small Areas, J. Stat. Appl.

& Prob,3. No.3, 355-361.

15. Singh, V.K. and Seth, S.K. (2015): An Efficient Family of Synthetic Estimators for Small Areas and Applications, J. of stat.

appl. & Prob. Lett.2, No. 1, 59-69.