Efficiency Of Audu & Adewara (2017) Two-Phase Factor-Type Estimators With Two Auxiliary Variables In Sample Survey

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22

1

Department of Mathematics, Usmanu Danfodiyo University, P.M.B. 2346, Sokoto, Nigeria

2

Department of Statistics, University of Ilorin, P.M.B 1515, Ilorin, Kwara State, Nigeria

Corresponding Author: Ahmed Audu, [email protected]

ABSTRACT: In this paper, efficiency of Audu & Adewara ([AA17]) two-phase factor-type estimators with two auxiliary variables for estimating finite population mean were examined using simulation. These estimators were obtained by incorporating some known functions of auxiliary variables X and Z in some existing factor-type estimators. Bias and Mean

square error (MSE) of these estimators, 1 d

FTAA

y

 and 2 d

FTAA

y were obtained using tailor’s series expansion. Audu & Adewara

([AA17]) revealed that although 1 d

FTAA

y and 2 d

FTAA

y , have minimum MSE and high PRE than all other related existing factor-type estimators considered using three life dataset but of these two, which is more efficient and most preferred. The

simulation results obtained in this study revealed that 2 d

FTAA

y is more efficient than 1 d

FTAA

y and hence, most preferred.

KEYWORDS: Factor-type estimator, Two-phase sampling, Mean square error (MSE), Efficiency.

1. INTRODUCTION

It has been established that when the population mean

X

of auxiliary variables

X

is not known but

X

is correlated to another auxiliary variable Z whose population mean

Z

is known, then information on Z can be used to improved the efficiency of estimator at hand ([CS12]). In this study, information on secondary auxiliary variable were incorporated into the work of Shukla ([Shu02]) in addition to power transformation with sole aim of improving Shukla ([Shu02]) factor-type estimator.

Shukla ([Shu02]) suggested a factor-type estimator for population mean under two-phase sampling as:









1 2

2

1 2

FTd

A C x

fBx

y

y

A

fB x

Cx











(1)





2

3 4

FTd I xy x y x

Bias y



YP

 



_

C C





C



_

(2)









2

1 3 2 4 2

FTd II x xy x y

Bias y



YP



_

  



C



 

C C



_

(3)





2 2 2 2

2 3

2

3

FTd _I y x xy x y

MSE y



Y

_





C





P C



 

P

C C



_

(4)





2 2 2 2

2 4

2

2

FTd II y x xy x y

MSE y



Y

_





C





P C



 

P

C C



_

(5)

Where:

1 2 3 4 1 2

1 2 2 1

1 1 1 1 1 1

, , ,

n N n N n n



 



 



 



 

 



1



2



A d d , B



d1



d4



, C



d2



d3



d4



,

1 , 2 , 3 , 4

A C fB A fB C

A fB C A fB C A fB C A fB C



 







 





        ,

P



   

3



1



2



4

In his work, it was observed that factor-type estimator

y

_FTd was more efficient than classical ratio estimator y_rd , if 2Cyx  P 0 under case I and if





1

2

C

_yx

1





P

0





 

under case II where

  

 _{1 2}1.

2. AUDU & ADEWARA ([AA17]) ESTIMATORS

Consider a preliminary large sample of size drawn from population Ω of size by SRSWOR and secondary sample of size drawn either of the following manners:

Case I: as a subset from the preliminary sample i.e. . Case II: as an independent sample from population i.e. .

Motivated by Choudhury and Singh ([CS12]) and Khan et. al ([KSS12]), Audu & Adewara ([AA17]) suggested the following estimators under two-phase sampling as:

 









1

1 1 2

2

1 2 1

d z z

FTAA

z z

A C x

fBx

a Z

b

y

y

A

fB x

Cx

a z

b





_

















_

_

 

_











(6)

 









2

2 1 2 1

2

1 2 2

d z z

FTAA

z z

A C x

fBx

a z

b

y

y

A

fB x

Cx

a z

b





_

















_

_

 

_











(7)

where , , and are assumed to be known as either real numbers or (linear or non-linear) functions,

0





₁



1

and

0





₂



1

3. PROPERTIES OF AUDU & ADEWARA ([AA17]) FACTOR-TYPE ESTIMATORS

(a). Under Case I ([AA17])

 



1



2





2 1



1



2

3 4 1 1

1 2

d

FTAA x xy x y z z z yz y z I

Bias y Y 



PC



C C 







C _

 







  

C C _

  

  (8)

 



1



2 2





2









3 3 2 4 1 3 1

1 / 2

d

FTAA x x yx z z z xz

II

Bias y





Y

_



 

P

C



P C



C







  

C

 





PC



_

(9)

 



1



2 2 2





2





2 3

2

1 1 1

2

d

FTAA y x yx z z z yz

I

MSE y





Y

_





C





C P P



C



  

C

 



C



_

(10)

and

 



1



2 2 2









2





2 2 2 3 3 1 1 2

d

FTAA y x yx z z z xz

II

MSE y Y _



C C P



P C 



P 

  

C

 

 PC _ (11)

(b). Under Case II ([AA17])

 













2 2 2 2 2

2 1 3 4 3 2

2 2 2 2 2 2 2 2 2

2 1 2

1 1

2 2

d

FTAA yz z yx x x z yz z I

z z z z

Bias y Y PC C PC C PC C C

C C



_

_

_{ }

_{  }

 

 





 





 _   



   

  _  _



 

(12)

 



2



2



2



2 2 2



2



2 2 2

2 1 2 2

1 1

2 2

d

FTAA z z z z z xz z

II

Bias y Y

 

 

C

 

 

C

  

P C C

 

 



  

 



 _  





and

 

















2 2 2 2 2

2 2 1 2 2

2 2 2

2 2 2 1

2 2

2 2

d

FTAA y x yx z z z xz

II

z z z yz xz x

MSE y Y C C P P C C PC

C C PC P C



_

_

_{  }

_{ }

  

 





 _    



   _ (15)

4. EFFICIENCY COMPARISON

 

2d

FTAA y

is said to be more efficient than

 

1d

FTAA

y



whenever:

(a). For case I:

 



1



2 2 2









2





2 2 2 3 3 1 1 2

d

FTAA y x yx z z z xz

II

MSE y Y _



C C P



P C 



P 

  

C

 

 PC _ <

 



1



2 2 2





2





2 3

2

1 1 1

2

d

FTAA y x yx z z z yz

I

MSE y





Y

_





C





C P P



C



  

C

 



C



_

and

(b). For case II:

 

















2 2 2 2 2

2 2 1 2 2

2 2 2

2 2 2 1

2 2

2 2

d

FTAA y x yx z z z xz

II

z z z yz xz x

MSE y Y C C P P C C PC

C C PC P C



_

_

_{  }

_{ }

  

 





 _    



   _

<

 



2



2 2 2





2





2 3

2

3 2 1

2

2

d

FTAA y x yx z z z yz xz

I

MSE y





Y

_





C





C P P



C



  

C

 



C



PC



_

5. EMPIRICAL STUDY USING SIMULATED DATA

In order to justify the results obtained in section three, a numerical simulation study is conducted to investigate the efficiency of these Audu & Adewara ([AA17]) factor-type estimators. The correlation coefficients between the study and auxiliary variables are assumed to be



xy  0.2,



xy  0.5,



xy  0.8 and



xy  0.99

using multivariate normal with the following parameters:

200, 1.51, 2.02, 3.05, y 0.241, x 0.387, z 0.239

N Y  X  Z  S  S  S 

Table 1: Bias, MSE and PRE of when



_xy



0.2

and



_xy



0.5

Estimators

Bias MSE PRE Bias MSE PRE

CASE I

6.92X10-4 10.75X10-4 107.1 23.2X10-4 5.41X10-4 157.8

4.81X10

-4

10.02X10-4 113.6 31.7X10-4 4.87X10-4 175.2

CASE II

21.2X10-4 8.52X10-4 133.7 13.9X10-4 5.08X10-4 168.1

16.1X10

-4

8.41X10-4 135.3 17.2X10-4 4.52X10-4 188.9

   

1d

_,

2d FTAA FTAA

y



y



0.2

xy







_xy



0.5

 

1d FTAA

y



 

2d FTAA

y



 

1d FTAA

y



 

2d FTAA

Table 2: Bias, MSE and PRE of when



_xy



0.8

and



_xy



0.99

Estimators



_xy



0.99

Bias MSE PRE Bias MSE PRE

CASE I

19.8X10-4 2.07X10-4 320.7 28.9X10-4 1.31X10-4 507.4

24.7X10-4 1.99X10-4 333.2 18.4X10-4 1.23X10-4 524.1 CASE II

3.82X10-4 2.84X10-4 233.6 11.9X10-4 1.58X10-4 421.1

8.42X10

-4

2.75X10-4 241.1 1.77X10-4 1.02X10-4 653.4

Table 3: Bias, MSE and PRE of when and

Estimators



_xy

 

0.2



_xy

 

0.5

Bias MSE PRE Bias MSE PRE

CASE I

2.13X10-4 9.03X10-4 125.9 23.2X10-4 4.41X10-4 193.9

2.34X10-4 7.79 X10-4 146.2 31.7X10-4 4.23X10-4 202.0 CASE II

2.24X10-4 7.93 X10-4 143.6 3.16X10-4 3.84X10-4 224.9

2.28X10-4 4.62 X10-4 246.2 5.31X10-4 2.34X10-4 249.3

Table 4: Bias, MSE and PRE of when and

Estimators



_xy

 

0.8



_xy

 

0.99

Bias MSE PRE Bias MSE PRE

CASE I

37.4X10

-4

3.51X10-4 279.5 42.9X10-4 1.58X10-4 421.3

45.1X10-4 3.33X10-4 294.8 45.4X10-4 1.52X10-4 438.2 CASE II

3.82X10-4 4.47X10-4 210.2 6.30X10-4 2.85X10-4 232.8

8.42X10-4 4.35X10-4 225.6 4.76X10-4 2.53X10-4 262.1

6. RESULTS AND DISCUSSION

Tables 1 - 4 show the biases, MSEs and PRE of 1 d

FTAA

y and 2 d

FTAA

y under simple random sampling scheme when the study and auxiliary variables are negatively and positively correlated with

0.2, 0.5, 0.8

xy xy xy



 



 



  and



_xy  0.99 coefficients respectively. These properties (Bias, MSE

and PRE) were computed under cases I and II. The simulation results of the analysis revealed that 2 d

FTAA y have

minimum MSEs and high PRE and hence, more efficient and most preferred to 1 d

FTAA y .

   

1d

_,

2d FTAA FTAA

y



y



0.8

xy





 

1d FTAA

y



 

2 d FTAA

y



 

1d FTAA

y



 

2 d FTAA

y



   

1 2

,

d d

FTAA FTAA

y



y





_xy

 

0.2



_xy

 

0.5

 

1d FTAA

y



 

2 d FTAA

y



 

1d FTAA

y



 

2 d FTAA

y



   

1d _, 2d FTAA FTAA

y y



_xy

 

0.8



_xy

 

0.99

 

1d FTAA

y



 

2 d FTAA

y



 

1d FTAA

y



 

2 d FTAA

REFERENCES

[AA17] Audu A., Adewara A. A. - Modified Factor-Type Estimators With Two Auxiliary Variable Under Two-Phase Sampling. Anale. Seria Informatica. XV(1), 63 -76, 2017.

[CS12] Choudhury S., Singh B. K. - Aclass of chain ratio-product type estimators with two auxiliary variables under double sampling scheme. Jour. of the Korean Stat. Soc. 41, 247-256, 2012.

[KSS12] Khan H., Shouket S., Sanaullah A. - Aclass of improved estimators for estimating population mean regarding partial information in double sampling. Global Journal of Science Frontier Research (Mathematics and Decision), 12(14), 32 – 44, 2012.