African Journal Shukla Et Al

8/2/2019 African Journal Shukla Et Al

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African Journal of Mathematics and Computer Science Research Vol. 5(4), pp. 78-89, 15 February, 2012Available online at http://www.academicjournals.org/AJMCSRDOI: 10.5897/AJMCSR11.159ISSN 2006-97312012 Academic Journals

Full Length Research Paper

Utilization of mixture of 1 X,X and 2X in imputation formissing data in post-stratification

D. Shukla1, D. S. Thakur2 and N. S. Thakur3

1Department of Mathematics and Statistics, Dr. H. S. Gour Central University, Sagar, (M.P.), India.

2School of Excellence, Sagar (M.P.), India.

3Centre for Mathematical Sciences (CMS), Banasthali University, Rajasthan, India.

Accepted 10 January, 2012

To estimate the population mean using auxiliary variable there are many estimators available in

literature like-ratio, product, regression, dual-to-ratio estimator and so on. Suppose that all theinformation of the main variable is present in the sample but only a part of data of the auxiliary variableis available. Then, in this case none of the aforementioned estimators could be used. This paperpresents an imputation based factor-type class of estimation strategy for population mean in presenceof missing values of auxiliary variables. The non-sampled part of the population is used as animputation technique in the proposed class. Some properties of estimators are discussed andnumerical study is performed with efficiency comparison to the non-imputed estimator. An optimumsub-class is recommended.

Key words: Imputation, non-response, post-stratification, simple random sampling without replacement(SRSWOR), respondents (R).

INTRODUCTION

In sampling theory, the problem of mean estimation of apopulation is considered by many authors like Srivastavaand Jhajj (1980, 1981), Sahoo (1984, 1986), Singh(1986), Singh et al. (1987), Singh and Singh (1991),Singh et al. (1994), Sahoo et al. (1995), Sahoo andSahoo (2001), and Singh and Singh (2001). Sometimesin survey situations, a small part of sample remains non-responded (or incomplete) due to many practicalreasons. Techniques and estimation procedures areneeded to develop for this purpose. The imputation is awell defined methodology by virtue of which this kind ofproblem could be partially solved. Ahmed et al. (2006),Rao and Sitter (1995), Rubin (1976) and Singh and Horn(2000) have given applications of various imputationprocedures. Hinde and Chambers (1990) studied thenon-response imputation with multiple source of non-response. The problem of non-response in samplesurveys immensely looked into by Hansen and Hurwitz(1946), Grover and Couper (1998), Jackway and Boyce

*Corresponding author. E-mail: [email protected].

(1987), Khare (1987), Khot (1994), Lessler and Kalsbeek(1992).

When the response and non-response part of thesample is assumed into two groups, it is closed to calupon as post-stratification. Estimation problem in samplesurvey, in the setup of post-stratification, under nonresponse situation is studied due to Shukla and Dubey(2001, 2004, and 2006). Some other useful contributionsto this area are by Holt and Smith (1979), Jagers et al(1985), Jagers (1986), Smith (1991), Agrawal and Panda(1993), Shukla and Trivedi (1999, 2001, 2006), Wywia(2001), Shukla et al. (2002, 2006). When a sample is fulof response over main variable but some of auxiliaryvalues are missing, it is hard to utilize the usuaestimators. Traditionally, it is essential to estimate thosemissing observations first by some specific estimationtechniques. One can think of utilizing the non-sampledpart of the population in order to get estimates of missingobservations in the sample. These estimates could beimputed into actual estimation procedures used for thepopulation mean. The content of this research work takesinto account the similar aspect for non-responding valuesof the sample assuming post-stratified setup and utilizing


2/12

the auxiliary source of data.

Symbols and setup

Let U = (U1, U2 , ........., UN) be a finite population of N

units with Y as a main variable and X the auxiliaryvariable. The population has two types of individuals likeN1 as number of "respondents (R)" and N2 "non-respondents (NR)", (N = N1+N2). Their populationproportions are expressed like W1 = N1/Nand W2 = N2/N.Quantities W1 and W2 could be guessed by past data or

by experience of the investigator. Further, let Y and X be the population means of Yand Xrespectively. Somesymbols are as follows:

R-group, Respondents group or group of those whoresponses in survey;

NR-group, Non-respondents group or group of those whodenied to response during survey;

1Y , Population mean of R-group of Y;

2Y , Population mean of NR-group of Y;

1X , Population mean of R-group of X;

2X , Population mean of NR-group of X;2

1YS , Population mean square of R-group of Y;2

2YS , Population mean square of NR-group of Y;2

1XS , Population mean square of R-group of X;2

2XS , Population mean square of NR-group of X;YC1 , Coefficient of variation of Yin R-group;

YC2 , Coefficient of variation of Yin NR-group;

XC1 , Coefficient of variation of Xin R-group;

XC2 , Coefficient of variation of Xin NR-group;

, Correlation coefficient in population between Xand Y;

N, Sample size from population of size Nby SRSWOR;

1n , Post-stratified sample size coming from R-group;

2n , Post-stratified sample size from NR-group;

1y , Sample mean of Ybased on n1 observations of R-

group;

2y , Sample mean of Ybased on n2 observations of NR-

group;

1x , Sample mean of X based on n1 observations of R-

group;

2x , Sample mean of Xbased on n2 observations of NR-

group;

1 . Correlation Coefficient of population of R-group;

2 , Correlation Coefficient of population of NR-group

Shukla et al. 79

Further, consider few more symbolic representations:

2

1

2

1

11

11

111

WnN

WnN

nWnED ;

2

2

2

2

22

21

111

WnN

WnN

nWnED

N

YNYNY 2211

; 2211

N

XNXNX

ASSUMPTIONS

Consider following in light of Figure 1 before formulatingan imputation based estimation procedure:

1) The sample of size nis drawn by SRSWOR and poststratified into two groups of size n1 and n2 (n1 + n2 = naccording to R and NR group respectively2) The information about Y variable in sample iscompletely available.

3) The sample means of both groups 1y and 2y are

known such that

2211

nynyny which is sample mean on n units.

4) The population means 1X and X are known.

5) The population size Nand sample size nare knownAlso, N1 andN2 are known by past data, past experienceor by guess of the investigator (N1 + N2 = N).

6) The sample mean of auxiliary information 1x is only

known for R-Group, but information about 2x of NR-group

is missing. Therefore

2211

n

xnxnx

could not be obtained due to absence o

2x .

7) Other population parameters are assumed known, in

either exact or in ratio from except the Y , 1Y and 2Y .

PROPOSED CLASS OF ESTIMATION STRATEGY

To estimate population meanY , in setup of Figure 1, a

problem to face is of missing observations related to 2x

therefore, usual ratio, product and regression estimatorsare not applicable. Singh and Shukla (1987) have

proposed a factor type estimator for estimating population

mean Y . Shukla et al. (1991), Singh and Shukla (1993)and Shukla (2002) have also discussed properties offactor-type estimators applicable for estimatingpopulation mean. But all these cannot be useful due tounknown information

2x . In order to solve this, an

imputation 4

*

2x is adopted as:

42

*x =

nN

XfXfnXN 21 1

(1)


3/12

80 Afr. J. Math. Comput. Sci. Res.

Population (N=N1 + N2)

R-group NR-group

N1 1Y 1X N2 2Y 2X

2

1YS

21X

S 2

2YS

22XS

Sample (n=n1+ n2)

R-group NR-group

n1 1y n2

1x 2y 2x

Figure 1. Setup to estimate population meanY .

The logic for this imputation is to utilize the non-sampledpart of the population of X for obtaining an estimate of

missing 2x and generate 2x for x as describe as

follows:

And

4x =

21

42211

NN

xNxN*

(2)

The proposed class of imputed factor-type estimator is:

[( yFT

)D]k

=

4

4

2211

xCXfBA

xfBXCA

N

yNyN

(3)

Where k0 and kis a constant and

A = (k 1) (k 2 ); B= (k 1) (k 4); C= (k 2) (k 3) (k 4); f= n/ N.

LARGE SAMPLE APPROXIMATION

Consider the following for large n:

422

311

222

111

1

1

1

1

eXx

eXx

eYy

eYy

(4)

where,1e ,

2e ,

3e and

4e are very small numbers 0ie

(i= 1,2,3,4).

Using the basic concept of SRSWOR and the concept ofpost-stratification of the sample ninto n1 and n2 (Cochran2005; Hansen et al., 1993; Sukhatme et al., 1984; Singhand Choudhary, 1986; Murthy, 1976), we get

0EEE

0EEE

0EEE

0EEE

244

133

222

111

nee

nee

nee

nee

(5)

Assume the independence of R-group and NR-grouprepresentation in the sample, the following expressioncould be obtained:

12121 EEE nee

1

2

1

1

11E nC

NnY

21

1

11E

YC

Nn

211

1Y

CN

D

(6)


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22222 eEE nEe

2

2

2

2

11E nC

NnY

222

1YC

ND

(7)

1

2

3

2

3EEE nee

2

11

1X

CN

D

(8)

and

2

22

2

4

1E XC

NDe

(9)

13131

EEE neeee

1111

1

11E nCC

NnXY

XY

CCN

D1111

1

(10)

0EEE212121

n,neeee(11)

0E 41 ee (12)

0,EEE 213232 nneeee (13)

XY

CCN

Dee222242

1E

(14)

0E 43 ee (15)The expressions (11), (12), (13) and (15) are true underthe assumption of independent representation of R-groupand NR-group units in the sample. This is introduced tosimplify mathematical expressions.

Shukla et al. 81

Theorem 1

The estimator [( yFT

)D]k

could be expressed under large

sample approximation in the form:

[( yFT

)D

]k

= 3Y[1 + s

1W

1e

1+ s

2W

2e

2][1 + (

3

3)e

3(

3

3)

3 2

3e + (

3

3) 2

3

3

3e

Proof

Rewrite 4

x as in Equation 2:

4x =

21

4

*

2211

NN

xNxN

Where,

42

*x =

nN

XfXfnXN 21 1

)4(

x =

nN

XfXfnXNNeXN

N

21

2311

)1(1

1

=

N

XfXfnXNpeXN 21311 11

=

N

eXNXfpnXpnfXpNXN 3112111 1

=

N

eXNXfpnXpnfNXpN 311211 1

= 3112121 1 erWr)f(pfrpfWpX

= 311 erWX (16)

Where,

p=nN

N

2 ; = p+ (W1- pf

2)r1pf(1-f) r2.

Now, the estimator [( yFT

)D]k

under approximation and

using (16) is

[( yFT

)D]k

=

4

4

2211

xCXfBA

xfBXCA

N

yNyN

=

XerWCXfBA

XerWfBXCA

N

eYNeYN

311

31122

211

111

=

311

311

2221111

erCWCfBA

erfBWCfBAeWseWsY

=

343

321

2221111

e

eeWseWsY

= 3 2221111 eWseWsY (1 + 3e3)(1 + 3e3)-1


5/12


Where,

1 = A + fB+ C; 2 = fBW1r1; 3 = A + fB+ C; 4 = CW1r1;

r1 =X

X1 ; r2 =X

X2 ; s1 =Y

Y1 ; s2 =Y

Y2 ; 3 =1

2

; 3 =

3

4

; 3 =

3

1

.

We can further express the above into following:

= 3Y [1 + s1W1e1+ s2W2e2] (1 + 3e3) (1 3e3+2

32

3e

3

3 33e .)

=3Y [1+s1W1e1+s2W2e2][1+(33)e3(33)32

3e+(3 3)

2

33

3e .]

BIAS AND MEAN SQUARED ERROR

Define E(.) for expectation, B(.) for bias and M(.) formean squared error, then the first order ofapproximations could be established for i,j= 1, 2, 3, .....as

0E 21 jiee when i+j>2

0E 31 jiee when i+j>2

0E 32 jiee when i+j>2 (17)

Theorem 2

The [( yFT

)D]k

is a biased estimator of Y with the amount

of bias to the first order of approximation:

YXXkDFT CWsCN

DCYy 111113133133 1

1B

Proof

YyykDFTkDFT

EB

Using theorem 1 and taking expectations

E[( yFT

)D]k

= 3YE[1 + (3 3)e3 (33) 32

3e

+ s1W1e1{1 + (33) e3 (33)

32

3e }

+ s2W2e2 {1+ (33) e3 (33)32

3e }]

= 3Y[1(33)3 E(2

3e )+ s1W1(33) E(e1e3) + s2W2(33)E(e2e3)]

=

21133331

1 XCN

DY

XY

CCN

DWs11111133

1

=

YXX

CWsCCN

DY11111311333

11

Therefore,

YyykDFTkDFT

EB

=

YXX CWsCN

DCY 1111131331331

1

Theorem 3

The mean squared error of [( yFT

)D]k

is

M [( yFT

)D]

=

2

2

2

2

2

2

2

3211113

2

12

2

1

2

111

2

3

2 12

11

YXYXYCWs

NDCCsJCJCsJ

NDY

Where

2

1

2

31 WJ , 333333332 12 J ; )(12( 33313 WJ

Proof

M [( yFT

)D]k

= E[{( y FT)D}kY ]2

Using Theorem 1, we can express

M[( yFT

)D]k

= E[ 3Y{1 + s1W1e1+ s2W2e2}{1+ (3 3)e3 (3 3)32

3e + (3 3)

2

3

3

3e

...}Y]2

Using large sample approximations of (17) we could

express

=2

Y E [(31)+3{(3 3)e3 (33)3 23e +(s1W1e1+ s2W2e2)+ (3 3)(s1W1e1 + s2W2e2)

e3}]2

=2

Y E [(3 1)2+ 2

3 {(3 3)

2 23e

+ 21s

2

1W

2

1e

+ 22s

22

W2

2e +2s1s2W1W2e1e2

+ 2(3 3)(s1W1e1 e3 + s2W2e2e3)}+ 23(3 1) {(3 3)e3 (3 3)32

3e

+ (s1W1e1 + s2W2e2) + (3 3) (s1W1e1e3 + s2W2e2e3)}]

Using (5), (11) and (12) we rewrite,

=2

Y [ (31)2+ 23 {(33)

2E( 23e )+ 21s

21W

E( 21e )+ 22s

22W E(

2

2e ) +2(33) s1W1 E(e1 e3)}

+ 23(3 1){(33)3 E(2

3e

)+ (3 3)s1W1 E(e1e3 )}]

2

11

2

1

2

1

2

11

2

33

2

3

2

3

2 111

YXC

NDWsC

NDY

XYYCC

NDWsC

NDWs

11111133

2

22

2

2

2

2

12

1

2

1133333

112

XC

ND

XYCC

NDWs

11111133

1

2

1

2

1

2

1

2

31

2

3

2 11

YCWs

NDY 12 2

133333333 XC

XYCCWs 111113333 12

22

22

22

232

1YCWs

ND

212

2

1

2

111

2

3

2 11 XY CJCsJ

NDY

2

2

2

2

2

2

2

3211113

12 YXY CWs

NDCCsJ


6/12

SOME SPECIAL CASES

The term A, B and C are functions of k. In particular,there are some special cases:

Case 1

When k= 1

A = 0; B= 0; C= 6; 1= -6; 2= 0; 3= - 6; 4= -6r1w1; 3 = 0; 3 =

11Wr

; 3 =1 ;

J1 = 2

2

1

W; J2 = 4

2

1

2

1 23

)(Wr ; J3 = 3

2

11 2

)(Wr ;

The estimator [( yFT

)D]k

along with bias and m.s.e. under

case i is:

[( yFT

)D]k =1

=

)(

x

X

N

yNyN4

2211

(18)

B[( yFT

)D]k =1

=Y-3[(1-)2+

N

D1

1

r12

1W C1X{r1C1X -s11C1Y}]

(19)

M[( yFT

)D]k=1

=2

Y -4[(1)22 + 21W

N

D1

1{2

2

1s

2

1YC + (3 - 2) 2

1r

2

1XC

+ 2(- 2)r1s11C1YC1X}+

ND1

2 2 2

2W

2

2s2

2YC ]

(20)

Case 2

When k= 2

A = 0; B= 2; C= 0; 1= -2f; 2 = -2fW1r1; 3 = -2f; 4 = 0;

3 =1

11

Wr ; 3 = 0;3 =; J1=22

1 W ; J2=2

1

2

1Wr ; J3 = )(Wr 12

2

11 ;

[( yFT

)D]k =2

=

X

x

N

yNyN)( 4

2211

(21)

B [( yFT

)D]k =2

=

XYCCsrWN

D)(Y 111112

11

11

(22)

M [( yFT

)D]k =2

=2

Y [( 1)2 + 21W

ND

11

{2

2

1s

2

1YC + r12 2

1XC + 2(21)s

1r1

1C

1YC

1X}

+

N

D1

22 22W

2

2s 2

2YC ]

(23)

Shukla et al. 83

Case 3

When k= 3

A = 2; B= 2; C= 0; 1= 2(1- f); 2 = -2fW1r1; 3= 2(1-f); 4=0; 3 =

f

rfW

1

11

3 = 0; 3 =f

f

1

1 ; J1=

2

2

1

2

1

1

f

Wf

; J2 = 2

2

1

2

1

2

1 f

rWf

; J3 =

21

2

1

1

12

f

ffrfW

;

[( yFT

)D]k =3

=

X)f(

xfX

N

yNyN)(

1

4

2211

(24)

B [( yFT

)D]k =3

=

XYCCsrWN

D)()f(fY 111112

11

1 111

(25)

M [( yFT

)D]k =3

=2

Y (1f)-2

[f2(1)2+

N

D1

1

2

1W {(1f)2 2

1s2

1YC + f2 2

1r2

1XC

+ 2(2ff 1)fr1s

1

1C

1YC

1X}+

N

D1

2(1f)2 22W

2

2s 2

2YC ]

(26)

Case 4

When k= 4;

A = 6; B= 0; C= 0 ; 1= 6; 2 = 0; 3 = 6; 4 = 0; 3 = 0; 3= 0; 3 =

1; J1=2

1W ; J2= 0; J3 = 0;

[( yFT

)D]k =4 =

N

yNyN 2211

(27)

B[( yFT

)D]k =4

= 0(28)

V[( yFT

)D]k =4

= 22

2

22

22

2

1

2

12

11

11YY CYW

NDCYW

ND

(29)

ESTIMATOR WITHOUT IMPUTATION

Throughout the discussion, the assumption is unknown

value of 2x . This is imputed by the term 42*

x , to provide

the generation of 4

x [Equations (1) and (2)]. Suppose

the 2x is known, then there is no need of imputation and

the proposed (2) and (3) reduces into:

N

xNxNx 2211(*)

(30)


7/12


*

(*)

kwFT xCX)fBA(

xfBX)CA(

N

yNyNy 2211

(31)

Where, kis a constant , 0 < k< and

A = (k

1) (k

2); B= (k

1) (k

4); C= (k

2) (k

3) (k

4); f= n/N.

Theorem 4

The estimator kwFT

y is biased for Y with the amount

of bias

X

'

YX

''

kwFTCrCsCrW

NDyB 11211111

2

1121

1

X

'

YX CrCsCrWND 222222222

22

1

Where,

CfBA/fB' 1 ; CfBA/C' 2 .

Proof

The estimator kwFT

y could be approximate like:

*

*

kwFTxCXfBA

xfBXCA

N

yNyNy 2211

N

eYNeYN 222111 11

422311

422311

11

11

eXNeXNCXfBAN

eXNeXNfBXCAN

422311

422311222111

erWerWCCfBA

erWerWfBCfBAeYWeYWY

4223111222111 1 erWerWeYWeYWY

' 142231121

erWerW'

Expanding above using binominal expansion, and

ignoring ljki ee terms for ( k+ l) > 2, (k, l= 0, 1, 2 ........),(i, j= 1, 2, 3, 4 ); the estimator result into

212222111121 11 eYWeYWYYy kwFT (32)

Where,

1 = 42231121 erWerW'' ; 2 = 2

422311212erWerW

'''

And

W1 r1 +W2 r2 = 1 holds.

Further, one can derive up to first order of approximationaccording to the following;

(i) E [1] = 0

(ii) E 21 =

222

2

2

2

2

2

11

2

1

2

1

2

21

11XX

''C

N

DrWC

N

DrW

(iii) E 2 =

222

2

2

2

2

2

11

2

1

2

1212

11 XX

'''C

NDrWC

NDrW

(iv) E 11e = YX'' CCN

DrW 111111211

(v) E 12 e = YX'' CCN

DrW 222222211

(vi) E 21e = 10under0 n

(vii) E 22 e = 10under0 n

The bias of estimator without imputation is

YyykwFTkwFT

EB

)1()1()( 212222111121 eYWeYWYE

212221111

EY)e(EYWeEYW

X

'

YX

''CrYCYCrW

ND 11111111

2

1121

1

X

'

YX CrYCYCrWND 221222222

22

1

X

'

YX

''CrCsCrW

NDY 21211111

2

1121

1

X

'

YX CrCsCrWN

D 222222222

22

1

Theorem 5

The mean squared error of the estimator kwFT

y is

XYXYkwFT CCTTCTCTWN

DYy 111212

1

2

2

2

1

2

1

2

11

2 21

M

XYXY CCSSCSCSWN

D22221

2

2

2

2

2

2

2

1

2

222

1

Where,

T1 = s1; T2 = 121 r'' ; S1 = s2; S2 = 221 r'' ;


8/12

Proof

2EM YyykwFTkwFT

[( yFT

)w]k

2212222111121 11E eYWeYWY

1111222222212121212 E2EEE eYYWeYWeYWY

2121211222 E2E2 eeYYWWeYYW

222

2

2

2

2

2

11

2

1

2

121

2 112

XX

''C

NDrWC

NDrWY

11 222

2

2

2

2

2

11

2

1

2

1

YY C

NDsWC

NDsW

XY

''CC

NDrWsW 1111112111

12

XY

''CC

NDrWsW 2222222122

12

XYXY CCTTCTCTWN

DY 11121212221212112 21

XYXY CCSSCSCSWN

D 222212

2

2

2

2

2

2

1

2

22 21

Remark 1

At k= 1, k= 2, k= 3 and k= 4, there are some specialcases of non-imputed estimators with the respective biasand mean squared error as laid down with the following

Case 5

When k= 1

A = 0 ; B= 0 ; C= 6 ; '1 = 0;'

2 = 1; T1 = s1;

T2 = -r1; S1 = s1; S2 =r2;

*kwFT x

X

N

yNyNy 2211

1

XYXkwFTCrCsCrW

NDYy

1111111

2

111

1B

XYXCrCsCrW

ND

2222222

2

22

1

XYXYkwFTCCrsCrCsW

NDYy 111112121212121121 21M

XYXY CCrsCrCsWN

D 222222

2

2

2

2

2

2

2

2

22 21

Case 6

When k= 2

A = 0; B= 2; C= 0; '1 = 1;'

2 = 0; T1 = s1; T2 = r1; S1= s2; S2 = r2.

Shukla et al. 85

X

x

N

yNyNy

kwFT

*

2221

2

YXYXkwFTCCrsW

NDCCrsW

NDYy 22222

2

2211111

2

112

11B

XYXY

kwFT CCrsCrCsW

NDYy

11111

2

1

2

1

2

1

2

1

2

11

2

2

21

M

XYXY CCrsCrCsWN

D 222222

2

2

2

2

2

2

2

2

22 21

Case 7

When k= 3

A = 2; B= 2; C= 0; '1 = -f/(1-f)-1; '2 =0; T1= s1; T2= r1 f(1- f)

-1

S1= s2; S2= r2 f(1- f)-1

;

Xf

xXN

yNyNykwFT )1(

*

22113

XYkwFTCCsrW

NDffYy

11111

2

11

1

3

11B

XYCCsrW

ND 22222

2

22

1

2

1

2

1

222

1

2

1

2

11

2

31

1M XYkwFT CrffCsWN

DYy XYCCsff 1111112

+ 22222221

YCsWN

D

XYX CCrsffCrff 222221222222 121

Case 8

When k= 4, then

A = 6 ; B= 0 ; C= 0; '1 = 0;'

2 = 0; T1= s1; T2= 0; S1= s2; S2= 0

N

yNyNy

kwFT

2211

4

0B4

kwFT

y

2

2

2

2

2

22

2

1

2

1

2

11

2

4

11V YYkwFT CsWN

DCsWN

DYy

NUMERICAL ILLUSTRATION

Consider two populations I and II given in Appendix Aand B. Both populations are divided into two parts as R-group and NR-group having size N1 and N2 respectively(N= N1 + N2). The population parameters are displayed inTables 1 to 6.

DISCUSSION

Using the imputation for 2x by the mixture of three


9/12


Table 1. Parameters of population - I (from Appendix A).

Parameter Entire population For R-group For NR-group

Size N= 180 N1 =100 N2= 80

Mean Y 03.159Y 60.1731 Y 81.1402 Y

Mean X 22.113X 45.1281 X 19.942 X

Mean square Y 18.22052 YS 36.25322

1 YS 90.12192

2 YS

Mean square X 61.19722 XS 86.23002

1 XS 179242

2 .S X

Coefficient of variation of Y 295.0YC 2899.01 YC 248.02 YC

Coefficient of variation of X 392.0X

C 373.01 XC 323.02 XC

Correlation coefficient 897.0XY 857.01 XY 956.02 XY

Table 2. Parameters of population - ii (from Appendix B).

Parameter Entire population For R-group For NR-group

Size N= 150 N1 = 90 N2= 60Mean Y 77.63Y 33.661 Y 92.592 Y

Mean X 2.29X 72.301 X 92.262 X

Mean Square Y 87.2992 YS 33.3492

1 YS 35.2062

2 YS

Mean Square X 43.1102 XS 67.112

2

1 XS 081002

2 .S X

Coefficient of Variation of Y 272.0YC 282.01 YC 2397.02 YC

Coefficient of Variation of X 3599.0XC 345.01 XC 3716.02 XC

Correlation coefficient 8093.0XY 8051.01 XY 8084.02 XY

Let samples of size n = 40 and n = 30 are drawn from population I and II respectively by SRSWOR and post-stratified into R and NR-groups. The sample values are in Tables 3 and 4.

Table 3. Sample values for population i.

Parameter Entire sample R-group NR-group

Size n= 40 n1 = 28 n2= 12

Fraction f =0.22 - -

Table 4. Sample values for population ii

Parameter Entire sample R-group NR-group

Size n= 30 n1 = 20 n2= 10

Fraction f =0.2 - -

population means is performed. The proposed class is inEquation (3) with bias in theorem 2 and mean squarederror in theorem 3. The class contains some specialimputed estimators for value k= 1, k= 2, k= 3 and k= 4.A non-imputed class of estimator is also developed whichhas bias and mean squared error derived in theorem 4and 5. This class also has some special cases. Thecomputation over two population is made whose

description is given in appendix. The two random sampleof size n = 40 and n = 30 are drawn from populations and II respectively and post-stratified into two groupsThe Tables 5 and 6 are presenting a numericacomparison between imputation and non-imputation classover the two populations in terms of their bias and m. se. The imputation technique (1) is effective because thereis not much increase in the mean squared error due to


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Shukla et al. 87

Table 5. Bias and M.S.E. comparisons of kDFT

y .

EstimatorPopulation I Population II

Bias M.S.E. Bias M.S.E.

[( yFT

)D

]k

=1

-2.8775 18.6000 -1.1874 9.8618

[( yFT

)D

]k=2

3.3127 232.4162 3.4360 41.4794

[( yFT

)D

]k=3

-4.0370 8.4710 -0.6377 6.3540

[( yFT

)D

]k=4

0 43.6500 0 9.2675

Table 6. Bias and MSE comparison of kwFT

y .

EstimatorPopulation I Population II

Bias M.S.E. Bias M.S.E.

1kwFTy 0.1433 12.9589 0.1095 6.0552

2kwFTy 0.3141 216.3024 0.1599 46.838

3kwFT

y 0.096 4.327 0.031 5.2423

4kwFT

y 0 43.65 0 9.2662

imputation. The k= 3 seems to be a good choice. The k=2 performs worst for both sots of data.

CONCLUSIONS

The technique of mixture of X , 1X , 2X performs welland the imputed factor-type estimator is very close tonon-imputed in terms of mean squared error when k= 1,2, and 3 holds. The choice k= 3 is better over the othertwo. Performance over population II is superior than I. itseems that factor type lass is able to replace the non-responded observation in a nice manner.

ACKNOWLEDGEMENT

Authors are thankful to the referee for his valuablesuggestions.

REFERENCES

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Shukla et al. 89

Appendix A. Population I (N= 180); R-group: (N1=100).

Y: 110 75 85 165 125 110 85 80 150 165 135 120 140 135 145X: 80 40 55 130 85 50 35 40 110 115 95 60 70 85 115

Y: 200 135 120 165 150 160 165 145 215 150 145 150 150 195 190

X: 150 85 80 100 25 130 135 105 185 110 95 75 70 165 160

Y: 175 160 165 175 185 205 140 105 125 230 230 255 275 145 125X: 145 110 135 145 155 175 80 75 65 170 170 190 205 105 85

Y: 110 110 120 230 220 280 275 220 145 155 170 195 170 185 195

X: 75 80 90 165 160 205 215 190 105 115 135 145 135 110 145

Y: 180 150 185 165 285 150 235 125 165 135 130 245 255 280 150X: 135 110 135 115 125 205 100 195 85 115 75 190 205 210 105

Y: 205 180 150 205 220 240 260 185 150 155 115 115 220 215 230

X: 110 105 110 175 180 215 225 110 90 95 85 75 175 185 190

Y: 210 145 135 250 265 275 205 195 180 115

X: 170 85 95 190 215 200 165 155 150 175

NR-group: (N2=80)

Y: 85 75 115 165 140 110 115 13.5 120 125 120 150 145 90 105

X: 55 40 65 115 90 55 60 65 70 75 80 120 105 45 65

Y: 110 90 155 130 120 95 100 125 140 155 160 145 90 90 95

X: 70 60 85 95 80 55 60 75 90 105 125 95 45 55 65

Y: 115 140 180 170 175 190 160 155 175 195 90 90 80 90 80

X: 75 105 120 115 125 135 110 115 135 145 45 55 50 60 50

Y: 105 125 110 120 130 145 160 170 180 `145 130 195 200 160 110

X: 65 75 70 80 85 105 110 115 130 95 65 135 130 115 55

Y: 155 190 150 180 200 160 155 170 195 200 150 165 155 180 200

X: 115 130 110 120 125 145 120 105 100 95 90 105 125 130 145

Y: 160 155 170 195 200

X: 120 115 120 135 150

Appendix B. Population II (N=150); R-group (N1=90).

Y: 90 75 70 85 95 55 65 80 65 50 45 55 60 60 95

X: 30 35 30 40 45 25 40 50 35 30 15 20 25 30 40

Y: 100 40 45 55 35 45 35 55 85 95 65 75 70 80 65

X: 50 10 25 25 10 15 10 25 35 55 35 40 30 45 40

Y: 90 95 80 85 55 60 75 85 80 65 35 40 95 100 55

X: 40 50 35 45 35 25 30 40 25 35 10 15 45 45 25

Y: 45 40 40 35 55 75 80 80 85 55 45 70 80 90 55

X: 15 15 20 10 30 25 30 40 35 20 25 30 40 45 30

Y: 65 60 75 75 85 95 90 90 45 40 45 55 60 65 60

X: 25 40 35 30 40 35 40 35 15 25 15 30 30 25 20

Y: 75 70 40 55 75 45 55 60 85 55 60 70 75 65 80

X: 25 20 35 30 45 10 30 25 40 15 25 30 35 30 45NR-group (N2=60)

Y: 40 90 95 70 60 65 85 55 45 60 65 60 55 55 45

X: 10 30 30 30 25 30 40 25 15 20 30 30 35 25 20

Y: 65 80 55 65 75 55 50 55 60 45 40 75 75 45 70

X: 35 45 30 30 40 15 15 20 30 15 10 40 45 10 30

Y: 65 70 55 35 35 50 55 35 55 60 30 35 45 55 65

X: 30 40 30 10 15 25 30 15 20 30 10 20 15 30 30

Y: 75 65 70 65 70 45 55 60 85 55 60 70 75 65 80

X: 30 35 40 25 45 10 30 25 40 15 25 30 35 30 45

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