An Almost Unbiased Estimator for Population Mean using ...An Almost Unbiased Estimator for Population Mean using Known Value of Population Parameter(s) ... S.B. Gupta and S. Malik:

J. Stat. Appl. Pro.3 , No. 2, 269-293 (2014) 269 Journal of Statistics Applications & Probability An International Journal

http://dx.doi.org/10.12785/jsap/030216

An Almost Unbiased Estimator for Population Mean using Known Value of

Population Parameter(s)

Rajesh Singh1, S.B. Gupta2 and Sachin Malik2, *

1 Department of Statistics, Banaras Hindu University, Varanasi-221005, India

2Department of Community Medicine, SRMS Institute of Medical Sciences, Bareilly- 243202, India

Received: 11 Feb. 2014, Revised: 22 May 2014, Accepted: 25 May 2014 Published: 1 Jul. 2014

Abstract: In this paper we have proposed an almost unbiased estimator using known value of some

population parameter(s) with known population proportion of an auxiliary variable. A class of estimators is

defined which includes [1], [2] and [3] estimators. Under simple random sampling without replacement

(SRSWOR) scheme the expressions for bias and mean square error (MSE) are derived. Numerical illustrations

are given in support of the present study.

Key words: Auxiliary information, proportion, bias, mean square error, unbiased estimator.

1. Introduction

It is well known that the precision of the estimates of the population mean or total of

the study variable y can be considering improved by the use of known information on an

auxiliary variable x which is highly correlated with the study variable y. Out of many

methods ratio, product and regression methods of estimation are good illustrations in this

context. Using known values of certain population’s parameters several authors have

proposed improved estimators including [4, 5, 6, 7, 8, 9, 10, 11, 12, 13].

In many practical situations, instead of existence of auxiliary variables there exit some

∗ Corresponding author e-mail: [email protected]

c 2014 NSP Natural Sciences Publishing Cor.

mailto:[email protected]

270 R. Singh, S.B. Gupta and S. Malik: An almost unbiased population parameter(s)

auxiliary attributes (say), which are highly correlated with the study variable y, such as

i. Amount of milk produced (y) and a particular breed of cow ( ).

ii. Sex ( ) and height of persons (y) and

iii. Amount of yield of wheat crop and a particular variety of wheat ( ) etc. (see [14]).

Many more situations can be encountered in practice where the information of the population

mean Y of the study variable y in the presence of auxiliary attributes assumes importance.

For these reasons various authors such as [15, 16, 17, 18, 19] have paid their attention

towards the improved estimation of population mean Y of the study variable y taking into

consideration the point biserial correlation between a variable and an attribute.

Let

N

1iiA and

n

1i

iφa denote the total number of units in the population and sample

possessing attribute respectively,N

AP and

n

ap denote the proportion of units in the

population and sample, respectively, possessing attribute .

To estimate Y , the usual estimator is given by

2y1SfyVar

(1)

Define,

Y

Yyey

,

P

Ppe

,yi,0eE i

,CfeE 2

y1

2

y ,CfeE 2

p1

2

φ

p.ypb1φy CCρfeeE


J. Stat. Appl. Pro. 3, No. 3, 269-293 (2014) / www.naturalspublishing.com/Journals.asp 271

Where,

,N

1

n

1f1

,Y

SC

2

2y2

y

,P

SC

2

2p2

p

and

SS

S

y

ypb is the point bi-serial correlation coefficient.

Here,

N

1i

2i

2y Yy

1N

1S

,

N

1i

2i

2 P1N

1S

and

N

1iiiy YNPy

1N

1S ,

N

1

n'

1f, 2

In order to have an estimate of the study variable y, assuming the knowledge of the

population proportion P, [1] proposed the following estimator

p

Pyt NGR

(2)

P

pyt NGP

(3)

Following [1] , we propose the following estimator

α

321

321S1

KKpK

KKPKyt

(4)

The Bias and MSE expressions of the estimator S1t up to the first order of approximation

are, respectively, given by



p1

2

12

p1S1 KαV2

1)Vα(αCfYtB

(5)

1p1

2

1

22

p

2

y1

2

S1 αK2VVαCCfYtMSE (6)

Also following [2], we propose the following estimator

5454

5454

β

S2KpKKPK

KpKKPKλexp

P

p2yt

(7)

The Bias and MSE expressions of the estimator S2t up to the first order of approximation are,

respectively, given by

2

KλVβK

8

V2λλ

2

1ββ

2

βλVCfYtB

p2

p

2

222

p1S2 (8)

2

λVβC2KβλV

4

VλβCCfYtMSE 22

pP2

2

2

222

p

2

y1

2

S2 (9)

, and are suitable chosen constants. Also1K ,

3K ,4K ,

5K are either real numbers or

function of known parameters of the auxiliary attributes such as pC , 2, pb and

PK .

2K is an integer which takes values +1 and -1 for designing the estimators and

54

42

321

11

KPK

PKV

KKPK

PKV

We see that the estimator’s tS1 and tS2 are biased estimators. In some applications bias is

disadvantageous. Following these estimators we have proposed almost unbiased estimator of

Y .



2. Almost unbiased estimator

Suppose, ytS0

α

321

321S1

KKpK

KKPKyt,

5454

5454

β

S2KpKKPK

KpKKPKλexp

P

p2yt,

Such thatS0t ,

S1t , WtS2 , where W denotes the set of all possible estimators for estimating

the population mean Y . By definition, the set W is a linear variety if

Wtwt2

0i

Siip (10)

Such that, 1w2

0i

i

and Rw i (11)

where, 0,1,2iwi denotes the constants used for reducing the bias in the class of

estimators, W denotes the set of those estimators that can be constructed from 0,1,2itSi

and R denotes the set of real numbers.

Expressing equation (10) in terms of e’s, we have

eeVeV

2

eV1we1Yt y11

22

1

1yp

2

eeVee

8

eV2

2

eV

2

eV

2

e1ew

y2

y

22

2

2

22

2

2 (12)

Subtracting Y from both sides of equation (12) and then taking expectation of both sides, we

get the bias of the estimator pt up to the first order of approximation, as



8

V2

2

1

2

VCwfYKV

2

V1CwfY)t(B

2

222

p21p1

2

12

p11p

2

KVK P2

P (13)

From (12), we have

2

eλVβeweαVweY)Y(t

φ2

φ2φ11yp

(14)

Squaring both sides of (14) and then taking expectation, we get the MSE of the estimator pt

up to the first order of approximation, as

p

22

p

2

y1

2

p QK2QCCfYtMSE

(15)

Which is minimum when

pKQ

(16)

where,

2

VwVwQ 2

211 (17)

Putting the value of pKQ in (15), we have optimum value of estimator as pt (optimum).

Thus the minimum MSE of pt is given by

2

pb

2

y1

2

p 1CfYtMSE.min

(18)

Which is same as that of traditional linear regression estimator.



from (11) and (17), we have only two equations in three unknowns. It is not possible to find

the unique values for ,s'w ii=0, 1, 2. In order to get unique values of ,s'w i

we shall impose

the linear restriction

2

0i

Sii 0tBw (19)

where, SitB denotes the bias in the ith estimator.

Equations (11), (17) and (19) can be written in the matrix form as

0

k

1

w

w

w

tBtB02

λVβαV0

111

p

2

1

0

S2S1

21 (20)

Using (20), we get the unique values of ,s'w ii=0, 1, 2 as

1121

1P2

2

11211

1P11

11211

1P11121121P1112110

XAAV

AKw

XXAAVV

AKXw

XAAVV

AKVXAAVVXAKXXAAVVw

1

p

2

211

P2P

2

222

p1

2

11

V

KX

2

VAX

2

KVK

8

V2

2

1

2

VA

KV2

V1A

Use of these ,s'w ii=0, 1, 2 remove the bias up to terms of order 1no at (10).


276 R. Singh, S.B. Gupta and S. Malik: An almost unbiased population parameter(s) 3.Empirical study

For empirical study we use the data sets earlier used by [20] (population 1) and [21]

(population 2) to verify the theoretical results.

Data statistics:

Population N n Y P yC pC pb 2

Population I 89 20 3.360 0.1236 0.60400 2.19012 0.766 6.2381

Population II 25 10 9.44 0.400 0.17028 1.27478 -0.387 4.3275

Table 3.1: Values of ,s'w i


,s'w i Population 1 Population 2

0w

-3.95624

1.124182

1w

5.356173

0.020794

2w

-0.39993

-0.14498


Table 3.2: PRE of different estimators of Y with respect to y

Choice of scalars

0w

1w

2w

1K

2K

3K

4K

5K

Estimator

PRE

(POPI)

PRE

(POPII)

1

0

0

y

100

100

0

1

0

1

1

0

1

NGRt

11.63

1.59

1

1

0

-1

NGPt

5.075

1.94

0

0

1

1

0

)0,1(1t

12.88

1.59

-1

0

)0,1(1t

5.43

1.95

1

0

1

1

)1,1(2t

73.59

0.84

1

0

1

-1

)1,1(2t

4.94

8.25

1

0

0

1

)1,0(2t

14.95

8.25

1

0

0

-1

)1,0(2t

73.48

5.58

0w

1w

2w

1

1

1

1

1

1

1

1

Pt optimum

241.98

117.61



4. Proposed estimators in two phase sampling

In some practical situations when P is not known a priori, the technique of two-phase

sampling is used. Let p'denote the proportion of units possessing attribute in the first phase

sample of size n' ; p denote the proportion of units possessing attribute in the second phase

sample of size nn' and y denote the mean of the study variable y in the second phase

sample.

In two-phase sampling the estimator pt will take the following form

Htht2

0i

idipd

(21)

Such that, 1h2

0i

i

and Rh i (22)

Where,

yt0d

m

321

3211d

KKpK

KKp'Kyt,

5454

5454

q

2dKpKKp'K

KpKKp'Kγexp

p'

p2yt,

The Bias and MSE expressions of the estimator 1dt and

2dt up to the first order of

approximation are, respectively, given by

2

pp31

2

p2

2

1

2

2

p1

2

1

2

p2

2

1

1d CkfmRCfRm2

Cf1)Rm(m

2

Cf1)Rm(mYtB

(23)

2

p3p1

2

p3

2

1

22

y1

2

1d Cfk2mRCfRmCfYtMSE (24)



2

p23

2

pp23

2

p2

22

pp2

2

p2

2

p1

2d qCγRfCkγRfCfqCkqf2

C1)fq(q

2

C1)fq(qYtB

(25)

2

p3

2

1

2

y1

2

2d CfLCfYtMSE (26)

Where,

21

54

42

321

11

γAqL

KPK2

PKR

KKPK

PKR

(27)

Expressing (21) in terms of e’s, we have

φy1

2

φ

2

1

φy1φy1φ1

2

φ

2

1

1ypd e'emR2

e'R1mmeemReemRemR

2

eR1mmwe1Yt

φyφyφy2φφ2

2

φ

φφ

2

φ

2

φ

φ2 eqee'eeeγRee'γR2

e'1qqe'eqqe'

2

e1qqqew

Subtracting Y from both sides of the above equation and then taking expectation of both

sides, we get the bias of the estimator pdt up to the first order of approximation, as

2d1dpd tBtBY)B(t (28)

Also,

φ2φ2φφ2φ1φ110pd eγRe'γRqe'qewemRe'mRweY)Y(t

(29)

c 2014 NSP Natural Sciences Publishing Cor

280 R. Singh, S.B. Gupta and S. Malik: An almost unbiased population parameter(s) Squaring both sides of (29) and then taking expectation, we get the MSE of the estimator tpd

up to the first order of approximation, as

2

pp32

2

p3

2

2

2

y1

2

pd Ckf2LCfLCfYtMSE (30)

Which is minimum when

p2 KL (31)

Where 22112 γRqhmRhL (32)

Putting the value of p2 KL in (30), we have optimum value of estimator as pdt (optimum).

Thus the minimum MSE of pdt is given by

2

pb31

2

y

2

pd ρffCYtmin.MSE

(33)

which is same as that of traditional linear regression estimator.

from (22) and (32), we have only two equations in three unknowns. It is not possible to find

the unique values for s,'h i 1=0, 1, 2. In order to get unique values of s,'h i

we shall impose

the linear restriction

2

0i

idi 0tBh (34)

where, idtB denotes the bias in the ith estimator.

Equations (22), (32) and (34) can be written in the matrix form as

0

k

1

h

h

h

tBtB0

γRqmR0

111

p

2

1

0

2d1d

21

(35)



Using (35), we get the unique values of s,'hi i=0, 1, 2 as

21211

1P2

21211

2P1

1

p

1

210

γRNNmRqN

NKh

γRNNmRqN

γRqKN

mR

kh

hh1h

where,

2

p23

2

pp23

2

p2

22

pp2

2

p2

2

p1

2

2

pp31

2

p2

2

1

2

2

p1

2

1

2

p2

2

1

1

qCγRfCkγRfCfqCkqf2

C1)fq(q

2

C1)fq(qN

CkfmRCfRm2

Cf1)Rm(m

2

Cf1)Rm(mN

Use of these s,'h i i=0, 1, 2 remove the bias up to terms of order 1no at (21).

5. Empirical Study

For empirical study we use the data sets earlier used by [20] (population 1) and [21]

(population 2) to verify the theoretical results.

Data statistics:

Pop. N n Y P p' yC pC pb n'

Pop.I 89 23 1322 0.1304 0.13336 0.69144 2.7005 0.408 45

Pop.II 25 7 7.143 0.294 0.308 0.36442 1.3470 -0.314 13



Table 5.1: PRE of different estimators of Y with respect to y

Choice of scalars

0w

1w

2w

1K

2K

3K

4K

5K

m

q

γ

Estimator

PRE

(POPI)

PRE

(POPII)

1

0

0

y

100

100

0

1

0

1

1

0

1

NGRt

11.13

8.85

1

1

0

-1

NGPt

7.48

12.15

0

0

1

1

0

1d(1,0)t

26.84

5.42

-1

0

1,0)1d(t

23.75

5.87

1

0

1

1

2d(1,1)t

82.55

1.23

1

0

1

-1

1)2d(1,t

8.56

8.46

1

0

0

1

2d(0,1)t

22.54

6.57

1

0

0

-1

1)2d(0,t

82.56

7.45

0w

1w

2w

1

1

1

1

1

1

1

1

tpd

optimum

112.55

106.89



6. Conclusion

In this paper, we have proposed an unbiased estimator pt and pdt using information on the

auxiliary attribute(s) in case of single phase and double phase sampling respectively.

Expressions for bias and MSE’s of the proposed estimators are derived up to first degree of

approximation. From theoretical discussion and empirical study we conclude that the

proposed estimatorspt and pdt under optimum conditions perform better than other estimators

considered in the article.


284 R. Singh, S.B. Gupta and S. Malik: An almost unbiased population parameter(s) Appendix A.

Some members of the proposed family of estimators -

Some members (ratio-type) of the class 1t

When ,0w0 ,1w1 0w2 ,1

1K 3K Estimators ( 1K2 ) Estimators ( 1K2 PRE’S

1K2

PRE’S

1K2

1 pC

p

p

1a1Cp

CPyt

p

p

1b1Cp

CPyt

134.99 72.50

1 2

)(p

)(Pyt

2

22a1

)(p

)(Pyt

2

22b1

111.62 89.34

2 pC

p2

p2

3a1C)(p

C)(Pyt

p2

p2

3b1C)(p

C)(Pyt

226.28 12.99

pC 2

)(pC

)(PCyt

2p

2p

4a1

)(pC

)(PCyt

2p

2p

4b1

126.66 77.93

1 pb

pb

pb

5a1p

Pyt

pb

pb

5b1p

Pyt

207.46 39.13

NP S

SNPp

SNPyt

2

6a1

SNPp

SNPyt

2

6b1 18.14 6.86

NP F

fNPp

fNPyt

2

7a1

fNPp

fNPyt

2

7b1 13.79 10.85


2

pbK

pb2

pb2

8a1Kp

KPyt

pb2

pb2

8b1Kp

KPyt

24.15 5.40

NP pbK

pb

pb

2

9a1KNPp

KNPyt

pb

pb

2

9b1KNPp

KNPyt

18.62 7.78

N 1

1Np

1NPyt 10a1

1Np

1NPyt 10b1

15.93 9.26

N pC

p

p

11a1CNp

CNPyt

p

p

11b1CNp

CNPyt

19.79 6.86

N pb

pb

pb

12a1Np

NPyt

pb

pb

12b1Np

NPyt

15.18 9.78

N S

SNp

SNPyt 13a1

SNp

SNPyt 13b1

12.34 10.96

N F

fNp

fNPyt 14a1

fNp

fNPyt 14b1

12.99 11.54

N g=1-f

gNp

gNPyt 15a1

gNp

gNPyt 15b1

15.81 9.34

N pbK

pb

pb

16a1KNp

KNPyt

pb

pb

16b1KNp

KNPyt

13.52 11.10

N pb

pb

pb

17a1np

nPyt

pb

pb

17b1np

nPyt

25.13 4.86

N S

pb

pb

18a1np

nPyt

pb

pb

18b1np

nPyt

14.98 8.81

N F

fnp

fnPyt 19a1

fnp

fnPyt 19b1

13.38 11.20




N g=1-f

gnp

gnPyt 20a1

gnp

gnPyt 20b1

29.13 3.68

N pbK

pb

pb

21a1Knp

KnPyt

pb

pb

21b1Knp

KnPyt

15.87 9.39

2 P

Pp

PPyt

2

222a1

Pp

PPyt

2

222b1

16.80 7.63

NP P

PNPp

PNPyt

2

23a1

PNPp

PNPyt

2

23b1 15.93 9.26

N P

PNp

PNPyt 24a1

PNp

PNPyt 24b1

13.23 11.32

N P

Pnp

PnPyt 25a1

Pnp

PnPyt 25b1

14.51 10.28

Appendix B.

Some members (product-type) of the class 1t

When ,0w0 ,1w1 0w2

,1

1K 3K Estimators ( 1K2 ) Estimators ( 1K2 ) PRE’S

1K2

PRE’S

1K2

1 pC

p

p

1c1CP

Cpyt

p

p

1d1CP

Cpyt

35.54 9.93


1 2

)(P

)(pyt

2

22c1

)(P

)(pyt

2

22d1

110.12 101.54

2 pC

p2

p2

3c1C)(P

C)(pyt

p2

p2

3d1C)(P

C)(pyt

6.09 0.127

pC 2

)(PC

)(pCyt

2p

2p

4c1

)(PC

)(pCyt

2p

2p

4d1

99.38 82.52

1 pb

pb

pb

5c1P

pyt

pb

pb

5d1P

pyt

0.00135 5.42

NP S

SNP

SNPpyt

26c1

SNP

SNPpyt

26d1 2.53 1.23

NP f

fNP

fNPpyt

27c1

fNP

fNPpyt

27d1 2.03 1.52

2 pbK

pb2

pb2

8c1KP

Kpyt

pb2

pb2

8d1Kp

KPyt

1.83 1.68

NP pbK

pb

2

pb

9c1KNP

KNPpyt

pb

2

pb

9d1KNP

KNPpyt

1.89 1.63

N 1

1NP

1Npyt 10c1

1NP

1Npyt 10d1

2.37 1.30

1 2

)(P

)(pyt

2

22c1

)(P

)(pyt

2

22d1

110.12 101.54

2 pC

p2

p2

3c1C)(P

C)(pyt

p2

p2

3d1C)(P

C)(pyt

6.09 0.127

pC 2

)(PC

)(pCyt

2p

2p

4c1

)(PC

)(pCyt

2p

2p

4d1

99.38 82.52




N pC

p

p

11c1CNP

CNpyt

p

p

11d1CNP

CNpyt

2.50 1.23

N pb

pb

pb

12c1NP

Npyt

pb

pb

12d1NP

Npyt

1.79 1.70

N S

SNP

SNpyt 13c1

SNP

SNpyt 13d1

2.16 1.44

N f

fNP

fNpyt 14c1

fNP

fNpyt 14d1

1.98 1.56

N g=1-f

gNP

gNpyt 15c1

gNP

gNpyt 15d1

2.34 1.32

N pbK

pb

pb

16c1KNP

KNpyt

pb

pb

16d1KNP

KNpyt

1.93 1.60

N pb

pb

pb

17c1nP

npyt

pb

pb

17d1nP

npyt

1.49 1.96

N S

pb

pb

18c1nP

npyt

pb

pb

18d1nP

npyt

2.65 1.14

N f

fnP

fnpyt 19c1

fnP

fnpyt 19d1

2.06 1.51

N g=1-f

gnP

gnpyt 20c1

gnP

gnpyt 20d1

3.29 0.84

N pbK

pb

pb

21c1KnP

Knpyt

pb

pb

21d1KnP

Knpyt

1.88 1.63



Appendix C.

Some members (product-type) of the class 2t

When ,0w0 ,0w1 1w2

4K 5K Estimators

( 1,1 )

PRE’S

1 pC

p

21C2Pp

Ppexp

P

p2yt

12.42

1 2

2

222Pp

Ppexp

P

p2yt

11.92

2 pC

p2

223

C2Pp

Ppexp

P

p2yt

16.29

pC 2

2p

p

242PpC

PpCexp

P

p2yt

12.53


2 P

PP

Ppyt

2

222c1

PP

Ppyt

2

222d1

2.99 0.97

NP P

PNP

PNPpyt

223c1

PNP

PNPpyt

223d1 2.37 1.30

N P

PNP

PNpyt 24c1

PNP

PNpyt 24d1

2.11 1.47

N P

PnP

Pnpyt 25c1

PnP

Pnpyt 25d1

2.49 1.23


1 pb

pb

252Pp

Ppexp

P

p2yt

13.86

NP S

S2PpNP

PpNPexp

P

p2yt 26

44.46

NP F

f2PpNP

PpNPexp

P

p2yt 27

61.84

2 pbK

pb2

228

K2Pp

Ppexp

P

p2yt

40.17

NP pbK

pb

29K2PpNP

PpNPexp

P

p2yt

48.09

N 1

2PpN

PpNexp

P

p2yt 210

54.10

N pC

p

211C2PpN

PpNexp

P

p2yt

44.84

N pb

pb

2122PpN

PpNexp

P

p2yt

56.48

N S

S2PpN

PpNexp

P

p2yt 213

62.40

N F

f2PpN

PpNexp

P

p2yt 214

65.67




N g=1-f

g2PpN

PpNexp

P

p2yt 215

54.47

N pbK

pb

216K2PpN

PpNexp

P

p2yt

63.21

N pb

pb

2172Ppn

Ppnexp

P

p2yt

38.59

N S

S2Ppn

Ppnexp

P

p2yt 218

52.19

N F

f2Ppn

Ppnexp

P

p2yt 219

63.74

N g=1-f

g2Ppn

Ppnexp

P

p2yt 220

35.33

N pbK

pb

221K2Ppn

Ppnexp

P

p2yt

54.68

2 P

P2Pp

Ppexp

P

p2yt

2

2222

47.53

NP P

P2PpNP

PpNPexp

P

p2yt 223

54.10

N P

P2PpN

PpNexp

P

p2yt 224

64.43

N P

P2Ppn

Ppnexp

P

p2yt 225

58.91


In addition to above estimators a large number of estimators can also be generated from the

proposed estimators just by putting different values of constants s'w i , s'hi ,K1 ,K2 ,K3,K4

,K5, and .

References

[1] Naik, V.D., Gupta, P.C. (1996): A note on estimation of mean with known population

proportion of an auxiliary character. Journal of the Indian Society of

Agricultural Statistics 48(2) 151-158.

[2] Singh, H. P., Solanki, R.S. (2012): Improved estimation of population mean in simple

random sampling using information on auxiliary attribute.Applied Mathematics

and Computation 218 (2012) 7798–7812.

[3] Sahai A., S. K. Ray S. K. (1980) “An efficient estimator using auxiliary information,”

Metrika, vol. 27, no. 4, pp. 271–275.

[4] Singh, H.P. and Tailor, R. (2003): Use of known correlation coefficient in estimating the

finite population mean. Statistics in Transition, 6,4,555-560.

[5] Kadilar,C. and Cingi,H. (2003): Ratio Estimators in Straitified Random Sampling.

Biometrical Journal 45 (2003) 2, 218-225.

[6] Shabbir, J. and Gupta, S. (2007): A new estimator of population mean in stratified

sampling, Commun. Stat. Theo. Meth.35: 1201–1209.

[7] Gupta, S. and Shabbir, J. (2008) : On improvement in estimating the population mean in

simple random sampling. Jour.Of Applied Statistics, 35, 5, 559-566.

[8] Khoshnevisan, M. Singh, R., Chauhan, P., Sawan, N. and Smarandache, F. (2007): A

general family of estimators for estimating population mean using known value

of some population parameter(s). Far east journal of theoretical statistics, 22(2),

181-191.

[9] Singh, R.,Cauhan, P., Sawan, N. and Smarandache,F. (2007): Auxiliary information and

a priory values in construction of improved estimators. Renaissance High press.



[10] Singh, R., Kumar, M. and Smarandache, F. (2008): Almost Unbiased Estimator for Estimating

Population Mean Using Known Value of Some Population Parameter(s). Pak. J. Stat.

Oper. Res., 4(2) pp63-76.

[11] Koyuncu, N. and Kadilar, C. (2009) : Family of estimators of population mean using two

auxiliary variables in stratified random sampling. Comm. In Stat. – Theory and Meth.,

38:14, 2398-2417.

[12] Diana G., Giordan M. Perri P.(2011): An improved class of estimators for the population mean.

Stat Methods Appl (2011) 20:123–140

[13] Upadhyaya LN, Singh HP, Chatterjee S, Yadav R (2011) Improved ratio and product exponential

type estimators. J Stat Theory Pract 5(2):285–302

[14] Jhajj, H. S., Sharma, M.K. and Grover, L.K. (2006): A family of estimators of population mean

using information on auxiliary attribute. Pakistan journal of statistics, 22(1), 43-50

(2006).

[15] Abd-Elfattah, A.M. El-Sherpieny, E.A. Mohamed, S.M. Abdou, O. F. (2010): Improvement in

estimating the population mean in simple random sampling using information on

auxiliary attribute. Appl. Mathe. and Compt. doi:10.1016/j.amc.2009.12.041.

[16] Grover L.,Kaur P.(2011): An improved exponential estimator of finite population mean in simple

random sampling using an auxiliary attribute.Applied Mathematics and Computation 218

(2011) 3093–3099.

[17] Malik, S. and Singh, R. (2013a) : An improved estimator using two auxiliary attributes. Appli.

Math. Compt., 219, 10983-10986.

[18] Malik, S. and Singh, R. (2013b) : A family of estimators of population mean using information

on point bi-serial and phi correlation coefficient. Int. Jour. Stat. Econ. 10(1), 75-89.

[19] Malik, S. and Singh, R. (2013c) : Dual to ratio cum product estimators of finite population mean

using auxiliary attribute(s) in stratified random sampling. WASJ, 28(9), 1193-1198

[20] Sukhatme, P.V. and Sukhatme, B.V.,(1970) : Sampling theory of surveys with applications. Iowa

State University Press, Ames, U.S.A.

[21] Mukhopadhyaya, P.(2000): Theory and methods of survey sampling. Prentice Hall of India, New

Delhi, India.


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