An Efficient Class of Two-Phase Exponential Ratio and

J. Stat. Appl. Pro. Lett. 2, No. 1, 71-88 (2015) 71

An Efficient Class of Two-Phase Exponential Ratio and

Product – Type Estimators for Estimating the Finite

Population Mean

Housila P. Singh, Neeraj Sharma and Tanveer A. Tarray*

School of Studies in Statistics, Vikram University Ujjain, M.P., 456010 India.

*E-mail: [email protected]

Received: 1 Dec. 2013, Revised: 5 Jan. 2014, Accepted: 9 Mar. 2014, Published online: 1 Jan. 2015

Abstract: This paper addresses the problem of estimating the population mean of the study variable

using auxiliary information in double sampling. We have suggested a class of estimators of population

mean in double sampling. It has been shown that the estimator due to Khatua and Mishra (2013) is a

member of the proposed class. We have obtained the bias and mean square error (MSE) of the proposed

class of estimators under large sample approximation. It is observed that the mean square error expression

of the estimator tdge obtained by Khatua and Mishra (2013) is not correct. So, we have obtained the correct

expression of the mean square error of the estimator tdge due to Khatua and Mishra (2013). We have

compared the proposed class of estimators with that of usual unbiased estimator, usual double sampling

ratio and product estimators, Singh and Vishwakarma (2007) estimators, usual regression estimator and

usual double sampling ratio estimator and shown that the proposed estimator is better than existing

estimators. Numerical illustrations are also given in support of the present study.

Keywords: Auxiliary variable, Exponential estimator, Mean squared error, Two- Phase sampling.

1. Introduction

In survey sampling, it is not uncommon to estimate the finite population mean Y of the study

variable of y. When information on auxiliary variable x, highly correlated with y, is readily

available on all units of population, it is well known that ratio estimator (for high positive

correlation), product estimator (for high negative correlation) and regression estimator (for high

correlation) can be used to increase the efficiency, incorporating the knowledge of population

mean X . However, in certain practical situation when the population mean of auxiliary variable

X is not known a priori, the technique of two-phase sampling is successfully used in practice.

The conventional method in such situation, a larger sample of size n' to furnish the good estimate

of the population mean X while a sub-sample size n is selected from the first phase sample 'n' to

observe the characteristic under study.

http://dx.doi.org/10.12785/jsapl/020107

Journal of Statistics Applications & Probability Letters An International Journal

© 2012 NSP

72 H. P. Singh et. al. : An Efficient Class of Two-Phase Exponential …

Consider a finite population U={1,2,3. ... ... ... .N}. Let y and x be two real variable assuming the

value yi and xi on the ith unit (i = 1,2,3.........N). Now consider y be the study variable and x be

the auxiliary variable. We consider simple random sampling without replacement (SRSWOR)

design to draw samples in each phase of two-phase sampling set-up. The first phase sample

s′(s′ ⊂ u) of fixed size n' is drawn to observe x only. The second phase sample s (s ⊂ s′) of

fixed size n' is drawn to observe y and x for given s, (n<n').

Let, si

ixn

1x

siiy

n

1y and

siix

n

1x

Now the usual two phase ratio, product and regression estimators are given by

tdr =y̅

x̅ x̅′, (1.1)

x

xytdp

, (1.2)

tdlr = y̅ + byx (x̅′ − x̅), (1.3)

where byx is the sample regression coefficient of 'y' on 'x', caiculated from the data based on

second phase sample of size 'n',

The mean square error (MSE) of the estimators given in (1.1), (1.2) and (1.3) to first order of

approximation are respectively given by

yx2x

2y

2dr C2CCYtMSE (1.4)

yx2x

2y

2dp C2CCYtMSE (1.5)

222y

2dlr 1CYtMSE (1.6)

where

N

1

n

1,

N

1

n

1

Cy2 =

Sy2

Y̅2, Cy2 =

Sy2

Y̅2 , xy

yx

yx CCxy

SC ,

Sy2 =

1

N−1∑ (Yi − Y̅)2,N

i=1 Sx2 =

1

N−1∑ (Xi − X̅)2,N

i=1

Syx =1

N−1∑ (Yi − Y̅)(Xi − X̅) = ρSySx

Ni=1 ,


ρ being the population correlation coefficient between 'y' and 'x'.

Singh and Vishwakarma (2007) have proposed modified exponential ratio and product estimate to

estimate finite population mean Y of study variable y in presence of auxiliary variable x. As

Singh and Vishwakarma (2007) assumed that the population mean X of auxiliary variable x is

not known, they used the two phase mechanism to estimate the population mean Y .

The modified exponential ratio and product estimator suggested by Singh and Vishwakarma

(2007) to estimate population mean Y under two phase sampling are given by

xx

xxexp.ytder (1.7)

xx

xxexp.ytdep (1.8)

The MSE of the estimators dert and dept to first order of approximation respectively are

given by

yx

2x2

y2

der C4

CCYtMSE (1.9)

yx

2x2

y2

dep C4

CCYtMSE (1.10)

Khatua and Mishra (2013) have suggested a generalized exponential estimator to estimate finite

population mean Y under two-phase sampling scheme with assumption that the population mean

auxiliary variable x, X is not known.

xx

xxexpxxddyt 21dge , (1.11)

where 1d and 2d are suitable chosen constants or suitable chosen constants or statistics and

21 dd are not necessarily equal to unity.

It is to be mentioned that the mean square error (MSE) expression of the estimator dget obtained

by Khatua and Mishra(2013) is not correct. The correct expression of the bias and MSE of the

Khatua and Mishra (2013) estimator dget are given in the following theorem.


Theoram 1.1 To the first degree of approximation, the bias and MSE of the Khatua and Mishra

(2013) estimator dget are respectively given by

1k21C2

Xdk43C

81dYtB 2

x22

x1dge (1.12)

and

2x2

2x1

2x21

2x

222

2x

2y

21

2dge

Ck21Xd

k43C8

1d2Ck21Xdd2

1CXdk21CC1dYtMSE

(1.13)

where xy CCk .

Proof. To obtain the bias and MSE of dget we write

0e1Yy , 1e1Xx , 1e1Xx

Such that

0eEeEeE 110

and

2y

20 CeE , 2

x21 CeE , 2

x21 CeE , 2

x10 kCeeE , 2x10 kCeeE ,

2x11 CeeE

Now expressing (1.11) in terms of e’s we have

11

1111210dge

e1e1

e1e1expe1e1Xdde1Yt

1

111111210

2

ee1

2

eeexpeeXdde1Y

...2

ee1

8

ee

2

ee1

2

ee1eeXdde1Y

2

112

11

1

111111210


...2

ee1

8

ee

2

ee1

2

ee1eee1Xd

...2

ee1

8

ee

2

ee1

2

ee1e1dY

2

112

11

1

11111102

2

112

11

1

111101

...8

ee...

2

ee1

2

ee1eee1Xd

...8

ee...

2

ee1

2

ee1e1dY

2111111

1102

2111111

01

...8

eee2e

4

ee

2

ee1eee1Xd

...8

eee2e

4

ee

2

ee1e1dY

2111

21

21

2111

1102

2111

21

221

2111

01

...8

ee2ee3ee

2

eee2eeeeeeeXd

...8

eee2eeee3

8

ee2ee3

2

eeee

2

eee1dY

112

12111

2111

21

1010112

1102

1021011

21

21101011

01

Neglecting terms of e’s having power greater than two we have

2

eee2eeeeeeeXd

8

ee2ee3

2

eeee

2

eee1dYt

2111

21

1010112

112

121101011

01dge

Subtracting Y from both side of the above expression we have


12

eee2eeeeeeeXd

8

ee2ee3

2

eeee

2

eee1dYYt

2111

21

1010112

112

121101011

01dge

(1.14)

Taking expectation of both sides of (1.14) we get the bias of dget to the first degree of

approximation as

1k21CXd

k43C8

1dY

YtEtB

2x2

2x1

dgedge

(1.15)

Squaring both sides of (1.14) and neglecting terms of e’s having power greater than two we have

2

eee2eeeeeeeXd2

8

ee2ee3

2

eeee

2

eee1d2

2

eee2eeeee

2

eeeeeeeeXdd2

1eeXd4

ee2ee3eeee

eeeeeee24

eee1dYYt

2111

21

1010112

112

121101011

01

2111

21

1010

211

10101121

211

222

112

121

1010

1010110

2112

021

22

dge

or


2

eee2eeeeeeeXd2

8

ee2ee3

2

eeee

2

eee1d2

eee2eeeee2eeXdd2

eeXdeeeeeee2eee21d1YYt

2111

21

1010112

112

121101011

01

2111

2110101121

211

22211

211010110

21

22

dge

(1.16)

Taking expectation of both sides of (1.16) we get the mean squared error of the estimator dget to

the first degree of approximation as

k21CXd

k438

C1d2

k21CXdd2

CXdk21CC1d1YtMSE

2x2

2x

1

2x21

2x

222

2x

2y

21

2dge

1321411321122211

21

2 AdAd2Add2AdAd1Y (1.17)

which is minimum when

2A

A

d

d

AA

AA

13

14

2

1

1213

1311 (1.18)

Solving (1.18) we get the optimum values of 21 dandd as

102

131211

2131412

1 dAAA2

AAA2d

(1.19)

202131211

141113

2 dAAA2

A2AAd

(1.20)

where ,


2x

2y11 Ck21C1A ,

2x

212 CXA ,

2x13 Ck21XA ,

k43

8

C1A

2x

14

Substituting 2010 d,d in place of 21 d,d in (1.17) we get the minimum MSE of dget as

2131211

142

132

13112

14122dge

AAA4

AA4AAAA41YtMSE.min (1.21)

which is correct expression of MSE of dget .

If we set 1d1 in (1.11), then the class of estimators dget reduces to:

xx

xxexpxxd1yt 21dge (1.22)

Putting 1d1 in (1.15) and (1.17) the bias and MSE of 1dget are respectively given by

k21Xd8k43C

8

YtB 2

2x1dge

(1.23)

and

13212221411

21dge AdAdA2A1YtMSE (1.24)

The MSE 1dget is minimized for

pt20

12

132 d

A2

Ad (1.25)

Thus the resulting (minimum) MSE of 1dget is given by


12

213

14112

1dgeA4

AA2A1YtMSE

22yS (1.26)

which is equal to the approximate variance of the usual regression estimator

xxˆyy lrd (1.27)

where 2xxy ssˆ is the sample regression coefficient of y on x.

we note that the MSE expression of the estimator of tdge(1):

12x21

2x21

2x

222

2x

2y

21

2)1(dge

d2k21CXdd2k21CXdd2

CXdk41C4

C1d1YtMSE

(1.28)

obtained by Khatua and Mishra (2013) is not corrected. So the other results of the paper are also

incorrect. This led authors to derive the correct expression of MSE of the estimator tdge(1) which is

given in 1.28.

2. The Suggested Class of Estimators

For estimating the population mean Y , we define a class of estimators as

xx

xxexp

x

xxxyt 21s

,

(2.1)

where 21, are suitably chosen constants such that the MSE of t is minimum, , are

suitably chosen classes scalars which may assume real number or the functions of xy C,C, etc.

It is to be noted that for 1,0, the proposed class of estimators st reduces to the class of

estimators reported by Khatua and Mishra (2013).

To obtain the bias and MSE of st we express st in terms of e’s we have

11

11

1

111210s

ee2

eeexp

e1

e1eeXe1Yt

.

(2.2)


We assume that 1e,1e 11 so that the right hand side of (2.2) is expandable. Expanding the

right hand side, multiplying out and neglecting terms of e’s having power greater than two we

have some members of the proposed class of estimators st are given in Table 2.1

Table 2.1 Some members of the proposed class of estimators st

S.No. Estimators Values of scalars

1 2

1 yt1 1 0 0 0

2 xxyt 2 1 0 1 0

3 xxyt3 1 0 -1 0

4 xxyt 4

Srivastava (1970) estimator 1 0 0

5

xx

xxexpyt5

Singh & Vishwakarma (2007) estimator

1 0 0 1

6

xx

xxexpyt6

Singh & Vishwakarma (2007) estimator

1 0 0 -1

7

xx

xxexpyt7 1 0 0

8

xx

xxexpxxddy8t 21

Khatua and Mishra (2013) estimator

1d 2d 0 1

9

x

xxxyt 219 1 2 -1 0

10

xx8

xxexp

x

xxxyt 2110 1 2 -1

-1/8

11

xx4

xxexp

x

xxxyt 2111 1 2 -1 -1/4


2

eeeeeeeeX

8

ee

4

ee

2

eeee

2

eee1Yt

211

1010112

211

221

21101011

01s

or

12

eeeeeeeeX

8

ee

4

ee

2

eeee

2

eee1YYt

211

1010112

211

221

21101011

01s

(2.3)

where 2 .

Taking expectation of both sides of (2.3) we get the bias of the proposed class of estimator t to the

first degree of approximation as

1

2

k2CX

8

2k4C1YtB

2x

2

2x

1s (2.4)

Squaring both sides of (2.3) and neglecting terms of e’s having power greater than two we have

2111010112

21

21

211

10101101

21110101121

211

222

21

2111

21

21

201010110

21

22

s

ee2

eeeeeeXd2

ee2ee82

eeee

2

eee1d2

eeeeee2eeXdd2

eeXdeeee2ee2

eeeee2eee21d1YYt

(2.5)

Taking expectation of both sides of (2.5) we get the mean squared error of st to the first degree of

approximation as

32413212221

21

2s AA2A2AA1YtMSE (2.6)


where

2

1k4CC1A

2x2

y1

2x

22 CXA ,

2x3 Ck2XA ,

2k4

8

C1A

2x

4

The MSE of st is minimum when

2A

A

AA

AA

3

4

2

1

23

31 (2.7)

Solving (2.7) we get the optimum values of 21 and as

102

331

2342

1AAA2

AAA2

(2.8)

202321

4132

AAA2

A2AA

(2.9)

Putting (2.8) and (2.9) in (2.6) we get the minimum MSE of st as

2321

423

231

2422

sAAA4

AA4AAAA41YtMSE.min (2.10)

For 11 , the proposed class of estimators st reduces to :

xx

xxexp

x

xxx1yt 21s (2.11)

Putting 11 in (2.4) and (2.6) we get the bias and MSE of 1st to the first degree of

approximation are respectively given by


k2X42k4

8

CYtB 2

2x

1s

(2.12)

3222241

21s AAA2A1YtMSE (2.13)

The 1stMSE is minimized for

)pt0(2

2

32 d

A2

A (2.14)

Thus resulting minimum MSE of 1st is given by

2

23

412

1sA4

AA2A1YtMSE.min

22yS (2.15)

which is equal to the approximate variance of the usual regression estimator lrdy in two phase

sampling.

3. Efficiency comparisons

For 0,0,0,1,,, 21 in (2.2.1), the class of estimators st reduces to the usual unbiased

estimator

yt 1s (3.1)

The variance of y is given by

2y

2y

2 SCyyMSEyvar (3.2)

From (2.1.14), (2.1.15), (2.1.6) and (2.3.2) we have

0kCY)t(MSE)y(MSE 22x

2dlr (3.3)

0)k1(CY)t(MSE)t(MSE 22x

2dlrdr (3.4)

0)k1(CY)t(MSE)t(MSE 2x

2dlrdp (3.5)


0)k21(C4

Y)t(MSE)t(MSE 22

x

2

dlrder

(3.6)

0)k21(C4

Y)t(MSE)t(MSE 22

x

2

dlrdep

(3.7)

It is clear from (3.3), to (3.7) that the usual two – phase regression estimator dlry is better than

sample mean y , ratio estimator tdr, the product estimator tdp in two phase sampling, two phase

exponential ratio estimator tder and the two phase exponential procedure estimator tdep.

Further from (2.10) and (2.15) we have

2321

423

231

242

2

23

412

sdlrAAA4

AA4AAAA4

A4

AA2AY)t(MSE.mintMSE

0

AAAA4

AA2AAA2Y23212

2

422321

2

(3.8)

which shows that the proposed class of estimator’s ts is more efficient than the regression

estimator tdlr. Thus the proposed class of estimator’s ts is better than usual unbiased estimator y ,

conventional two phase ratio estimator tdr two – phase product estimator tdp and the two phase

regression estimator tdlr.

4. Empirical Study

To have tangible idea about the performance of the members of the proposed class of estimators ts

over usual unbiased estimator y we have computed the percent relative efficiency (PRE) of the

member of the suggested class of estimator’s ts with respect to y by using the formula:

100AA2A2AA1

Cy,tPRE

32413212221

21

2y

s

(4.1)

For better eye view, we have considered two natural population data set with positive correlation

between y and x and two population data sets with negative correlation between y and x earlier

considered by Khatua and Mishra (20103). The descriptions are given below:

Population I: Murthy (1967), P. 228

y: Output, x: Fixed Capital, N =80, n=70 ,n=30


,00179.0,02083.0 64.5182Y , 1255.0C2y , 5635.0C2

x , 2503.0Cyx ,

9413.0 , 44419.0k

Population II: Das (1988)

y: No. of agricultural labors for 1971, x: No of agricultural labors for 1961, N =278, n=70 ,

n=30

,01069.0,0297.0 0 6 8 0.39Y , 0883.2C2y , 6237.2C2

x , 6883.1Cyx ,

7213.0 , 64348.0k

Population III: Steel and Torrie (1960), P. 282

y: Log of leaf burn in sacs , x:Chlorine percentage, N =30, n=12 ,n=4

,05.0,21667.0 6 8 6 0.0Y , 2306.0C2y , 5614.0C2

x , 1798.0Cyx ,

4996.0 , 32027.0k

Population IV: Gujurati(1999), P. 259

y: Year to year percentage change in the index of hourly earnings, x: The unemployment rate

(%), N =12, n=8 ,n=5,

,041667.0,11667.0

066.4Y , 0977.0C2y , 0535.0C2

x , 0519.0Cyx ,

718.0 , 970.0k

We have computed the percent relative efficiencies of the different members of the proposed

class of estimator’s tdge with respect to usual unbiased estimator y and findings are shown in

Tables 2.4.1 and 2.4.2.


Table 2.4.1 Percent relative efficiency with respect to mean per estimator ( y ) when ρ > 0

Table 2.4.2 Percent relative efficiency with respect to mean per estimator ( y ) when ρ < 0

Tables 2.4.1 and 2.4.2 clearly show that the modified two-phase estimator is more efficient than

mean per unit estimator, two- phase ratio estimator, two –phase ratio type exponential estimator,

two-phase product estimator, two-phase product type exponential and two-phase regression

estimator. Tables 2.4.1 and 2.4.2 also exhibits that the new generated estimators are more

efficient than the existing once. Thus the proposed estimators are to be preferred in practice.

References

[1] Bhal S. and Tuteja R.K. (1991) Ratio and product type exponential estimator, Infor. Optimiz. Sci. 12, 159-163.

[2]Cochran W.G. (1977) Sampling Technique, 3rd Edition. New York: John Wiley and Sons, USA.

[3]Das A.K. (1977) Contribution to the theorey of sampling strategies bases on auxiliary information. Unpublised

Ph.D. Thesis.

Populatio

n

Estimators

drt dert dlrt dget 9t 10t 11t

1

68.57 493.04 526.58 530.93 1406.10 1948.66 3427.95

2

130.02 146.34 149.98 156.69 172.39 179.84 189.72

Populatio

n

Estimators

prt dept dlrt dget 9t 10t 11t

3

59.76 115.15 123.76 126.25 127.78 129.96 132.96

4

149.47 133.95 149.56 149.65 149.81 149.79 149.77


[4]Khatua P. and Mishra G.(2013)A modified generalized two-phase exponential estimator to estimate finite population

mean. IOSR J. Math. Vol 8 No 6 pp. 43-47.

[5]Gujurati D. (1999) Essential of Economics, 2th Edition Mc Graw-Hill International.

[6]Muthy M.N. (1977)Sampling theory and methods. Statist. Pub. Soc. Calcutta.

[7] Solanki R. S., Singh H.P. and Pal S. K. (2013) Improved estimation of finite population mean sample surveys. J.

Adv. Com. Vol 1 No 2 pp. 70-78.

[8]Singh H.P., Pal S. K. and Solanki R. S. (2013) Improved estimation of finite population variance using quartiles.

Istatistik - J. Tur. Stat. Asso.,Vol6 No 3 pp. 166-121.

[9]Singh H.P. and Vishwakaram G.K. (2007) Modified exponential ratio and product estimators for finite

population mean in double sampling. Aust. J. Stat. Vol 36 No 3 pp.217-225.

[10]Steel R.G.D. and Torrie J.H. (1960) Principle and procedures of statistics, MC – Grew Hill, Newyork.

[11]Sukhatme P.V., Sukhatme B.V., and Ahsok C. (1984) Sampling Theory of surveys with applications. Ind. Soc.

Agri. Stat., New Delhi.

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