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Rajesh Singh ■ Florentin Smarandache (editors)

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SAMPLING STRATEGIES

FOR FINITE POPULATION USING AUXILIARY INFORMATION

The Educational Publisher Columbus, 2015

Sampling Strategies for Finite Population Using Auxiliary Information

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SAMPLING STRATEGIES FOR FINITE POPULATION USING AUXILIARY INFORMATION

Papers by Sachin Malik, Rajesh Singh, Florentin Smarandache, B. B. Khare, P. S. Jha, Usha Srivastava, Habib Ur. Rehman.


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The Educational Publisher

Zip Publishing 1313 Chesapeake Ave.

Columbus, Ohio 43212, USA Email: [email protected]

ISBN 978-1-59973-348-7 © The Authors, The Editors, The Publisher, 2015.


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Rajesh Singh Department of Statistics, BHU, Varanasi (U.P.), India

Editor

Florentin Smarandache Chair of Department of Mathematics, University of New Mexico, Gallup, USA

Editor

SAMPLING STRATEGIES

FOR FINITE POPULATION USING AUXILIARY INFORMATION

The Educational Publisher

Columbus, 2015


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Contents

Foreword ................................................................................................................................... 7

A Generalized Family Of Estimators For Estimating Population Mean Using Two Auxiliary Attributes ................................................................................................................. 9

Abstract .................................................................................................................................. 9 Keywords ............................................................................................................................... 9 1. Introduction ....................................................................................................................... 9 2. Some Estimators in Literature ......................................................................................... 10 3. The Suggested Class of Estimators ................................................................................. 12 4. Empirical Study ............................................................................................................... 14 5. Double Sampling ............................................................................................................. 15 6. Estimator tpd in Two-Phase Sampling ............................................................................. 17 7. Conclusion ....................................................................................................................... 20 References ............................................................................................................................ 20

A General Procedure of Estimating Population Mean Using Information on Auxiliary Attribute.................................................................................................................................. 21

Abstract ................................................................................................................................ 21 Keywords ............................................................................................................................. 21 1. Introduction ...................................................................................................................... 21 2. Proposed Estimator .......................................................................................................... 22 3. Members of the family of estimator of t and their Biases and MSE ................................ 25 4. Empirical study ................................................................................................................ 28 Conclusion ............................................................................................................................ 29 References ............................................................................................................................ 29

Estimation of Ratio and Product of Two Population Means Using Auxiliary Characters in the Presence of Non Response .......................................................................................... 31

Abstract ................................................................................................................................ 31 Keywords ............................................................................................................................. 31 Introduction .......................................................................................................................... 31 Estimation of Ratio and product of two population means .................................................. 31

Case 1. The Case of Complete Response: ........................................................................ 31 Case 2. Incomplete Response in the Sample due to Non-response: ................................. 34

References ............................................................................................................................ 36

On The Use of Coefficient of Variation and 21 , in Estimating Mean of a Finite Population ............................................................................................................................... 39

Abstract ................................................................................................................................ 39 Keywords ............................................................................................................................. 39 Introduction .......................................................................................................................... 39 Estimators and their Mean Square Error .............................................................................. 39 References ............................................................................................................................ 43


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A Study of Improved Chain Ratio-cum-Regression type Estimator for Population Mean in the Presence of Non- Response for Fixed Cost and Specified Precision ....................... 45

Abstract ................................................................................................................................ 45 Keywords ............................................................................................................................. 45 Introduction .......................................................................................................................... 45 The Estimators...................................................................................................................... 46 Mean Square Errors of the Study Estimator......................................................................... 48 An Empirical Study .............................................................................................................. 51 Conclusion ............................................................................................................................ 53 References ............................................................................................................................ 53


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Foreword

The present book aims to present some improved estimators using auxiliary and attribute information in case of simple random sampling and stratified random sampling and in some cases when non-response is present.

This volume is a collection of five papers, written by seven co-authors (listed in the order of the papers): Sachin Malik, Rajesh Singh, Florentin Smarandache, B. B. Khare, P. S. Jha, Usha Srivastava and Habib Ur. Rehman.

The first and the second papers deal with the problem of estimating the finite population mean when some information on two auxiliary attributes are available. In the third paper, problems related to estimation of ratio and product of two population mean using auxiliary characters with special reference to non-response are discussed.

In the fourth paper, the use of coefficient of variation and shape parameters in each stratum, the problem of estimation of population mean has been considered. In the fifth paper, a study of improved chain ratio-cum-regression type estimator for population mean in the presence of non-response for fixed cost and specified precision has been made.

The authors hope that the book will be helpful for the researchers and students that are working in the field of sampling techniques.


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A Generalized Family Of Estimators For Estimating Population Mean Using Two Auxiliary Attributes

1Sachin Malik, †1Rajesh Singh and 2Florentin Smarandache 1Department of Statistics, Banaras Hindu University

Varanasi-221005, India

2Chair of Department of Mathematics, University of New Mexico, Gallup, USA

† Corresponding author, [email protected]

Abstract This paper deals with the problem of estimating the finite population mean when some information on two auxiliary attributes are available. A class of estimators is defined which includes the estimators recently proposed by Malik and Singh (2012), Naik and Gupta (1996) and Singh et al. (2007) as particular cases. It is shown that the proposed estimator is more efficient than the usual mean estimator and other existing estimators. The study is also extended to two-phase sampling. The results have been illustrated numerically by taking empirical population considered in the literature.

Keywords Simple random sampling, two-phase sampling, auxiliary attribute, point bi-serial correlation, phi correlation, efficiency.

1. Introduction There are some situations when in place of one auxiliary attribute, we have

information on two qualitative variables. For illustration, to estimate the hourly wages we can use the information on marital status and region of residence (see Gujrati and Sangeetha (2007), page-311). Here we assume that both auxiliary attributes have significant point bi-serial correlation with the study variable and there is significant phi-correlation (see Yule (1912)) between the auxiliary attributes. The use of auxiliary information can increase the precision of an estimator when study variable Y is highly correlated with auxiliary variables X. In survey sampling, auxiliary variables are present in form of ratio scale variables (e.g. income, output, prices, costs, height and temperature) but sometimes may present in the form of qualitative or nominal scale such as sex, race, color, religion, nationality and geographical region. For example, female workers are found to earn less than their male counterparts do or non-white workers are found to earn less than whites (see Gujrati and Sangeetha (2007), page 304). Naik and Gupta (1996) introduced a ratio estimator when the study variable and the auxiliary attribute are positively correlated. Jhajj et al. (2006) suggested a family of estimators for the population mean in single and two-phase sampling when the study variable

mailto:[email protected]


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and auxiliary attribute are positively correlated. Shabbir and Gupta (2007), Singh et al. (2008), Singh et al. (2010) and Abd-Elfattah et al. (2010) have considered the problem of estimating population mean Y taking into consideration the point biserial correlation between auxiliary attribute and study variable.

2. Some Estimators in Literature In order to have an estimate of the study variable y, assuming the knowledge of the

population proportion P, Naik and Gupta (1996) and Singh et al. (2007) respectively, proposed following estimators:

1

11 p

Pyt (2.1)

2

22 P

pyt (2.2)

11

113 pP

pPexpyt

(2.3)

22

224 Pp

Ppexpyt (2.4)

The Bias and MSE expression’s of the estimator’s it (i=1, 2, 3, 4) up to the first order of approximation are, respectively, given by

11 pb2p11 K1CfYtB

(2.5)

C2

ppb1222

KfYtB (2.6)

22

pb

2p

13 K41

2

CfYtB

(2.7)

22

pb

2p

14 K41

2

CfYtB

(2.8)

MSE 11 pb

2p

2y1

21 K21CCfYt

(2.9)

MSE 21 pb

2p

2y1

22 K21CCfYt (2.10)


11

MSE

21 pb2p

2y1

23 K

41CCfYt

(2.11)

MSE

22 pb2p

2y1

24 K

41CCfYt

(2.12)

where, ,PYy1N

1S ,P1N

1S , N1-

n1 f

N

1ijjiiy

2N

1ijji

21 jj

),2,1j(;PS

C,YS

C,SS

S

jjp

yy

y

ypb

j

j

j

j

.CC

K,CC

K2

221

11p

ypbpb

p

ypbpb

21

21

21 sss

and pp1n

1sn

1i2i21i1

be the sample phi-covariance and phi-

correlation between 1 and 2 respectively, corresponding to the population phi-covariance

and phi-correlation

N

1i2i21i1 PP

1N1S

21

.SS

Sand

21

21

Malik and Singh (2012) proposed estimators t5 and t6 as

21

2

2

1

15 p

PpPyt

(2.13)

21

22

22

11

116 Pp

PpexppPpPexpyt

(2.14)

where 121 ,, and 2 are real constants.

The Bias and MSE expression’s of the estimator’s 5t and 6t up to the first order of approximation are, respectively, given by

kk

22Ck

22CfY)t(B 21pb2

2222

ppb11

212

p15 2211 (2.15)


12

K

4K

24CK

24CfY)t(B 21

pb2

222

ppb1

212

p16 2211 (2.16)

K2K2CK2CCfY)t(MSE 21pb222

2ppb1

21

2p

2y1

25 2211 (2.17)

KβK2ββ

4βCKβ

4βCCfY)MSE(t

1211 pb2φ21

222

ppb1

212

p2y1

26

(2.18)

3. The Suggested Class of Estimators Using linear combination of ,0,1,2it i we define an estimator of the form

Htwt3

0iiip

(3.1)

Such that, 1w3

0ii

and Rw i (3.2)

Where,

yt0 , 21 α

423

423

α

211

2111 LpL

LPLLpLLPLyt

and 21 β

827227

827627

β

615211

616152 )LP(L)Lp(L

)LP(L)Lp(Lexp)Lp(L)LP(L)L(Lp)LP(Lexpt

where 0,1,2iw i denotes the constants used for reducing the bias in the class of estimators, H denotes the set of those estimators that can be constructed from 0,1,2it i and R denotes the set of real numbers (for detail see Singh et. al (2008)). Also, 1,2,...,8iLi are either real numbers or the functions of the known parameters of the

auxiliary attributes.

Expressing tp in terms of e’s, we have

1 2

1

2

α α0 1 1 1 2 2

β1p 0 2 1 1 1 1

β12 2 2 2

w w 1 φ e 1 φ e

t Y 1 e w exp θ e 1 θ e

exp θ e 1 θ e

(3.3)

where,


13

827

272

625

151

413

232

211

111

LPL2PLθ

LPL2PLθ

LPLPLφ

LPLPLφ

After expanding, Subtracting Y from both sides of the equation (3.3) and neglecting the term having power greater than two, we have

222111222211110p eθβeθβweφαeφαweYYt

(3.4)

Squaring both sides of (3.4) and then taking expectations, we get MSE of the estimator pt up

to the first order of approximation, as

52413212221

21

2p T2wT2wTw2wTwTwfYtMSE

(3.5)

where,

2321

43512

2321

53421

LLLLLLLw

LLLLLLLw

(3.6)

and

2ppb22

2ppb115

2ppb22

2ppb114

2pφ2211

2pφ1212

2p222

2p113

2pφ2121

2p

22

22

2p

21

212

2pφ2121

2p

22

22

2p

21

211

2211

2211

2221

211

221

CkθβCkθβL

CkφαCkφαL

CkβθφαCkθφβαCθβαCθα1βL

Ckθφβ2βcβθcβθL

Ckφφα2αCαφCαφL

(3.7)


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4. Empirical Study

Data: (Source: Government of Pakistan (2004))

The population consists rice cultivation areas in 73 districts of Pakistan. The variables

are defined as:

Y= rice production (in 000’ tonnes, with one tonne = 0.984 ton) during 2003,

1P = production of farms where rice production is more than 20 tonnes during the year 2002, and

2P = proportion of farms with rice cultivation area more than 20 ha during the year 2003.

For this data, we have

N=73, Y =61.3, 1P =0.4247, 2P =0.3425, 2yS =12371.4, 2

1S =0.225490, 2

2S =0.228311,

1pb =0.621, 2pb =0.673,

=0.889.

Table 4.1: PRE of different estimators of Y with respect to y .

CHOICE OF SCALERS, when 0w 0 1w1 0w2

1α 2α 1L 2L 3L 4L PRE’S

0 1 1 0 179.77

1 0 1 0 162.68

1 1 1 1 1 1 156.28

-1 1 1 0 1 0 112.97

1 1 1pC 1pb 2pC 2pb 178.10

1 1 1NP 1pbK 2NP 2pbK 110.95

-1 1 1NP f 2NP f 112.78

-1 1 N 1pbK N 2pbK 112.68

-1 1 1NP 1P 2NP 2P 112.32

1 1 n 1P n 2P 115.32

-1 1 N 1pb N

2pb 112.38

-1 1 n 1P n 2P

113.00

-1 1 N 1P N 2P 112.94

When, 0w 0 0w1 1w2


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1β 2β 5L 6L 7L 8L PRE’S

1 0 1 0 1 0 141.81

0 1 1 0 1 0 60.05

1 -1 1 0 1 0 180.50

1 -1 1 1 1 1 127.39

1 -1 1 1 1 0 170.59

1 -1 1pC 1pb 2pC 2pb 143.83

1 -1 1NP 1pbK 2NP 2pbK 179.95

1 -1 1NP f 2NP f 180.52

1 -1 N 1pbK N 2pbK 180.56

1 -1 1NP 1P 2NP 2P 180.53

1 -1 n 1P n 2P 179.49

1 -1 N 1pb N

2pb 180.55

1 -1 n 1P n 2P

180.36

1 -1 N 1P N 2P 180.57

When, 0w 0 0w1 1w2 also 11,2,...,8iLi

1α 2α 1β 1β2 ptPRE =183.60

5. Double Sampling It is assumed that the population proportion P1 for the first auxiliary attribute 1 is

unknown but the same is known for the second auxiliary attribute 2 . When P1 is unknown, it is some times estimated from a preliminary large sample of size non which only the attribute 1 is measured. Then a second phase sample of size n (n< n ) is drawn and Y is observed.

Let ).2,1j(,n1p

n

1ijij

The estimator’s t1, t2, t3 and t4 in two-phase sampling take the following form

1

'1

1d pp

yt (5.1)


16

'

2

22d p

Pyt

(5.2)

1'1

1'1

3d ppppexpyt

(5.3)

2'2

2'2

4d PpPpexpyt

(5.4)

The bias and MSE expressions of the estimators td1, td2, td3 and td4 up to first order of

approximation, are respectively given as

11 pb

2p31d k1CfYtB

(5.5)

22 pb

2p22d K1CfYtB

(5.6)

2

2pb

2p

33d K14

CfYtB

(5.7)

2pb

22p

34d K14

CfYtB

(5.8)

MSE 11 pb

2P3

2y1

21d K21CfCfYt (5.9)

MSE 22 kp

2p2

2y1

22d K21CfCfYt (5.10)

MSE

1

1pb

2p

32y1

23d K41

4C

fCfYt (5.11)

MSE

1

1pb

2p

32y1

24d K41

4

CfCfYt

(5.12)

where,

2n

1ijji

2 p1n

1SJ

, ,p1n

1S2n

1i

'jji'

2'!

j

,N1

n1f '2 .

n1

n1f '3


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The estimator’s t5 and t6, in two-phase sampling, takes the following form

1m

1

'1

5d ppyt

2m

'2

2

pP

(5.13)

6dt1n

1'1

1'1

ppppexpy

2n

2'2

2'2

PpPpexp

(5.14)

Where 1m , 2m , 21 n and n are real constants.

The Bias and MSE expression’s of the estimator’s d5t and d6t up to the first order of approximation are, respectively, given by

2211 pb22

222

P2pb11

212

p3d5 km2

m2

mCfKm2

m2

mCfYtB (5.15)

211 pb22

22

22ppb

1121

36d K2n

8n

8nfCK

2n

8n

8nfYtB

(5.16)

2211 pb2

22

22p2pb1

21

2p3

2y15d Km2mCfKm2mCfCfYtMSE (5.17)

)18.5( CKn4

nfCKn4

nfCfYtMSE 2ppb2

22

22ppb1

21

32y1

26d 2211

6. Estimator tpd in Two-Phase Sampling Using linear combination of ,0,1,2it di we define an estimator of the form

Htht3

0idiipd

(6.1)

Such that, 1h3

0ii

and Rh i (6.2)

where,

yt0 , 21 m

423

423

m

211

211d1 Lp'L

LPLLpLLp'Lyt

and 21 n

827227

827627

n

615211

61615d2 )LP(L)Lp'(L

)LP(L)Lp'(Lexp)Lp(L)Lp'(L)L(Lp)Lp'(Lexpt

where 0,1,2ih i denotes the constants used for reducing the bias in the class of estimators,

H denotes the set of those estimators that can be constructed from 0,1,2it di and R


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denotes the set of real numbers (for detail see Singh et. al. (2008)). Also, 1,2,...,8iLi are either real numbers or the functions of the known parameters of the auxiliary attributes.

Expressing tpd in terms of e’s, we have

211 -m22

m11

m11100p e'φ1eφ1e'φ1hhe1Yt

21 n2222

n11111112 e'θ1e'θexpee'θ1ee'θexph

(6.3)

After expanding, subtracting Y from both sides of the equation (6.3) and neglecting the terms having power greater than two, we have

222111111222211111110pd e'θneθne'θnhe'φmeφme'φmheYYt

(6.4)

Squaring both sides of (6.4) and then taking expectations, we get MSE of the estimator pt up

to the first order of approximation, as

52413212221

21

2pd R2hR2hRh2hRhRhYtMSE

(6.5)

where,

2321

43512

2321

53421

RRRRRRRh

RRRRRRRh

(6.6)

and

2ppb222

2ppb3115

2ppb222

2ppb3114

2pφ21111

2p222223

2p2

22

22

2p3

21

212

2p2

22

22

2p3

21

211

2211

2211

12

21

21

CkfθnCkfθnR

CkfφmCkfφmR

Ckfθφmn-CθφfnmR

CfnθCfnθR

CfmφCfmφR

(6.7)

Data: (Source: Singh and Chaudhary (1986), p. 177).

The population consists of 34 wheat farms in 34 villages in certain region of India. The variables are defined as:

y = area under wheat crop (in acres) during 1974.

1p = proportion of farms under wheat crop which have more than 500 acres land during 1971. and


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2p = proportion of farms under wheat crop which have more than 100 acres land during 1973.

For this data, we have

N=34, Y =199.4, 1P =0.6765, 2P =0.7353, 2yS =22564.6, 2

1S =0.225490, 2

2S =0.200535,

1pb =0599, 2pb =0.559,

=0.725.

Table 6.1: PRE of different estimators of Y with respect to y

CHOICE OF SCALERS, when 0h 0 1h1 0h2

1m 2m 1L 2L 3L 4L PRE’S

0 1 1 0 108.16 1 0 1 0 121.59 1 1 1 1 1 1 142.19 1 1 1 0 1 0 133.40 1 1

1pC 1pb 2pC 2pb 144.78

1 1 1NP 1pbK 2NP 2pbK 136.90

1 1 1NP f 2NP f 133.30

1 1 N 1pbK N 2pbK 135.73

1 1 1NP 1P 2NP 2P 137.09

1 1 n 1P n 2P 138.23

1 1 N 1pb N

2pb 135.49

1 1 n 1P n 2P

138.97

1 1 N 1P N 2P 135.86

When, 0h 0 0h1 1h2

1n 2n 5L 6L 7L 8L PRE’S

1 0 1 0 1 0 130.89 0 -1 1 0 1 0 108.93 1 -1 1 0 1 0 146.63 1 -1 1 1 1 1 121.68 1 -1 1 1 1 0 127.24 1 -1

1pC 1pb 2pC 2pb 123.43

1 -1 1NP 1pbK 2NP 2pbK 145.49

1 -1 1NP f 2NP f 146.57

1 -1 N 1pbK N 2pbK 145.84

1 -1 1NP 1P 2NP 2P 145.43

1 -1 n 1P n 2P 145.03


20

1 -1 N 1pb N

2pb 145.92

1 -1 n 1P n 2P

144.85

1 -1 N 1P N 2P 145.80

When, 0h 0 0h1 1h2 also 11,2,...,8iLi 1m 2m 1n 1n 2 pdtPRE =154.28

7. Conclusion

In this paper, we have suggested a class of estimators in single and two-phase sampling by using point bi serial correlation and phi correlation coefficient. From Table 4.1 and Table 6.1, we observe that the proposed estimator tp and tpd performs better than other estimators considered in this paper.

References 1. 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

2. Government of Pakistan, 2004, Crops Area Production by Districts (Ministry of Food, Agriculture and Livestock Division, Economic Wing, Pakistan).

3. Gujarati, D. N. and Sangeetha, 2007, Basic econometrics. Tata McGraw – Hill. 4. Jhajj, H.S., Sharma, M.K. and Grover, L.K., 2006 , A family of estimators of population

mean using information on auxiliary attribute. Pak. Journ. of Stat., 22(1), 43-50. 5. Malik, S. And Singh, R. ,2012, A Family Of Estimators Of Population Mean Using

Information On Point Bi-Serial And Phi-Correlation Coefficient. Intern. Jour. Stat. And Econ. (accepted).

6. Naik,V.D and Gupta, P.C., 1996, A note on estimation of mean with known population proportion of an auxiliary character. Jour. Ind. Soc. Agri. Stat., 48(2), 151-158.

7. Shabbir, J. and Gupta, S., 2007, On estimating the finite population mean with known population proportion of an auxiliary variable. Pak. Journ. of Stat., 23 (1), 1-9.

8. Singh, D. and Chaudhary, F. S., 1986, Theory and Analysis of Sample Survey Designs (John Wiley and Sons, NewYork).

9. Singh, R., Cauhan, P., Sawan, N. and Smarandache, F., 2007, Auxiliary information and a priori values in construction of improved estimators. Renaissance High press.

10. Singh, R. Chauhan, P. Sawan, N. Smarandache, F., 2008, Ratio estimators in simple random sampling using information on auxiliary attribute. Pak. J. Stat. Oper. Res. 4(1) 47–53.

11. Singh, R., Kumar, M. and Smarandache, F., 2010, Ratio estimators in simple random sampling when study variable is an attribute. WASJ 11(5): 586-589.

12. Yule, G. U., 1912, On the methods of measuring association between two attributes. Jour. of The Royal Soc. 75, 579-642.


21

A General Procedure of Estimating Population Mean Using Information on Auxiliary Attribute

1Sachin Malik, †1Rajesh Singh and 2Florentin Smarandache 1Department of Statistics, Banaras Hindu University

Varanasi-221005, India

2Chair of Department of Mathematics, University of New Mexico, Gallup, USA

† Corresponding author, [email protected]

Abstract This paper deals with the problem of estimating the finite population mean when some information on auxiliary attribute is available. It is shown that the proposed estimator is more efficient than the usual mean estimator and other existing estimators. The results have been illustrated numerically by taking empirical population considered in the literature.

Keywords Simple random sampling, auxiliary attribute, point bi-serial correlation, ratio estimator, efficiency.

1. Introduction The use of auxiliary information can increase the precision of an estimator when

study variable y is highly correlated with auxiliary variable x. There are many situations when auxiliary information is available in the form of attributes, e.g. sex and height of the persons, amount of milk produced and a particular breed of cow, amount of yield of wheat crop and a particular variety of wheat (see Jhajj et. al. (2006)). Consider a sample of size n drawn by simple random sampling without replacement

(SRSWOR) from a population of size N. Let iy and i denote the observations on variable y

and respectively for thi unit ( i =1, 2,......, N). Let i =1; if the thi unit of the population possesses attribute = 0; otherwise.

Let A=

N

1ii and a=

n

1ii , denote the total number of units in the population and sample

respectively possessing attribute . Let P=A/N and p=a/n denote the proportion of units in the population and sample respectively possessing attribute . Naik and Gupta (1996) introduced a ratio estimator NGt when the study variable and the auxiliary attribute are

positively correlated. The estimator NGt is given by


22

pPyt NG

(1.1)

with MSE

y222

y1NG RS2SRSf)t(MSE (1.2)

,Nn

nNf where 1

,PYR ,Yy

nN1S

N

1i

2i

2y

2N

1ii

2 P1N

1S

, .YyP1N

1S iN

1iiy

(for details see Singh et al. (2008))

Jhajj et. al. (2006) suggested a family of estimators for the population mean in single and two phase sampling when the study variable and auxiliary attribute are positively correlated. Shabbir and Gupta (2007), Singh et. al. (2008) and Abd-Elfattah et. al. (2010) have considered the problem of estimating population mean Y taking into consideration the point biserial correlation coefficient between auxiliary attribute and study variable.

The objective of this article is to suggest a generalised class of estimators for population mean Y and analyse its properties. A numerical illustration is given in support of the present study.

2. Proposed Estimator

Let mAi*i , m being a suitably chosen scalar, that takes values 0 and 1. Then

NmPpmApq , and

,P)1Nm(Q

where .b and B,NBQ,

nbq

n

1ii

N

1ii

Motivated by Bedi (1996), we define a family of estimators for population mean Y as

QqpPbwywt 21

(2.1)

where 1w , 2w and are suitably chosen scalars.

To obtain the Bias and MSE of the estimator t, we write


23

0e1Yy , 1e1Pp , ,e1Ss 222

3yy e1Ss , 123 e1e1b

such that 0)e(E i , i=0,1,2,3 and

2y

20 C

N1

n1)e(E

, 2

p21 C

N1

n1)e(E

,

pypb10 CC

N1

n1)ee(E

, ,C

N1

n1)ee(E 03p21

,C

N1

n1)ee(E

pb

12p31

Expressing (2.1) in terms of e’s , we have

1Nme

1e1e1eR

we1wYt 11231201

(2.2)

We assume that 12 e and 11Na

e1

, so that ( 21 e ) 1 and

1Nme

1 1 are expandable.

Expanding the right hand side of (2.2) and retaining terms up to second powers of e’s ,we have

1Nmee

1Nme

21

1Nmee1w[YYt 10

2

211

01

]11Nm

eeeeee

Rw

21

213112

(2.3)

Taking expectation of both sides of (2.3) , we get the bias of t to the first degree of approximation as :

2

p12py111 Cf1Nm21Cf

1Nmw1wY)t(B

(2.4) C1Nm

CCfR

w 2p03p

pb

12p12

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


24

1Nmee4

1Nm

e12e

1Nme2

e21wYYt 102

212

01

021

22

1Nme2

eeeeeeeR

ww21eR

w21

21103112121

222

2

21101

011Nm2

e11Nm

ee1Nm

ee1w2

1Nmeeeeee

Rw2

21

213112 (2.5)

Taking expectation of both sides of (2.5), we get the MSE of t to the first degree of approximation as:

m52

m41

m3212

22

m1

21

2 Aw2Aw2Aww2AwAw1Y)t(MSE

(2.6)

where,

k4

1Nm12

1NmC

Cf1A2p2

y1m

1

2p1

2

2 CfR

A

03p12pb

p2p1

m3 C

Ck

1Nm2Cf

RA

k

1Nm21f

1Nm1A 1

m4

03p12pb

p2p

1m

5 CC

1NmC

fR

A

where , .CC

kp

ypb

The MSE(t) is minimised for

(2.7) wAAA

AAAAw 102m

32m

1

m5

m3

m42

1

(2.8) wAAA

AAAAw 202m

32m

1

m5

m1

m4

m3

2


25

3. Members of the family of estimator of t and their Biases and MSE

Table 3.1: Different members of the family of estimators of t

Choice of scalars Estimator

1w 2w m 1

0

0

0

yt1

1w 0

0

0 ywt 12

Searls (1964) type estimator

1w 0

m

Qqywt 13

1w 0

0

Ppywt 4

1

0

-1

0

pPyt5

, Naik and Gupta (1996) estimator

1

1

-1

0

pPpPbyt 6

Singh et. al. (2008) estimator

1w

2w 0

0 pPbwywt 217

1w 1

0

0 pPbywt 18

w

w

0

0 pPbywt 9

1

1

0

0 pPbyt10

Regression estimator

The estimator yt1 is an unbiased estimator of the population mean Y and has the variance

2y11 SftVar (3.1)


26

To, the first degree of approximation the biases and MSE’s of s't i , i=1,2,.......,10 are respectively given by

1wYtB 12 (3.2)

2

ppypb1113 C1Nm2

1CC1Nm

wf1wYtB (3.3)

2ppypb1114 C

21CCfw1wYtB

(3.4)

pypb

2p15 CCCfYtB (3.5)

03p

pb

12ppypb

2p16 CC

R)CCC(fYtB

(3.6)

03p

pb

12p1

1217 CCf

R1wYtB

(3.7)

03p

pb

12p118 CCf

R1wYtB

(3.8)

03p

pb

12p19 CC

Rwf1wYtB

(3.9)

03p

pb

12p110 CCf

RYtB

(3.10)

The corresponding MSE’s will be

0

041001

21

22 Aw2Aw1YtMSE (3.11)

m41

m1

21

23 Aw2Aw1YtMSE

(3.12)

041

01

21

24 Aw2Aw1YtMSE

(3.13)

0

140

112

5 A2A1YtMSE

(3.14)


27

0

150

140

1320

112

6 A2A2A2AA1YtMSE

(3.15)

(3.16) Aw2Aw2Aww2AwAw1YtMSE 0

0520

041003212

22

001

21

27

005

004

00312

001

21

28 A2AAw2AAw1YtMSE (3.17)

0

050

040032

001

229 AAw2A2AAw1YtMSE (3.18)

0

050

040032

001

210 AAA2AA1YtMSE (3.19)

The MSE’s of the estimaors of ti, i=2,3,4,7,8,9 will be minimised respectively, for

0

01

004

1A

Aw

(3.20)

m1

m4

1A

Aw

(3.21)

01

04

1A

Aw

(3.22)

2)0()0(3

)0()0(12

)0()0(5

)0()0(1

)0()0(4

)0()0(3

2

2003

0012

005

003

0042

1

AAA

AAAAw

AAA

AAAAw

(3.23)

0

01

004

003

1 AAA

w

(3.24)

0

032001

005

0)0(4

A2AAAA

w

(3.25)

Thus the resulting minimum MSE of ti , i= 2,3,4,7,8,9 are, respectively given by


28

001

20042

2A

A1YtMSE .min

(3.26)

m1

2m42

3A

A1YtMSE .min

(3.27)

01

2042

4A

A1YtMSE .min

(3.28)

2003

0012

2005

001

005

003

200422

7AAA

AAAA2AA1YtMSE .min

(3.29)

001

2004

0030

0522

8A

AAA2A1YtMSE .min

(3.30)

0

032001

2005

0042

9A2AA

AA1YtMSE .min

(3.31)

4. Empirical study

The data for the empirical study is taken from natural population data set considered by Sukhatme and Sukhatme (1970):

y = Number of villages in the circles and

= A circle consisting more than five villages

190.2C,6040.0C,766.0,1236.0P,36.3Y,89N pypb

2744.2,475.146 ,810.3,1619.6 03124004

In the Table 4.1 percent relative efficiencies (PRE’s) of various estimators are computed with respect to y .


29

Table 4.1: PRE of different estimators of Y with respect to y .

Estimator PRE’s yt1

100.00

2t

101.41 3t

90.35

4t

6.92 5t

11.64

6t

7.38 7t

100.44

8t

243.39 9t

243.42

10t

241.98

Conclusion

The MSE values of the members of the family of the estimator t have been obtained using (2.6). These values are given in Table 4.1. When we examine Table 4.1, we observe the superiority of the proposed estimators t2, t7, t8, t9 and t10 over usual unbiased estimator t1, t3, t4, Naik and Gupta (1996) estimator t5 and Singh et. al. (2008) estimator t6. From this result we can infer that the proposed estimators t8 and t9 are more efficient than the rest of the estimators considered in this paper for this data set.

We would also like to remark that the value of the min. MSE(t10), which is equal to the value of the MSE of the regression estimator is 241.98. From Table 4.1 we notice that the value of MSE of the estimators t8 and t9 are less than this value, as shown in Table 4.1. Finally, we can say that the proposed estimators t8 and t9 are more efficient than the regression estimator for this data set.

References 1. 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

2. Bedi, P. K. (1996). Efficient utilization of auxiliary information at estimation stage. Biom. Jour, 38:973–976.

3. Jhajj, H.S., Sharma, M.K. and grover, L.K. (2006) : A family of estimators of population mean using information on auxiliary attribute. Pak. Journ. of Stat., 22(1), 43-50.

4. Naik,V.D. and Gupta,P.C.(1996): A note on estimation of mean with known population proportion of an auxiliary character. Journ. of the Ind. Soc. of Agr. Stat., 48(2), 151-158.

5. Searls, D.T. (1964): The utilization of known coefficient of variation in the estimation procedure. Journ. of the Amer. Stat. Assoc., 59, 1125-1126.


30

6. Singh, R. Chauhan, P. Sawan, N. Smarandache, F. (2008): Ratio estimators in simple random sampling using information on auxiliary attribute. Pak. J. Stat. Oper. Res. 4(1) 47–53.

7. Shabbir, J. and Gupta, S.(2007): On estimating the finite population mean with known population proportion of an auxiliary variable. Pak. Journ. of Stat., 23 (1), 1-9.

8. Sukhatme, P.V. and Sukhatme, B.V. (1970): Sampling theory of surveys with applications. Iowa State University Press, Ames, U.S.A.


31

Estimation of Ratio and Product of Two Population Means Using Auxiliary Characters in the Presence of Non Response

B. B. Khare Department of Statistics, Banaras Hindu University, Varanasi (U.P), India

Email: [email protected]

Abstract The auxiliary information is used in increasing the efficiency of the estimators for the parameters of the populations such as mean, ratio, and product of two population means. In this context, the estimation procedure for the ratio and product of two population means using auxiliary characters in special reference to the non response problem has been discussed.

Keywords Auxiliary variable, MSE, non response, SRS, efficiency.

Introduction The use of auxiliary information in sample surveys in the estimation of population mean, ratio, and product of two population means has been studied by different authors by using different estimation procedures. The review work in this topic has been given by Tripathi et al. (1994) and Khare (2003). In the present context the problems of estimation of ratio and product of two population means have been considered in different situations especially in the presence of non response.

Estimation of Ratio and product of two population means

Case 1. The Case of Complete Response: Singh (1965,69), Rao and Pareira (1968), Shahoo and Shahoo (1978), Tripathi (1980), Ray and Singh (1985) and Khare (1987) have proposed estimators of ratio and product of two population means using auxiliary characters with known mean. Singh (1982) has proposed the case of double sampling for the estimation of ratio and product of two population mean. Khare (1991(a)) has proposed a class of estimators for R and P using double sampling scheme, which are given as follows:

uvfR ,* and uwgP ,* (1)



32

such that RRf 1, , PPg 1, , 11,1 Rf and 11,1 Pg , where

2

1

yyv , 21yyw

and

1

1

xxu . Here 1y , 2y and 1x denote the sample mean of study characters 1y , 2y and

auxiliary character 1x based on a sub sample of size )( nn and 1x is sample mean of 1x based on a larger sample of size n drawn by using SRSWOR method of sampling from the population of size N . The first partial derivatives of uvf , and uwg , with respect to

wv and are denoted by uvf ,1 and uwg ,1 respectively. The function uvf , and uwg , also satisfied some regularity conditions for continuity and existence of the functions. The sample size for first phase and second phase sample which may be from the first phase sample or independent of first phase sample drawn from the remaining part of the population ( nN ). Singh et al. (1994) have extended the class of estimators proposed by Khare (1991(a)) and proposed a new class of estimator for R, which is given as follows:

vuhRRg ,ˆ (2)

where 2

1ˆyyR ,

xxu

and 2

2

x

x

ssv

, where 2, xsx and 2, xsx are sample mean and sample

mean square of auxiliary character based on n and n n units respectively.

Srivastava et al. (1988,89) have suggested chain ratio estimators for R and P . Which are given as follows:

4

'4

'3

3*1

ˆYy

yyRR and

'

4

4

3

3*2

ˆyY

yyRR (3)

21

4

'4

'3

3*1

ˆ

Yy

yyPP and

21

4

'4

3

3*2

ˆ

Yy

YyPP (4)

Further Singh et al. (1994) have given a general class of estimators

vuRhRh ,,ˆˆ and vuPhPh ,,ˆˆ , (5)

such that RRh 1,1, and PPh 1,1, , where

'

3

3

yyu and

4

'4

Yyv . The functions

vuRh ,,ˆ and vuPh ,,ˆ satisfy the regularity conditions.

Khare (1991(b)) have proposed the class of estimators for using multi-auxiliary characters with known means. which are given as follows:

uhRuuuhRR pˆ...,ˆ

21* and uRgR ,ˆ** , (6)


33

such that 1eh and ReRg ,ˆ , where uh and uRg ,ˆ satisfying some responding conditions.

Further, Khare (1993(a)) has proposed a class of estimators for R using multi-auxiliary characters with unknown means, the class of estimators is given as follows:

uRgRm ,ˆ* , (7)

such that 1, eRg , where i

ii x

xu

, puuuu ..., 21 , ix and ix are sample mean based on n

and n n units for auxiliary characters ix , .,...2,1 pi

Similarly, Khare (1992) have proposed class of estimators for P using p auxiliary characters with known and unknown population mean and studied their properties.

Further, Khare (1990) has proposed a generalized class of estimator for a combination of product and ratio of some population means using multi-auxiliary characters. The parametric combination is given by:

kmmm

m

YYYYYYYY,...,,,

,...,,,

321

321

, (8)

which is the product of first m population means mYYYY ,...,,, 321 divided by product of mk

population means kMmm YYYY ,...,,, 321 respectively. The conventional estimator for is given by

kmmm

m

yyyyyyyy,...,,,

,...,,,ˆ321

321

, (9)

It is important to note that for ;2,1 km R

;2,2 km P

;1,1 km 1Y

21;1 YYkm , 21Y ,

321;3 YYYkm , 31Y ,

4321 ,;4,2 YYYYkm , 221 RY ,

Using p auxiliary characters pxxx ..., , , 21 with known population means pXXX ..., , , 21 the

class of estimators * is given by:

uh ˆ* , (10)


34

such that 1eh , where puuuu ,..., 21 and piXx

ui

ii ...,,2,1, .

The function uhuuuh p ,..., 21 satisfied the following regularity conditions:

a) Whatever be the sample chosen u , assume values in abounded closed convex sub set G of p dimensional real space containing the point eu .

b) In G , the function uh is continuous and bounded. c) The first and second partial derivatives of uh exists and are continuous and bounded

inG . For two auxiliary variables it is found that the lower bond of the variance of the class of estimators * is same as given by the estimators proposed by Singh (1969) and Shah and Shah (1978). Hence it is remarked that the class of estimators * will attain lower bound for mean square error if the specified and regularity conditions are satisfied.

Further, Khare (1993b) have proposed the class of two phase sampling estimators for the combination of product and ratio of some population means using multi-auxiliary characters with unknown population means, which is given as follows:

vh ˆ** , (11)

where pvvvv ,..., 21 , i

ii x

xv

, pi ,...2,1 .

Such that 1eh and vh satisfies some regularly conditions.

Case 2. Incomplete Response in the Sample due to Non-response: In case of non-response on some units selected in the sample, Hansen and Hurwitz (1946) have suggested the method of sub sampling from non-respondents and proposed the estimator for population mean. Further, Khare et al. (2014) have proposed some new estimators in this situation of sub sampling from non-respondents.

Khare & Pandey (2000) and Khare & Sinha (2010) have proposed the class of estimators for ratio and product of two population means using auxiliary character with known population mean in the presence of non-response on the study characters, which is given as follows:

ii uhRR ** and ii uhPP ** , 2,1i , (12)

such that 11 h , where *2

*1*

yyR , *

2*1

* yyP , Xxu

*

1 , Xxu 2 and *

1y , *2y and *x are

sample means for 1y , 2y and x characters proposed by Hansen and Hurwitz (1946) based on rn 1 units and x is the sample mean based on n units. Khare & Sinha (2012) have proposed a combined class of estimators for ratio and product of two population mean in the presence of non-response with known population mean X . This is a more general class of estimators for R and P under some specified and regularity conditions. Khare et al. (2013 (a)) have proposed an improved class of estimators for R. In this case, the improved class of estimators for R using auxiliary character with known population mean X in the presence of non response is given as follows:


35

ii uvgR , 2,1i , (13)

such that RRg 1, , 1,1 Rg and 1,1, 21

12 RgRRg , where *2

*1

yyv , *

1 xu and

xu 2 . The function iuvg , 2,1i assumes positive values in a real line containing the

point 1,R . The function iuvg , is assumed to be continuous and bounded in a real line and its first and second order partial derivatives exists. The first partial derivative of iuvg , 2,1i at the point 1,R with respect to iuv and is denoted by 1,1 Rg and 1,2 Rg .

The second order partial derivative of iuvg , 2,1i with respect to ii uuvv and , and , at the

point 1,R is denoted by 1,11 Rg , 1,12 Rg and 1,22 Rg respectively. Some members of the class of estimators iR are given as follows:

iivuwC

01 , iuwvwC 212 , iivuwvwC

213 , 2,1i , (14)

where 0w , 1w , 2w , 1w , 2w , i and i 2,1i , are constants. Further the class of estimator proposed by Khare and Sinha (2013) is more efficient than the estimator proposed by Khare and Pandey (2000).

Further, Khare and Sinha (2002(a, b)) have proposed two phase sampling estimators for ratio and product of two population means in the presence of non-response. Khare and Sinha (2004(a,b)) have proposed a more general class of two phase sampling estimators for R and P. which are given as follows:

ii uvgT , , 2,1i , (15)

such that RRg 1, and 11,1 Rg , where *2

*1

yyv ,

xxu

*

1 , xxu

2 and x is sample mean

based on n n units. The function iuvg , satisfy some regularly conditions.

ii uwgT ,* , 2,1i , (16)

such that 11, Pg and 11,1 Pg , where *2

*1 yyw ,

xxu

*

1 , xxu

2 and iuwg , satisfy

some regularly conditions.

Khare et al. (2012) have proposed two generalized chain type estimators 1gT and 2gT for R

using auxiliary characters in the presence of non-response, which are given as follows:

2

'

*

1

1

ˆ

Zz

xxRTg and

2

'2

1

ˆ

Zz

xxRTg , (17)

where *2

*1ˆ

yyR and 21,

and 21 ,

are suitable constants. It has been observed that

due to use of additional auxiliary character with known population mean along with the main auxiliary character, the proposed class of estimators 1gT and 2gT are more efficient than the


36

corresponding generalized estimators for R using the main auxiliary character only in the case of two phase sampling in the presence of non response for fixed sample sizes ( nn , ) and also for fixed cost ( 0CC ). It is also seen that less cost is incurred for 1gT and 2gT than the

cost incurred in the generalized estimator for R in the case of two phase sampling in the presence of non response for specified precision ( 0VV ). Further, generalized chain estimators for ratio and product of two population means have been improved by putting RkR ˆ

1*

and PkP ˆ

1* in place of R and P in the proposed

estimators of R and P . Further, Khare et al. (2013 (b)) have proposed the improved class of chain type estimators for ratio of two population means using two auxiliary characters in the presence of non-response. The class of estimators is given as follows:

vuRfR ici ,,ˆ , 2,1i , (18)

such that 11,1, Rf and 11,1,1 Rf , where *2

*1ˆ

yyR ,

xxu

*

1 , xxu

2 and Zzv

. The

function vuRf i ,,ˆ , 2,1i satisfies some regularity conditions.

Khare and Sinha (2007) have proposed estimator for R using multi-auxiliary characters with known population mean in the presence of non-response. The class of estimators it is given as follows:

2,1),(ˆ iugRt iii , (19)

such that 1)( ii eg , where iu and ie denote the column vectors ),...,,( 21 ipii uuu and

)1,...,1,1( , j

jj X

xu

*

1 and j

jj X

xu 2 pj ...,,2,1 .

An improved under class of estimators for R using multi-auxiliary variables using double sampling scheme in the presence of non-response has been proposed by Khare and Sinha (2012) and studies their properties.

Khare and Sinha (2014) have extended the class of estimator proposed by Khare and Sinha (2012) and proposed a wider class of two phase sampling estimators for R using multi-auxiliary characters in the presence of non-response.

References 1. Hansen, M. H. and Hurwitz, W. N. (1946): The problem of non-response in sample

surveys. Jour. Amer. Stat. Assoc., 41, 517-529. 2. Khare, B. B. (1987): On modified class of estimators of ratio and product of two population

means using auxiliary character. Proc. Math. Soc, B.H.U. 3, 131-137. 3. Khare, B. B. (1990): A generalized class of estimators for combination of products and

ratio of some population means using multi-auxiliary characters. J. Stat. Res., 24, 1-8. 4. Khare, B. B. (1991 (a)): Determination of sample sizes for a class of two phase sampling

estimators for ratio and product of two population means using auxiliary character. Metron (Italy), XLIX, (1-4), 185-197.


37

5. Khare, B. B. (1991 (b)): On generalized class of estimators for ratio of two population means using multi-auxiliary characters. Aligarh J. Stat. 11, 81-90.

6. Khare, B. B. (1992): On class of estimators for product of two population means using multi-auxiliary characters with known and unknown means. Ind. J. Appl. Stat., 1, 56-67.

7. Khare, B. B. (1993(a)): A class of two phase sampling estimators for the combination of product and ratio of several population means using multi-auxiliary characters. Proc. Nat. Acad. Sci., India, 63 (4), Pt. II, 391-397.

8. Khare, B. B. (1993(b)): On a class of two phase sampling estimators for ratio of two population means using multi-auxiliary characters. Proc. Nat. Acad. Sci., India, 63(a), III, 513-520.

9. Khare, B. B. (2014): Estimation of population parameters using the technique of sub sampling from non respondents in sample surveys- A Review. Proc. Nat. Acad. Sci. Sec A, 84 (3), 337-343.

10. Khare, B. B. (2014-15): Applications of statistics in bio-medical sciences. Prajna, Special Issue on Science & Technology, Vol. - 60 (2),.

11. Khare, B. B. and Pandey, S. K. (2000): A class of estimators for ratio of two population means using auxiliary character in presence of non-response. J. Sc. Res., 50, 115-124.

12. Khare, B. B. and Sinha, R. R. (2002a): Estimation of the ratio of two populations means using auxiliary character with unknown population mean in presence of non response. Prog. Maths. Vol.- 36. No. (1, 2), 337-348.

13. Khare, B. B. and Sinha, R. R. (2002b): On class of two phase sampling estimators for the product of two population means using auxiliary character in presence of non response. Proc. of Vth international symposium on optimization and Statistics held at AMU, Aligarah, 221-232.

14. Khare, B. B. and Sinha, R. R. (2004 (a)): Estimation of finite population ratio using two phase sampling in presence of non response. Aligarh J. Stat. 24, 43-56.

15. Khare, B. B. and Sinha, R. R. (2004 (b)): On the general class of two phase sampling estimators for the product of two population means using the auxiliary characters in the presence of non-response. Ind.J. Appl. Statistics., 8, 1-14.

16. Khare, B. B. and Sinha, R. R. (2007): Estimation of the ratio of the two population means using multi- auxiliary characters in presence of non-response. In “Statistical techniques in life testing, reliability, sampling theory and quality control” edited by B. N. Pandey Narosa publishing house, New Delhi, 163-171.

17. Khare, B. B. and Sinha, R. R. (2010): On class of estimators for the product of two population means using auxiliary character in presence of non-response. Inter. Trans. Appl. Sci., 2(4), 841-846.

18. Khare, B. B. and Sinha, R. R. (2012 (a)): Combined class of estimators for ratio and product of two population means in presence of non-response. Int. Jour. Stats. and Eco., 8(S12), 86-95.

19. Khare, B. B. and Sinha, R. R. (2012 (b)): Improved classes of ratio of two population means with double sampling the non-respondents. Statistika-Statistics & Economy Jour. 49(3), 75-83.

20. Khare, B. B. and Sinha, R. R. (2014): A class of two phase sampling estimator for ratio of two populations means using multi-auxiliary characters in the presence of non response. Stat. in Trans. New series, 15, (3), 389-402.

21. Khare, B. B. and Srivastava, S. R. (1999): A class of estimators for ratio of two population means and means of two populations using auxiliary character. J. Nat. Acad. Math. 13, 100-104.

22. Khare, B. B. and Srivastava, S. Rani. (1998): Combined generalized chain estimators for ratio and product of two population means using auxiliary characters. Metron (Italy) LVI (3-4), 109-116.


38

23. Khare, B. B., Jha, P. S. and Kumar, K. (2014): Improved generalized chain estimators for ratio and product of two population means using two auxiliary characters in the presence of non-response. International J. Stats & Economics, 13(1), 108-121.

24. Khare, B. B., Pandey, S. K. and Kumar, A. (2013 (a)): Improved class of estimators for ratio of two population means using auxiliary character in presence of non-response. Proc. Nat. Acad. Sci. India, 83(1), 33-38.

25. Khare, B. B., Kumar, K. and Srivastava, U. (2013 (b)): Improved classes of chain type estimators for ratio of two population means using two auxiliary characters in the presence of non-response. Int. Jour. Adv. Stats. & Prob. 1 (3), 53-63.

26. Khare, B. B., Srivastava, U. and Kumar K. (2012): Chain type estimators for ratio of two population means using auxiliary characters in the presence of non response. J. Sc. Res., BHU, 56, 183-196.

27. Khare, B. B., Srivastava, U. and Kumar K. (2013): Generalized chain type estimators for ratio of two population means using two auxiliary characters in the presence of non-response. International J. Stats & Economics, 10(1), 51-64.

28. Rao, J. N. K. and Pareira, N. P. (1968): On double ratio estimators. Sankhya Ser. A., 30, 83-90.

29. Ray, S. K. and Singh, R. K. (1985): Some estimators for the ratio and product of population parameters. Jour. Ind. Soc. Agri. Stat., 37, 1-10.

30. Shah, S. M. and Shah, D. N. (1978): Ratio cum product estimators for estimating ratio (product) of two population parameters. Sankhya Ser. C., 40, 156-166.

31. Singh, M. P. (1969): Comparison of some ratio cum product estimators. Sankhya, Ser. B, 31, 375-378.

32. Singh, R. K. (1982b): On estimating ratio and product of population parameters. Calcutta Stat. Assoc., 20, 39-49.

33. Singh, V. K., Singh, Hari P. and Singh, Housila P. (1994): Estimation of ratio and product of two finite population means in two phase sampling. J. Stat. Plan. Inf. 41, 163-171.

34. Singh, V. K., Singh, Hari P., Singh, Housila P. and Shukla, D. (1994): A general class of chain type estimators for ratio and product of two population means of a finite population. Commun. Stat.- TM. 23 (5), 1341-1355.

35. Srivastava, Rani S., Khare, B. B. and Srivastava, S.R. (1988): On generalized chain estimator for ratio and product of two population means using auxiliary characters. Assam Stat. Review, 2(1), 21-29.

36. Srivastava, Rani S., Srivastava, S. R. and Khare, B. B. (1989): Chain ratio type estimator for ratio of two population means using auxiliary characters. Commun. Stat. Theory Math.(USA), 18(10), 3917-3926.

37. Tripathi, T. P. (1980): A general class of estimators for population ratio. Sankhya Ser. C., 42, 63-75.

38. Tripathi, T. P., Das, A. K. and Khare, B. B. (1994): Use of auxiliary information in sample surveys - A review. Aligarh J. Stat., 14, 79-134.


39

On The Use of Coefficient of Variation and 21 , in Estimating Mean of a Finite Population

1B. B. Khare, 1P. S. Jha and 2U. Srivastava

1Department of Statistics, B.H.U, Varanasi-221005 2Statistics Section, MMV, B.H.U, Varanasi-221005

[email protected]

Abstract In this paper the use of coefficient of variation and shape parameters in each stratum, the problem of estimation of population of mean has been considered. The expression of mean squared error of the proposed estimator is derived and its properties are discussed.

Keywords Auxiliary information, MSE, coefficient of variation, stratum, shape parameter.

Introduction The use of prior information about the population parameters such as coefficient of

variation, mean and skewness and kurtosis are very useful in the estimation of the population parameter of the study character. In agricultural and biological studies information about the coefficient of variation and the shape parameters are often available. If these parameters remain essentially unchanged over the time than the knowledge about them in such case it may profitably be used to produce optimum estimates of the parameters (Sen and Gerig (1975)). Searls (1964, 67) and Hirano (1972) have proposed the use of coefficient of variation in the estimation the population mean. Searl and Intarapanich (1990) have suggested the use of kurtosis in the estimation of variance. Sen (1978) has proposed the estimator for population mean using the known value of coefficient of variation.

In Stratified random sampling, the theory has been developed to provide the optimum estimator 1T of the population mean based on sample mean from each stratum. We extend it

by constructing an estimator 2T using the coefficient of variation iC and shape parameter

),...2,1( , 21 Kiii from each stratum and discuss its usefulness. We also define estimators

43 T and T when the coefficients of variation are unknown but shape parameters are known and when neither the coefficients of variation are known nor the shape parameters are known.

Estimators and their Mean Square Error Let iN denotes the size of the ith stratum and in denotes the size of the sample to be

selected from the ith stratum and h be the number of strata with



40

h

ii

h

ii nnNN

11 and , (1)

where N and n denote the number of units in the population and sample respectively.

Let ijy be the jth unit of the ith stratum. Then the population mean NY can be

expressed as

h

iiiYp

1NY , (2)

where NN

p ii and iY is the population mean for the ith stratum.

Let in units be selected from the ith stratum and the corresponding sampling mean and

sample variance be denoted by iy and 2is respectively. Then the estimate of NY is given by

h

iii yp

11T (3)

and the

h

i i

iin

p1

22

1)V(T

(if f.p.c is ignored), (4)

where 2i is the population variance of y in the ith stratum.

Case 1: Coefficient of variation and the shape parameters are known. We defined

h

iiiiiii sCyp

1

212 })1({T (5)

and expectation of 2T is given by

)1(

)})1(

21(

8)1(

{

)})(

81)(1({

)})(

811()1({)E(T

12

1 4

2

1 4

2

2

iN

h

i i

iiii

h

i i

iiii

h

i i

iiiiii

nOY

nnnYp

sVYp

sVYYp


41

(6)

where i2 is the measure of kurtosis in the ith stratum.

The bias in 2T is of order in

1 and will be negligible for large in ’s.

The mean square error of the estimator is

h

iii

iiiiii

i

iii

nOCC

np

1 2/322

2

112

22

2 )1()}1(4

)1()1({)/MSE(T

. (7)

Minimising (7) with respect to i , we get the optimum value of i is given by

144

12

212

12i

iiii

iiiopt CC

C

, (8)

where i1 is the measure of kurtosis in the ith stratum.

On putting the optimum value of opti from (8) in (7) and on simplification we get

h

i iiii

ii

i

ii

nO

Cnp

1 2/32112

1222

min2 )1(})2()1(

1{)MSE(T

. (9)

The value of opti will be less than one for ii C21 , which implies that the

distribution is near normal, poison, negative binomial and Neyman type I. The value of opti

will be equal to one for ii C21 , which is true for gamma and exponential distribution.

The value of opti will be greater than one for ii C21 , which is likely to the distribution

of lognormal or inverse Gaussian. It is easy to see that 2T will always be more efficient than

1T if ii C21 or ii C21 , justifying the use of 2T in the case of near normal, poison,

negative binomial, Neyman type I and lognormal or inverse Gaussian distribution. 2T is

equally efficient 1T , if ii C21 and so for in gamma or exponential distribution one may

use 1T or 2T . This shows that proposed estimator 2T is uniformly superior to the estimator

1T , though a comparatively high efficiency may be seen in near normal, poison, negative binomial than lognormal or inverse Gaussian distribution.


42

Case 2: s'iC are unknown, s'1i and s'2i are known. When s'iC are unknown, we use their estimates ic based on a larger sample of size

in from a previous occasion. Now we define an estimator 3T for NY given by

h

iiiiiii scyp

1

213 })1({T (10)

The mean square error of the estimator 3T as given by

h

iiiiiiiiiiii

i

iii cVCnCC

np

1

22

221

1222

2 )}(4)1{()1()1()/MSE(T

,

(11)

where )1( })1{()( 2

2

21

2

12

32

2

i

i

ii

i

ii n

CnC

cV

.

The optimum value of i is given by

21

212

212

i)2())(41(

)(4)1(

iiiiiii

iiiiiopt CcVCn

cVCn

. (12)

It is easy to see that

h

i iiiiiii

iiiii

i

iiopt CcVCn

cVCnn

p1 2

12

12

212

22

mini3)2())(41(

)(4)1()/MSE(T

. (13)

It may be remarked that (13) differs from (9) by a single term )(4 2iii cVCn both in

numerator and denominator. The nature of the estimator 3T is similar to 2T and its MSE will

converge to )( 2TMSE for 0)(

2

i

i

CcV

.

Case 3: s'iC , s'1i and s'2i are unknown: When s'iC , s'1i and s'2i are not known then they can be estimated on the basis

of a larger sample of size ii nn ... from the past data and we may have the estimator for the

population mean NY given by

h

iiiiiii scyp

1

214 })ˆ1(ˆ{T , (14)


43

where 1ˆˆˆ4ˆ4

1ˆˆ2ˆˆ

212

12i

iiii

iiiopt

CC

C

.

It is easy to see that the )MSE(T4 will be same as )MSE(T3 because after estimating the unknown parameters in the constant iopt , the MSE will remains unchanged up to the

terms of O ( 1n ) (Srivastava and Jhajj (1983)).

References 1. Searls, D. T. (1964): The utilization of coefficient of variation in the estimation procedure.

Jour. of Amer. Stat. Assoc., 59, 1125-1126. 2. Searls, D. T. (1967): A note on the use of a approximately known coefficient of variation.

The Amer. Statistician, 21, 20-21. 3. Sen, A. R. and Gerig, T. M. (1975): Estimation of a population mean having equal

coefficient of variation on succession occasions. Bull. Int. Stat. Inst., 46, 314-22. 4. Sen, A. R. (1978): Estimation of the population mean when the coefficient of variation is

known. Commu. Stat. Theory Meth., A7, 1, 657-672. 5. Srivastava, S.K. and Jhajj, H.S. (1983): A class of estimators of the population means using

multi- auxiliary information. Calcutta Stat. Assoc. Bull, 32, 47-56. 6. Searls, D. T. and Intarapanich R (1990): A note on an estimator for variance that utilized

the kurtosis. Amer. Stat., 44(4), 295-296. 7. Hirano K (1972): Using some approximately known coefficient of variation in estimating

mean. Proc. Inst. Stat. Math, 20(2), 61-64.


44


45

A Study of Improved Chain Ratio-cum-Regression type Estimator for Population Mean in the Presence of Non- Response for Fixed

Cost and Specified Precision 1B. B. Khare , 1Habib Ur Rehman and 2U. Srivastava

1Department Of Statistics, B.H.U, Varanasi-221005

2Statistics Section, MMV , B.H.U, Varanasi-221005 [email protected]

Abstract In this paper, a study of improved chain ratio-cum regression type estimator for population mean in the presence of non-response for fixed cost and specified precision has been made. Theoretical results are supported by carrying out one numerical illustration.

Keywords Simple random sampling, non response, fixed cost, precision.

Introduction In the field of socio, economics, researches and agricultures the problem arises due to non-response which friendly occur due to not at home, lack of interest, call back etc. In this expression a procedure of sub sampling from non respondents was suggested by Hansen and Hurwitz (1946). The use of auxiliary information in the estimators of the population parameters have helped in increased the efficiency of the proposed estimator. Using auxiliary character with known population mean of the estimators have been proposed by Rao (1986,90) and Khare and Srivastava (1996,1997). Further, Khare and Srivastava (1993,1995),Khare et al. (2008), Singh and Kumar (2010), Khare and Kumar (2009) and Khare and Srivastava(2010) have proposed different types of estimators for the estimation of population mean in the presence of non-response in case of unknown population mean of the auxiliary character.

In the present paper, we have studied an improved chain ratio-cum-regression type estimator for population mean in the presence of non-response have proposed by Khare and Rehman (2014) in the case of fixed cost and specified precision. In the present study we have obtained the optimum size of first phase sample ( n ) and second phase sample ( n ) is drawn from the population of size N by using SRSWOR method of sampling in case of fixed cost and also in case of specified precision 0VV . The expression for the minimum MSE of the

estimator has been obtained for the optimum values of n and n in case of fixed cost 0CC . The expression for minimum cost for the estimator has also been obtained in



46

case of specified precision 0VV . An empirical study has been considered to observe the properties of the estimator in case of fixed cost and also in case of specified precision.

The Estimators LetY , X and Z denote the population mean of study character y , auxiliary character x and additional auxiliary character z having jth value jY , jX and jZ : Nj ,...,3,2,1 .

Supposed the population of size N is divided in 1N responding units and 2N not responding unit. According to Hansen and Hurwitz a sample of size n is taken from population of size N by using simple random sampling without replacement (SRSWOR) scheme of sampling and it has been observed that 1n units respond and 2n units do not respond. Again by making

extra effort, a sub sample of size )( 12

knr is drawn from 2n non-responding unit and collect information on r units for study character y . Hence the estimator for Y based on rn 1 units on study character y is given by:

22

11* y

nny

nny (1)

where 1n and 2n are the responding and non-responding units in a sample of size n selected from population of size N by SRSWOR method of sampling. 1y and 2y are the means based on 1n and r units selected from 2n non-responding units by SRSWOR methods of sampling.

Similarly we can also define estimator for population mean X of auxiliary character x based on 1n and r unit respectively, which is given as;

22

11* x

nnx

nnx (2)

Variance of the estimators *y and *x are given by

2)2(

22* )1()( yy SnkWS

nfyV

(3)

and

2)2(

22* )1()( xx S

nkW

SnfxV

(4)

whereNnf 1 ,

NN

W 22 , ),( 2

)2(2

yy SS and ),( 2)2(

2xx SS are population mean squares of y and

x for entire population and non-responding part of population. In case when the population means of the auxiliary character is unknown, we select a larger first phase sample of size n units from a population of size N units by using simple random sample without replacement (SRSWOR) method of sampling and estimate X by x based on these units n . Further second phase sample of size n (i.e. n < n ) is drawn from n units by using SRSWOR method of sampling and variable y under investigation is measured 1n responding and 2n non-responding units. Again a sub sample of size r ( 1,/2 kkn ) is drawn from 2n non-responding units and collect information on r units by personal interview.


47

In this case two phase sampling ratio, product and regression estimators for population mean Y using one auxiliary character in the presence of non-response have been proposed by Khare and Srivastava (1993,1995) which are given as follows:

**

1 xxyT

(5)

***2 xxbyT

(6)

where 22

11* x

nnx

nnx ,

n

jjx

nx

1

1 ,2

1

ˆ1 , *ˆ

nyx

jxj

Sx x b

n S

,

n

iix xx

ns

1

22

11

yxS and 2ˆxS are estimates of yxS and 2

xS based on rn 1 units.

The conventional and alternative two phase sampling ratio type estimators suggested by Khare and Srivastava (2010) which are as follows:

xxyT

**

3 ,

xxyT *

4 (7)

where and are constants. Singh and Kumar (2010) have proposed difference type estimator using auxiliary character in the presence of non-response which is given as follows:

2'1

**

5

xx

xxyT (8)

where 1 and 2 are constants. In case when X is not known than we may use an additional auxiliary character z with known population mean Z with the assumption that the variable z is less correlated to y than x i.e, ( yz yx ), x and z are variables such that z is more cheaper than x .

Following Chand (1975), some estimators have been proposed by Kiregyera (1980,84), Srivasatava et al. (1990) and Khare & Kumar (2011). In the case of non-response on the study character, the chain regression type and generalized chain type estimators for the population mean in the presence of non-response have been proposed by Khare & Kumar (2010) and Khare et al. (2011). An improved chain ratio-cum-regression type estimator for population mean in the presence of non-response have been proposed by Khare & Rehman (2014), which is given as follows:

zZbxxbzZ

xxyT xzyx

qp

*'

'*

'*

6 (9)

where p and q are constants. yxb and xzb are regression coefficients. Z and z population

mean and sample mean based on first phase sample of size nunits selected from population of size N by SRSWOR method.


48

Mean Square Errors of the Study Estimator Using the large sample approximations, the expressions for the mean square errors of the estimator proposed by Khare & Rehman (2014) up to the terms of order )( 1n are given by

* 2 2 2 2 2 2 2 26

1 1( ) 2 2 2x yx x yx yx yx yx xMSE T V y Y p C b X C Y pC XYb C XYb pCn n

2 2 2 2 2 2 2 21 1 2 2 2z xz z yz xz yz xz zY q C b Z C Y qC YZb C YZqb C

n N

2 2 2 2 2 2 2 2 2

(2) (2) (2) (2) (2)1

2 2 2x yx x yx yx yx yx xW K

Y p C b X C Y pC XYb C XYb pCn

(10) The optimum values of p and q and the values of regression coefficient are given as follows:

2)2(

22

2)2()2(

22

111

111

xx

xyxyxxyxyx

opt

CYnkW

CYnn

CbXCYnkW

CbXCYnnp

(11)

2

2

z

zxzyzopt CY

CbZCYq

, (12)

x

yyxyx C

CX

Yb

and

z

xxzyx C

CZ

Xb (13)

Mean square errors of the estimators 1T , 2T , 3 ,T 4T and 5T are given as follows:

)2(

2)2(

222*min1 2)1(211)()( yxxyxx CC

nkWCC

nnYyVTMSE

(14)

* 2 2 2 2 222 min (2)(2)

( 1)1 1( ) ( ) 2 yx y yxxW kMSE T V y Y C B C B C

n n n

(15)

2

)2(22

2

)2(2

2*min3 )1(11

)1(11

)()(

xx

yxyx

CnkW

Cnn

CnkW

Cnn

YyVTMSE (16)

222*min4

11)()( yyxCnn

YyVTMSE

(17)

and

2222

min5 )1(11)( yyxy Cnn

CnfYTMSE

2)2(

2)2(

2 )1()1(

yyx CnkW

(18)

where * 2 2 22(2)

( 1)( ) y yW kfV y Y C C

n n

and

yx y

x

Y CB

XC


49

Determination of knn and , for the Fixed Cost 0CC Let us assume that 0C be the total cost (fixed) of the survey apart from overhead cost. The expected total cost of the survey apart from overhead cost is given as follows:

kWeWeenneeC 2

312121 )( , (19)

where 1e : the cost per unit of obtaining information on auxiliary character x at the first phase.

2e : the cost per unit of obtaining information on additional auxiliary character z at the first phase.

1e : the cost per unit of mailing questionnaire/visiting the unit at the second phase.

2e : the cost per unit of collecting, processing data obtained from 1n responding units.

3e : the cost per unit of obtaining and processing data (after extra efforts) for the sub sampling units.

The expression for, 6( )MSE T can be expressed in terms of 210 ,, DDD and 3D which are the

coefficients n1 ,

n 1 ,

nk and

N1 respectively. The expression of 6( )MSE T

is given as follows:

ND

nDk

nD

nDTMSE 3210

min6 )(

, (20)

For obtaining the optimum values of n , n , k for the fixed cost 0C C , we define a function which is given as:

0min6 )( CCTMSE , (21) where is the Lagrange’s multiplier. We differentiating with respect to n , n , k and equating zero, we get optimum values of

,n n and k .which are given as follows:

)( 21

1

eeDnopt

, (22)

opt

optopt

kWeWee

DkDn

23121

20

, (23)

and

)( 1212

230

WeeDWeDkopt

, (24)

where

optopt k

WeWeeDkDeeDC

2312120211

0)(1

, (25)


50

The minimum value of )( 6TMSE for the optimum values of ,n n and k in the expression )( 6TMSE , we get:

ND

kWeWeeDkDeeD

CTMSE

optopt

3

2

2312120211

0min6 )()(1)(

,

(26)

Now neglecting the term of O ( 1N ), we have

2

2312120211

0min6 )()(1)(

optopt k

WeWeeDkDeeDC

TMSE

(27)

Determination of knn and , for the Specified Precision 0VV

Let 0V be the specified variance of the estimator 6T which is fixed in advance, so we have

ND

nkD

nD

nDV 3210

0

, (28)

To find the optimum values of n , n , k and minimum expected total cost, we define a function which is give as follows:

))(()( 0min62

312121 VTMSEk

WeWeennee

, (29)

where is the Lagrange’s multiplier. After differentiating with respect to n , n , k and equating to zero, we find the optimum

value of nn , and k which are given as;

)( 21

1

eeDnopt

, (30)

opt

optopt

kWeWee

DkDn

23121

20 , (31)

and

)( 1212

320

WeeDeWDkopt

, (32)

where

NDV

kWeWeeDkDeeD

optopt

30

2

2312120211 )()(

, (33)

The minimum expected total cost incurred on the use of 6T for the specified variance 0V will

be given as follows:


51

NDV

kWeWeeDkDeeD

Copt

opt

30

2

2312120211

min6

)()(

, (34)

Now neglecting the terms of O ( 1N ), we have

0

2

2312120211

min6

)()(

V

kWeWeeDkDeeD

Copt

opt

, (35)

An Empirical Study To illustrate the results we use the data considered by Khare and Sinha (2007).The description of the population is given below: The data on physical growth of upper socio-economic group of 95 schoolchildren of Varanasi under an ICMR study, Department of Pediatrics, B.H.U., during 1983-84 has been taken under study. The first 25% (i. e. 24 children) units have been considered as non-responding units. Here we have taken the study variable )(y , auxiliary variable )(x and the

additional auxiliary variable )(z are taken as follows: y : weight (in kg.) of the children.

x : skull circumference (in cm) of the children. z : chest circumference (in cm) of the children.

The values of the parameters of the zxy and , characters for the given data are given as follows:

19.4968 , Y 51.1726Z , 55.8611X , 0.15613,yC 0.03006zC , .05860xC ,

(2) 0.12075,yC (2) 0.02478zC , (2) 0.05402,xC 0.328yz , 0.846,yx 0.297,xz

(2) 0.570,xz 2 0.25,W

1 0.74, 95, 35W N n

Table 1. Relative efficiency (in %) of the estimators with respect to *y (for the fixed cost

0CC =Rs.220, 1c=Rs. 0.90, 2c =Rs. 0.10, 1c =Rs. 2, 2c =Rs. 4, 3c =Rs. 25).

Estimators optk optn optn Efficiency

*y 2.68 --- 30 100 (0.3843)*

1T 2.89 58 23 117 (0.3272)

2T 2.03 74 19 131 (0.2941)


52

3T 2.61 81 20 155 (0.2473)

4T 1.06 76 14 136 (0.2819)

5T 2.68 81 20 157 (0.2453)

6T 2.67 68 21 166 (0.2315)

*Figures in parenthesis give the MSE (.).

From table 1, we obtained that for the fixed cost 0CC the study estimator 6T is more

efficient in comparison to the estimators ,*y 1T , 2T , 3T , 4T and 5T .

Table 2. Expected cost of the estimators for the specified variance 2356.00 V : ( 1c=Rs. 0.90,

2c =Rs. 0.10, 1c =Rs. 2, 2c =Rs. 5, 3c =Rs. 25)

Estimators optk optn optn Expected Cost

(in Rs.)

*y 2.68 --- 61 502

1T 2.89 107 40 418

2T 2.03 115 25 332

3T 2.61 88 20 246

4T 1.06 92 16 275

5T 2.68 87 21 244

6T 2.67 69 20 231

From table 2, we obtained that for the specified variance the study estimator 6T has less cost

in comparison to the cost incurred in the estimators ,*y 1T , 2T , 3T , 4T and 5T .


53

Conclusion The information on additional auxiliary character and optimum values of increase the efficiency of the study estimators in comparison to corresponding estimators in case of the fixed cost 0CC and specified precision 0VV .

References 1. Chand, L. (1975): Some ratio-type estimators based on two or more auxiliary variables.

Ph.D. Thesis submitted to Iowa State University, Ames, IOWA. 2. Hansen, M. H. and Hurwitz, W. N. (1946): The problem of non-response in sample

surveys. J. Amer. Statist. Assoc., 41, 517-529. 3. Kiregyera, B. (1980): A chain ratio type estimator in finite population double sampling

using two auxiliary variables. Metrika, 27, 217-223. 4. Kiregyera, B. (1984): Regression - type estimators using two auxiliary variables and model

of double sampling from finite populations. Metrika, 31, 215-226. 5. Khare, B. B. and Srivastava, S. (1993): Estimation of population mean using auxiliary

character in presence of non-response. Nat. Acad. Sci. Letters, India, 16(3), 111-114. 6. Khare, B. B. and Srivastava, S. (1995): Study of conventional and alternative two phase

sampling ratio, product and regression estimators in presence of non-response. Proc. Nat. Acad. Sci., India, Sect. A 65(a) II, 195-203.

7. Khare, B. B. and Srivastava, S. (1996): Transformed product type estimators for population mean in presence of softcore observations. Proc. Math. Soc. B. H.U., 12, 29-34.

8. Khare, B. B. and Srivastava, S. (1997): Transformed ratio type estimators for population mean in presence. Commun. Statist. Theory Meth., 26(7), 1779-1791.

9. Khare, B.B. and Sinha, R.R. (2007): Estimation of the ratio of the two populations means using multi-auxiliary characters in the presence of non-response. In “Statistical Technique in Life Testing, Reliability, Sampling Theory and Quality Control”. Edited by B.N. Pandey, Narosa publishing house, New Delhi, 163-171.

10. Khare, B.B., Kumar Anupam, Sinha R.R., Pandey S.K.(2008): Two phase sampling estimators for population mean using auxiliary character in presence of non-response in sample surveys. Jour. Scie. Res., BHU, Varanasi, 52: 271-281.

11. Khare, B.B. and Kumar, S. (2009): Transformed two phase sampling ratio and product type estimators for population mean in the presence of nonresponse. Aligarh J. Stats., 29, 91-106.

12. Khare, B.B. and Srivastava, S. (2010): Generalized two phase sampling estimators for the population mean in the presence of nonresponse. Aligarh. J. Stats., 30, 39-54.

13. Khare, B. B. and Kumar, S. (2010): Chain regression type estimators using additional auxiliary variable in two phase sampling in the presence of non response. Nat. Acad. Sci. Letters, India, 33, No. (11 & 12), 369-375.

14. Khare, B.B. and Kumar, S. (2011): A generalized chain ratio type estimator for population mean using coefficient of variations of the study variable. Nat. Acad. Sci. Letters, India, 34(9-10), 353-358.

15. Khare, B.B., Srivastava, U. and Kamlesh Kumar (2011): Generalized chain estimators for the population mean in the presence of non-response. Proc. Nat. Acad. Sci., India, 81(A), pt III. 231-238.

16. Khare, B.B., and Rehman, H. U. (2014): An improved chain ratio-cum-regression type estimator for population mean in the presence of non-response. Int. J. Agri. Stat. Sci., 10(2), 281-284.


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17. Rao,P.S.R.S.(1986):Ratio estimation with sub-sampling the non-respondents. Survey Methodology. 12(2):217-230.

18. Rao, P.S.R.S.(1990): Ratio and regression Estimators with Sub-sampling of the non-respondents.In: Liepine, Guner E.,Uppuluri VRR, editors. Data Quality Control theory and Pragmatics, Marcel Dekker, New York. pp.191-208.

19. Singh, H.P. and Kumar, S. (2010): Estimation of mean in presence of non-response using two phase sampling scheme. Statistical papers, 51,559-582.

20. Srivastava, S. R., Khare, B.B. and Srivastava, S.R .(1990): A generalised chain ratio estimator for mean of finite population. Jour. Ind. Soc. Agri. Stat., 42(1), 108-117.


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The present book aims to present some improved estimators using auxiliary and attribute information in case of simple random sampling and stratified random sampling and in some cases when non-response is present.

This volume is a collection of five papers, written by seven co-authors (listed in the order of the papers): Sachin Malik, Rajesh Singh, Florentin Smarandache, B. B. Khare, P. S. Jha, Usha Srivastava and Habib Ur. Rehman.

The first and the second papers deal with the problem of estimating the finite population mean when some information on two auxiliary attributes are available. In the third paper, problems related to estimation of ratio and product of two population mean using auxiliary characters with special reference to non-response are discussed.

In the fourth paper, the use of coefficient of variation and shape parameters in each stratum, the problem of estimation of population mean has been considered. In the fifth paper, a study of improved chain ratio-cum-regression type estimator for population mean in the presence of non-response for fixed cost and specified precision has been made.

The authors hope that the book will be helpful for the researchers and students that are working in the field of sampling techniques.

sampling strategies for finite population using auxiliary ...fs.unm.edu/samplingstrategies.pdf ·...

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