Research ArticleA Study on the Chain Ratio-Type Estimator ofFinite Population Variance

Yunusa Olufadi1 and Cem Kadilar2

1 Department of Statistics and Mathematical Sciences Kwara State University PMB 1530 Malete Ilorin Nigeria2 Department of Statistics Hacettepe University Beytepe 06800 Ankara Turkey

Correspondence should be addressed to Yunusa Olufadi olufadi14yunusyahoocom

Received 12 August 2013 Revised 15 January 2014 Accepted 15 January 2014 Published 24 February 2014

Academic Editor Shein-chung Chow

Copyright copy 2014 Y Olufadi and C Kadilar This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We suggest an estimator using two auxiliary variables for the estimation of the unknown population variance The bias and themean square error of the proposed estimator are obtained to the first order of approximations In addition the problem is extendedto two-phase sampling scheme After theoretical comparisons as an illustration a numerical comparison is carried out to examinethe performance of the suggested estimator with several estimators

1 Introduction

Variations are present everywhere in our daily life It is thelaw of nature that no two things or individuals are exactlyalike For instance a physician needs a full understandingof variations in the degree of human blood pressure bodytemperature andpulse rate for adequate prescriptionAman-ufacturer needs constant knowledge of the level of variationsin peoplersquos reaction to his product to be able to knowwhetherto reduce or increase his price or improve the quality of hisproduct An agriculturist needs an adequate understandingof the variations in climatic factors especially from place toplace (or time to time) to be able to plan on when how andwhere to plant his crop

It is well known that the use of auxiliary information insample survey designs results in efficient estimators of pop-ulation parameters such as variance under some realisticconditions For example when information is available on theauxiliary variable that is positively correlated with the studyvariable the ratio estimator is a suitable estimator for theestimation of the population variance

Let 119875 be a finite population consisting of 119873 units 1198751 1198752

119875119873 The units of this finite population are identifiable in

the sense that they are uniquely labeled from 1 to 119873 and thelabel on each unit is known Let 119910 be the character understudy taking the value 119910

119894on the units 119875

119894(119894 = 1 2 119873)

and assume a sample of size 119899 is drawn by the simple randomsampling without replacement (SRSWOR)

Suppose in a survey problem that we are interested inestimating the population variance 1198782

119910 Isaki [1] presented the

ratio estimator for the population variance using the auxil-iary information The problem of estimating the populationvariance using information on single auxiliary variable hasalso been discussed by various authors including Prasad andSingh [2 3] Biradar and Singh [4] Rueda Garcia and ArcosCebrian [5] Arcos et al [6] Kadilar and Cingi [7] and Singhet al [8]

Themean square error (MSE) of the classical estimator ofthe population variance 1198782

119910 which we denote as 119905

0 is119881(119905

0) =

1198784

1199101198600 Quite often information onmany auxiliary variables is

available in the survey which can be utilized to increase theprecision of the estimate The ratio estimator of populationvariance for a single auxiliary variable denoted as 119905

1suggested

by Isaki [1] and the two-phase sampling (TPS) estimator of 1199051

denoted as 119905lowast1are as follows

1199051= 1199042

119910

1198782

1199091

1199042

1199091

MSE (1199051) = 1198784

119910(1198600+ 1198601minus 21198603)

Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2014 Article ID 723982 5 pageshttpdxdoiorg1011552014723982

2 Journal of Probability and Statistics

119905lowast

1= 1199042

119910

119904lowast2

1199091

1199042

1199091

MSE (119905lowast1) = MSE (119905

1) minus 1198784

119910(119860lowast

1minus 2119860lowast

3)

(1)

Following Olkin [9] Isaki [1] also presented the ratioestimator of variance using two auxiliary variables as follows

1199052= 1198821

1199042

119910

1199042

1199091

1198782

1199091+1198822

1199042

119910

1199042

1199091

1198782

1199092

MSE (1199052) = 1198784

119910(1198621+1198822

11198622minus 211988211198623)

119905lowast

2= 1198721

1199042

119910

1199042

1199091

119904lowast2

1199091+1198722

1199042

119910

1199042

1199092

119904lowast2

1199092(TPS approach of 119905

2)

MSE (119905lowast2) = 1198784

119910(1198631+1198722

11198632minus 211987211198633)

(2)

where119882119894and119872

119894 for 119894 = 1 2 are weights chosen tominimize

the MSE of 1199052and 119905lowast2 Furthersum119882

119894= 1 andsum119872

119894= 1 where

1199042

1199091=1

119899

119899

sum

119894=1

(1199091119894minus 1198831)2

119904lowast2

1199092=1

119899

119899

sum

119894=1

(1199092119894minus 1198832)2

119904lowast2

1199091=

1

1198991015840

1198991015840

sum

119894=1

(1199091119894minus 1199091015840

1)2

119904lowast2

1199092=

1

1198991015840

1198991015840

sum

119894=1

(1199092119894minus 1199091015840

2)2

1199091015840

1=

1

1198991015840

1198991015840

sum

119894=1

1199091119894 119909

1015840

2=

1

1198991015840

1198991015840

sum

119894=1

1199092119894

1198621= 1198600+ 1198602minus 21198604 119862

lowast

1= 119860lowast

2minus 2119860lowast

4

1198631= 1198621minus 119862lowast

1 119862

2= 1198601+ 1198602minus 21198605

119862lowast

2= 119860lowast

1+ 119860lowast

2minus 2119860lowast

5 119863

2= 1198622minus 119862lowast

2

1198623= 1198602+ 1198603minus 1198604minus 1198605 119862

lowast

3= 119860lowast

2+ 119860lowast

3minus 119860lowast

4minus 119860lowast

5

1198633= 1198623minus 119862lowast

3

1198600=1

119899(120582400

minus 1) 1198601=1

119899(120582040

minus 1)

1198602=1

119899(120582004

minus 1) 1198603=1

119899(120582220

minus 1)

1198604=1

119899(120582202

minus 1) 1198605=1

119899(120582022

minus 1)

119860lowast

1=

1

1198991015840(120582040

minus 1) 119860lowast

2=

1

1198991015840(120582004

minus 1)

119860lowast

3=

1

1198991015840(120582220

minus 1) 119860lowast

4=

1

1198991015840(120582202

minus 1)

119860lowast

5=

1

1198991015840(120582022

minus 1)

120582 =120583119886119887119888

1205831198862

2001205831198872

0201205831198882

002

120583119886119887119888

=1

119873 minus 1

119873

sum

119894=1

(119910119894minus 119884)119886

(1199091119894minus 1198831)119887

(1199092119894minus 1198832)119888

(3)

where 119886 119887 and 119888 are nonnegative integersSeveral authors (Srivastava et al [10] Upadhyaya et al

[11] and Singh et al [12]) adopted TPS procedure proposedby Chand [13] and have suggested some chain ratio-typeestimators for estimating populationmean119884of119910 In the samevein Gupta et al [14] and Singh et al [8] proposed the fol-lowing classes of estimators under the assumption that thepopulation variance of the first auxiliary variable 1198782

1199091is not

known but the population variance of another auxiliaryvariable119883

2closely related to119883

1is available TheMSEs of the

estimators suggested by Gupta et al [14] and Singh et al [8]are respectively given by

1199053= 1199042

119910(1199042

1199091

119904lowast2

1199091

)

1198681

(119904lowast2

1199092

1198782

1199092

)

1198682

MSEmin (1199053) = 1199044

119910[1198600minus 120601(

1198602

3

1198601

) minus (1198602

4

11989910158401198602

)]

1199054= 1199042

119910(1199042

1199091

119904lowast2

1199091

)

1198691

(119904lowast2

1199092

1198782

1199092

)

1198692

(1199042

1199092

1198782

1199092

)

1198693

MSEmin (1199054) = 1199044

1199101198600[1 minus 120575120574

lowast2

012minus 120579120588lowast2]

(4)

where 1198681 1198682 and 119869

119894for 119894 = 1 2 3 are constants chosen to

minimize theMSE of 1199053and 1199054 120601 = (1119899minus1119899

1015840) 120579 = 119899119899

1015840 120575 =((1198991015840minus119899)119899

1015840) 120574lowast2012

= (11986021198602

3minus2119860311986041198605+11986011198602

4)1198600(11986011198602minus

1198602

5) 120588lowast2 = 119860

2

411986001198602

In most studies several variables are considered simul-taneously either to explain or estimate (predict) the studyvariable In most cases information on several auxiliaryvariables closely related to the study variable may be easilyobtained on all units in the population For example whileconducting an educational survey the investigator may beinterested in studying characteristics such as age genderhours spent on studying per day sitting position parentrsquos edu-cational level parentrsquos income relationship with lectures andaccess to facilities (eg library internet laboratory) amongothers With the main aim of suggesting a more efficientestimator we propose in this paper under SRSWOR a chainratio-type estimator for estimating the population variancewhen information on two auxiliary variables is available Inaddition the problem is extended to the case of TPS

2 The Suggested Estimator

Following Abu-Dayyeh et al [15] we define an estimator forestimating the population variance 1198782

119910 as follows

119905 = 1199042

119910(1198782

1199091

1199042

1199091

)

1205721

(1198782

1199092

1199042

1199092

)

1205722

(5)

where 1205721and 1205722are real constants to be determined such that

the MSE of 119905 is minimum

Journal of Probability and Statistics 3

To determine the bias and MSE of 119905 we define

1199042

119910= 1198782

119910(1 + 119896

0) 119904

2

1199091= 1198782

1199091(1 + 119896

1) 119904

2

1199092= 1198782

1199092(1 + 119896

2)

(6)

such that

119864 (1198960) = 119864 (119896

1) = 119864 (119896

2) = 0

119864 (1198962

0) = 119860

0 119864 (119896

2

1) = 119860

1

119864 (1198962

2) = 119860

2 119864 (119896

01198961) = 119860

3

119864 (11989601198962) = 119860

4 119864 (119896

11198962) = 119860

5

(7)

Now expressing 119905 in terms of 119896rsquos we have

119905 = 1198782

119910(1 + 119896

0) (1 + 119896

1)minus1205721

(1 + 1198962)minus1205722

= 1198782

119910(1 + 119896

0) (1 minus 120572

11198961+1205721(1205721+ 1)

21198962

1)

times (1 minus 12057221198962+1205722(1205722+ 1)

21198962

2)

(8)

We assume that |1198961| lt 1 and |119896

2| lt 1 so that (1 + 119896

1)minus1

and (1 + 1198962)minus1 are expandable in terms of 119896rsquos By expanding

the right hand side of (8) multiplying and neglecting termsinvolving power of 119896rsquos greater than two we have

119905 minus 1198782

119910= 1198782

119910(1198960minus 12057211198961minus 12057221198962+ 1205721120572211989611198962minus 120572111989601198961

minus120572211989601198962+1205721(1205721+ 1)

21198962

1+1205722(1205722+ 1)

21198962

2)

(9)

Taking expectations on both sides of (9) we get the biasof 119905 to the first degree of approximation as

119861 (119905) = 1198782

119910(1205722

1

21198601+1205722

2

21198602+ 120572112057221198605minus 12057211198603minus 12057221198604)

(10)

Squaring both sides of (9) and neglecting terms of 119896rsquosinvolving power greater than two we have

(119905 minus 1198782

119910)2

= 1198784

119910(1198962

0+ 21205721120572211989611198962minus 2120572111989601198961

minus 2120572211989601198962+ 1205722

11198962

1+ 1205722

21198962

2)

(11)

Taking expectations on both sides of (11) we get the MSEof 119905 to the first order of approximation as

MSE (119905) = 1198784

119910(1198600+ 1205722

11198601+ 1205722

21198602minus 212057211198603

minus 212057221198604+ 2120572112057221198605)

(12)

The optimal values of 1205721and 120572

2in (12) could be obtained by

differentiating (12) with respect to 1205721and 120572

2and equalizing

to zero After a little algebraic simplification we have

120572lowast

1=11986021198603minus 11986041198605

11986011198602minus 1198602

5

120572lowast

2=11986011198604minus 11986031198605

11986011198602minus 1198602

5

(13)

We can obtain the minimumMSE of 119905 by simply substitutingthe optimal equations of 120572

1and 120572

2in (12)

3 Suggested Estimator in TPS

In certain practical situations when 1198782119909is not also known the

technique of TPS sometimes referred to as double samplingis used This scheme requires the collection of informationon 1199091and 119909

2in the first phase sample 1199041015840 of size 1198991015840 (1198991015840 lt 119873)

and on 119910 for the second phase sample 119904 of size 119899 (119899 lt 1198991015840) The

estimator 119905lowast in TPS will take the following form

119905lowast= 1199042

119910(119904lowast2

1199091

1199042

1199091

)

1205723

(119904lowast2

1199092

1199042

1199092

)

1205724

(14)

To obtain the bias and MSE of 119905lowast we write

1199042

119910= 1198782

119910(1 + 119896

0) 119904

2

1199091= 1198782

1199091(1 + 119896

1)

119904lowast2

1199091= 1198782

1199091(1 + 119896

lowast

1) 119904

2

1199092= 1198782

1199092(1 + 119896

2)

119904lowast2

1199092= 1198782

1199092(1 + 119896

lowast

2)

(15)

Note that

119864 (119896lowast

1) = 119864 (119896

lowast

2) = 0 119864 (119896

lowast2

1) = 119860

lowast

1

119864 (119896lowast2

2) = 119860

lowast

2 119864 (119896

1119896lowast

1) = 119860lowast

1

119864 (1198962119896lowast

2) = 119860lowast

2 119864 (119896

0119896lowast

1) = 119860lowast

3

119864 (1198960119896lowast

2) =

1

1198991015840(120582202

minus 1) = 119860lowast

4

119864 (1198961119896lowast

1) = 119864 (119896

2119896lowast

1) = 119864 (119896

lowast

1119896lowast

2) = 119860lowast

5

(16)

Expressing 119905lowast in terms of 119896rsquos and following the procedure

explained in Section 2 we get the bias and MSE of the esti-mator 119905lowast respectively as

119861 (119905lowast) = 1198782

119910(1205723(1205723+ 1)

21198643+1205724(1205724+ 1)

21198644

+120572312057241198645minus 12057231198641minus 12057241198642)

MSE (119905lowast) = MSE (119905) minus 1198784119910(1205722

3119860lowast

1+ 1205722

4119860lowast

2+ 212057231205724119860lowast

5

+ 21205723119860lowast

3+ 21205724119860lowast

4)

(17)

where1198641= 1198603minus 119860lowast

3 119864

2= 1198604minus 119860lowast

4

1198643= 1198601minus 119860lowast

1 119864

4= 1198602minus 119860lowast

2

1198645= 1198605minus 119860lowast

5

(18)

4 Journal of Probability and Statistics

Minimization of (17) with respect to 1205723and 120572

4 yields their

optimum values as

120572lowast

3=11986411198644minus 11986421198645

11986431198644minus 1198642

5

120572lowast

4=11986421198643minus 11986411198645

11986431198644minus 1198642

5

(19)

Substitution of 120572lowast3and 120572lowast

4in (17) gives the minimum value of

the MSE of 119905lowast

4 Efficiency Comparisons

In this section we considered the theoretical comparisonsof the performances of the suggested estimators (119905 and 119905

lowast)with respect to the traditional estimator (119905

0) Isaki [1] ratio

estimators 1199051 119905lowast1 1199052 and 119905

lowast

2(for single and double auxiliary

variables) Gupta et al [14] estimator (1199053) and Singh et al [8]

estimator (1199054) which are investigated We have the following

conditions(i) MSE (119905) minusMSE (119905

0) lt 0 997904rArr 119867

1lt 0

(ii) MSE (119905) minusMSE (1199051) lt 0 997904rArr 119867

2lt 0

(iii) MSE (119905) minusMSE (1199052) lt 0 997904rArr 119867

3+ 1198674lt 0

(iv) MSE (119905) minusMSE (1199053) lt 0 997904rArr 119867

5lt 1198676

(v) MSE (119905) minusMSE (1199054) lt 0 997904rArr 119867

5lt 1198677

(vi) MSE (119905lowast) minusMSE (119905lowast1) lt 0

997904rArr MSE (119905) minusMSE (1199051) lt 1198784

1199101198678

(vii) MSE (119905lowast) minusMSE (119905lowast2) lt 0

997904rArr MSE (119905) minusMSE (1199052) lt 1198784

1199101198679

(20)

where1198671= 1205722

11198601+ 1205722

21198602+ 2120572112057221198605minus 212057211198603minus 212057221198604

1198672= (1205722

1minus 1)119860

1+ 1205722

21198602minus 2 (120572

1minus 1)119860

3

minus 212057221198604+ 2120572112057221198605

1198673= (1205721minus1198821) [(1205721+1198821) 1198601minus 21198603]

+ (1205722+1198821) [(1205722minus1198821) 1198602minus 21198604] + 2119860

4

1198674= 2 (120572

11205722+1198822

1minus1198821)1198605minus (1 minus 2119882

1) 1198602

1198675= 120572lowast2

11198601+ 120572lowast2

21198602minus 2120572lowast

11198603minus 2120572lowast

21198604+ 2120572lowast

1120572lowast

21198605

1198676= 120601(

1198602

3

1198601

) minus (1198602

4

11989910158401198602

)

1198677= 120575120574lowast2

012minus 120579120588lowast2

1198678= (1205722

3minus 1)119860

lowast

1+ 1205722

4119860lowast

2

minus 2 (1205723minus 1)119860

lowast

3+ 21205724119860lowast

4+ 212057231205724119860lowast

5

1198679= 119867 + 119862

lowast

1+1198722

1119862lowast

2+ 21198721119862lowast

3

(21)

Table 1 The MSE and PRE of the different estimators with respectto 1199050

Estimators MSE PRE1199050

01267 1001199051

01241 103119905lowast

101251 101

1199052(opt) 00586 217119905lowast

2(opt) 00859 1471199053

00543 2331199054

00479 265119905(opt) 00451 281119905lowast

(opt) 00774 164

5 Numerical Illustration

In this section we illustrate the performance of variousestimators of the population variance 1198782

119910 by considering the

data about 119884 output 119883 number of workers and 119885 fixedcapital given in Murthy [16] The data summary is brieflypresented as follows

119873 = 80 119899 = 10 120582400

= 22667

120582040

= 36500 120582004

= 28664

120582220

= 23377 120582202

= 22208

120582022

= 31400

(22)

The MSE and percent relative efficiency (PRE) of variousestimators of 1198782

119910 with respect to the conventional estimator

1199050 have been computed and presented in Table 1 Note that

for the calculation of the MSE of 119905lowast we take 1198991015840 = 25 and alsonote that the minimum MSE of 119905

2and 119905lowast2is obtained using

MSEmin(1199052) = 1198784

119910(1198621minus 1198622

31198622) and MSEmin(119905

lowast

2) = 119878

4

119910(1198631minus

1198632

31198632)

Table 1 reveals that the suggested estimator 119905 has thesmallest MSE and thus the highest PRE among other esti-mators considered in this study The suggested estimator inTPS 119905

lowast also provides a sufficient improvement in varianceestimation compared to the existing ones (119905lowast

1and 119905lowast2) It is also

observed fromTable 1 that the TPS estimators are less efficientthan their corresponding

6 Conclusion

We have developed a new estimator for estimating the finitepopulation variance under SRSWOR which is found to bemore efficient than the traditional estimator Isaki [1] ratioestimators (using single and double auxiliary variables)Gupta et al [14] estimator and Singh et al [8] estimator whencertain conditions as outlined in Section 4 are satisfiedThistheoretical inference is also supported by the result of anapplication with original data In future we hope to extendthe estimators suggested here for the development of a newestimator in the stratified random sampling

Journal of Probability and Statistics 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C T Isaki ldquoVariance estimation using auxiliary informationrdquoJournal of the American Statistical Association vol 78 no 381pp 117ndash123 1983

[2] B Prasad and H P Singh ldquoSome improved ratio-type estima-tors of finite population variance in sample surveysrdquo Commu-nications in Statistics vol 19 no 3 pp 1127ndash1139 1990

[3] B Prasad and H P Singh ldquoUnbiased estimators of finite popu-lation variance using auxiliary information in sample surveysrdquoCommunications in Statistics vol 21 no 5 pp 1367ndash1376 1992

[4] R S Biradar and H P Singh ldquoAn alternative to ratio estimatorof population Variancerdquo Assam Statistical Review vol 8 no 2pp 18ndash33 1994

[5] M Rueda Garcia and A Arcos Cebrian ldquoRepeated substitu-tion method the ratio estimator for the population variancerdquoMetrika vol 43 no 2 pp 101ndash105 1996

[6] A Arcos M Rueda M D Martınez S Gonzalez and YRoman ldquoIncorporating the auxiliary information available invariance estimationrdquo Applied Mathematics and Computationvol 160 no 2 pp 387ndash399 2005

[7] C Kadilar andH Cingi ldquoImprovement in estimating the popu-lation mean in simple random samplingrdquo Applied MathematicsLetters vol 19 no 1 pp 75ndash79 2006

[8] H P Singh S Singh and J M Kim ldquoEfficient use of auxiliaryvariables in estimating finite population variance in two-phasesamplingrdquoCommunications of the Korean Statistical Society vol17 no 2 pp 165ndash181 2010

[9] I Olkin ldquoMultivariate ratio estimation for finite populationsrdquoBiometrika vol 45 pp 154ndash165 1958

[10] S R Srivastava S R Srivastava and B B Khare ldquoChain ratiotype estimator for ratio of two populationmeans using auxiliarycharactersrdquo Communications in Statistics vol 18 no 10 pp3917ndash3926 1989

[11] L N Upadhyaya K S Kushwaha and H P Singh ldquoA modifiedchain ratio-type estimator in two-phase sampling using multiauxiliary informationrdquo Metron vol 48 no 1ndash4 pp 381ndash3931990

[12] V K Singh H P Singh H P Singh and D Shukla ldquoA generalclass of chain estimators for ratio and product of two means ofa finite populationrdquo Communications in Statistics vol 23 no 5pp 1341ndash1355 1994

[13] L Chand Some ratio-type estimators based on two or more aux-iliary variables [PhD thesis] Iowa State University Ames IowaUSA 1975

[14] R K Gupta S Singh and N S Mangat ldquoSome chain ratio typeestimators for estimating finite population variancerdquo AligarhJournal of Statistics vol 12-13 pp 65ndash69 1992-1993

[15] W A Abu-Dayyeh M S Ahmed R A Ahmed and H AMuttlak ldquoSome estimators of a finite population mean usingauxiliary informationrdquo Applied Mathematics and Computationvol 139 no 2-3 pp 287ndash298 2003

[16] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 3

To determine the bias and MSE of 119905 we define

1199042

119910= 1198782

119910(1 + 119896

0) 119904

2

1199091= 1198782

1199091(1 + 119896

1) 119904

2

1199092= 1198782

1199092(1 + 119896

2)

(6)

such that

119864 (1198960) = 119864 (119896

1) = 119864 (119896

2) = 0

119864 (1198962

0) = 119860

0 119864 (119896

2

1) = 119860

1

119864 (1198962

2) = 119860

2 119864 (119896

01198961) = 119860

3

119864 (11989601198962) = 119860

4 119864 (119896

11198962) = 119860

5

(7)

Now expressing 119905 in terms of 119896rsquos we have

119905 = 1198782

119910(1 + 119896

0) (1 + 119896

1)minus1205721

(1 + 1198962)minus1205722

= 1198782

119910(1 + 119896

0) (1 minus 120572

11198961+1205721(1205721+ 1)

21198962

1)

times (1 minus 12057221198962+1205722(1205722+ 1)

21198962

2)

(8)

We assume that |1198961| lt 1 and |119896

2| lt 1 so that (1 + 119896

1)minus1

and (1 + 1198962)minus1 are expandable in terms of 119896rsquos By expanding

the right hand side of (8) multiplying and neglecting termsinvolving power of 119896rsquos greater than two we have

119905 minus 1198782

119910= 1198782

119910(1198960minus 12057211198961minus 12057221198962+ 1205721120572211989611198962minus 120572111989601198961

minus120572211989601198962+1205721(1205721+ 1)

21198962

1+1205722(1205722+ 1)

21198962

2)

(9)

Taking expectations on both sides of (9) we get the biasof 119905 to the first degree of approximation as

119861 (119905) = 1198782

119910(1205722

1

21198601+1205722

2

21198602+ 120572112057221198605minus 12057211198603minus 12057221198604)

(10)

Squaring both sides of (9) and neglecting terms of 119896rsquosinvolving power greater than two we have

(119905 minus 1198782

119910)2

= 1198784

119910(1198962

0+ 21205721120572211989611198962minus 2120572111989601198961

minus 2120572211989601198962+ 1205722

11198962

1+ 1205722

21198962

2)

(11)

Taking expectations on both sides of (11) we get the MSEof 119905 to the first order of approximation as

MSE (119905) = 1198784

119910(1198600+ 1205722

11198601+ 1205722

21198602minus 212057211198603

minus 212057221198604+ 2120572112057221198605)

(12)

The optimal values of 1205721and 120572

2in (12) could be obtained by

differentiating (12) with respect to 1205721and 120572

2and equalizing

to zero After a little algebraic simplification we have

120572lowast

1=11986021198603minus 11986041198605

11986011198602minus 1198602

5

120572lowast

2=11986011198604minus 11986031198605

11986011198602minus 1198602

5

(13)

We can obtain the minimumMSE of 119905 by simply substitutingthe optimal equations of 120572

1and 120572

2in (12)

3 Suggested Estimator in TPS

In certain practical situations when 1198782119909is not also known the

technique of TPS sometimes referred to as double samplingis used This scheme requires the collection of informationon 1199091and 119909

2in the first phase sample 1199041015840 of size 1198991015840 (1198991015840 lt 119873)

and on 119910 for the second phase sample 119904 of size 119899 (119899 lt 1198991015840) The

estimator 119905lowast in TPS will take the following form

119905lowast= 1199042

119910(119904lowast2

1199091

1199042

1199091

)

1205723

(119904lowast2

1199092

1199042

1199092

)

1205724

(14)

To obtain the bias and MSE of 119905lowast we write

1199042

119910= 1198782

119910(1 + 119896

0) 119904

2

1199091= 1198782

1199091(1 + 119896

1)

119904lowast2

1199091= 1198782

1199091(1 + 119896

lowast

1) 119904

2

1199092= 1198782

1199092(1 + 119896

2)

119904lowast2

1199092= 1198782

1199092(1 + 119896

lowast

2)

(15)

Note that

119864 (119896lowast

1) = 119864 (119896

lowast

2) = 0 119864 (119896

lowast2

1) = 119860

lowast

1

119864 (119896lowast2

2) = 119860

lowast

2 119864 (119896

1119896lowast

1) = 119860lowast

1

119864 (1198962119896lowast

2) = 119860lowast

2 119864 (119896

0119896lowast

1) = 119860lowast

3

119864 (1198960119896lowast

2) =

1

1198991015840(120582202

minus 1) = 119860lowast

4

119864 (1198961119896lowast

1) = 119864 (119896

2119896lowast

1) = 119864 (119896

lowast

1119896lowast

2) = 119860lowast

5

(16)

Expressing 119905lowast in terms of 119896rsquos and following the procedure

explained in Section 2 we get the bias and MSE of the esti-mator 119905lowast respectively as

119861 (119905lowast) = 1198782

119910(1205723(1205723+ 1)

21198643+1205724(1205724+ 1)

21198644

+120572312057241198645minus 12057231198641minus 12057241198642)

MSE (119905lowast) = MSE (119905) minus 1198784119910(1205722

3119860lowast

1+ 1205722

4119860lowast

2+ 212057231205724119860lowast

5

+ 21205723119860lowast

3+ 21205724119860lowast

4)

(17)

where1198641= 1198603minus 119860lowast

3 119864

2= 1198604minus 119860lowast

4

1198643= 1198601minus 119860lowast

1 119864

4= 1198602minus 119860lowast

2

1198645= 1198605minus 119860lowast

5

(18)

4 Journal of Probability and Statistics

Minimization of (17) with respect to 1205723and 120572

4 yields their

optimum values as

120572lowast

3=11986411198644minus 11986421198645

11986431198644minus 1198642

5

120572lowast

4=11986421198643minus 11986411198645

11986431198644minus 1198642

5

(19)

Substitution of 120572lowast3and 120572lowast

4in (17) gives the minimum value of

the MSE of 119905lowast

4 Efficiency Comparisons

In this section we considered the theoretical comparisonsof the performances of the suggested estimators (119905 and 119905

lowast)with respect to the traditional estimator (119905

0) Isaki [1] ratio

estimators 1199051 119905lowast1 1199052 and 119905

lowast

2(for single and double auxiliary

variables) Gupta et al [14] estimator (1199053) and Singh et al [8]

estimator (1199054) which are investigated We have the following

conditions(i) MSE (119905) minusMSE (119905

0) lt 0 997904rArr 119867

1lt 0

(ii) MSE (119905) minusMSE (1199051) lt 0 997904rArr 119867

2lt 0

(iii) MSE (119905) minusMSE (1199052) lt 0 997904rArr 119867

3+ 1198674lt 0

(iv) MSE (119905) minusMSE (1199053) lt 0 997904rArr 119867

5lt 1198676

(v) MSE (119905) minusMSE (1199054) lt 0 997904rArr 119867

5lt 1198677

(vi) MSE (119905lowast) minusMSE (119905lowast1) lt 0

997904rArr MSE (119905) minusMSE (1199051) lt 1198784

1199101198678

(vii) MSE (119905lowast) minusMSE (119905lowast2) lt 0

997904rArr MSE (119905) minusMSE (1199052) lt 1198784

1199101198679

(20)

where1198671= 1205722

11198601+ 1205722

21198602+ 2120572112057221198605minus 212057211198603minus 212057221198604

1198672= (1205722

1minus 1)119860

1+ 1205722

21198602minus 2 (120572

1minus 1)119860

3

minus 212057221198604+ 2120572112057221198605

1198673= (1205721minus1198821) [(1205721+1198821) 1198601minus 21198603]

+ (1205722+1198821) [(1205722minus1198821) 1198602minus 21198604] + 2119860

4

1198674= 2 (120572

11205722+1198822

1minus1198821)1198605minus (1 minus 2119882

1) 1198602

1198675= 120572lowast2

11198601+ 120572lowast2

21198602minus 2120572lowast

11198603minus 2120572lowast

21198604+ 2120572lowast

1120572lowast

21198605

1198676= 120601(

1198602

3

1198601

) minus (1198602

4

11989910158401198602

)

1198677= 120575120574lowast2

012minus 120579120588lowast2

1198678= (1205722

3minus 1)119860

lowast

1+ 1205722

4119860lowast

2

minus 2 (1205723minus 1)119860

lowast

3+ 21205724119860lowast

4+ 212057231205724119860lowast

5

1198679= 119867 + 119862

lowast

1+1198722

1119862lowast

2+ 21198721119862lowast

3

(21)

Table 1 The MSE and PRE of the different estimators with respectto 1199050

Estimators MSE PRE1199050

01267 1001199051

01241 103119905lowast

101251 101

1199052(opt) 00586 217119905lowast

2(opt) 00859 1471199053

00543 2331199054

00479 265119905(opt) 00451 281119905lowast

(opt) 00774 164

5 Numerical Illustration

In this section we illustrate the performance of variousestimators of the population variance 1198782

119910 by considering the

data about 119884 output 119883 number of workers and 119885 fixedcapital given in Murthy [16] The data summary is brieflypresented as follows

119873 = 80 119899 = 10 120582400

= 22667

120582040

= 36500 120582004

= 28664

120582220

= 23377 120582202

= 22208

120582022

= 31400

(22)

The MSE and percent relative efficiency (PRE) of variousestimators of 1198782

119910 with respect to the conventional estimator

1199050 have been computed and presented in Table 1 Note that

for the calculation of the MSE of 119905lowast we take 1198991015840 = 25 and alsonote that the minimum MSE of 119905

2and 119905lowast2is obtained using

MSEmin(1199052) = 1198784

119910(1198621minus 1198622

31198622) and MSEmin(119905

lowast

2) = 119878

4

119910(1198631minus

1198632

31198632)

Table 1 reveals that the suggested estimator 119905 has thesmallest MSE and thus the highest PRE among other esti-mators considered in this study The suggested estimator inTPS 119905

lowast also provides a sufficient improvement in varianceestimation compared to the existing ones (119905lowast

1and 119905lowast2) It is also

observed fromTable 1 that the TPS estimators are less efficientthan their corresponding

6 Conclusion

We have developed a new estimator for estimating the finitepopulation variance under SRSWOR which is found to bemore efficient than the traditional estimator Isaki [1] ratioestimators (using single and double auxiliary variables)Gupta et al [14] estimator and Singh et al [8] estimator whencertain conditions as outlined in Section 4 are satisfiedThistheoretical inference is also supported by the result of anapplication with original data In future we hope to extendthe estimators suggested here for the development of a newestimator in the stratified random sampling

Journal of Probability and Statistics 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C T Isaki ldquoVariance estimation using auxiliary informationrdquoJournal of the American Statistical Association vol 78 no 381pp 117ndash123 1983

[2] B Prasad and H P Singh ldquoSome improved ratio-type estima-tors of finite population variance in sample surveysrdquo Commu-nications in Statistics vol 19 no 3 pp 1127ndash1139 1990

[3] B Prasad and H P Singh ldquoUnbiased estimators of finite popu-lation variance using auxiliary information in sample surveysrdquoCommunications in Statistics vol 21 no 5 pp 1367ndash1376 1992

[4] R S Biradar and H P Singh ldquoAn alternative to ratio estimatorof population Variancerdquo Assam Statistical Review vol 8 no 2pp 18ndash33 1994

[5] M Rueda Garcia and A Arcos Cebrian ldquoRepeated substitu-tion method the ratio estimator for the population variancerdquoMetrika vol 43 no 2 pp 101ndash105 1996

[6] A Arcos M Rueda M D Martınez S Gonzalez and YRoman ldquoIncorporating the auxiliary information available invariance estimationrdquo Applied Mathematics and Computationvol 160 no 2 pp 387ndash399 2005

[7] C Kadilar andH Cingi ldquoImprovement in estimating the popu-lation mean in simple random samplingrdquo Applied MathematicsLetters vol 19 no 1 pp 75ndash79 2006

[8] H P Singh S Singh and J M Kim ldquoEfficient use of auxiliaryvariables in estimating finite population variance in two-phasesamplingrdquoCommunications of the Korean Statistical Society vol17 no 2 pp 165ndash181 2010

[9] I Olkin ldquoMultivariate ratio estimation for finite populationsrdquoBiometrika vol 45 pp 154ndash165 1958

[10] S R Srivastava S R Srivastava and B B Khare ldquoChain ratiotype estimator for ratio of two populationmeans using auxiliarycharactersrdquo Communications in Statistics vol 18 no 10 pp3917ndash3926 1989

[11] L N Upadhyaya K S Kushwaha and H P Singh ldquoA modifiedchain ratio-type estimator in two-phase sampling using multiauxiliary informationrdquo Metron vol 48 no 1ndash4 pp 381ndash3931990

[12] V K Singh H P Singh H P Singh and D Shukla ldquoA generalclass of chain estimators for ratio and product of two means ofa finite populationrdquo Communications in Statistics vol 23 no 5pp 1341ndash1355 1994

[13] L Chand Some ratio-type estimators based on two or more aux-iliary variables [PhD thesis] Iowa State University Ames IowaUSA 1975

[14] R K Gupta S Singh and N S Mangat ldquoSome chain ratio typeestimators for estimating finite population variancerdquo AligarhJournal of Statistics vol 12-13 pp 65ndash69 1992-1993

[15] W A Abu-Dayyeh M S Ahmed R A Ahmed and H AMuttlak ldquoSome estimators of a finite population mean usingauxiliary informationrdquo Applied Mathematics and Computationvol 139 no 2-3 pp 287ndash298 2003

[16] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Probability and Statistics

Minimization of (17) with respect to 1205723and 120572

4 yields their

optimum values as

120572lowast

3=11986411198644minus 11986421198645

11986431198644minus 1198642

5

120572lowast

4=11986421198643minus 11986411198645

11986431198644minus 1198642

5

(19)

Substitution of 120572lowast3and 120572lowast

4in (17) gives the minimum value of

the MSE of 119905lowast

4 Efficiency Comparisons

In this section we considered the theoretical comparisonsof the performances of the suggested estimators (119905 and 119905

lowast)with respect to the traditional estimator (119905

0) Isaki [1] ratio

estimators 1199051 119905lowast1 1199052 and 119905

lowast

2(for single and double auxiliary

variables) Gupta et al [14] estimator (1199053) and Singh et al [8]

estimator (1199054) which are investigated We have the following

conditions(i) MSE (119905) minusMSE (119905

0) lt 0 997904rArr 119867

1lt 0

(ii) MSE (119905) minusMSE (1199051) lt 0 997904rArr 119867

2lt 0

(iii) MSE (119905) minusMSE (1199052) lt 0 997904rArr 119867

3+ 1198674lt 0

(iv) MSE (119905) minusMSE (1199053) lt 0 997904rArr 119867

5lt 1198676

(v) MSE (119905) minusMSE (1199054) lt 0 997904rArr 119867

5lt 1198677

(vi) MSE (119905lowast) minusMSE (119905lowast1) lt 0

997904rArr MSE (119905) minusMSE (1199051) lt 1198784

1199101198678

(vii) MSE (119905lowast) minusMSE (119905lowast2) lt 0

997904rArr MSE (119905) minusMSE (1199052) lt 1198784

1199101198679

(20)

where1198671= 1205722

11198601+ 1205722

21198602+ 2120572112057221198605minus 212057211198603minus 212057221198604

1198672= (1205722

1minus 1)119860

1+ 1205722

21198602minus 2 (120572

1minus 1)119860

3

minus 212057221198604+ 2120572112057221198605

1198673= (1205721minus1198821) [(1205721+1198821) 1198601minus 21198603]

+ (1205722+1198821) [(1205722minus1198821) 1198602minus 21198604] + 2119860

4

1198674= 2 (120572

11205722+1198822

1minus1198821)1198605minus (1 minus 2119882

1) 1198602

1198675= 120572lowast2

11198601+ 120572lowast2

21198602minus 2120572lowast

11198603minus 2120572lowast

21198604+ 2120572lowast

1120572lowast

21198605

1198676= 120601(

1198602

3

1198601

) minus (1198602

4

11989910158401198602

)

1198677= 120575120574lowast2

012minus 120579120588lowast2

1198678= (1205722

3minus 1)119860

lowast

1+ 1205722

4119860lowast

2

minus 2 (1205723minus 1)119860

lowast

3+ 21205724119860lowast

4+ 212057231205724119860lowast

5

1198679= 119867 + 119862

lowast

1+1198722

1119862lowast

2+ 21198721119862lowast

3

(21)

Table 1 The MSE and PRE of the different estimators with respectto 1199050

Estimators MSE PRE1199050

01267 1001199051

01241 103119905lowast

101251 101

1199052(opt) 00586 217119905lowast

2(opt) 00859 1471199053

00543 2331199054

00479 265119905(opt) 00451 281119905lowast

(opt) 00774 164

5 Numerical Illustration

In this section we illustrate the performance of variousestimators of the population variance 1198782

119910 by considering the

data about 119884 output 119883 number of workers and 119885 fixedcapital given in Murthy [16] The data summary is brieflypresented as follows

119873 = 80 119899 = 10 120582400

= 22667

120582040

= 36500 120582004

= 28664

120582220

= 23377 120582202

= 22208

120582022

= 31400

(22)

The MSE and percent relative efficiency (PRE) of variousestimators of 1198782

119910 with respect to the conventional estimator

1199050 have been computed and presented in Table 1 Note that

for the calculation of the MSE of 119905lowast we take 1198991015840 = 25 and alsonote that the minimum MSE of 119905

2and 119905lowast2is obtained using

MSEmin(1199052) = 1198784

119910(1198621minus 1198622

31198622) and MSEmin(119905

lowast

2) = 119878

4

119910(1198631minus

1198632

31198632)

Table 1 reveals that the suggested estimator 119905 has thesmallest MSE and thus the highest PRE among other esti-mators considered in this study The suggested estimator inTPS 119905

lowast also provides a sufficient improvement in varianceestimation compared to the existing ones (119905lowast

1and 119905lowast2) It is also

observed fromTable 1 that the TPS estimators are less efficientthan their corresponding

6 Conclusion

We have developed a new estimator for estimating the finitepopulation variance under SRSWOR which is found to bemore efficient than the traditional estimator Isaki [1] ratioestimators (using single and double auxiliary variables)Gupta et al [14] estimator and Singh et al [8] estimator whencertain conditions as outlined in Section 4 are satisfiedThistheoretical inference is also supported by the result of anapplication with original data In future we hope to extendthe estimators suggested here for the development of a newestimator in the stratified random sampling

Journal of Probability and Statistics 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C T Isaki ldquoVariance estimation using auxiliary informationrdquoJournal of the American Statistical Association vol 78 no 381pp 117ndash123 1983

[2] B Prasad and H P Singh ldquoSome improved ratio-type estima-tors of finite population variance in sample surveysrdquo Commu-nications in Statistics vol 19 no 3 pp 1127ndash1139 1990

[3] B Prasad and H P Singh ldquoUnbiased estimators of finite popu-lation variance using auxiliary information in sample surveysrdquoCommunications in Statistics vol 21 no 5 pp 1367ndash1376 1992

[4] R S Biradar and H P Singh ldquoAn alternative to ratio estimatorof population Variancerdquo Assam Statistical Review vol 8 no 2pp 18ndash33 1994

[5] M Rueda Garcia and A Arcos Cebrian ldquoRepeated substitu-tion method the ratio estimator for the population variancerdquoMetrika vol 43 no 2 pp 101ndash105 1996

[6] A Arcos M Rueda M D Martınez S Gonzalez and YRoman ldquoIncorporating the auxiliary information available invariance estimationrdquo Applied Mathematics and Computationvol 160 no 2 pp 387ndash399 2005

[7] C Kadilar andH Cingi ldquoImprovement in estimating the popu-lation mean in simple random samplingrdquo Applied MathematicsLetters vol 19 no 1 pp 75ndash79 2006

[8] H P Singh S Singh and J M Kim ldquoEfficient use of auxiliaryvariables in estimating finite population variance in two-phasesamplingrdquoCommunications of the Korean Statistical Society vol17 no 2 pp 165ndash181 2010

[9] I Olkin ldquoMultivariate ratio estimation for finite populationsrdquoBiometrika vol 45 pp 154ndash165 1958

[10] S R Srivastava S R Srivastava and B B Khare ldquoChain ratiotype estimator for ratio of two populationmeans using auxiliarycharactersrdquo Communications in Statistics vol 18 no 10 pp3917ndash3926 1989

[11] L N Upadhyaya K S Kushwaha and H P Singh ldquoA modifiedchain ratio-type estimator in two-phase sampling using multiauxiliary informationrdquo Metron vol 48 no 1ndash4 pp 381ndash3931990

[12] V K Singh H P Singh H P Singh and D Shukla ldquoA generalclass of chain estimators for ratio and product of two means ofa finite populationrdquo Communications in Statistics vol 23 no 5pp 1341ndash1355 1994

[13] L Chand Some ratio-type estimators based on two or more aux-iliary variables [PhD thesis] Iowa State University Ames IowaUSA 1975

[14] R K Gupta S Singh and N S Mangat ldquoSome chain ratio typeestimators for estimating finite population variancerdquo AligarhJournal of Statistics vol 12-13 pp 65ndash69 1992-1993

[15] W A Abu-Dayyeh M S Ahmed R A Ahmed and H AMuttlak ldquoSome estimators of a finite population mean usingauxiliary informationrdquo Applied Mathematics and Computationvol 139 no 2-3 pp 287ndash298 2003

[16] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C T Isaki ldquoVariance estimation using auxiliary informationrdquoJournal of the American Statistical Association vol 78 no 381pp 117ndash123 1983

[2] B Prasad and H P Singh ldquoSome improved ratio-type estima-tors of finite population variance in sample surveysrdquo Commu-nications in Statistics vol 19 no 3 pp 1127ndash1139 1990

[3] B Prasad and H P Singh ldquoUnbiased estimators of finite popu-lation variance using auxiliary information in sample surveysrdquoCommunications in Statistics vol 21 no 5 pp 1367ndash1376 1992

[4] R S Biradar and H P Singh ldquoAn alternative to ratio estimatorof population Variancerdquo Assam Statistical Review vol 8 no 2pp 18ndash33 1994

[5] M Rueda Garcia and A Arcos Cebrian ldquoRepeated substitu-tion method the ratio estimator for the population variancerdquoMetrika vol 43 no 2 pp 101ndash105 1996

[6] A Arcos M Rueda M D Martınez S Gonzalez and YRoman ldquoIncorporating the auxiliary information available invariance estimationrdquo Applied Mathematics and Computationvol 160 no 2 pp 387ndash399 2005

[7] C Kadilar andH Cingi ldquoImprovement in estimating the popu-lation mean in simple random samplingrdquo Applied MathematicsLetters vol 19 no 1 pp 75ndash79 2006

[8] H P Singh S Singh and J M Kim ldquoEfficient use of auxiliaryvariables in estimating finite population variance in two-phasesamplingrdquoCommunications of the Korean Statistical Society vol17 no 2 pp 165ndash181 2010

[9] I Olkin ldquoMultivariate ratio estimation for finite populationsrdquoBiometrika vol 45 pp 154ndash165 1958

[10] S R Srivastava S R Srivastava and B B Khare ldquoChain ratiotype estimator for ratio of two populationmeans using auxiliarycharactersrdquo Communications in Statistics vol 18 no 10 pp3917ndash3926 1989

[11] L N Upadhyaya K S Kushwaha and H P Singh ldquoA modifiedchain ratio-type estimator in two-phase sampling using multiauxiliary informationrdquo Metron vol 48 no 1ndash4 pp 381ndash3931990

[12] V K Singh H P Singh H P Singh and D Shukla ldquoA generalclass of chain estimators for ratio and product of two means ofa finite populationrdquo Communications in Statistics vol 23 no 5pp 1341ndash1355 1994

[13] L Chand Some ratio-type estimators based on two or more aux-iliary variables [PhD thesis] Iowa State University Ames IowaUSA 1975

[14] R K Gupta S Singh and N S Mangat ldquoSome chain ratio typeestimators for estimating finite population variancerdquo AligarhJournal of Statistics vol 12-13 pp 65ndash69 1992-1993

[15] W A Abu-Dayyeh M S Ahmed R A Ahmed and H AMuttlak ldquoSome estimators of a finite population mean usingauxiliary informationrdquo Applied Mathematics and Computationvol 139 no 2-3 pp 287ndash298 2003

[16] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of