Research ArticleAn Improved Class of Chain Ratio-Product Type Estimators inTwo-Phase Sampling Using Two Auxiliary Variables
Gajendra K Vishwakarma and Manish Kumar
Department of Applied Mathematics Indian School of Mines Dhanbad Jharkhand 826004 India
Correspondence should be addressed to Manish Kumar manishstats88gmailcom
Received 13 September 2013 Accepted 23 January 2014 Published 6 March 2014
Academic Editor Zhidong Bai
Copyright copy 2014 G K Vishwakarma and M Kumar This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
This paper presents a technique for estimating finite populationmean of the study variable in the presence of two auxiliary variablesusing two-phase sampling scheme when the regression line does not pass through the neighborhood of the origin The propertiesof the proposed class of estimators are studied under large sample approximation In addition bias and efficiency comparisons arecarried out to study the performances of the proposed class of estimators over the existing estimators It has also been shown that theproposed technique has greater applicability in survey research An empirical study is carried out to demonstrate the performanceof the proposed estimators
1 Introduction
The use of auxiliary information for estimating populationmean of the study variable has greater applicability in surveyresearch It is utilized at the estimation stage and designstage to obtain an improved estimator compared to thosenot utilizing auxiliary information The use of ratio andproduct strategies in survey sampling solely depends uponthe knowledge of populationmean119883 of the auxiliary variable119883
The ratio estimator was developed by Cochran [1] toestimate the population mean 119884 of the study variable 119884
by using information on auxiliary variable 119883 positivelycorrelated with 119884 The ratio estimator is most effective whenthe relationship between 119884 and 119883 is linear through theorigin and the variance of 119884 is proportional to 119883 Robson [2]defined a product estimator that was revisited by Murthy [3]The product estimator is used when the auxiliary variable 119883is negatively correlated with the study variable 119884
When the population mean 119883 of the auxiliary variable119883 is not known before the start of a survey then a first-phase sample of size 119899
1015840 is selected from the population ofsize 119873 on which only the auxiliary variable 119883 is measuredin order to furnish a good estimate of 119883 And then a
second-phase sample of size 119899 is selected from the first-phasesample of size 119899
1015840 on which both the study variable 119884 andthe auxiliary variable 119883 are measured This procedure ofselecting the samples from the given population is knownas two-phase sampling (or double sampling) The conceptof double sampling was first introduced by Neyman [4]Some contribution to two-phase sampling has been madeby Sukhatme [5] Hidiroglou and Sarndal [6] Fuller [7]Hidiroglou [8] Singh and Vishwakarma [9] and Sahoo et al[10]
We can use either one or two (ormore than two) auxiliaryvariables while estimating population mean of the studyvariable keeping this fact Chand [11] introduced chain ratioestimators This led various authors including Kiregyera [12]Singh and Upadhyaya [13] Prasad et al [14] Singh et al [15]Singh and Choudhury [16] and Vishwakarma and Gangele[17] to modify the chain type estimators and discuss theirproperties
When the population mean 119885 of another auxiliary vari-able 119885 which has a positive correlation with X (ie 120588
119883119885gt 0)
is known and if 120588119884119883
gt 120588119884119885
gt 0 then it is advisable to estimate119883 by119883 = 119909
1015840(1198851199111015840) which would provide a better estimate of
119883 as compared to 1199091015840
Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2014 Article ID 939701 6 pageshttpdxdoiorg1011552014939701
2 Journal of Probability and Statistics
The usual chain type ratio and product estimators of 119884under double sampling scheme using two auxiliary variables119883 and 119885 are given respectively by
119910dc119877
= 1199101199091015840
119909
119885
1199111015840
119910dc119875
= 119910119909
1199091015840
1199111015840
119885
(1)
Singh and Choudhury [16] suggested the following expo-nential chain type ratio and product estimators of 119884 underdouble sampling scheme using two auxiliary variables119883 and119885
119910dcRe = 119910 exp
(11990910158401199111015840)119885 minus 119909
(11990910158401199111015840)119885 + 119909
119910dcPe = 119910 exp
119909 minus (11990910158401199111015840)119885
119909 + (11990910158401199111015840)119885
(2)
where1199091015840 and 1199111015840 are the samplemeans of119883 and119885 respectivelybased on the first-phase sample of size 119899
1015840 drawn from thepopulation of size 119873 with the help of Simple RandomSampling Without Replacement (SRSWOR) scheme Also 119910and 119909 are the samplemeans of119884 and119883 respectively based onthe second-phase sample of size 119899 drawn from the first-phasesample of size 1198991015840 with the help of SRSWOR scheme
2 Proposed Estimator
It has been theoretically established that in general thelinear regression estimator is more efficient than the ratio(product) estimator except when the regression line of 119884 on119883 passes through the neighborhood of the origin in whichthe efficiencies of these estimators are almost equal Howeverowing to stronger intuitive appeal survey statisticians favourthe use of ratio and product estimators Further we note thatin many practical situations the regression line does not passthrough the neighborhood of the origin In these situationsthe ratio estimator does not perform well as the linearregression estimator Considering this fact Singh and RuizEspejo [18] made an attempt to improve the performance ofthese estimators and suggested the following ratio-producttype estimator for populationmean119884 under double samplingscheme using single auxiliary variable119883
119910119889
RP = 119910[1205721199091015840
119909+ (1 minus 120572)
119909
1199091015840] (3)
where 120572 is a real constantWe propose the following exponential chain ratio-
product type estimator for population mean 119884 under doublesampling scheme using two auxiliary variables119883 and 119885
119910dcRPe = 119910[120572 exp
(11990910158401199111015840)119885 minus 119909
(11990910158401199111015840)119885 + 119909
+ (1 minus 120572) exp119909 minus (119909
10158401199111015840)119885
119909 + (11990910158401199111015840)119885
]
(4)
where 120572 is a real constant to be determined such that theMean Square Error (MSE) of the proposed estimator 119910dc
RPe isminimum For 120572 = 1 119910dc
RPe rarr 119910dcRe whereas for 120572 = 0
119910dcRPe rarr 119910
dcPe
Remark It is noted that the proposed estimator in (4) is aspecial case of the class of estimators 119910class = 119910119867(119909 119911
1015840)
proposed by Srivastava [19] where 119867(sdot) is a parametricfunction such that 119867(119909
1015840| 1199041 119885) = 1 and satisfies certain
regularity conditions defined in Srivastava [19]
3 Bias and MSE of the Proposed Estimator
To obtain the Bias and Mean Square Error (MSE) of theproposed estimator 119910dc
(ii) the Percentage Relative Efficiencies (PREs) of differ-ent suggested estimators of 119884 with respect to 119910 usingthe formula
PRE (sdot 119910) =119881 (119910)
MSE (sdot)times 100 (28)
Journal of Probability and Statistics 5
Table 1 Absolute Relative Bias (ARB) of different estimators of 119884
Estimators Population I Population II119910 00000 00000119910dc119877
00369 00042119910dc119875
00564 00513119910dcRe 00068 00079
119910dcPe 00165 00198
119910119889
RP 00222 00008119910dcRPe 00058 00243
Table 2 Percentage Relative Efficiencies (PREs) of different estima-tors of 119884 with respect to 119910
Estimators Population I Population II119910 100 100119910dc119877
13691 73081119910dc119875
lowast lowast
119910dcRe 18436 25955
119910dcPe lowast lowast
119910119889
RP 13395 15696119910dcRPe 18927 76330lowastData is not applicable
6 Conclusion
It is observed from Table 1 that
(i) for population I
ARB (119910) lt ARB (119910dcRPe) lt ARB (119910
dcRe) lt ARB (119910
dcPe)
lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dc119875)
(29)
(ii) for population II
ARB (119910) lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dcRe)
lt ARB (119910dcPe) lt ARB (119910
dcRPe) lt ARB (119910
dc119875)
(30)
From Table 2 we see that the Percentage Relative Effi-ciency (PRE) of the proposed estimator 119910dc
RPe for populationsI and II is more as compared to all other existing estimatorsthat is usual unbiased estimator 119910 chain type ratio estimator119910dc119877 chain type product estimator 119910dc
119875 exponential chain type
ratio estimator 119910dcRe exponential chain type product estimator
119910dcPe and ratio-product type estimator 119910119889RPFinally from Tables 1 and 2 we conclude that the
proposed estimator 119910dcRPe (based on two auxiliary variables 119883
and119885) is amore appropriate estimator in comparison to otherexisting estimators as it has appreciable efficiency as well aslower relative bias
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the editor Professor Zhidong Baiand the learned referee for their comments leading to theimprovement of the paper
References
[1] W G Cochran ldquoThe estimation of the yields of the cerealexperiments by sampling for the ratio of grain to total producerdquoThe Journal of Agricultural Science vol 30 pp 262ndash275 1940
[2] D S Robson ldquoApplications of multivariate polykays to the the-ory of unbiased ratio-type estimationrdquo Journal of the AmericanStatistical Association vol 52 pp 511ndash522 1957
[3] M N Murthy ldquoProduct method of estimationrdquo The IndianJournal of Statistics A vol 26 pp 69ndash74 1964
[4] J Neyman ldquoContribution to the theory of sampling humanpopulationsrdquo Journal of American Statistical Association vol 33pp 101ndash116 1938
[5] B V Sukhatme ldquoSome ratio-type estimators in two-phasesamplingrdquo Journal of the American Statistical Association vol57 pp 628ndash632 1962
[6] M A Hidiroglou and C E Sarndal ldquoUse of auxiliary informa-tion for two phase samplingrdquo Survey Methodology vol 24 pp11ndash20 1998
[7] W A Fuller ldquoTwo-phase samplingrdquo in Proceedings of theAnnual Meeting of the Survey Methods Section of the StatisticalSociety of Canada pp 23ndash30 2000
[8] M A Hidiroglou ldquoDouble samplingrdquo Survey Methodology vol27 pp 143ndash154 2000
[9] H P Singh andGKVishwakarma ldquoModified exponential ratioand product estimators for finite population mean in doublesamplingrdquo Austrian Journal of Statistics vol 36 no 3 pp 217ndash225 2007
[10] L N Sahoo G Mishra and S R Nayak ldquoOn two differ-ent classes of estimators in two-phase sampling using multi-auxiliary variablesrdquo Model Assisted Statistics and Applicationsvol 5 no 1 pp 61ndash68 2010
[11] L Chand Some ratio type estimators based on two or moreauxiliary variables [PhD dissertation] Iowa State UniversityAmes Iowa USA 1975
[12] B Kiregyera ldquoA chain ratio-type estimator in finite populationdouble sampling using two auxiliary variablesrdquoMetrika vol 27no 4 pp 217ndash223 1980
[13] G N Singh and L N Upadhyaya ldquoA class of modified chain-type estimators using two auxiliary variables in two phasesamplingrdquoMetron vol 53 no 3-4 pp 117ndash125 1995
[14] B Prasad R S Singh and H P Singh ldquoSome chain ratio-typeestimators for ratio of two populationmeans using two auxiliarycharacters in two phase samplingrdquo Metron vol 54 no 1-2 pp95ndash113 1996
[15] S Singh H P Singh and L N Upadhyaya ldquoChain ratio andregression type estimators for median estimation in surveysamplingrdquo Statistical Papers vol 48 no 1 pp 23ndash46 2007
[16] B K Singh and S Choudhury ldquoExponential chain ratio andproduct type estimators for finite population mean underdouble sampling schemerdquo Global Journal of Science FrontierResearch vol 12 no 6 2012
[17] G K Vishwakarma and R K Gangele ldquoA class of chain ratio-type exponential estimators in double sampling using two
6 Journal of Probability and Statistics
auxiliary variatesrdquo Applied Mathematics and Computation vol227 pp 171ndash175 2014
[18] H P Singh andMRuiz Espejo ldquoDouble sampling ratio-productestimator of a finite population mean in sample surveysrdquoJournal of Applied Statistics vol 34 no 1-2 pp 71ndash85 2007
[19] S K Srivastava ldquoA generalized estimator for the mean of afinite population using multi-auxiliary informationrdquo Journal ofAmerican Statistical Association vol 66 pp 404ndash407 1971
[20] W G Cochran Sampling Techniques John Wiley amp Sons NewYork NY USA 1977
[21] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967
The usual chain type ratio and product estimators of 119884under double sampling scheme using two auxiliary variables119883 and 119885 are given respectively by
119910dc119877
= 1199101199091015840
119909
119885
1199111015840
119910dc119875
= 119910119909
1199091015840
1199111015840
119885
(1)
Singh and Choudhury [16] suggested the following expo-nential chain type ratio and product estimators of 119884 underdouble sampling scheme using two auxiliary variables119883 and119885
119910dcRe = 119910 exp
(11990910158401199111015840)119885 minus 119909
(11990910158401199111015840)119885 + 119909
119910dcPe = 119910 exp
119909 minus (11990910158401199111015840)119885
119909 + (11990910158401199111015840)119885
(2)
where1199091015840 and 1199111015840 are the samplemeans of119883 and119885 respectivelybased on the first-phase sample of size 119899
1015840 drawn from thepopulation of size 119873 with the help of Simple RandomSampling Without Replacement (SRSWOR) scheme Also 119910and 119909 are the samplemeans of119884 and119883 respectively based onthe second-phase sample of size 119899 drawn from the first-phasesample of size 1198991015840 with the help of SRSWOR scheme
2 Proposed Estimator
It has been theoretically established that in general thelinear regression estimator is more efficient than the ratio(product) estimator except when the regression line of 119884 on119883 passes through the neighborhood of the origin in whichthe efficiencies of these estimators are almost equal Howeverowing to stronger intuitive appeal survey statisticians favourthe use of ratio and product estimators Further we note thatin many practical situations the regression line does not passthrough the neighborhood of the origin In these situationsthe ratio estimator does not perform well as the linearregression estimator Considering this fact Singh and RuizEspejo [18] made an attempt to improve the performance ofthese estimators and suggested the following ratio-producttype estimator for populationmean119884 under double samplingscheme using single auxiliary variable119883
119910119889
RP = 119910[1205721199091015840
119909+ (1 minus 120572)
119909
1199091015840] (3)
where 120572 is a real constantWe propose the following exponential chain ratio-
product type estimator for population mean 119884 under doublesampling scheme using two auxiliary variables119883 and 119885
119910dcRPe = 119910[120572 exp
(11990910158401199111015840)119885 minus 119909
(11990910158401199111015840)119885 + 119909
+ (1 minus 120572) exp119909 minus (119909
10158401199111015840)119885
119909 + (11990910158401199111015840)119885
]
(4)
where 120572 is a real constant to be determined such that theMean Square Error (MSE) of the proposed estimator 119910dc
RPe isminimum For 120572 = 1 119910dc
RPe rarr 119910dcRe whereas for 120572 = 0
119910dcRPe rarr 119910
dcPe
Remark It is noted that the proposed estimator in (4) is aspecial case of the class of estimators 119910class = 119910119867(119909 119911
1015840)
proposed by Srivastava [19] where 119867(sdot) is a parametricfunction such that 119867(119909
1015840| 1199041 119885) = 1 and satisfies certain
regularity conditions defined in Srivastava [19]
3 Bias and MSE of the Proposed Estimator
To obtain the Bias and Mean Square Error (MSE) of theproposed estimator 119910dc
(ii) the Percentage Relative Efficiencies (PREs) of differ-ent suggested estimators of 119884 with respect to 119910 usingthe formula
PRE (sdot 119910) =119881 (119910)
MSE (sdot)times 100 (28)
Journal of Probability and Statistics 5
Table 1 Absolute Relative Bias (ARB) of different estimators of 119884
Estimators Population I Population II119910 00000 00000119910dc119877
00369 00042119910dc119875
00564 00513119910dcRe 00068 00079
119910dcPe 00165 00198
119910119889
RP 00222 00008119910dcRPe 00058 00243
Table 2 Percentage Relative Efficiencies (PREs) of different estima-tors of 119884 with respect to 119910
Estimators Population I Population II119910 100 100119910dc119877
13691 73081119910dc119875
lowast lowast
119910dcRe 18436 25955
119910dcPe lowast lowast
119910119889
RP 13395 15696119910dcRPe 18927 76330lowastData is not applicable
6 Conclusion
It is observed from Table 1 that
(i) for population I
ARB (119910) lt ARB (119910dcRPe) lt ARB (119910
dcRe) lt ARB (119910
dcPe)
lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dc119875)
(29)
(ii) for population II
ARB (119910) lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dcRe)
lt ARB (119910dcPe) lt ARB (119910
dcRPe) lt ARB (119910
dc119875)
(30)
From Table 2 we see that the Percentage Relative Effi-ciency (PRE) of the proposed estimator 119910dc
RPe for populationsI and II is more as compared to all other existing estimatorsthat is usual unbiased estimator 119910 chain type ratio estimator119910dc119877 chain type product estimator 119910dc
119875 exponential chain type
ratio estimator 119910dcRe exponential chain type product estimator
119910dcPe and ratio-product type estimator 119910119889RPFinally from Tables 1 and 2 we conclude that the
proposed estimator 119910dcRPe (based on two auxiliary variables 119883
and119885) is amore appropriate estimator in comparison to otherexisting estimators as it has appreciable efficiency as well aslower relative bias
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the editor Professor Zhidong Baiand the learned referee for their comments leading to theimprovement of the paper
References
[1] W G Cochran ldquoThe estimation of the yields of the cerealexperiments by sampling for the ratio of grain to total producerdquoThe Journal of Agricultural Science vol 30 pp 262ndash275 1940
[2] D S Robson ldquoApplications of multivariate polykays to the the-ory of unbiased ratio-type estimationrdquo Journal of the AmericanStatistical Association vol 52 pp 511ndash522 1957
[3] M N Murthy ldquoProduct method of estimationrdquo The IndianJournal of Statistics A vol 26 pp 69ndash74 1964
[4] J Neyman ldquoContribution to the theory of sampling humanpopulationsrdquo Journal of American Statistical Association vol 33pp 101ndash116 1938
[5] B V Sukhatme ldquoSome ratio-type estimators in two-phasesamplingrdquo Journal of the American Statistical Association vol57 pp 628ndash632 1962
[6] M A Hidiroglou and C E Sarndal ldquoUse of auxiliary informa-tion for two phase samplingrdquo Survey Methodology vol 24 pp11ndash20 1998
[7] W A Fuller ldquoTwo-phase samplingrdquo in Proceedings of theAnnual Meeting of the Survey Methods Section of the StatisticalSociety of Canada pp 23ndash30 2000
[8] M A Hidiroglou ldquoDouble samplingrdquo Survey Methodology vol27 pp 143ndash154 2000
[9] H P Singh andGKVishwakarma ldquoModified exponential ratioand product estimators for finite population mean in doublesamplingrdquo Austrian Journal of Statistics vol 36 no 3 pp 217ndash225 2007
[10] L N Sahoo G Mishra and S R Nayak ldquoOn two differ-ent classes of estimators in two-phase sampling using multi-auxiliary variablesrdquo Model Assisted Statistics and Applicationsvol 5 no 1 pp 61ndash68 2010
[11] L Chand Some ratio type estimators based on two or moreauxiliary variables [PhD dissertation] Iowa State UniversityAmes Iowa USA 1975
[12] B Kiregyera ldquoA chain ratio-type estimator in finite populationdouble sampling using two auxiliary variablesrdquoMetrika vol 27no 4 pp 217ndash223 1980
[13] G N Singh and L N Upadhyaya ldquoA class of modified chain-type estimators using two auxiliary variables in two phasesamplingrdquoMetron vol 53 no 3-4 pp 117ndash125 1995
[14] B Prasad R S Singh and H P Singh ldquoSome chain ratio-typeestimators for ratio of two populationmeans using two auxiliarycharacters in two phase samplingrdquo Metron vol 54 no 1-2 pp95ndash113 1996
[15] S Singh H P Singh and L N Upadhyaya ldquoChain ratio andregression type estimators for median estimation in surveysamplingrdquo Statistical Papers vol 48 no 1 pp 23ndash46 2007
[16] B K Singh and S Choudhury ldquoExponential chain ratio andproduct type estimators for finite population mean underdouble sampling schemerdquo Global Journal of Science FrontierResearch vol 12 no 6 2012
[17] G K Vishwakarma and R K Gangele ldquoA class of chain ratio-type exponential estimators in double sampling using two
6 Journal of Probability and Statistics
auxiliary variatesrdquo Applied Mathematics and Computation vol227 pp 171ndash175 2014
[18] H P Singh andMRuiz Espejo ldquoDouble sampling ratio-productestimator of a finite population mean in sample surveysrdquoJournal of Applied Statistics vol 34 no 1-2 pp 71ndash85 2007
[19] S K Srivastava ldquoA generalized estimator for the mean of afinite population using multi-auxiliary informationrdquo Journal ofAmerican Statistical Association vol 66 pp 404ndash407 1971
[20] W G Cochran Sampling Techniques John Wiley amp Sons NewYork NY USA 1977
[21] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967
(ii) the Percentage Relative Efficiencies (PREs) of differ-ent suggested estimators of 119884 with respect to 119910 usingthe formula
PRE (sdot 119910) =119881 (119910)
MSE (sdot)times 100 (28)
Journal of Probability and Statistics 5
Table 1 Absolute Relative Bias (ARB) of different estimators of 119884
Estimators Population I Population II119910 00000 00000119910dc119877
00369 00042119910dc119875
00564 00513119910dcRe 00068 00079
119910dcPe 00165 00198
119910119889
RP 00222 00008119910dcRPe 00058 00243
Table 2 Percentage Relative Efficiencies (PREs) of different estima-tors of 119884 with respect to 119910
Estimators Population I Population II119910 100 100119910dc119877
13691 73081119910dc119875
lowast lowast
119910dcRe 18436 25955
119910dcPe lowast lowast
119910119889
RP 13395 15696119910dcRPe 18927 76330lowastData is not applicable
6 Conclusion
It is observed from Table 1 that
(i) for population I
ARB (119910) lt ARB (119910dcRPe) lt ARB (119910
dcRe) lt ARB (119910
dcPe)
lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dc119875)
(29)
(ii) for population II
ARB (119910) lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dcRe)
lt ARB (119910dcPe) lt ARB (119910
dcRPe) lt ARB (119910
dc119875)
(30)
From Table 2 we see that the Percentage Relative Effi-ciency (PRE) of the proposed estimator 119910dc
RPe for populationsI and II is more as compared to all other existing estimatorsthat is usual unbiased estimator 119910 chain type ratio estimator119910dc119877 chain type product estimator 119910dc
119875 exponential chain type
ratio estimator 119910dcRe exponential chain type product estimator
119910dcPe and ratio-product type estimator 119910119889RPFinally from Tables 1 and 2 we conclude that the
proposed estimator 119910dcRPe (based on two auxiliary variables 119883
and119885) is amore appropriate estimator in comparison to otherexisting estimators as it has appreciable efficiency as well aslower relative bias
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the editor Professor Zhidong Baiand the learned referee for their comments leading to theimprovement of the paper
References
[1] W G Cochran ldquoThe estimation of the yields of the cerealexperiments by sampling for the ratio of grain to total producerdquoThe Journal of Agricultural Science vol 30 pp 262ndash275 1940
[2] D S Robson ldquoApplications of multivariate polykays to the the-ory of unbiased ratio-type estimationrdquo Journal of the AmericanStatistical Association vol 52 pp 511ndash522 1957
[3] M N Murthy ldquoProduct method of estimationrdquo The IndianJournal of Statistics A vol 26 pp 69ndash74 1964
[4] J Neyman ldquoContribution to the theory of sampling humanpopulationsrdquo Journal of American Statistical Association vol 33pp 101ndash116 1938
[5] B V Sukhatme ldquoSome ratio-type estimators in two-phasesamplingrdquo Journal of the American Statistical Association vol57 pp 628ndash632 1962
[6] M A Hidiroglou and C E Sarndal ldquoUse of auxiliary informa-tion for two phase samplingrdquo Survey Methodology vol 24 pp11ndash20 1998
[7] W A Fuller ldquoTwo-phase samplingrdquo in Proceedings of theAnnual Meeting of the Survey Methods Section of the StatisticalSociety of Canada pp 23ndash30 2000
[8] M A Hidiroglou ldquoDouble samplingrdquo Survey Methodology vol27 pp 143ndash154 2000
[9] H P Singh andGKVishwakarma ldquoModified exponential ratioand product estimators for finite population mean in doublesamplingrdquo Austrian Journal of Statistics vol 36 no 3 pp 217ndash225 2007
[10] L N Sahoo G Mishra and S R Nayak ldquoOn two differ-ent classes of estimators in two-phase sampling using multi-auxiliary variablesrdquo Model Assisted Statistics and Applicationsvol 5 no 1 pp 61ndash68 2010
[11] L Chand Some ratio type estimators based on two or moreauxiliary variables [PhD dissertation] Iowa State UniversityAmes Iowa USA 1975
[12] B Kiregyera ldquoA chain ratio-type estimator in finite populationdouble sampling using two auxiliary variablesrdquoMetrika vol 27no 4 pp 217ndash223 1980
[13] G N Singh and L N Upadhyaya ldquoA class of modified chain-type estimators using two auxiliary variables in two phasesamplingrdquoMetron vol 53 no 3-4 pp 117ndash125 1995
[14] B Prasad R S Singh and H P Singh ldquoSome chain ratio-typeestimators for ratio of two populationmeans using two auxiliarycharacters in two phase samplingrdquo Metron vol 54 no 1-2 pp95ndash113 1996
[15] S Singh H P Singh and L N Upadhyaya ldquoChain ratio andregression type estimators for median estimation in surveysamplingrdquo Statistical Papers vol 48 no 1 pp 23ndash46 2007
[16] B K Singh and S Choudhury ldquoExponential chain ratio andproduct type estimators for finite population mean underdouble sampling schemerdquo Global Journal of Science FrontierResearch vol 12 no 6 2012
[17] G K Vishwakarma and R K Gangele ldquoA class of chain ratio-type exponential estimators in double sampling using two
6 Journal of Probability and Statistics
auxiliary variatesrdquo Applied Mathematics and Computation vol227 pp 171ndash175 2014
[18] H P Singh andMRuiz Espejo ldquoDouble sampling ratio-productestimator of a finite population mean in sample surveysrdquoJournal of Applied Statistics vol 34 no 1-2 pp 71ndash85 2007
[19] S K Srivastava ldquoA generalized estimator for the mean of afinite population using multi-auxiliary informationrdquo Journal ofAmerican Statistical Association vol 66 pp 404ndash407 1971
[20] W G Cochran Sampling Techniques John Wiley amp Sons NewYork NY USA 1977
[21] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967
(ii) the Percentage Relative Efficiencies (PREs) of differ-ent suggested estimators of 119884 with respect to 119910 usingthe formula
PRE (sdot 119910) =119881 (119910)
MSE (sdot)times 100 (28)
Journal of Probability and Statistics 5
Table 1 Absolute Relative Bias (ARB) of different estimators of 119884
Estimators Population I Population II119910 00000 00000119910dc119877
00369 00042119910dc119875
00564 00513119910dcRe 00068 00079
119910dcPe 00165 00198
119910119889
RP 00222 00008119910dcRPe 00058 00243
Table 2 Percentage Relative Efficiencies (PREs) of different estima-tors of 119884 with respect to 119910
Estimators Population I Population II119910 100 100119910dc119877
13691 73081119910dc119875
lowast lowast
119910dcRe 18436 25955
119910dcPe lowast lowast
119910119889
RP 13395 15696119910dcRPe 18927 76330lowastData is not applicable
6 Conclusion
It is observed from Table 1 that
(i) for population I
ARB (119910) lt ARB (119910dcRPe) lt ARB (119910
dcRe) lt ARB (119910
dcPe)
lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dc119875)
(29)
(ii) for population II
ARB (119910) lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dcRe)
lt ARB (119910dcPe) lt ARB (119910
dcRPe) lt ARB (119910
dc119875)
(30)
From Table 2 we see that the Percentage Relative Effi-ciency (PRE) of the proposed estimator 119910dc
RPe for populationsI and II is more as compared to all other existing estimatorsthat is usual unbiased estimator 119910 chain type ratio estimator119910dc119877 chain type product estimator 119910dc
119875 exponential chain type
ratio estimator 119910dcRe exponential chain type product estimator
119910dcPe and ratio-product type estimator 119910119889RPFinally from Tables 1 and 2 we conclude that the
proposed estimator 119910dcRPe (based on two auxiliary variables 119883
and119885) is amore appropriate estimator in comparison to otherexisting estimators as it has appreciable efficiency as well aslower relative bias
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the editor Professor Zhidong Baiand the learned referee for their comments leading to theimprovement of the paper
References
[1] W G Cochran ldquoThe estimation of the yields of the cerealexperiments by sampling for the ratio of grain to total producerdquoThe Journal of Agricultural Science vol 30 pp 262ndash275 1940
[2] D S Robson ldquoApplications of multivariate polykays to the the-ory of unbiased ratio-type estimationrdquo Journal of the AmericanStatistical Association vol 52 pp 511ndash522 1957
[3] M N Murthy ldquoProduct method of estimationrdquo The IndianJournal of Statistics A vol 26 pp 69ndash74 1964
[4] J Neyman ldquoContribution to the theory of sampling humanpopulationsrdquo Journal of American Statistical Association vol 33pp 101ndash116 1938
[5] B V Sukhatme ldquoSome ratio-type estimators in two-phasesamplingrdquo Journal of the American Statistical Association vol57 pp 628ndash632 1962
[6] M A Hidiroglou and C E Sarndal ldquoUse of auxiliary informa-tion for two phase samplingrdquo Survey Methodology vol 24 pp11ndash20 1998
[7] W A Fuller ldquoTwo-phase samplingrdquo in Proceedings of theAnnual Meeting of the Survey Methods Section of the StatisticalSociety of Canada pp 23ndash30 2000
[8] M A Hidiroglou ldquoDouble samplingrdquo Survey Methodology vol27 pp 143ndash154 2000
[9] H P Singh andGKVishwakarma ldquoModified exponential ratioand product estimators for finite population mean in doublesamplingrdquo Austrian Journal of Statistics vol 36 no 3 pp 217ndash225 2007
[10] L N Sahoo G Mishra and S R Nayak ldquoOn two differ-ent classes of estimators in two-phase sampling using multi-auxiliary variablesrdquo Model Assisted Statistics and Applicationsvol 5 no 1 pp 61ndash68 2010
[11] L Chand Some ratio type estimators based on two or moreauxiliary variables [PhD dissertation] Iowa State UniversityAmes Iowa USA 1975
[12] B Kiregyera ldquoA chain ratio-type estimator in finite populationdouble sampling using two auxiliary variablesrdquoMetrika vol 27no 4 pp 217ndash223 1980
[13] G N Singh and L N Upadhyaya ldquoA class of modified chain-type estimators using two auxiliary variables in two phasesamplingrdquoMetron vol 53 no 3-4 pp 117ndash125 1995
[14] B Prasad R S Singh and H P Singh ldquoSome chain ratio-typeestimators for ratio of two populationmeans using two auxiliarycharacters in two phase samplingrdquo Metron vol 54 no 1-2 pp95ndash113 1996
[15] S Singh H P Singh and L N Upadhyaya ldquoChain ratio andregression type estimators for median estimation in surveysamplingrdquo Statistical Papers vol 48 no 1 pp 23ndash46 2007
[16] B K Singh and S Choudhury ldquoExponential chain ratio andproduct type estimators for finite population mean underdouble sampling schemerdquo Global Journal of Science FrontierResearch vol 12 no 6 2012
[17] G K Vishwakarma and R K Gangele ldquoA class of chain ratio-type exponential estimators in double sampling using two
6 Journal of Probability and Statistics
auxiliary variatesrdquo Applied Mathematics and Computation vol227 pp 171ndash175 2014
[18] H P Singh andMRuiz Espejo ldquoDouble sampling ratio-productestimator of a finite population mean in sample surveysrdquoJournal of Applied Statistics vol 34 no 1-2 pp 71ndash85 2007
[19] S K Srivastava ldquoA generalized estimator for the mean of afinite population using multi-auxiliary informationrdquo Journal ofAmerican Statistical Association vol 66 pp 404ndash407 1971
[20] W G Cochran Sampling Techniques John Wiley amp Sons NewYork NY USA 1977
[21] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967
Table 1 Absolute Relative Bias (ARB) of different estimators of 119884
Estimators Population I Population II119910 00000 00000119910dc119877
00369 00042119910dc119875
00564 00513119910dcRe 00068 00079
119910dcPe 00165 00198
119910119889
RP 00222 00008119910dcRPe 00058 00243
Table 2 Percentage Relative Efficiencies (PREs) of different estima-tors of 119884 with respect to 119910
Estimators Population I Population II119910 100 100119910dc119877
13691 73081119910dc119875
lowast lowast
119910dcRe 18436 25955
119910dcPe lowast lowast
119910119889
RP 13395 15696119910dcRPe 18927 76330lowastData is not applicable
6 Conclusion
It is observed from Table 1 that
(i) for population I
ARB (119910) lt ARB (119910dcRPe) lt ARB (119910
dcRe) lt ARB (119910
dcPe)
lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dc119875)
(29)
(ii) for population II
ARB (119910) lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dcRe)
lt ARB (119910dcPe) lt ARB (119910
dcRPe) lt ARB (119910
dc119875)
(30)
From Table 2 we see that the Percentage Relative Effi-ciency (PRE) of the proposed estimator 119910dc
RPe for populationsI and II is more as compared to all other existing estimatorsthat is usual unbiased estimator 119910 chain type ratio estimator119910dc119877 chain type product estimator 119910dc
119875 exponential chain type
ratio estimator 119910dcRe exponential chain type product estimator
119910dcPe and ratio-product type estimator 119910119889RPFinally from Tables 1 and 2 we conclude that the
proposed estimator 119910dcRPe (based on two auxiliary variables 119883
and119885) is amore appropriate estimator in comparison to otherexisting estimators as it has appreciable efficiency as well aslower relative bias
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the editor Professor Zhidong Baiand the learned referee for their comments leading to theimprovement of the paper
References
[1] W G Cochran ldquoThe estimation of the yields of the cerealexperiments by sampling for the ratio of grain to total producerdquoThe Journal of Agricultural Science vol 30 pp 262ndash275 1940
[2] D S Robson ldquoApplications of multivariate polykays to the the-ory of unbiased ratio-type estimationrdquo Journal of the AmericanStatistical Association vol 52 pp 511ndash522 1957
[3] M N Murthy ldquoProduct method of estimationrdquo The IndianJournal of Statistics A vol 26 pp 69ndash74 1964
[4] J Neyman ldquoContribution to the theory of sampling humanpopulationsrdquo Journal of American Statistical Association vol 33pp 101ndash116 1938
[5] B V Sukhatme ldquoSome ratio-type estimators in two-phasesamplingrdquo Journal of the American Statistical Association vol57 pp 628ndash632 1962
[6] M A Hidiroglou and C E Sarndal ldquoUse of auxiliary informa-tion for two phase samplingrdquo Survey Methodology vol 24 pp11ndash20 1998
[7] W A Fuller ldquoTwo-phase samplingrdquo in Proceedings of theAnnual Meeting of the Survey Methods Section of the StatisticalSociety of Canada pp 23ndash30 2000
[8] M A Hidiroglou ldquoDouble samplingrdquo Survey Methodology vol27 pp 143ndash154 2000
[9] H P Singh andGKVishwakarma ldquoModified exponential ratioand product estimators for finite population mean in doublesamplingrdquo Austrian Journal of Statistics vol 36 no 3 pp 217ndash225 2007
[10] L N Sahoo G Mishra and S R Nayak ldquoOn two differ-ent classes of estimators in two-phase sampling using multi-auxiliary variablesrdquo Model Assisted Statistics and Applicationsvol 5 no 1 pp 61ndash68 2010
[11] L Chand Some ratio type estimators based on two or moreauxiliary variables [PhD dissertation] Iowa State UniversityAmes Iowa USA 1975
[12] B Kiregyera ldquoA chain ratio-type estimator in finite populationdouble sampling using two auxiliary variablesrdquoMetrika vol 27no 4 pp 217ndash223 1980
[13] G N Singh and L N Upadhyaya ldquoA class of modified chain-type estimators using two auxiliary variables in two phasesamplingrdquoMetron vol 53 no 3-4 pp 117ndash125 1995
[14] B Prasad R S Singh and H P Singh ldquoSome chain ratio-typeestimators for ratio of two populationmeans using two auxiliarycharacters in two phase samplingrdquo Metron vol 54 no 1-2 pp95ndash113 1996
[15] S Singh H P Singh and L N Upadhyaya ldquoChain ratio andregression type estimators for median estimation in surveysamplingrdquo Statistical Papers vol 48 no 1 pp 23ndash46 2007
[16] B K Singh and S Choudhury ldquoExponential chain ratio andproduct type estimators for finite population mean underdouble sampling schemerdquo Global Journal of Science FrontierResearch vol 12 no 6 2012
[17] G K Vishwakarma and R K Gangele ldquoA class of chain ratio-type exponential estimators in double sampling using two
6 Journal of Probability and Statistics
auxiliary variatesrdquo Applied Mathematics and Computation vol227 pp 171ndash175 2014
[18] H P Singh andMRuiz Espejo ldquoDouble sampling ratio-productestimator of a finite population mean in sample surveysrdquoJournal of Applied Statistics vol 34 no 1-2 pp 71ndash85 2007
[19] S K Srivastava ldquoA generalized estimator for the mean of afinite population using multi-auxiliary informationrdquo Journal ofAmerican Statistical Association vol 66 pp 404ndash407 1971
[20] W G Cochran Sampling Techniques John Wiley amp Sons NewYork NY USA 1977
[21] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967
auxiliary variatesrdquo Applied Mathematics and Computation vol227 pp 171ndash175 2014
[18] H P Singh andMRuiz Espejo ldquoDouble sampling ratio-productestimator of a finite population mean in sample surveysrdquoJournal of Applied Statistics vol 34 no 1-2 pp 71ndash85 2007
[19] S K Srivastava ldquoA generalized estimator for the mean of afinite population using multi-auxiliary informationrdquo Journal ofAmerican Statistical Association vol 66 pp 404ndash407 1971
[20] W G Cochran Sampling Techniques John Wiley amp Sons NewYork NY USA 1977
[21] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967
