ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Review Article
Volume 4 Issue 6 - 2016
Modified Ratio cum Product Estimator for Estimation of Finite Population Mean with Known Correlation Coefficient
Jambulingam Subramani* and Master Ajith S
Department of Statistics, Ramanujan School of Mathematical Sciences, Pondicherry University, India
Received: September 26, 2016 | Published: November 15, 2016
*Corresponding author: Jambulingam Subramani, Department of Statistics, Ramanujan School of Mathematical Sciences, Pondicherry University, RV Nagar, Kalapet, Pondicherry, India, Email:
Citation: Subramani J, Ajith MS (2016) Modified Ratio cum Product Estimator for Estimation of Finite Population Mean with Known Correlation Coefficient. Biom Biostat Int J 4(6): 00113. DOI: 10.15406/bbij.2016.04.00113

Abstract

In this paper, a modified ratio cum product estimator for the estimation of finite population mean of the study variable using the known correlation coefficient of the auxiliary variable is introduced. The bias and mean squared error of the proposed estimator are also obtained. The relative performance of the proposed estimator along with some existing estimators is accessed for certain labeled and natural populations. The results show that the proposed estimator is to be more efficient than the existing estimators.

Keywords: Bias; Mean squared error; Natural population; Simple random sampling; Linear regression estimator

Introduction

In sampling theory, a wide variety of techniques is used to obtain efficient estimators for the population mean. The commonly used method to obtain the estimator for population mean is simple random sampling without replacement (SRSWOR) when there is no auxiliary variable available. There are methods that use the auxiliary information of the study characteristics. If there exists an auxiliary variable X which is correlated with the study variable Y, then a number of estimators such as ratio, product, modified ratio, modified product, regression estimators and their modifications are widely available for estimation of population mean of the study variable Y.

Consider a finite population U={ U 1 , U 2 , U 3 .... U N } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyvai abg2da9maacmaabaGaamyvamaaBaaajuaibaGaaGymaaqcfayabaGa aiilaiaadwfadaWgaaqcfasaaiaaikdaaKqbagqaaiaacYcacaWGvb WaaSbaaKqbGeaacaaIZaaajuaGbeaacaGGUaGaaiOlaiaac6cacaGG UaGaamyvamaaBaaajuaibaGaamOtaaqcfayabaaacaGL7bGaayzFaa aaaa@48A0@ of N distinct and identifiable units. Let Y be the study variable which takes the values Y={ Y 1 , Y 2 , Y 3 ,... Y N } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywai abg2da9maacmaabaGaamywamaaBaaajuaibaGaaGymaaqcfayabaGa aiilaiaadMfadaWgaaqcfasaaiaaikdaaKqbagqaaiaacYcacaWGzb WaaSbaaKqbGeaacaaIZaaajuaGbeaacaGGSaGaaiOlaiaac6cacaGG UaGaamywamaaBaaajuaibaGaamOtaaqcfayabaaacaGL7bGaayzFaa aaaa@48B2@ . Here the problem is to estimate the population mean Y ¯ = 1 N i=1 N Y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaraGaeyypa0ZaaSaaaeaacaaIXaaabaGaamOtaaaadaaeWaqaaiaa dMfadaWgaaqcfasaaiaadMgaaKqbagqaaaqcfasaaiaadMgacqGH9a qpcaaIXaaabaGaamOtaaqcfaOaeyyeIuoaaaa@42EF@ on the basis of a random sample selected from the population U.

Before discussing further the various estimators, the notations to be used in this article are listed here.

N                             -               Population size
n                              -               Sample size         
f= n N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzai abg2da9maalaaabaGaamOBaaqaaiaad6eaaaaaaa@3A4B@                    -               Sampling fraction
Y                              -               Study variable
X                              -               Auxiliary variable
X ¯ , Y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiway aaraGaaiilaiqadMfagaqeaaaa@3920@                        -               Population means
x ¯ , y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiEay aaraGaaiilaiqadMhagaqeaaaa@395F@                         -               Sample means
S x , S y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uam aaBaaajuaibaGaamiEaaqcfayabaGaaiilaiaadofadaWgaaqcfasa aiaadMhaaKqbagqaaaaa@3C99@      -              Population standard deviations
S x , S y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uam aaBaaajuaibaGaamiEaaqcfayabaGaaiilaiaadofadaWgaaqcfasa aiaadMhaaKqbagqaaaaa@3C99@                     -               Sample standard deviations
C x , C y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qam aaBaaajuaibaGaamiEaaqcfayabaGaaiilaiaadoeadaWgaaqcfasa aiaadMhaaKqbagqaaaaa@3C79@                    -               Coefficient of variations
ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyWdi haaa@3844@                            -               Correlation coefficient between x and y
β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIXaaajuaGbeaaaaa@39BD@                            -               Coefficient of skewness
β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39BE@                           -               Coefficient of kurtosis
B(.)                         -               Bias of estimators
MSE(.)                   -               Mean squared error of estimators

In simple random sampling without replacement, the estimator y ¯ srs MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyEay aaraWaaSbaaKqbGeaacaWGZbGaamOCaiaadohaaKqbagqaaaaa@3B5E@ is an unbiased estimator for the population mean Y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaraaaaa@377B@ and its variance is given by                           

V( y ¯ srs )=δ S y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvam aabmaabaGabmyEayaaraWaaSbaaKqbGeaacaWGZbGaamOCaiaadoha aKqbagqaaaGaayjkaiaawMcaaiabg2da9iabes7aKjaadofadaqhaa qcfasaaiaadMhaaeaacaaIYaaaaaaa@434F@                                                                                                                   (1)        

Where δ=( 1f n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq Maeyypa0ZaaeWaaeaadaWcaaqaaiaaigdacqGHsislcaWGMbaabaGa amOBaaaaaiaawIcacaGLPaaaaaa@3E4E@

Cochran [1], use auxiliary information for the estimation of population mean of the variable under study and proposed the ratio estimator of the population mean Y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaraaaaa@377B@ of the study variable,

Y ^ R = y ¯ x ¯ X ¯ = R ^ X ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aajaWaaSbaaKqbGeaacaWGsbaajuaGbeaacqGH9aqpdaWcaaqaaiqa dMhagaqeaaqaaiqadIhagaqeaaaaceWGybGbaebacqGH9aqpceWGsb GbaKaaceWGybGbaebaaaa@403E@

The bias and mean squared error of the ratio estimator are given by

B( Y ¯ R )=δ Y ¯ [ C x 2 ρ C x C y ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqam aabmaabaGabmywayaaryaataWaaSbaaKqbGeaacaWGsbaajuaGbeaa aiaawIcacaGLPaaacqGH9aqpcqaH0oazceWGzbGbaebadaWadaqaai aadoeadaWgaaqcfasaaKqbaoaaDaaajuaibaGaamiEaaqaaiaaikda aaaajuaGbeaacqGHsislcqaHbpGCcaWGdbWaaSbaaKqbGeaacaWG4b aajuaGbeaacaWGdbWaaSbaaKqbGeaacaWG5baajuaGbeaaaiaawUfa caGLDbaaaaa@4D58@

MSE( Y ¯ R )=δ Y ¯ 2 [ C y 2 + C x 2 2ρ C x C y ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai aadofacaWGfbWaaeWaaeaaceWGzbGbaeHbambadaWgaaqcfasaaiaa dkfaaKqbagqaaaGaayjkaiaawMcaaiabg2da9iabes7aKjqadMfaga qeamaaCaaabeqcfasaaiaaikdaaaqcfa4aamWaaeaacaWGdbWaa0ba aKqbGeaacaWG5baabaGaaGOmaaaajuaGcqGHRaWkcaWGdbWaa0baaK qbGeaacaWG4baabaGaaGOmaaaajuaGcqGHsislcaaIYaGaeqyWdiNa am4qamaaBaaajuaibaGaamiEaaqcfayabaGaam4qamaaBaaajuaiba GaamyEaaqcfayabaaacaGLBbGaayzxaaaaaa@54C0@                                     (2)

The linear regression estimator and its variance are given by

y ¯ lr = y ¯ +b( X ¯ x ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyEay aaraWaaSbaaKqbGeaacaWGSbGaamOCaaqcfayabaGaeyypa0JabmyE ayaaraGaey4kaSIaamOyamaabmaabaGabmiwayaaraGaeyOeI0Iabm iEayaaraaacaGLOaGaayzkaaaaaa@42C4@

V( y ¯ lr )=δ S y 2 ( 1 ρ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvam aabmaabaGabmyEayaaraWaaSbaaKqbGeaacaWGSbGaamOCaaqcfaya baaacaGLOaGaayzkaaGaeyypa0JaeqiTdqMaam4uamaaDaaajuaiba GaamyEaaqaaiaaikdaaaqcfa4aaeWaaeaacaaIXaGaeyOeI0IaeqyW di3aaWbaaeqajuaibaGaaGOmaaaaaKqbakaawIcacaGLPaaaaaa@4969@                                                              (3)

where b is the regression coefficient Y on X

Murthy [2] proposed the product estimator to estimate the population mean of the study variable when there is a negative correlation between the study variable Y and auxiliary variable X as

Y ¯ p = y ¯ X ¯ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaryaataWaaSbaaKqbGeaacaWGWbaajuaGbeaacqGH9aqpceWG5bGb aebadaqhaaqcfasaaiqadIfagaqeaaqaaiqadIhagaqeaaaaaaa@3DDB@

The bias and the mean squared error of the product estimator are given by

B( Y ¯ p )=δ Y ¯ [ ρ C x C y ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqam aabmaabaGabmywayaaraWaaSbaaKqbGeaacaWGWbaajuaGbeaaaiaa wIcacaGLPaaacqGH9aqpcqaH0oazceWGzbGbaebadaWadaqaaiabeg 8aYjaadoeadaWgaaqcfasaaiaadIhaaKqbagqaaiaadoeadaWgaaqc fasaaiaadMhaaKqbagqaaaGaay5waiaaw2faaaaa@4834@     

MSE( Y ¯ p )=δ Y ¯ 2 [ C y 2 + C x 2 +2ρ C x C y ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai aadofacaWGfbWaaeWaaeaaceWGzbGbaeHbambadaWgaaqaaiaadcha aeqaaaGaayjkaiaawMcaaiabg2da9iabes7aKjqadMfagaqeamaaCa aabeqaaiaaikdaaaWaamWaaeaacaWGdbWaa0baaeaacaWG5baabaGa aGOmaaaacqGHRaWkcaWGdbWaa0baaeaacaWG4baabaGaaGOmaaaacq GHRaWkcaaIYaGaeqyWdiNaam4qamaaBaaabaGaamiEaaqabaGaam4q amaaBaaabaGaamyEaaqabaaacaGLBbGaayzxaaaaaa@506B@                                                                                 (4)

Singh and Tailor [3] introduced the modified ratio estimator for the population mean with known population correlation coefficient ρ of the auxiliary variable and is given by

Y ¯ MR = y ¯ ( X ¯ +ρ x ¯ +ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaryaataWaaSbaaKqbGeaacaWGnbGaamOuaaqcfayabaGaeyypa0Ja bmyEayaaraWaaeWaaeaadaWcaaqaaiqadIfagaqeaiabgUcaRiabeg 8aYbqaaiqadIhagaqeaiabgUcaRiabeg8aYbaaaiaawIcacaGLPaaa aaa@451C@

The bias and mean squared error of this modified ratio estimator are given by

B( Y ¯ MR )=δ Y ¯ [ θ 2 C x 2 θρ C x C y ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqam aabmaabaGabmywayaaryaataWaaSbaaKqbGeaacaWGnbGaamOuaaqc fayabaaacaGLOaGaayzkaaGaeyypa0JaeqiTdqMabmywayaaraWaam WaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakaadoeadaqh aaqcfasaaiaadIhaaeaacaaIYaaaaKqbakabgkHiTiabeI7aXjabeg 8aYjaadoeadaWgaaqcfauaaiaadIhaaKqbagqaaiaadoeadaWgaaqc fasaaiaadMhaaKqbagqaaaGaay5waiaaw2faaaaa@5273@

MSE( Y ¯ MR )=δ Y ¯ 2 [ C y 2 + θ 2 C x 2 2θρ C x C y ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai aadofacaWGfbWaaeWaaeaaceWGzbGbaeHbambadaWgaaqcfasaaiaa d2eacaWGsbaajuaGbeaaaiaawIcacaGLPaaacqGH9aqpcqaH0oazce WGzbGbaebadaahaaqabKqbGeaacaaIYaaaaKqbaoaadmaabaGaam4q amaaDaaajuaibaGaamyEaaqaaiaaikdaaaqcfaOaey4kaSIaeqiUde 3aaWbaaeqajuaibaGaaGOmaaaajuaGcaWGdbWaa0baaKqbGeaacaWG 4baabaGaaGOmaaaajuaGcqGHsislcaaIYaGaeqiUdeNaeqyWdiNaam 4qamaaBaaajuaibaGaamiEaaqcfayabaGaam4qamaaBaaajuaibaGa amyEaaqcfayabaaacaGLBbGaayzxaaaaaa@5A98@     

where     θ= X X +ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXj abg2da9maalaaabaGabmiwayaapaaabaGabmiwayaapaGaey4kaSIa eqyWdihaaaaa@3DCD@

The modified product estimator with known correlation coefficient of the auxiliary variable when there is a negative correlation between the study variable Y and auxiliary variable X is given as

Y ^ Mp = y ( x +ρ X +ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadMfaga WdgaqcamaaBaaajuaibaGaamytaiaadchaaKqbagqaaiabg2da9iqa dMhagaWdamaabmaabaWaaSaaaeaaceWG4bGba8aacqGHRaWkcqaHbp GCaeaaceWGybGba8aacqGHRaWkcqaHbpGCaaaacaGLOaGaayzkaaaa aa@451D@   

The bias and mean squared error of the modified product estimator are given by

B ( Y ^ Mp )=δ Y [ θρ C x C y ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkeaqa aaaaaaaaWdbiaacckadaqadaqaaiqadMfagaWdgaqcamaaBaaajuai baGaamytaiaadchaaKqbagqaaaGaayjkaiaawMcaaiabg2da9iabes 7aKjqadMfagaWdamaadmaabaGaeqiUdeNaeqyWdiNaam4qamaaBaaa juaibaGaamiEaaqcfayabaGaam4qamaaBaaajuaibaGaamyEaaqcfa yabaaacaGLBbGaayzxaaaaaa@4C00@

MSE( Y ^ Mp )=δ Y ^ [ C y 2 + θ 2 C x 2 +2θρ C x C y ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamytaiaadofacaWGfbWaaeWaaeaaceWGzbGba8GbaKaadaWg aaqcfasaaiaad2eacaWGWbaajuaGbeaaaiaawIcacaGLPaaacqGH9a qpcqaH0oazceWGzbGba8GbaKaadaWadaqaaiaadoeadaqhaaqcfasa aiaadMhaaeaacaaIYaaaaKqbakabgUcaRiabeI7aXnaaCaaabeqcfa saaiaaikdaaaqcfaOaam4qamaaDaaajuaibaGaamiEaaqaaiaaikda aaqcfaOaey4kaSIaaGOmaiabeI7aXjabeg8aYjaadoeadaWgaaqcfa saaiaadIhaaKqbagqaaiaadoeadaWgaaqcfasaaiaadMhaaKqbagqa aaGaay5waiaaw2faaaaa@5927@                                                (6)

where θ= X X +ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXj abg2da9maalaaabaGabmiwayaapaaabaGabmiwayaapaGaey4kaSIa eqyWdihaaaaa@3DCD@

In literature, several estimators are available with auxiliary variables. However the problem is that the best estimator in terms of bias and efficiency are not fully addressed. In this paper, we attempt to solve such type of problems. The existing estimators are biased but the percentage relative efficiency is better than that of simple random sampling, ratio and product estimators. These points are motivated us to introduce a new class of improved ratio cum product estimators for the estimation of the population mean of the study variable.

Proposed Estimators

For estimating population mean Y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaraaaaa@377B@  we have proposed a class of ratio cum product estimators [4] for the population mean by using the known population correlation coefficient of the auxiliary variable and is given by

Y ^ pr =α λ 1 y ( X +ρ x +ρ )+( 1α ) λ 2 y ( x +ρ X +ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmywayaapyaajaWaaSbaaKqbGeaacaWGWbGaamOCaaqcfaya baGaeyypa0JaeqySdeMaeq4UdW2aaSbaaKqbGeaacaaIXaaajuaGbe aaceWG5bGba8aadaqadaqaamaalaaabaGabmiwayaapaGaey4kaSIa eqyWdihabaGabmiEayaapaGaey4kaSIaeqyWdihaaaGaayjkaiaawM caaiabgUcaRmaabmaabaGaaGymaiabgkHiTiabeg7aHbGaayjkaiaa wMcaaiabeU7aSnaaBaaajuaibaGaaGOmaaqcfayabaGabmyEayaapa WaaeWaaeaadaWcaaqaaiqadIhagaWdaiabgUcaRiabeg8aYbqaaiqa dIfagaWdaiabgUcaRiabeg8aYbaaaiaawIcacaGLPaaaaaa@5D43@                                                                              (9)

Here, λ 1 = S y S y + γ 1 C y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4UdW2aaSbaaKqbGeaacaaIXaaajuaGbeaacqGH9aqpdaWc aaqaaiaadofadaWgaaqcfasaaiaadMhaaKqbagqaaaqaaiaadofada WgaaqcfasaaiaadMhaaKqbagqaaiabgUcaRiabeo7aNnaaBaaajuai baGaaGymaaqcfayabaGaam4qamaaBaaajuaibaGaamyEaaqcfayaba aaaaaa@4725@ and λ 2 = S y S y + γ 2 C y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWc aaqaaiaadofadaWgaaqcfasaaiaadMhaaKqbagqaaaqaaiaadofada WgaaqcfasaaiaadMhaaKqbagqaaiabgUcaRiabeo7aNnaaBaaajuai baGaaGOmaaqcfayabaGaam4qamaaBaaajuaibaGaamyEaaqcfayaba aaaaaa@4727@ , γ 1 =B( Y ^ MR ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4SdC2aaSbaaKqbGeaacaaIXaaajuaGbeaacqGH9aqpcaWG cbWaaeWaaeaaceWGzbGba8GbaKaadaWgaaqcfasaaiaad2eacaWGsb aajuaGbeaaaiaawIcacaGLPaaacaGGSaaaaa@4167@ γ 2 =B( Y ^ Mp ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4SdC2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcaWG cbWaaeWaaeaaceWGzbGba8GbaKaadaWgaaqcfasaaiaad2eacaWGWb aajuaGbeaaaiaawIcacaGLPaaaaaa@40D6@

Bias and Mean Squared Error of the Proposed Estimators

The detailed derivation of the bias and mean squared error are given in the appendix whereas the procedures to obtain the bias and mean squared error of the proposed estimators are briefly outlined below:

Consider e θ = y Y Y , e 1 = x X X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyzamaaBaaajuaibaGaeqiUdehajuaGbeaacqGH9aqpdaWc aaqaaiqadMhagaWdaiabgkHiTiqadMfagaWdaaqaaiqadMfagaWdaa aacaGGSaGaamyzamaaBaaajuaibaGaaGymaaqcfayabaGaeyypa0Za aSaaaeaaceWG4bGba8aacqGHsislceWGybGba8aaaeaaceWGybGba8 aaaaaaaa@4743@ , θ= X X +ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiUdeNaeyypa0ZaaSaaaeaaceWGybGba8aaaeaaceWGybGb a8aacqGHRaWkcqaHbpGCaaaaaa@3DED@

E( e θ )=E( e 1 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyramaabmaabaGaamyzamaaBaaajuaibaGaeqiUdehajuaG beaaaiaawIcacaGLPaaacqGH9aqpcaWGfbWaaeWaaeaacaWGLbWaaS baaKqbGeaacaaIXaaajuaGbeaaaiaawIcacaGLPaaacqGH9aqpcaaI Waaaaa@4404@  , E( e 0 2 )=δ Y 2 C y 2  , E( e 1 2 ) = δ X 2 C x 2 , E( e 0 e 1 )= δρ C x C y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyramaabmaabaGaamyzamaaBaaajuaibaGaaGimaaqcfaya baWaaWbaaeqajuaibaGaaGOmaaaaaKqbakaawIcacaGLPaaacqGH9a qpcqaH0oazceWGzbGba8aadaahaaqabKqbGeaacaaIYaaaaKqbakaa doeadaqhaaqcfasaaiaadMhaaeaacaaIYaaaaKqbakaacckacaGGSa GaaiiOaiaadweadaqadaqaaiaadwgadaWgaaqcfasaaiaaigdaaKqb agqaamaaCaaabeqcfasaaiaaikdaaaaajuaGcaGLOaGaayzkaaGaai iOaiabg2da9iaacckacqaH0oazceWGybGba8aadaahaaqabKqbGeaa caaIYaaaaKqbakaadoeadaqhaaqcfasaaiaadIhaaeaacaaIYaaaaK qbakaacYcacaGGGcGaamyramaabmaabaGaamyzamaaBaaajuaibaGa aGimaaqcfayabaGaamyzamaaBaaajuaibaGaaGymaaqcfayabaaaca GLOaGaayzkaaGaeyypa0JaaiiOaiabes7aKjabeg8aYjaadoeadaWg aaqcfasaaiaadIhaaKqbagqaaiaadoeadaWgaaqcfasaaiaadMhaaK qbagqaaaaa@6DE4@

Substitute these values in equation (9) and neglecting the high order expressions, we get

B( Y ^ Pr )=E( Y ^ Pr Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOqamaabmaabaGabmywayaapyaajaWaaSbaaKqbGeaaciGG qbGaaiOCaaqcfayabaaacaGLOaGaayzkaaGaeyypa0Jaamyramaabm aabaGabmywayaapyaajaWaaSbaaKqbGeaaciGGqbGaaiOCaaqcfaya baGaeyOeI0IabmywayaapaaacaGLOaGaayzkaaaaaa@457B@

B( Y ^ Pr )= Y ( α λ 1 +( 1α ) λ 2 1 )+δ Y { α λ 1 θ 2 C x 2 θρ C x C y ( α λ 1 ( 1α ) λ 2 ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOqamaabmaabaGabmywayaapyaajaWaaSbaaKqbGeaaciGG qbGaaiOCaaqcfayabaaacaGLOaGaayzkaaGaeyypa0Jabmywayaapa WaaeWaaeaacqaHXoqycqaH7oaBdaWgaaqcfasaaiaaigdaaKqbagqa aiabgUcaRmaabmaabaGaaGymaiabgkHiTiabeg7aHbGaayjkaiaawM caaiabeU7aSnaaBaaajuaibaGaaGOmaaqcfayabaGaeyOeI0IaaGym aaGaayjkaiaawMcaaiabgUcaRiabes7aKjqadMfagaWdamaacmaaba GaeqySdeMaeq4UdW2aaSbaaKqbGeaacaaIXaaajuaGbeaacqaH4oqC daahaaqabKqbGeaacaaIYaaaaKqbakaadoeadaqhaaqcfasaaiaadI haaeaacaaIYaaaaKqbakabgkHiTiabeI7aXjabeg8aYjaadoeadaWg aaqcfasaaiaadIhaaKqbagqaaiaadoeadaWgaaqcfasaaiaadMhaaK qbagqaamaabmaabaGaeqySdeMaeq4UdW2aaSbaaKqbGeaacaaIXaaa juaGbeaacqGHsisldaqadaqaaiaaigdacqGHsislcqaHXoqyaiaawI cacaGLPaaacqaH7oaBdaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjk aiaawMcaaaGaay5Eaiaaw2haaaaa@7A16@

MSE( Y ^ Pr )= Y 2 ( A1 ) 2 +δ Y 2 { C y 2 ( α λ 1 +( 1α ) λ 2 ) 2 + θ 2 C x 2 ( 3 α 2 λ 1 2 + ( 1+α ) 2 λ 2 2 2α λ 1 )+2θρ C x C y ( α λ 1 ( 1α ) λ 2 )2( α 2 λ 1 2 ( 1α ) 2 λ 2 2 ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamytaiaadofacaWGfbWaaeWaaeaaceWGzbGba8GbaKaadaWg aaqcfasaaiGaccfacaGGYbaajuaGbeaaaiaawIcacaGLPaaacqGH9a qpceWGzbGba8aadaahaaqabKqbGeaacaaIYaaaaKqbaoaabmaabaGa amyqaiabgkHiTiaaigdaaiaawIcacaGLPaaadaahaaqabKqbGeaaca aIYaaaaKqbakabgUcaRiabes7aKjqadMfagaWdamaaCaaabeqcfasa aiaaikdaaaqcfa4aaiWaaeaacaWGdbWaaSbaaKqbGeaacaWG5baaju aGbeaadaahaaqabKqbGeaacaaIYaaaaKqbaoaabmaabaGaeqySdeMa eq4UdW2aaSbaaKqbGeaacaaIXaaajuaGbeaacqGHRaWkdaqadaqaai aaigdacqGHsislcqaHXoqyaiaawIcacaGLPaaacqaH7oaBdaWgaaqc fasaaiaaikdaaKqbagqaaaGaayjkaiaawMcaamaaCaaabeqcfasaai aaikdaaaqcfaOaey4kaSIaeqiUde3aaWbaaeqajuaibaGaaGOmaaaa juaGcaWGdbWaaSbaaKqbGeaacaWG4baajuaGbeaadaahaaqabKqbGe aacaaIYaaaaKqbaoaabmaabaGaaG4maiabeg7aHnaaCaaabeqcfasa aiaaikdaaaqcfaOaeq4UdW2aaSbaaKqbGeaacaaIXaaajuaGbeaada ahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRmaabmaabaGaaGymaiab gUcaRiabeg7aHbGaayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaaa qcfaOaeq4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaadaahaaqabKqb GeaacaaIYaaaaKqbakabgkHiTiaaikdacqaHXoqycqaH7oaBdaWgaa qcfasaaiaaigdaaKqbagqaaaGaayjkaiaawMcaaiabgUcaRiaaikda cqaH4oqCcqaHbpGCcaWGdbWaaSbaaKqbGeaacaWG4baajuaGbeaaca WGdbWaaSbaaKqbGeaacaWG5baajuaGbeaadaqadaqaaiabeg7aHjab eU7aSnaaBaaajuaibaGaaGymaaqcfayabaGaeyOeI0YaaeWaaeaaca aIXaGaeyOeI0IaeqySdegacaGLOaGaayzkaaGaeq4UdW2aaSbaaKqb GeaacaaIYaaajuaGbeaaaiaawIcacaGLPaaacqGHsislcaaIYaWaae WaaeaacqaHXoqydaahaaqabKqbGeaacaaIYaaaaKqbakabeU7aSnaa BaaajuaibaGaaGymaaqcfayabaWaaWbaaeqajuaibaGaaGOmaaaaju aGcqGHsisldaqadaqaaiaaigdacqGHsislcqaHXoqyaiaawIcacaGL PaaadaahaaqabKqbGeaacaaIYaaaaKqbakabeU7aSnaaBaaajuaiba GaaGOmaaqcfayabaWaaWbaaeqajuaibaGaaGOmaaaaaKqbakaawIca caGLPaaaaiaawUhacaGL9baaaaa@BAEC@

MSE( Y ^ Pr )= Y 2 ( A1 ) 2 +δ Y 2 { A 2 C y 2 + θ 2 C x 2 ( A 2 +( A+B )( B1 ) )2θρ C x C y B( 2A1 ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamytaiaadofacaWGfbWaaeWaaeaaceWGzbGba8GbaKaadaWg aaqcfasaaiGaccfacaGGYbaajuaGbeaaaiaawIcacaGLPaaacqGH9a qpceWGzbGba8aadaahaaqabKqbGeaacaaIYaaaaKqbaoaabmaabaGa amyqaiabgkHiTiaaigdaaiaawIcacaGLPaaadaahaaqabKqbGeaaca aIYaaaaKqbakabgUcaRiabes7aKjqadMfagaWdamaaCaaabeqcfasa aiaaikdaaaqcfa4aaiWaaeaacaWGbbWaaWbaaeqajuaibaGaaGOmaa aajuaGcaWGdbWaaSbaaKqbGeaacaWG5baajuaGbeaadaahaaqabKqb GeaacaaIYaaaaKqbakabgUcaRiabeI7aXnaaCaaabeqcfasaaiaaik daaaqcfaOaam4qamaaBaaajuaibaGaamiEaaqcfayabaWaaWbaaeqa juaibaGaaGOmaaaajuaGdaqadaqaaiaadgeadaahaaqabKqbGeaaca aIYaaaaKqbakabgUcaRmaabmaabaGaamyqaiabgUcaRiaadkeaaiaa wIcacaGLPaaadaqadaqaaiaadkeacqGHsislcaaIXaaacaGLOaGaay zkaaaacaGLOaGaayzkaaGaeyOeI0IaaGOmaiabeI7aXjabeg8aYjaa doeadaWgaaqcfasaaiaadIhaaKqbagqaaiaadoeadaWgaaqcfasaai aadMhaaKqbagqaaiaadkeadaqadaqaaiaaikdacaWGbbGaeyOeI0Ia aGymaaGaayjkaiaawMcaaaGaay5Eaiaaw2haaaaa@7A51@

where A=( α λ 1 +( 1α ) λ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeacq GH9aqpdaqadaqaaiabeg7aHjabeU7aSnaaBaaajuaibaGaaGymaaqc fayabaGaey4kaSYaaeWaaeaacaaIXaGaeyOeI0IaeqySdegacaGLOa GaayzkaaGaeq4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaaaiaawIca caGLPaaaaaa@47B8@ , B=( α λ 1 ( 1α ) λ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkeacq GH9aqpdaqadaqaaiabeg7aHjabeU7aSnaaBaaajuaibaGaaGymaaqc fayabaGaeyOeI0YaaeWaaeaacaaIXaGaeyOeI0IaeqySdegacaGLOa GaayzkaaGaeq4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaaaiaawIca caGLPaaaaaa@47C4@ and θ= X X +ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXj abg2da9maalaaabaGabmiwayaapaaabaGabmiwayaapaGaey4kaSIa eqyWdihaaaaa@3DCD@

The optimal value of α is determined by minimizing the MSE ( Y ^ Pr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaKqbacbaaaaaaaaape qaaiqadMfagaWdgaqcamaaBaaajuaibaGaciiuaiaackhaaKqbagqa aaaa@3A3B@ with respect to α. For this differentiate MSE with respect to α and equate to zero [5].

MSE α =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaaeaacqGHciITcaWGnbGaam4uaiaadweaaeaacqGHciIT cqaHXoqyaaGaeyypa0JaaGimaaaa@3F49@ , and we get the value of α, as

α= ( λ 2 1 )( λ 2 λ 1 )δ[ C y 2 λ 2 ( λ 1 λ 2 ) θ 2 C x 2 ( λ 1 + λ 2 2 )+θρ C x C y ( λ 1 + λ 2 4 λ 2 2 ) ] ( λ 1 λ 2 ) 2 +δ[ ( λ 1 λ 2 ) 2 C y 2 + θ 2 C x 2 ( 3 λ 1 2 + λ 2 2 )4θρ C x C y ( λ 2 2 λ 1 2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg7aHj abg2da9maalaaabiqaaq7adaqadaqaaiabeU7aSnaaBaaajuaibaGa aGOmaaqcfayabaGaeyOeI0IaaGymaaGaayjkaiaawMcaamaabmaaba Gaeq4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGHsislcqaH7oaB daWgaaqcfasaaiaaigdaaKqbagqaaaGaayjkaiaawMcaaiabgkHiTi abes7aKnaadmaabaGaam4qamaaBaaajuaibaGaamyEaaqcfayabaWa aWbaaeqajuaibaGaaGOmaaaajuaGcqaH7oaBdaWgaaqcfasaaiaaik daaKqbagqaamaabmaabaGaeq4UdW2aaSbaaKqbGeaacaaIXaaajuaG beaacqGHsislcqaH7oaBdaWgaaqcfasaaiaaikdaaKqbagqaaaGaay jkaiaawMcaaiabgkHiTiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqc faOaam4qamaaBaaajuaibaGaamiEaaqcfayabaWaaWbaaeqajuaiba GaaGOmaaaajuaGdaqadaqaaiabeU7aSnaaBaaajuaibaGaaGymaaqc fayabaGaey4kaSIaeq4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaada ahaaqabKqbGeaacaaIYaaaaaqcfaOaayjkaiaawMcaaiabgUcaRiab eI7aXjabeg8aYjaadoeadaWgaaqcfasaaiaadIhaaKqbagqaaiaado eadaWgaaqcfasaaiaadMhaaKqbagqaamaabmaabaGaeq4UdW2aaSba aKqbGeaacaaIXaaajuaGbeaacqGHRaWkcqaH7oaBdaWgaaqcfasaai aaikdaaKqbagqaaiabgkHiTiaaisdacqaH7oaBdaWgaaqcfasaaiaa ikdaaKqbagqaamaaCaaabeqcfasaaiaaikdaaaaajuaGcaGLOaGaay zkaaaacaGLBbGaayzxaaaabaWaaeWaaeaacqaH7oaBdaWgaaqcfasa aiaaigdaaKqbagqaaiabgkHiTiabeU7aSnaaBaaajuaibaGaaGOmaa qcfayabaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGOmaaaajuaG cqGHRaWkcqaH0oazdaWadaqaamaabmaabaGaeq4UdW2aaSbaaKqbGe aacaaIXaaajuaGbeaacqGHsislcqaH7oaBdaWgaaqcfasaaiaaikda aKqbagqaaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaaaqcfa Oaam4qamaaBaaajuaibaGaamyEaaqcfayabaWaaWbaaeqajuaibaGa aGOmaaaajuaGcqGHRaWkcqaH4oqCdaahaaqabKqbGeaacaaIYaaaaK qbakaadoeadaWgaaqcfasaaiaadIhaaKqbagqaamaaCaaabeqcfasa aiaaikdaaaqcfa4aaeWaaeaacaaIZaGaeq4UdW2aaSbaaeaadaWgaa qcfasaaiaaigdaaKqbagqaamaaCaaabeqcfasaaiaaikdaaaqcfaOa ey4kaSIaeq4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaadaahaaqabK qbGeaacaaIYaaaaaqcfayabaaacaGLOaGaayzkaaGaaGinaiabeI7a Xjabeg8aYjaadoeadaWgaaqcfasaaiaadIhaaKqbagqaaiaadoeada WgaaqcfasaaiaadMhaaKqbagqaamaabmaabaGaeq4UdW2aaSbaaKqb GeaacaaIYaaajuaGbeaadaahaaqabKqbGeaacaaIYaaaaKqbakabgk HiTiabeU7aSnaaBaaajuaibaGaaGymaaqcfayabaWaaWbaaeqajuai baGaaGOmaaaaaKqbakaawIcacaGLPaaaaiaawUfacaGLDbaaaaaaaa@D424@

Efficiency comparison

The efficiencies of the proposed estimators with that of the existing estimators are obtained algebraically and are as follows:

Comparison of proposed estimator and simple random sampling (SRSWOR) estimator

The proposed estimator is more efficient than simple random sampling estimator,

V( y lr )MSE( Y ^ Pr )   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOvamaabmaabaGabmyEayaapaWaaSbaaKqbGeaacaWGSbGa amOCaaqcfayabaaacaGLOaGaayzkaaGaeyyzImRaamytaiaadofaca WGfbWaaeWaaeaaceWGzbGba8GbaKaadaWgaaqcfasaaiGaccfacaGG YbaajuaGbeaaaiaawIcacaGLPaaacaGGGcGaaiiOaaaa@488E@ if

C y 2 { ( A1 ) 2 + δ 2 { θ 2 C x 2 ( A 2 +( A+B )( B1 ) )2θρ C x C y B( 2A1 ) } } δ 2 ( 1 A 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4qamaaDaaajuaibaGaamyEaaqaaiaaikdaaaqcfaOaeyyz Im7aaSaaaeaadaGadaqaamaabmaabaGaamyqaiabgkHiTiaaigdaai aawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiab es7aKnaaCaaabeqcfasaaiaaikdaaaqcfa4aaiWaaeaacqaH4oqCda ahaaqabKqbGeaacaaIYaaaaKqbakaadoeadaqhaaqcfasaaiaadIha aeaacaaIYaaaaKqbaoaabmaabaGaamyqamaaCaaabeqcfasaaiaaik daaaqcfaOaey4kaSYaaeWaaeaacaWGbbGaey4kaSIaamOqaaGaayjk aiaawMcaamaabmaabaGaamOqaiabgkHiTiaaigdaaiaawIcacaGLPa aaaiaawIcacaGLPaaacqGHsislcaaIYaGaeqiUdeNaeqyWdiNaam4q amaaBaaajuaibaGaamiEaaqcfayabaGaam4qamaaBaaajuaibaGaam yEaaqcfayabaGaamOqamaabmaabaGaaGOmaiaadgeacqGHsislcaaI XaaacaGLOaGaayzkaaaacaGL7bGaayzFaaaacaGL7bGaayzFaaaaba GaeqiTdq2aaWbaaeqajuaibaGaaGOmaaaajuaGdaqadaqaaiaaigda cqGHsislcaWGbbWaaWbaaeqajuaibaGaaGOmaaaaaKqbakaawIcaca GLPaaaaaaaaa@75FB@

Comparison of proposed estimator and linear regression estimator

The proposed estimator is more efficient than linear regression estimator,

V( y lr )MSE( Y ^ Pr )   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOvamaabmaabaGabmyEayaapaWaaSbaaKqbGeaacaWGSbGa amOCaaqcfayabaaacaGLOaGaayzkaaGaeyyzImRaamytaiaadofaca WGfbWaaeWaaeaaceWGzbGba8GbaKaadaWgaaqcfasaaiGaccfacaGG YbaajuaGbeaaaiaawIcacaGLPaaacaGGGcGaaiiOaaaa@488E@ if

C y 2 { ( A1 ) 2 +δ[ θ 2 C x 2 ( A 2 +( A+B )( B1 ) )2θ C x C y ( B( 2A1 )1 ) ] } δ( 1 ρ 2 A 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4qamaaDaaajuaibaGaamyEaaqaaiaaikdaaaqcfaOaeyyz Im7aaSaaaeaadaGadaqaamaabmaabaGaamyqaiabgkHiTiaaigdaai aawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiab es7aKnaadmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGca WGdbWaaSbaaKqbGeaacaWG4baajuaGbeaadaahaaqabKqbGeaacaaI YaaaaKqbaoaabmaabaGaamyqamaaCaaabeqcfasaaiaaikdaaaqcfa Oaey4kaSYaaeWaaeaacaWGbbGaey4kaSIaamOqaaGaayjkaiaawMca amaabmaabaGaamOqaiabgkHiTiaaigdaaiaawIcacaGLPaaaaiaawI cacaGLPaaacqGHsislcaaIYaGaeqiUdeNaam4qamaaBaaajuaibaGa amiEaaqcfayabaGaam4qamaaBaaajuaibaGaamyEaaqcfayabaWaae WaaeaacaWGcbWaaeWaaeaacaaIYaGaamyqaiabgkHiTiaaigdaaiaa wIcacaGLPaaacqGHsislcaaIXaaacaGLOaGaayzkaaaacaGLBbGaay zxaaaacaGL7bGaayzFaaaabaGaeqiTdq2aaeWaaeaacaaIXaGaeyOe I0IaeqyWdi3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHsislcaWGbb WaaWbaaeqajuaibaGaaGOmaaaaaKqbakaawIcacaGLPaaaaaaaaa@791D@

Comparison of proposed estimator and ratio estimator

The proposed estimator is more efficient than ratio estimator

MSE( Y ^ P )MSE( Y ^ Pr )   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamytaiaadofacaWGfbWaaeWaaeaaceWGzbGba8GbaKaadaWg aaqcfasaaiaadcfaaKqbagqaaaGaayjkaiaawMcaaiabgwMiZkaad2 eacaWGtbGaamyramaabmaabaGabmywayaapyaajaWaaSbaaKqbGeaa ciGGqbGaaiOCaaqcfayabaaacaGLOaGaayzkaaGaaiiOaiaacckaaa a@4903@  if

C y 2 { ( A1 ) 2 +δ{ C x 2 ( θ 2 ( A 2 +( A+B )( B1 ) )1 )2ρ C x C y ( θB( 2A1 )1 )+1 } } δ( 1 A 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4qamaaDaaajuaibaGaamyEaaqaaiaaikdaaaqcfaOaeyyz Im7aaSaaaeaadaGadaqaamaabmaabaGaamyqaiabgkHiTiaaigdaai aawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiab es7aKnaacmaabaGaam4qamaaDaaajuaibaGaamiEaaqaaiaaikdaaa qcfa4aaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbaoaa bmaabaGaamyqamaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSYaae WaaeaacaWGbbGaey4kaSIaamOqaaGaayjkaiaawMcaamaabmaabaGa amOqaiabgkHiTiaaigdaaiaawIcacaGLPaaaaiaawIcacaGLPaaacq GHsislcaaIXaaacaGLOaGaayzkaaGaeyOeI0IaaGOmaiabeg8aYjaa doeadaWgaaqcfasaaiaadIhaaKqbagqaaiaadoeadaWgaaqcfasaai aadMhaaKqbagqaamaabmaabaGaeqiUdeNaamOqamaabmaabaGaaGOm aiaadgeacqGHsislcaaIXaaacaGLOaGaayzkaaGaeyOeI0IaaGymaa GaayjkaiaawMcaaiabgUcaRiaaigdaaiaawUhacaGL9baaaiaawUha caGL9baaaeaacqaH0oazdaqadaqaaiaaigdacqGHsislcaWGbbWaaW baaeqajuaibaGaaGOmaaaaaKqbakaawIcacaGLPaaaaaaaaa@7AC6@

Comparison of proposed estimator and product estimator

The proposed estimator is more efficient than ratio estimator, [6]

MSE( Y ^ P )MSE( Y ^ Pr )   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamytaiaadofacaWGfbWaaeWaaeaaceWGzbGba8GbaKaadaWg aaqcfasaaiaadcfaaKqbagqaaaGaayjkaiaawMcaaiabgwMiZkaad2 eacaWGtbGaamyramaabmaabaGabmywayaapyaajaWaaSbaaKqbGeaa ciGGqbGaaiOCaaqcfayabaaacaGLOaGaayzkaaGaaiiOaiaacckaaa a@4903@ if

C y 2 { ( A1 ) 2 +δ{ C x 2 ( θ 2 ( A 2 +( A+B )( B1 ) )1 )2ρ C x C y ( θB( 2A1 )+1 ) } } δ( 1 A 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4qamaaDaaajuaibaGaamyEaaqaaiaaikdaaaqcfaOaeyyz Im7aaSaaaeaadaGadaqaamaabmaabaGaamyqaiabgkHiTiaaigdaai aawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiab es7aKnaacmaabaGaam4qamaaDaaajuaibaGaamiEaaqaaiaaikdaaa qcfa4aaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbaoaa bmaabaGaamyqamaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSYaae WaaeaacaWGbbGaey4kaSIaamOqaaGaayjkaiaawMcaamaabmaabaGa amOqaiabgkHiTiaaigdaaiaawIcacaGLPaaaaiaawIcacaGLPaaacq GHsislcaaIXaaacaGLOaGaayzkaaGaeyOeI0IaaGOmaiabeg8aYjaa doeadaWgaaqcfasaaiaadIhaaKqbagqaaiaadoeadaWgaaqcfasaai aadMhaaKqbagqaamaabmaabaGaeqiUdeNaamOqamaabmaabaGaaGOm aiaadgeacqGHsislcaaIXaaacaGLOaGaayzkaaGaey4kaSIaaGymaa GaayjkaiaawMcaaaGaay5Eaiaaw2haaaGaay5Eaiaaw2haaaqaaiab es7aKnaabmaabaGaaGymaiabgkHiTiaadgeadaahaaqabKqbGeaaca aIYaaaaaqcfaOaayjkaiaawMcaaaaaaaa@791E@

Comparison of proposed estimator and modified ratio estimator

The proposed estimator is more efficient than modified ratio estimator, [7]

MSE( Y ^ MR )MSE( Y ^ Pr )   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamytaiaadofacaWGfbWaaeWaaeaaceWGzbGba8GbaKaadaWg aaqcfasaaiaad2eacaWGsbaajuaGbeaaaiaawIcacaGLPaaacqGHLj YScaWGnbGaam4uaiaadweadaqadaqaaiqadMfagaWdgaqcamaaBaaa juaibaGaciiuaiaackhaaKqbagqaaaGaayjkaiaawMcaaiaacckaca GGGcaaaa@49D7@

C y 2 { ( A1 ) 2 +δ{ θ 2 C x 2 ( A 2 +( A+B )( B1 )1 )2θρ C x C y ( θB( A1 )1 )1 } } δ( 1 A 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4qamaaDaaajuaibaGaamyEaaqaaiaaikdaaaqcfaOaeyyz Im7aaSaaaeaadaGadaqaamaabmaabaGaamyqaiabgkHiTiaaigdaai aawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiab es7aKnaacmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGca WGdbWaa0baaKqbGeaacaWG4baabaGaaGOmaaaajuaGdaqadaqaaiaa dgeadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRmaabmaabaGaam yqaiabgUcaRiaadkeaaiaawIcacaGLPaaadaqadaqaaiaadkeacqGH sislcaaIXaaacaGLOaGaayzkaaGaeyOeI0IaaGymaaGaayjkaiaawM caaiabgkHiTiaaikdacqaH4oqCcqaHbpGCcaWGdbWaaSbaaKqbGeaa caWG4baajuaGbeaacaWGdbWaaSbaaKqbGeaacaWG5baajuaGbeaada qadaqaaiabeI7aXjaadkeadaqadaqaaiaadgeacqGHsislcaaIXaaa caGLOaGaayzkaaGaeyOeI0IaaGymaaGaayjkaiaawMcaaiabgkHiTi aaigdaaiaawUhacaGL9baaaiaawUhacaGL9baaaeaacqaH0oazdaqa daqaaiaaigdacqGHsislcaWGbbWaaWbaaeqajuaibaGaaGOmaaaaaK qbakaawIcacaGLPaaaaaaaaa@7A42@

Comparison of proposed estimator and modified product estimator

The proposed estimator is more efficient than modified product estimator,

MSE( Y ^ Mp )MSE( Y ^ Pr )   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamytaiaadofacaWGfbWaaeWaaeaaceWGzbGba8GbaKaadaWg aaqcfasaaiaad2eacaWGWbaajuaGbeaaaiaawIcacaGLPaaacqGHLj YScaWGnbGaam4uaiaadweadaqadaqaaiqadMfagaWdgaqcamaaBaaa juaibaGaciiuaiaackhaaKqbagqaaaGaayjkaiaawMcaaiaacckaca GGGcaaaa@49F5@  if

C y 2 { ( A1 ) 2 +δ{ θ 2 C x 2 ( A 2 +( A+B )( B1 )1 )2θρ C x C y ( B( 2A1 )1 ) } } δ( 1 A 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4qamaaDaaajuaibaGaamyEaaqaaiaaikdaaaqcfaOaeyyz Im7aaSaaaeaadaGadaqaamaabmaabaGaamyqaiabgkHiTiaaigdaai aawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiab es7aKnaacmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGca WGdbWaa0baaKqbGeaacaWG4baabaGaaGOmaaaajuaGdaqadaqaaiaa dgeadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRmaabmaabaGaam yqaiabgUcaRiaadkeaaiaawIcacaGLPaaadaqadaqaaiaadkeacqGH sislcaaIXaaacaGLOaGaayzkaaGaeyOeI0IaaGymaaGaayjkaiaawM caaiabgkHiTiaaikdacqaH4oqCcqaHbpGCcaWGdbWaaSbaaKqbGeaa caWG4baajuaGbeaacaWGdbWaaSbaaKqbGeaacaWG5baajuaGbeaada qadaqaaiaadkeadaqadaqaaiaaikdacaWGbbGaeyOeI0IaaGymaaGa ayjkaiaawMcaaiabgkHiTiaaigdaaiaawIcacaGLPaaaaiaawUhaca GL9baaaiaawUhacaGL9baaaeaacqaH0oazdaqadaqaaiaaigdacqGH sislcaWGbbWaaWbaaeqajuaibaGaaGOmaaaaaKqbakaawIcacaGLPa aaaaaaaa@77A0@

Numerical Study

In this section, we consider the four natural populations population 1 Khoshnevisan et al. [8], Population 2 Cochran [9] (page 325) population 3 and 4 Singh and Chaudhary [10], (page 177) and are used to compare the percentage relative efficiency of proposed estimator with that of the existing estimators such as SRSWOR sample mean, linear regression estimator, ratio estimator, product estimator, modified ratio estimators, and modified product estimators.

Conclusion

We have proposed a class of modified ratio cum product estimators for finite population [11] mean of the study variable Y with known correlation coefficient of the auxiliary variable X. The bias and mean squared error of the proposed estimators are obtained and compared with that of the simple random sampling without replacement, regression, ratio, product, modified ratio, modified product estimators by both algebraically and numerically. We support this theoretical result with numerical examples. We have shown that the proposed estimator is more efficient than other existing estimators under the optimum values of α. Table 1&2 shows that the bias and MSE of the proposed estimators are smaller than the other competing estimators. Table 3 shows that the percentage relative efficiency of the proposed estimator with respect to the existing estimators,

Parameters

Population  1

Population 2

Population 3

N

20

10

34

n

8

3

3

  Y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaraaaaa@377A@

19.55

101.1

856.4117

  X ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiway aaraaaaa@3779@

18.8

58.8

208.8823

 ρ

-0.9199

0.6515

0.4491

 Sy

6.9441

15.4448

733.1407

 Cy

0.3552

0.1527

0.8561

  Sx

7.4128

7.9414

150.5059

Cx  

0.3943

0.1351

0.7205

  β1

3.0613

0.2363

2.9123

  β2

0.5473

2.2388

0.9781

 θ

1.0514

0.989

0.9978

 γ1

0.4506

0.1072

625915

γ2  

-0.1986

0.3136

71.947

 λ1

0.9774

0.9989

0.9319

 λ2

1.0102

0.9969

0.9225

*α 

0.1055

0.8717

0.7614

Table 1: The computed values of constants and parameters from different populations.

Estimator

Population 1

Population 2

Population 3

Population 4

Bias

MSE

Bias

MSE

Bias

MSE

`Bias

MSE

Proposed

1.14e-06

0.5463

1.13e-15

31.9319

-0.00274

109092.8

-0.0015

66145.84

y ¯ srs MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyEay aaraWaaSbaaKqbGeaacaWGZbGaamOCaiaadohaaKqbagqaaaaa@3B5E@

-

3.6166

-

55.6603

-

163356.4

-

91690.37

y ¯ lr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyEay aaraWaaSbaaKqbGeaacaWGSbGaamOCaaqcfayabaaaaa@3A5F@

-

0.5561

-

32.0343

-

130408.9

-

73197.27

Y ¯ R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaryaataWaaSbaaKqbGeaacaWGsbaajuaGbeaaaaa@3947@

0.4168

15.4595

0.1132

35.0447

63.0193

155580.6

35.3721

87325.9

Y ¯ p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaryaataWaaSbaaKqbGeaacaWGWbaajuaGbeaaaaa@3965@

-0.1889

0.6869

0.3171

163.283

72.0984

402564.2

40.4681

225955.4

Y ¯ MR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaryaataWaaSbaaKqbGeaacaWGnbGaamOuaaqcfayabaaaaa@3A19@

0.4506

16.3099

0.1072

34.7991

62.5915

155359.1

35.1319

87201.54

Y ¯ MP MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaryaataWaaSbaaKqbGeaacaWGnbGaamiuaaqabaaaaa@3989@

-0.1986

0.7774

0.3163

161.6321

71.947

401824.2

40.3832

225540

Table 2: Bias and MSE of proposed and Existing Estimators from different population.

Estimators

Population  1

Population 2

Population 3

Population 4

y ¯ srs MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyEay aaraWaaSbaaKqbGeaacaWGZbGaamOCaiaadohaaKqbagqaaaaa@3B5E@  

661.981

174.3092

149.6356

138.6185

y ¯ lr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyEay aaraWaaSbaaKqbGeaacaWGSbGaamOCaaqcfayabaaaaa@3A5F@  

101.802

100.3205

119.4555

110.6604

Y ¯ R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaryaataWaaSbaaKqbGeaacaWGsbaajuaGbeaaaaa@3947@  

2829.752

109.7481

142.5216

132.0283

Y ¯ p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaryaataWaaSbaaKqbGeaacaWGWbaajuaGbeaaaaa@3965@  

125.7468

511.3465

368.7708

341.6196

  Y ¯ MR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaryaataWaaSbaaKqbGeaacaWGnbGaamOuaaqcfayabaaaaa@3A19@

2985.408

108.979

142.183

131.8322

Y ¯ MP MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyway aaryaataWaaSbaaKqbGeaacaWGnbGaamiuaaqabaaaaa@3989@  

142.2864

506.1764

367.6544

340.9739

Table 3:  Percentage Relative Efficiency of the Proposed Estimator.

In fact, the PRE is ranging from

  1. 138.6185 to 661.9810 in case of SRSWOR sample mean
  2. 100.3205 to 119.4555 in case of Linear Regression Estimator
  3. 109.7481 to 2829.7520 in case of Ratio estimator
  4. 125.7468 to 511.3465 in case of Product estimator
  5. 108.9790 to 2985.4080 in case of Modified Ratio estimator and
  6. 142.2864 to 506.1764 in case of Product estimator

From this, we have observed that the proposed estimator is performed better than that of other existing estimators and hence we recommend the proposed estimators for the practical problems.

References

  1. Cochran WG (1940) The estimation of the yields of the cereal experiments by sampling for the ratio of grain to total produce, The Journal of Agricultural Science 30(2): 262-275.
  2. Murthy MN (1964) Product method of estimation. Sankhyā: The Indian Journal of Statistics 26(1): 69-74.
  3. Singh HP, Tailor R (2003) Use of known correlation coefficient in estimating the finite population means, Statistics in Transition 6(4): 555-560.
  4. Ekaette Inyang Enang, Victoria Matthew Akpan, Emmanuel John Ekpenyong (2014) Alternative ratio estimator of population mean in simple random sampling, Journal of Mathematics Research 6(3).
  5. HousilaP Singh, Surya K Pal, Vishal Mehta (2016) A generalized class of dual to product-cum-dual to ratiotype estimators of finite population mean in sample surveys. Appl Math Inf Sci Lett 4(1): 25-33.
  6. J Subramani G Kumarapandiyan (2012) A class of almost unbiased modified ratio estimators for population mean with known population parameters. Elixir Statistics 44: 7411-7415.
  7. Murthy MN (1967) Sampling theory and methods. Statistical Publishing Society, Calcutta, India.
  8. Khoshnevisan M, Singh R, Chauhan P, Sawan N, Smarandache F (2007) A general family of estimators for estimating population mean using known value of some population parameter(s). Far East Journal of Theoretical Statistics 22: 181-191.
  9. Cochran WG (1977) Sampling Techniques. (3rd edn), Wiley Eastern Limited, India, pp. 448.
  10. Singh D, Chaudhary FS (1986) Theory and analysis of sample survey designs.(1st edn), New Age International Publisher, India, pp. 332.
  11. Subramani J (2013) "Generalized modified ratio estimator for estimation of finite population mean," Journal of Modern Applied Statistical Methods 12(2): pp.121-155.
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