This paper presents a family of dual to ratio-cum-product estimators for the finite population mean. Under simple random sampling without replacement (SRSWOR) scheme, expressions of the bias and mean-squared error (MSE) up to the first order of approximation are derived. We show that the proposed family is more efficient than usual unbiased estimator, ratio estimator, product estimator, Singh estimator (1967), Srivenkataramana (1980) and Bandyopadhyaya estimator (1980) and Singh et al. (2005) estimator. An empirical study is carried out to illustrate the performance of the constructed estimator over others

Chauhan, Pankaj

Kumar, Mukesh

Sawan, Nirmala

Singh, Rajesh

Smarandache, Florentin

English

Rajesh Singh

Mukesh Kumar

Pankaj Chauhan

Nirmala Sawan

Florentin Smarandache

CiteSeerX

A GENERAL FAMILY OF DUAL TO RATIO-CUM- PRODUCT ESTIMATOR IN SAMPLE SURVEYS

Biblioteka Nauki - repozytorium artykuÅÃ³w

A General Family of Dual to Ratio-Cum-Product Estimator in Sample Surveys

University of New Mexico

University of New Mexico 
UNM Digital Repository 
Mathematics and Statistics Faculty and Staff 
Publications Academic Department Resources 
12-2011 
A GENERAL FAMILY OF DUAL TO RATIO-CUM-PRODUCT 
ESTIMATOR IN SAMPLE SURVEYS 
Florentin Smarandache 
Rajesh Singh 
Mukesh Kumar 
Pankaj Chauhan 
Nirmala Sawan 
Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp 
 Part of the Applied Statistics Commons, Design of Experiments and Sample Surveys Commons, 
Mathematics Commons, Other Statistics and Probability Commons, and the Statistical Methodology 
Commons 
STATISTICS IN TRANSITION-new series, December 2011 
 
587 
STATISTICS IN TRANSITION-new series, December 2011 
Vol. 12, No. 3, pp. 587—594 
A GENERAL FAMILY OF DUAL TO RATIO-CUM-
PRODUCT ESTIMATOR IN SAMPLE SURVEYS 
 
Rajesh Singh, Mukesh Kumar1, Pankaj Chauhan, Nirmala Sawan2, 
Florentin Smarandache3 
ABSTRACT 
This paper presents a family of dual to ratio-cum-product estimators for the finite 
population mean. Under simple random sampling without replacement 
(SRSWOR) scheme, expressions of the bias and mean-squared error (MSE) up to 
the first order of approximation are derived. We show that the proposed family is 
more efficient than usual unbiased estimator, ratio estimator, product estimator, 
Singh estimator (1967), Srivenkataramana (1980) and Bandyopadhyaya estimator 
(1980) and Singh et al. (2005) estimator. An empirical study is carried out to 
illustrate the performance of the constructed estimator over others. 
Key words: Family of estimators, auxiliary variables, bias, mean-squared error. 
1. Introduction 
It is common practice to use the auxiliary variable for improving the precision 
of the estimate of a parameter. Out of many ratio and product methods of 
estimation are good examples in this context. When the correlation between the 
study variate and auxiliary variates is positive (high), ratio method of estimation 
is quite effective. On the other hand, when this correlation is negative (high), 
product method of estimation can be employed effectively. Let U be a finite 
population consisting of N units U1,U2,…,UN. Let y and (x, z) denote the study 
variate and auxiliary variates taking the values yi and (xi, zi), respectively, on the 
unit Ui (i=1,2…,N), where x is positively correlated with y and z is negatively 
correlated with y. We wish to estimate the population mean ∑
=
=
N
1i
iyN
1Y  of y, 
                                                          
1 Department of Statistics, BHU, Varanasi(U.P.), India (rsinghstat@yahoo.com). 
2 School of Statistics, DAVV, Indore (M.P.), India. 
3 Department of Mathematics, University of New Mexico, Gallup, USA, (smarand@unm.edu). 
588              R. Singh, M. Kumar, P. Chauhan, N. Sawan, F. Smarandache: A general … 
 
 
assuming that the population means ( )Z,X  of (x, z) are known. Assume that a 
simple random sample of size n is drawn without replacement from U. The 
classical ratio and product estimators for estimating Y  are: 
X
x
yyR =                        (1.1) 
and  
Z
zyyP =                        (1.2) 
respectively, where ∑
=
=
n
1i
iyn
1y , ∑
=
=
n
1i
ixn
1x  and ∑
=
=
n
1i
izn
1z are the sample 
means of y, x and z respectively.  
 
Using the transformation ( )( )nN
nxXNx i*i −
−
= , and 
)nN(
)nzZN(z i*i −
−
= , (i=1,2…N) 
Srivenkataramana (1980) and Bandyopadhyaya (1980) suggested a dual to ratio 
and product estimator as: 
X
xyy
*
*
R =                       (1.3) 
and    *
*
P z
Zyy =                        (1.4) 
where 
)nN(
xnXNx*
−
−
=  and 
)nN(
znZNz*
−
−
= . 
 
In some survey situations, information on a secondary auxiliary variate z, 
correlated negatively with the study variate y, is readily available. Let Z be the 
known population mean of z. For estimating Y , Singh (1967) considered a ratio-
cum-product estimator 
 












=
Z
z
x
XyyRP                     (1.5) 
 
Using a simple transformation ( ) i*i gxXg1x −+=  and i*i gzZ)g1(z −+= , 
i=1,2…,N, with ( )nN
ng
−
= , Singh et al. (2005) proposed a dual to usual ratio-
cum-product estimator 
STATISTICS IN TRANSITION-new series, December 2011 
 
589 












= *
*
*
RP z
Z
X
xyy                     (1.6) 
where xgX)g1(x* −+=  and zgZ)g1(z* −+= . 
 
To the first degree of approximation 
 
( ) 2yCYyV θ=         (1.7) 
( ) ( )[ ]yx2x2y2R K21CCYyMSE −+θ=           (1.8) 
( ) ( )[ ]yz2x2y2P K21CCYyMSE ++θ=           (1.9) 
( ) ( ) ( )[ ]K21CK21CCYyMSE 2xyz2x2y2RP −+++θ=         (1.10) 
         ( ) ( )[ ]yx2x2y2*R K2ggCCYyMSE −+θ=             (1.11) 
( ) ( )[ ]yz2z2y2*P K2ggCCYyMSE ++θ=             (1.12) 
( ) ( ) ( )[ ]yxzx2xyz2z2y2*RP K2gK2ggCK2ggCCYyMSE −−+++θ=     (1.13) 
 
where MSE (.) stands for mean square error (MSE) of  (.). 
 
n
f1−
=θ ,
N
nf = ,
Y
S
C yy = , X
S
C xx = , Z
SC zz = ,
x
y
yxyx C
C
K ρ= ,
z
y
yzyz C
C
K ρ= , 
x
z
xzzx C
CK ρ= , zxyx KKK += , 
xy
yx
yx SS
S
=ρ , 
zy
yz
yz SS
S
=ρ , 
zx
xz
xz SS
S
=ρ , 
∑
=
−
−
=
N
1i
2
i
2
y )Yy()1N(
1S , ∑
=
−
−
=
N
1i
2
i
2
x )Xx()1N(
1S , ∑
=
−
−
=
N
1i
2
i
2
z )Zz()1N(
1S , 
∑
=
−−
−
=
N
1i
iiyx )Xx)(Yy()1N(
1S ,       ∑
=
−−
−
=
N
1i
iiyz )Zz)(Yy()1N(
1S , 
and  ∑
=
−−
−
=
N
1i
iixz )Zz)(Xx()1N(
1S . 
 
In this paper, under SRSWOR, we present a family of dual to ratio-cum-
product estimator for estimating the population mean Y . We obtain the first 
order approximation of the bias and the MSE for this family of estimators. 
Numerical illustrations are given to show the performance of the constructed 
estimator over other estimators. 
 
590              R. Singh, M. Kumar, P. Chauhan, N. Sawan, F. Smarandache: A general … 
 
 
2. The suggested family of estimators 
We define a family of estimators of Y as 
21
*
*
z
Z
X
xyt
αα












=                 (2.1) 
where αi′s (i=1,2) are unknown constants to be suitably determined. 
 
To obtain the bias and MSE of t, we write 
( )0e1Yy += , ( )1e1Xx += , ( )2e1Zz +=   
such that 
E (eo)= E (e1)= E (e2)= 0 
and  
( ) 2y20 CeE θ= , ( ) 2x21 CeE θ= , ( ) 2z22 CeE θ= ,  
( ) xyxy10 CCeeE θρ= , ( ) zyyz20 CCeeE θρ= , ( ) zxxz21 CCeeE θρ= . 
 
Expressing t in terms of e’s, we have 
( )( ) ( ) 21 210 ge1ge1e1Yt α−α −−+=            (2.2) 
 
We assume that |ge1|<1, |ge2|<1, so that ( ) 11ge1 α− and ( ) 22ge1 α−−  are 
expandable. Expanding the right hand side of (2.2) and retaining terms up to 
second powers of e’s (up to the first order of approximation), we have  
 
20222
2
1
211
101110 egegeeg2
)1(egegee1[Yt α+α+−αα+α−α−+=  
  ]eg
2
)1(eeg 22
222
21
2
21
−αα
+αα−            (2.3) 
Taking expectations of both sides in (2.3) and then subtracting Y  from both 
sides, we get the bias of the estimator t, up to the first order of approximation, as 
 
)Yt(E)t(B −=  
        
}]
2
)1(K{gCgC
2
)1()CKCK[(gY 2xz1
2
z2
2
x
112
xyx1
2
zyz2
−α
−αα−
−αα
+α−αθ=
                         (2.4) 
where 
z
x
xzxz C
CK ρ= . 
 
STATISTICS IN TRANSITION-new series, December 2011 
 
591 
From (2.3), we have  
( ) [ ])ee(geYYt 22110 α−α−≅−             (2.5) 
 
Squaring both sides of (2.5), we have  
( ) )]eeee(g2}ee2ee{ge[YYt 20210121212222212122022 α−α−αα−α+α+=−
                        (2.6) 
 
Taking expectations of both sides of (2.6), we get the MSE of t to the first 
degree of approximation as 
    
)]K2gK2g(gC)K2g(CgC[Y)t(MSE yx1zx21
2
1
2
xyz2
2
z2
2
y
2 α−αα−α++αα+θ=  
                        (2.7) 
 
Minimization of (2.7) with respect to 1α  and 2α  yields their optimum values 
as  
( )







ρ−
−−
=α
ρ−
−
=α
)1(g
)KKK(
1g
KKK
2
xz
yxxzyz
2
2
xz
yzzxyx
1
              (2.8) 
 
Substitution of (2.8) in (2.7) yields the minimum value of MSE (t) as 
)1(CY)t(MSE.min 2 xz.y
2
y
2 ρ−θ=             (2.9) 
where     
)1(
2
2
xz
xzyzyx
2
yz
2
yx2
xz.y ρ−
ρρρ−ρ+ρ
=ρ  is the multiple correlation coefficient of 
y on x and z. 
 
Remark 2.1:  For ( 1α , 2α )=(1,0), the estimator t reduces to the ‘dual to ratio’ 
estimator 






=
X
xyy
*
*
R  
 
While for ),( 21 αα =(0,1) it reduces to the ‘dual to product’ estimator  






= *
*
P z
Zyy  
It coincides with the estimator in Singh et al. (2005) when ),( 21 αα =(1,1). 
 
592              R. Singh, M. Kumar, P. Chauhan, N. Sawan, F. Smarandache: A general … 
 
 
Remark 2.2: The optimum values of αio’s (i=1,2) are functions of unknown 
population parameters such as Kyx, Kyz, Kyx. The values of these unknown 
population parameters can be guessed quite accurately from the past data or 
experiences gathered in due course of time, for instance, see Srivastava (1967), 
Reddy (1973), Prasad (1989,p.391), Murthy (1967,pp96-99), Singh et. al. (2007) 
and Singh et. al. (2009). Also the prior values of Kyx, Kyz, Kzx may be either 
obtained on the basis of the information from the most recent survey or by 
conducting a pilot survey, see, Lui (1990,p.3805). 
 From (1.7) to (1.13) and (2.9) it can be shown that the proposed estimator t at 
(2.1) is more efficient than usual unbiased estimator y , usual ratio estimator Ry , 
product estimator Py , Singh (1967) ratio-cum product estimator RPy , 
Srivenkataramana (1980) and Bandyopadhyaya (1980) dual to ratio estimator 
*
Ry , dual to product estimator 
*
Py  and Singh et al. (2005) dual to ratio-cum-
product estimator *RPy at its optimum conditions. 
3. Empirical study 
In this section we illustrate the performance of the constructed estimator t 
over various other estimators *RP
*
P
*
RRPPR y,y,y,y,y,y,y  through natural data 
earlier used by Singh (1969, p.377). 
y: number of females employed 
x: number of females in service 
z: number of educated females 
46.7Y = , 31.5=X , 00.179=Z , 5046.02 =yC , 5737.0
2 =xC , 0633.0
2 =zC ,
7737.0=yxρ 2070.0−=yzρ , ,0033.0−=xzρ  N=61 and n=20. 
Table 3.1:  Range of 1α  and 2α  for which t is better than 
*
RPy  
 
                                                1α  
0.94 1 1.25 1.39( )opt(1α ) 1.5 1.7 1.85 
2.7  100.81 104.67 116.46 118.52 117.4 109.7 100.6 
3.1  101.01 105.02 116.8 118.9 117.8 110.13 100.9 
3.34( )opt(2α ) 101.14 105.08 116.96 119.03 117.8 110.19 101.0 
3.4 101.2 105.08 116.95 119.02 117.8 110.19 101.04 
3.8 101.02 104.8 116.71 118.76 117.6 109.9 100.8 
4.5 100.04 103.8 115.38 117.38 116.2 108.7 99.8 
5.0 98.83 102.5 113.77 115.71 114.6 107.3 98.6 
2α
STATISTICS IN TRANSITION-new series, December 2011 
 
593 
The percent relative efficiencies (PRE’s) of the different estimators with 
respect to the proposed estimator t are computed by the formula  
 
PRE (t,.) = 100*
)t(MSE
(.)MSE  
 
and presented in table 3.2. 
 Table 3.2: PRE’s of various estimators of Y with respect to y  
Estimator PRE 
y  100 
Ry  203.43 
Py  123.78 
RPy  213.64 
*
Ry  214.78 
*
Py  104.35 
*
RPy  235.68 
*
optt  278.21 
 
4. Conclusion 
(i) Table 3.1 shows that there is a wide scope of choosing 1α  and 2α  for 
which our proposed estimator ‘t’ performs better than *RPy . 
(ii)  Table 3.2 shows clearly that the proposed estimator ‘t’ is more efficient 
than all other estimators *P
*
RRPPR y,y,y,y,y,y  and 
*
RPy  with considerable gain 
in efficiency. 
I this paper we have presented a family of dual to ratio-cum-product 
estimators for the finite population mean. For future research the family suggested 
here can be adapted to double sampling scheme using  Kumar and Bahl (2006) 
estimator. 
  
  
594              R. Singh, M. Kumar, P. Chauhan, N. Sawan, F. Smarandache: A general … 
 
 
REFERENCES 
BANDYOPADHYAYA, S. (1980): Improved ratio and product estimators, 
Sankhya, Series C 42,45-49. 
KUMAR, M., BAHL, S., 2006, Class of dual to ratio estimators for double 
sampling, Statistical Papers, 47, 319-326.  
LUI, K. J. (1990): Modified product estimators of finite population mean in finite 
sampling. Communications in Statistics, Theory and Methods, 19(10), 3799-
3807. 
MURTHY, M. N. (1967): Sampling Theory and Methods. Statistical Publishing 
Society, Calcutta, India. 
PRASAD, B. (1989): Some improved ratio type estimators of population mean 
and ratio in finite population sample surveys. Communications in Statistics, 
Theory and Methods, 18, 379-392.  
REDDY, V. N. (1973): On ratio and product methods of estimation. Sankhya, 
Series B 35(3), 307-316. 
SINGH, H. P., SINGH, R., ESPEJO, M.R., PINED, D.M., NADARAJAH, S. 
(2005): On the efficiency of a dual to ratio-cum-product estimator in sample 
surveys. Mathematical Proceedings of the Royal Irish Academy, 105 A(2), 
51-56. 
SINGH, M. P. (1967): Ratio-cum-product method of estimation. Metrika 12,  
34-72. 
SINGH, M. P. (1969): Comparison of some ratio-cum-product estimators. 
Sankhya Series B, 31, 375-378. 
SINGH, R., CAUHAN, P., SAWAN, N., and SMARANDACHE, F. (2007): 
Auxiliary Information and a Priory Values in Construction of Improved 
Estimators. Renaissance High Press. 
SINGH, R., SINGH, J. and SMARANDACHE, F. (2009): Some Studies in 
Statistical Inference , Sampling Techniques and Demography. ProQuest 
Information & Learning. 
SRIVASTAVA, S. K. (1967): An estimator using auxiliary information. Cal. Stat. 
Assoc. Bull., 16, 121-132. 
SRIVENKATARAMANA, T (1980): A dual to ratio estimator in sample surveys. 
Biometrika 67, 199-204. 
 


A GENERAL FAMILY OF DUAL TO RATIO-CUM-PRODUCT ESTIMATOR IN SAMPLE SURVEYS

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A GENERAL FAMILY OF DUAL TO RATIO-CUM-PRODUCT ESTIMATOR IN SAMPLE SURVEYS

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A GENERAL FAMILY OF DUAL TO RATIO-CUM-PRODUCT ESTIMATOR IN SAMPLE SURVEYS

Abstract

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