Review Article
Volume 5 Issue 2 - February 2018
DOI: 10.19080/BBOAJ.2018.04.555659
Biostat Biometrics Open Acc J
Copyright © All rights are reserved by Etebong P C
Improved Family of Ratio Estimators of Finite
Population Variance in Stratified Random Sampling
Etebong P C*
Department of Mathematics and Statistics, University of Uyo, Nigeria
Submission: December 19, 2017; Published: February 23, 2018
*Corresponding author: Etebong P Clement, Department of Mathematics and Statistics, University of Uyo, Nigeria, Email:
Biostat Biometrics Open Acc J 5(2): BBOAJ.MS.ID.555659 (2018) 0048
Introduction
In sampling theory population information of the auxiliary
variable such as the total, mean and variance are often used
to provide additional information which help to increase the
efficiency of the estimation of the population parameter(s)
of interest. When the information on an auxiliary variable is
known, one can use the ratio, product and regression estimators
to improve the performance of the estimator of the study
variable. When the correlation between the study variable and
the auxiliary variable is positive, ratio method of estimation
is quite effective. Many authors like [5-19] among others have
proposed different ratio estimators (of total or mean) in sample
surveys.
The estimation of the finite population variance has been of
great significance in various fields such as Industry, Agriculture,
Medical and Biological sciences. Various authors such as [4,20-
35] have used auxiliary information to improve the efficiency
of the estimator of population variance of the study variable.
In this paper, the problem of constructing efficient estimator
of the population variance is considered and a new family of
exponential ratio estimators of population variance in stratified
random sampling that is as efficient as the general regression
estimator is introduced.
Basic notations and definitions
Consider a finite population ( ) 1, 2
, ,
N
ππ π Π= … of size (N). Let
(X)and (Y) denote the auxiliary and study variables taking
values X
i
and Y
i
respectively on the i-th unit ( 1, 2, , )
i
i N π = … of
the population. Let the population be divided into K strata with
N
k
units in the k
th
stratum from which a simple random sample
of size n
k
is taken without replacement. The total population
size be
1
K
k
k
N N
=
=
∑
and the sample size
1
,
K
k
k
n n
=
=
∑
respectively.
Associated with the ith element of the h
th
stratum are y
ki
and x
ki
with
0
ki
x >
being the covariate; where y
hi
is the y value of the
ith element in stratum k, and x
ki
is the x value of the ith element
in stratum , 1, 2, , kk K = … , and 1, 2, . ,
k
i N = … For the k
th
stratum, let
k
k
N
W
N
= be the stratum weights and
k
k
k
n
f
N
= the sample fraction.
Let the k
th
stratum means of the study variable Y and auxiliary
variable X ( ) 1 1
;
k k n n
k i ki k k i ki k
y y n x x n
= =
=Σ =Σ
be the unbiased estimator of
the population stratum means ( ) 1 1
;
k k
N N
k i ki k k i ki k
Y y N X x N
= =
=Σ =Σ
of Y
and X respectively, based on n
k
observations.
Let,
( )
2 2
1 ,
y
ky ky kS
s S e = +
( )
2 2
1
x
kx kx kS
s S e = +
So that,
( ) ( )
0,
x y
kS kS
Ee Ee = =
( ) ( )
2
2
1,
x
kS k
Ee x γ β = −
( ) ( )
22
1
x y
kS kS k
Ee e γ λ = − ,
Where,
( ) ( )
2 2
2 2
1 1
1 1 1 1 1
; ;
1 1
k k
N N k
k kx i ki k ky i ki k
k k k k k
f
S x X S y Y
n n N N N
γ
= =
−
= = − = Σ − = Σ −
− −
( ) ( )
1
1
1
,
k
N
kxy i ki k ki k
k
S x X y Y
N
=
−
= Σ − −
( ) ( ) ( )
2
2 40 20
, , ,
k k k
y yy yy β µ µ =
( ) ( ) ( )
2
2 40 20
, , ,
k k k
x xx xx β µ µ =
( ) ( )
22 20
, , ,
yx k k
yx yx λ µ µ =
Biostatistics and Biometrics
Open Access Journal
ISSN: 2573-2633
Abstract
This paper introduces a new family of exponential ratio estimators of population variance in stratified random sampling and studies its
properties. Based on Bahl & Tuteja [1], Kadillar & Cingi [2] and Solanki et al. [3], membership of the new family of estimators is identified.
Analytical and numerical results show that under certain prescribed conditions, the new estimator has equal optimal efficiency with the
regression estimator of population variance but always fares better than the classical ratio estimator of population variance by Isaki [4] and
every identified existing estimator of its family.
Keywords: Optimum estimator; Large sample approximation; Optimal efficiency; Regression estimator