Ratio and Product Type Estimators Using Stratified Ranked Set Sampling

International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 4 Issue 7 || September. 2016 || PP-20-22
www.ijmsi.org 20 | Page
Ratio and Product Type Estimators Using Stratified Ranked Set
Sampling
Dr. (Prof.) Shashi Bahl1
, Prem Parkash2
1
Department of Statistics, M.D.U., Rohtak-124001
2
Department of Statistics, M.D.U., Rohtak-124001
ABSTRACT: In this paper, we propose a class of estimators for estimating the population mean of variable of
interest using information on an auxiliary variable in stratified ranked set sampling. The bias and Mean
Squared Error of proposed class of estimators are obtained to first degree of approximation. It has been shown
that these methods are highly beneficial to the estimation based on Stratified Simple Random Sampling.
Theoretically, it is shown that these suggested estimators are more efficient than the estimators in Stratified
Random Sampling.
Keywords: Stratified ranked set sampling, Ratio and product type estimators, Ranked set sampling, Auxiliary
variables, Mean squared error, Population mean, Efficiency.
I. INTRODUCTION
Ranked set sampling is cost effective sampling procedure to the commonly used simple random sampling. RSS
was first introduced by Mc Intyre in 1952. Samawi(1996) introduced the concept of Stratified Ranked Set
Sampling to increase the efficiency of estimator of population mean. In this paper we propose estimators based
on the modified ratio and product estimators.
In stratified ranked set sampling, for the hth
stratum of the population, first choose rh independent samples each
of size rh , h = 1, 2, . . . , L. Rank each sample, and use RSS scheme to obtain L independent RSS samples of
size rh , one from each stratum. Let ∑L
h=1= r. This complete one cycle of stratified ranked set sample. The cycle
may be repeated m times until n = mr elements have been obtained. A modification of the above procedure is
suggested here to be used for the estimation of the ratio using stratified ranked set sample. For the h th stratum,
first choose rh independent samples each of size rh of independent bivariate elements from the hth
subpopulation
(Stratum) , h = 1, 2, . . . , L. Rank each sample with respect to one of the variables say Y or X. Then use the RSS
sampling scheme to obtain L independent RSS samples of size rh one from each stratum. This complete one
cycle of stratified ranked set sample. The cycle may be repeated m times until n = mr bivariate elements have
been obtained. We will use the following notation for the stratified ranked set sample when the ranking is on the
variable X. For the k th
cycle and the h th
stratum, the SRSS is denoted by {(Yh(1)k, Xh(1)k),(Yh(2)k, Xh(2)k), . . .
,(Yh(rh)k, Xh(rh)k) : k = 1, 2, . . . , m; h = 1, 2, . . . , L} , where Yh[i]k is the i th Judgment ordering in the i th
set for
the study variable and Xh(i)k is the i th
order statistic in the i th
set for the auxiliary variable.
II. SOME EXISTING ESTIMATORS AND NOTATIONS
Samawi and Siam (2003) gives the combined ratio estimator of the population mean Y using SRSS is as
2.1
The combined product estimator of the population mean Y using SRSS can be also defined as
2.2
Where and
To the first degree of approximation, the Biases and MSEs of and are respectively given by
B( - 2.3
B( 2.4
B( 2.5

Ratio And Product Type Estimators Using Stratified Ranked Set Sampling
And Mean Square Error is given by
MSE( 2.6
And
MSE( 2.7
Where nh = mrh , , , and ,
, ( ( , where
, is the mean of hth
stratum for the variable Y and is the mean of hth
stratum for the
variable X.
III. PROPOSED ESTIMATORS
We propose ratio and product type estimators for population mean using SRSS as
3.1
3.2
Where , and are the stratified ranked set sampling means
for variable Y and X.
To obtain the bias and MSE of , we put
and , So that E(
V( ( =
V( (
Expressing in terms of e’s, we obtain
1
3.3
Taking expectation on both sides we get the bias of as
B 3.4
Squaring both sides of ( ), neglecting terms e’s having power greater than two and then taking expectation , we
get the MSE of to the first degree of approximation as
MSE( )= -2
+ 3.5
1
3.7
Taking expectation on both sides we get the bias of as
B 3.8
and MSE is

MSE( )= +2
+ 3.9
Now we propose Exponential type ratio and product estimators for population mean using SRSS as
3.10
3.11
3.12
The bias and MSE of is given by
3.13
And
MSE ( )=
+ ) 3.14
- = 3.15
The bias and MSE of is given by
3.16
MSE ( )=
+ ) 3.16
IV. EFFICIENCY COMPARISONS
In this section ,we have compared the proposed ratio and product estimators to the SRSS
1.
i.e.
2 .
i.e.
3 .
i.e.
4.
i.e.
The present paper deals with ratio and product type estimators of finite population mean in stratified ranked set
sampling . The bias and MSE of proposed estimators under large approximation are derived. The proposed
estimators for the population mean using SRSS are asymptotically more efficient than the usual estimators in
SRSS .

REFERENCES
[1]. McIntyre, G.A.(1952), A method of unbiased selective sampling using ranked sets, Australian Journal of Agricultural Research, 3,
385-390.
[2]. Samawi, H.M., and Muttlak, H.A. (1996): Estimation of ratio using rank set sampling. The Biometrical Journal, 38, 753-764.
[3]. Samawi, H. M. and Siam, M. I., Ratio estimation using stratified ranked set sample, Metron- International Journal of Statistics,61,
1, pp.75-90, 2003.
[4]. Kowalczyk, B. 2004. “Ranked Set Sampling and its Applications in Finite Population Studies”. Statistics in Transition. 6(7):1031-
1046.
[5]. Chen, Z., Z. Bai, and B.K. Sinha. 2004. Ranked Set Sampling: Theory and Applications. Springer: New York, NY.
[6]. Cochran, W.G. 1977. Sampling Techniques, Third Edition. Wiley Eastern Limited.
[7]. Dell, T.R. and J.L. Clutter. 1972. “Ranked Set Sampling Theory with Order Statistics Background”. Biometrics. 28:545 -555.
[8]. Kadilar, C. and H. Cingi. 2003. “Ratio Estimators in Stratified Random Sampling”. Biometrical Journal. 45(2):218-225.

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