Rajesh Singh Department of Mathematics, SRM University Delhi NCR, Sonepat- 131029, India
Sachin Malik Department of Statistics, Banaras Hindu University Varanasi-221005, India
Florentin Smarandache Department of Mathematics, University of New Mexico Gallup, NM 87301, USA
An Improved Suggestion in Stratified Random Sampling Using Two Auxiliary Variables
Published in: Sachin Malik, Neeraj Kumar, Florentin Smarandache (Editors) USES OF SAMPLING TECHNIQUES & INVENTORY CONTROL WITH CAPACITY CONSTRAINTS Pons Editions, Brussels, Belgium, 2016 ISBN 978-1-59973-484-2 pp. 9 - 18
Abstract In this paper, we suggest an estimator using two auxiliary variables in stratified random sampling following Malik and Singh [12]. The propose estimator has an improvement over mean per unit estimator as well as some other considered estimators. Expressions for bias and MSE of the estimator are derived up to first degree of approximation. Moreover, these theoretical findings are supported by a numerical example with original data. Key words: Study variable, auxiliary variable, stratified random sampling, bias and mean squared error.
1. Introduction The problem of estimating the population mean in the presence of an auxiliary variable has been widely discussed in finite population sampling literature. Out of many ratio, product and regression methods of estimation are good examples in this context. Diana [2] suggested a class of estimators of the population mean using one auxiliary variable in the stratified random sampling and examined the MSE of the estimators up to the kth order of approximation. Kadilar and Cingi [3], Singh et al. [7], Singh and Vishwakarma [8],Koyuncu and Kadilar [4] proposed estimators in stratified random sampling. Singh [9] and Perri [6] suggested some ratio cum product estimators in simple random sampling. Bahl and Tuteja [1] and Singh et al. [11] suggested some exponential ratio type estimators. In this chapter, we suggest some exponentialtype estimators using the auxiliary information in the stratified random sampling.
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Uses of Sampling Techniques & Inventory Control with Capacity Constraints
L
Consider a finite population of size Nand is divided into Lstrata such that
N h 1
h
N where
is the size of h th stratum (h=1,2,...,L). We select a sample of size n h from each stratum by L
simple random sample without replacement sampling such that n h n , where n h is the h 1
stratum sample size. A simple random sample of size nh is drawn without replacement from the hth stratum such that
Let (yhi, xhi, zhi) denote the observed values of y, x, and z on
the ith unit of the hth stratum, where i=1, 2, 3...Nh. To obtain the bias and MSE, we write L
L
L
h 1
h 1
h 1
y st w h y h Y1 e 0 , x st w h x h X1 e1 , z st w h z h Z1 e 2 Such that, Ee 0 Ee 0 Ee 0 0 L
Vrst w h 1
r s t h
r
s
E yh Y x h X zh Z r
s
Y X Z
t
t
where, L
y st w h y h , y h h 1
L
1 nh
nh
y hi , Y h i 1
Y Y st w h Y h , w h h 1
1 Nh
nh
Y
hi
i 1
Nh N
and 2
V( y st ) Y V200
(1.1)
Similar expressions for X and Z can also be defined.
10
An Improved Suggestion in Stratified Random Sampling Using Two Auxiliary Variables
L
L
2 And E e 0
W f S 2 h h
h 1
Y
2
2 yh
V200 ,
E e12
W f S h 1
X
L
E e 22
W f S 2 h h
h 1
Z
2 h h
Ee 0 e1
V002 ,
2
W f S h 1
V020 ,
2
L
2 zh
W f S
2 h h
2 h h
h 1
2 yxh
YX
L
Ee 0 e 2
2 xh
V110 ,
L
2 yzh
YZ
W f S
and E e1e 2
V101 ,
2 h h
h 1
2 xzh
XZ
V011 ,
where ,
y
Nh
S 2 yh
i 1
Nh
S2zh
z
i 1
Nh
Syzh
2
Yh , Nh 1
h
2
Zh , N h 1 h
z
h
i 1
Nh
S 2 xh
x
i 1
Nh
Syxh
Xh N h 1
h
x
h
i 1
Zh y h Y h , Nh 1
Nh
Sxzh i 1
x
2
Xh yh Yh N h 1 h
X h z h Zh Nh 1
And,
fh
1 1 n h Nh
2. Estimators in literature In order to have an estimate of the study variable y, assuming the knowledge of the population proportion P, Naik and Gupta [5] and Singh et al. [11] respectively proposed following estimators
X t 1 y st x st
X x st t 2 y st exp X x st
(2.1)
(2.2)
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Uses of Sampling Techniques & Inventory Control with Capacity Constraints
The MSE expressions of these estimators are given as
MSE t1 Y V200 V020 2V110 2
(2.3)
2 V MSEt 2 Y V200 020 V110 4
(2.4)
When the information on the two auxiliary variables is known, Singh [10] proposed some ratio cum product estimators in simple random sampling to estimate the population mean of the study variable y. Motivated by Singh [10] and Singh et al. [7], Singh and kumar propose some estimators in stratified sampling as X x st Z z st t 3 y st exp exp X x st Z z st
(2.5)
x st X z st Z t 4 y st exp exp x st X z st Z
(2.6)
X x st z st Z t 5 y st exp exp X x st z st Z
(2.7)
x st X Z z st t 6 y st exp exp x st X Z z st
(2.8)
The MSE equations of these estimators can be written as 2 V V V MSE(t 3 ) Y V200 020 002 V110 - V101 011 4 4 2
(2.9)
2 V V V MSE(t 4 ) Y V200 020 002 V110 V101 011 4 4 2
(2.10)
2 V V V MSE(t 5 ) Y V200 020 002 V110 V101 011 4 4 2
(2.11)
12
An Improved Suggestion in Stratified Random Sampling Using Two Auxiliary Variables
2 V V V MSE(t 6 ) Y V200 020 002 V110 - V101 011 4 4 2
(2.12)
When there are two auxiliary variables, the regression estimator of Y will be
t 7 y st b1h X x st b 2h Z z st
Where b1h
s yx 2
sx
and b 2h
s yz 2
sz
(2.13)
. Here s 2x and s 2z are the sample variances of x and z respectively,
s yx and s yz are the sample covariance’s between y and x and between z respectively. The MSE expression of this estimator is:
L
MSE t 7 Wh2 f h S2yh 1 ρ 2yxh ρ 2yzh 2ρ yxhρ yzh ρ xzh
(2.14)
h 1
3. The proposed estimator Following Malik and Singh [12], we propose an estimator using information on two auxiliary attributes as m1
X x st Z z st t p y st exp exp X x st Z z st
m2
b1h X x st b 2h Z z st
(3.1)
Expressing equation (3.1) in terms of e’s, we have m2 e m1 e 2 1 exp b1h Xe1 b 2h Ze 2 t p Y 1 e 0 exp 2 e 2 e 1 2
m e m 2e 2 m e m m e e m 2e 2 m e e m e e Y 1 e 0 1 1 1 1 2 2 1 2 1 2 2 2 2 0 2 1 0 1 2 4 2 4 4 2 2 b1h e1 X b2h e2 Z (3.2) Squaring both sides of (3.2) and neglecting the term having power greater than two, we have
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Uses of Sampling Techniques & Inventory Control with Capacity Constraints
t
p
2 me m e Y Y e 0 1 1 2 2 b1h e1 X b 2h e 2 Z 2 2
2
(3.3)
Taking expectations of both the sides of (3.3), we have the mean squared error of t p up to the first degree of approximation as
MSE(t p ) Y V200 P1 P2 YP3 2
(3.4)
Where,
m1B 2h V011 m 2 B1h V011 m 2 B 2h V002
m12 V020 m 22 V002 m1m 2 V011 P1 m1V110 m 2 V101 4 4 2 2 P2 B1h V020 B 22h V002 2B1h B 2h V011 P3 2B1h V110 2B2h V101 m1B1h V020
L
Where, B1h
Wh2f h ρ yxhSyhSxh h 1
L
W f S h 1
2 h h
(3.5)
L
and B 2h
2 xh
W f ρ 2 h h
h 1
yzh
S yhSzh
L
W f S h 1
2 h h
2 zh
The optimum values of m1 and m 2 will be
2 4 B1h V011V002 B2h V011 B1h V020 V002 B 2h V011V002 m1 2 Y V020 V002 V011 2 4 B1h V011V020 B 2h V011 B1h V011V020 B 2h V002 V020 m2 2 Y V020 V002 V011
(3.6)
Putting optimum values of m1 and m 2 from (3.6), we obtained min MSE of proposed estimator t p . 4. Efficiency comparison In this section, the conditions for which the proposed estimator t p is better than y st , t 1 , t 2 , t 3 ,
t 4 , t 5 , t 6 , and t 7 . The variance is given by
14
An Improved Suggestion in Stratified Random Sampling Using Two Auxiliary Variables
2
V( y st ) Y V200
(4.1)
To compare the efficiency of the proposed estimator with the existing estimator, from (4.1) and (2.3), (2.4), (2.9), (2.10), (2.11), (2.12) and (2.14), we have 2
V( y st ) - MSE(t p ) Y P1 P2 YP3 0
(4.2)
MSE(t 1 ) - MSE(t p ) Y V020 2V110 Y P1 P2 YP3 0 2
2
(4.3)
2 2 V MSE(t 2 ) - MSE(t p ) Y 020 V110 - Y P1 P2 YP3 0 4
(4.4)
2 2 V V V MSE(t 3 ) - MSE(t p ) Y 020 002 V110 - V101 011 - Y P1 P2 YP3 0 4 2 4
2 2 V V V MSE(t 4 ) - MSE(t p ) Y 020 002 V110 V101 011 - Y P1 P2 YP3 0 4 2 4
2 2 V V V MSE(t 5 ) - MSE(t p ) Y 020 002 V110 V101 011 - Y P1 P2 YP3 0 4 2 4
2 2 V V V MSE(t 6 ) - MSE(t p ) Y 020 002 V110 V101 011 - Y P1 P2 YP3 0 4 2 4
(4.5)
(4.6)
(4.7)
(4.8)
Using (4.2) - (4.8), we conclude that the proposed estimator outperforms than the estimators considered in literature. 5. Empirical study In this section, we use the data set in Koyuncu and Kadilar [4]. The population statistics are given in Table 3.2.1. In this data set, the study variable (Y) is the number of teachers, the first auxiliary variable (X) is the number of students, and the second auxiliary variable (Z) is the number of classes in both primary and secondary schools. Table 5.1: Data Statistics of Population N1=127
N2=117
N3=103
N4=170
N5=205
N6=201
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Uses of Sampling Techniques & Inventory Control with Capacity Constraints
n1=31
n2=21
n3=29
n4=38
n5=22
n6=39
= 883.835 810.585 = 703.74 = 424.66
= 644
= 1033.467
= 403.654
=711.723
= 413 = 267.03
573.17 = 393.84
=30486.751
=15180.760
=27549.697
=18218.931
=8997.776
=23094.141
=20804.59
=9211.79
=14309.30
=9478.85
= 5569.95
=12997.59
=25237153.52
=9747942.85
=28294397.04
= 14523885.53
=3393591.75
=15864573.97
= 0.936
= 0.996
=0.994
= 0.983
= 0.989
= 0.965
= 555.5816
= 365.4576
=612.9509281
= 458.0282
= 260.8511
= 397.0481
= 498.28 = 498.28
= 318.33
= 431.36
= 227.20
= 313.71
= 480688.2
= 230092.8
= 623019.3
= 364943.4
= 101539
= 277696.1
= 15914648
= 5379190
= 164900674.56
= 8041254
= 2144057
= 8857729
= 0.978914
= 0.9762
= 0.983511
= 0.982958
= 0.964342
= 0.982689
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An Improved Suggestion in Stratified Random Sampling Using Two Auxiliary Variables
We have computed the pre relative efficiency (PRE) of different estimators of Y st with respect to y st and complied in table 5.2: Table 5.2: Percent Relative Efficiencies (PRE) of estimator
S.No.
Estimators
PRE’S
1
y st
100
2
t1
1029.46
3
t2
370.17
4
t3
2045.43
5
t4
27.94
6
t5
126.41
7
t6
77.21
8
t7
2360.54
9
tp
4656.35
6. Conclusion In this paper, we proposed a new estimator for estimating unknown population mean of study variable using information on two auxiliary variables. Expressions for bias and MSE of the estimator are derived up to first degree of approximation. The proposed estimator is compared with usual mean estimator and other considered estimators. A numerical study is carried out to
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Uses of Sampling Techniques & Inventory Control with Capacity Constraints
support the theoretical results. In the table 5.2, the proposed estimator performs better than the usual sample mean and other considered estimators. References [1] Bahl, S. and Tuteja, R.K. (1991): Ratio and product type exponential estimator. Infrm.andOptim. Sci., XII, I, 159-163. [2] Diana, G. (1993). A class of estimators of the population mean in stratified random sampling. Statistica53(1):59–66. [3] Kadilar,C. and Cingi,H. (2003): Ratio Estimators in Straitified Random Sampling. Biometrical Journal 45 (2003) 2, 218-225. [4] Koyuncu, N. and Kadilar, C. (2009) : Family of estimators of population mean using two auxiliary variables in stratified random sampling. Comm. In Stat. – Theory and Meth., 38:14, 23982417. [5] Naik,V.D and Gupta, P.C., 1996: A note on estimation of mean with known population proportion of an auxiliary character. Jour. Ind. Soc. Agri. Stat., 48(2), 151-158. [6] Perri, P. F. (2007). Improved ratio-cum-product type estimators.Statist. Trans. 8:51–69. [7] Singh, R., Kumar, M., Singh,R.D.,and Chaudhary, M.K.(2008): Exponential Ratio Type Estimators in Stratified Random Sampling. Presented in International Symposium On Optimisation and Statistics ( I.S.O.S).-2008 held at A.M.U ., Aligarh, India, during 29-31 Dec 2008. [8] Singh, H., P. and Vishwakarma, G. K. (2008): A family of estimators of population mean using auxiliary information in stratified sampling. Communication in Statistics Theory and Methods, 37(7), 1038-1050. [9] Singh, M. P. (1965): On the estimation of ratio and product of population parameter. Sankhya 27 B, 321-328. [10] Singh, M.P. (1967): Ratio-cum-product method of estimation. Metrika 12, 34- 72. [11] Singh, R.,Chauhan, P., Sawan, N. and Smarandache,F. (2007): Auxiliary information and a priory values in construction of improved estimators. Renaissance High press. [12] Malik and Singh (2013): An improved estimator using two auxiliary attributes. Applied Mathematics and Computation, 219(2013), 10983-10986
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