Sampling Algorithm of Order Statistics for Conditional ...

Improved Exponential Type Estimators for Finite Population Mean in Stratified Random Sampling Nursel KOYUNCU Hacettepe University, Department of Statistics, Beytepe, Ankara, Turkey [email protected]

Abstract In this paper new exponential type estimators are suggested for estimating the population mean in stratified random sampling. The biases and mean square errors of the suggested estimators are obtained. Also an empirical study is carried out to show the properties of the proposed estimators.

Keywords: Ratio estimator; Exponential estimator; Mean square error; Efficiency; Stratified random sampling.

1.

Introduction

In sampling theory the role of auxiliary information has a great importance. It is well known that using suitable auxiliary information such as population total, mean, skewness, correlation, attribute etc. can improve precision of estimates. Many authors have used this information in their ratio, product and exponential type estimators to get more efficient estimator under different sampling design. In stratified random sampling scheme the authors including Diana (1993), Singh and Vishwakarma (2008), Koyuncu and Kadilar (2009, 2010), Tailor et al. (2012), Koyuncu (2013), Yadav et al. (2014) etc. have suggested estimators using auxiliary information. In this paper we have tried to generalize Singh et al. (2008) and Yadav et al. (2014) estimators and suggested two exponential type estimators in stratified random sampling. Consider a finite population U  u1 , u 2 ,..., u N  of size N and let y and x, respectively, be the study and auxiliary variable associated with each unit u j  j  1, 2, ...., N  of the population. Assume that the population of size, N, is stratified into L strata with h-th stratum containing N h units, where h  1, 2, ..., L such that

L

N h 1

h

 N . A simple random

sample of size nh is drawn without replacement from the h-th stratum such that L

n h 1

h

 n . Let  y hi , xhi  denote the observed values of y and x on the i-th unit of the h-th nh

Nh L y hi y , y st  Wh y h , and Yh   hi , i 1 n h i 1 N h h 1 L N Y  Yst  WhYh be the sample and population means of y, respectively, where Wh  h is N h 1 the stratum weight.

stratum, where i  1, 2, ..., N h . Moreover, let y h  

Pak.j.stat.oper.res. Vol.XII No.3 2016 pp429-441

Nursel KOYUNCU

To obtain the bias and MSE, let us define e0   y st  Y  Y and e1  xst  X  X . Using these notations, E e0   E e1   0 , L

Vrs  W h 1

r s h





E  y h  Yh  xh  X h  . Y rXs r

s

(1)

Using (1), we can write L

 

Ee

2 0

Wh2 h S yh2 h 1

Y2

L

 

 V20 , E e

2 1

Wh2 h S xh2 h 1

X2

L

E e0 e1  

 V02

W h 1

2 h

 h S xyh

XY

 V11

where

 y Nh

S

2 yh



i 1

hi  Yh 

Nh 1 n and f h  h . Nh

 x Nh

2

,

S

2 xh



i 1

hi  X h 

Nh 1

 x Nh

2

, S xyh 

i 1

hi

 X h  y hi  Yh  Nh 1

, h 

1 fh , nh

Singh et al. (2008) have suggested estimators of population mean using auxiliary information, given by  X  x1   (2) t1  y st exp  1  X 1  x1   X  x2   t 2  y st exp  2  X 2  x2  L

x1  Wh x h  C xh  ,

where

h 1

(3) X 1  Wh X h  C xh  , L

h 1

L

x 2  Wh x h   2 h x 

and

h 1

X 2  Wh X h   2 h x  . L

h 1

To generalize t1 and t 2 estimators in a class we can define  X  x st k1  y st exp   X  x st  2 Ast

  

(4)

L

where Ast   Wh Ah and Ah may be some population information of the auxiliary h 1

variable for h-th stratum such as S xh , coefficient of variation C xh , skewness 1h x  , kurtosis  2 h x  , correlation coefficient  h  xy  . Note that t1 and t 2 estimators are member of k1 if we take Ah as C xh and  2 h x  respectively as given in Table2. Expressing (4) in terms of e’s, we have

k

1

430

  X 3X 2 X   2  Y   Y  e1  e  e  e e  1 0 0 1 2 2  X  Ast  2  X  Ast  8  X  Ast   

(5)


Improved Exponential Type Estimators for Finite Population Mean in Stratified Random Sampling

Squaring both sides of (5) and neglecting the terms of e’s having greater than two, we can write

k

1

  2 X2 X    Y   Y 2 e02  e2  e0 e1  2 1  X  Ast   4  X  Ast  

(6)

Taking expectation of (6) we get the MSE of k1 , under the first order of approximation is given by L L   Y2 Y L  2 2 2 2 MSE k1   Wh2 h S yh  W  S  W  S  h h xyh  2  h h xh X  Ast  h1   4X  Ast  h1  h 1    L Y2 Y 2 2   Wh2 h  S yh  S  S xh xyh 2   X  A   h 1 4 X  A st  st    L  2 1 2 2  (7)  Wh2h  S yh  Rk1S xh  Rk1S xyh  4   h 1 Y where Rk1  .  X  Ast  Secondly Singh et al. (2008) have suggested estimators using more auxiliary information  X  x3   (8) t 3  y st exp  3 X  x 3   3  X  x4   t 4  y st exp  4  X 4  x4 

(9) X 3  Wh X h  2 h x   C xh ,

L

L

where x3  Wh xh  2 h x   C xh , h 1

h 1

x4  Wh xh C xh   2 h x  , X 4  Wh X h C xh   2 h x  . L

L

h 1

h 1

To generalize t 3 and t 4 estimators let us define following class

 A*  a * k 2  y st exp  * st * st  Ast  a st  2 Bst

  

(10)

L

L

L

h 1

h 1

h 1

where Ast*  Wh X h Ah , ast*  Wh xh Ah , Bst   Wh Bh , Ah and Bh may be some population information of the auxiliary variable for h-th stratum such as S xh , coefficient of variation C xh , skewness 1h x  , kurtosis  2 h x  , correlation coefficient  h  xy  . To obtain the bias and MSE, let us define e1* 

a

* st

  W A  x L

A

* st

* st

A

h 1

h

h

h

* st

A

 Xh 

r s t E  yh  Yh   xh  X h     Whr  s Ahs  r *s t Y Ast X h 1 L

,

Vr*,s ,t


431

Nursel KOYUNCU

Using these notations we can write following notations L

 

* E e02  V200  V20

L

E e e   * 0 1

W

2 h

h 1

E  e1*2  

Ah h S xyh

YAst*

W h 1

2 h

Ah2 h S xh2 *2 st

A

*  V110

L

L

E  e1e1*  

Wh2 Ahh S x2 h 1

XAst* L

E e0 e1  

W h 1

2 h

h S xyh

*  V020

 

* ,  V011

E e12 

W h 1

2 h

h S xh2

X2

*  V101  V11 .

XY

Expressing (10) in terms of e’s, we have

 

2  Ast* Ast* 3 * k2  Y  Y   e   2 Ast*  Bst 1 8 A*  B st st 





*  V002  V02 ,



 

2



2

e1*2  e0 

Ast* 1 * * 3 Ast e0 e1  2 8 Ast*  Bst





2

 e0 e1*2   

(11)

Squaring both sides of (11) and neglecting the terms of e’s having greater than two, we can write

k

  2 Ast* 1 2 2 *2 *2 *   Y  Y e  A e  e e  2  0 4 A*  C 2 st 1  Ast*  Cst  1 0   st st   

(12)

Taking expectation both sides of (12) then we get corrected MSE of Singh et al. (2008) as follows: L L  L 2 2 2 2 2 W  S W A  S Wh2 Ah h S xyh   *2 * h h h xh   h h yh A A st h 1 h 1 MSE  k2   Y 2  h 1 2   * st 2 *2 * Y Ast YAst*  Ast  Cst   4  Ast  Cst      L Y 1 Y2  2 2 2 2   Wh h  S yh  * AS  AS  h 1  Ast  Cst  h xyh 4  Ast*  Cst 2 h xh  

1  2   Wh2h S yh  Rk 2 S xyh  Rk22 S xh2  4   h 1 YA where Rk 2  * h .  Ast  Bst  L

432

(13)


     


Note that t 3 and t 4 estimators are member of k 2 for appropriate values of Ah and Bh as given in Table2. Also we have corrected the MSE of t3 and t4 estimators as given in Appendix with detail. Yadav et al. (2014) have suggested ratio and product exponential estimators are given by  X  x st  a   (14) t RC  y st exp   X  a  1x st   x st  X b  t PC  y st exp   X  b  1x st

  

(15)

 V2  L b a 2 MSEmin tPC  MSEmin t RC  Y 2V20 1  11   Wh2 h S yh 1  c2   V02V20  h 1

 

 

L

where  c 

W h 1

L

W h 1

2 h

2 h

(16)

h S xyh

h S xh2

.

L

W h 1

2 h

h S yh2

In sampling literature many authors have suggested estimators using auxiliary information. Koyuncu and Kadilar (2010) defined a general combined class in stratified random sampling to get a compact expression for the asymptotic variance and avoid tedious calculations, given by

t c  g  y st , u st 

(17)

where u st  x st X and g  y st , u st  is a function of y st and u st . To study the properties of t c we assume following regularity conditions:

 y st , u st 

assumes the value in a closed convex subset R2 of two

1.

The point

2.

The function g  y st , u st  is continuous and bounded in R2 ,

3.

g Y ,1  Y and g 0 Y ,1  1 , where g 0 Y ,1 denotes the first order partial derivative of g with respect to y st ,

4.

The first and second order partial derivatives of g  y st , u st  exist and are

dimensional real space containing the point Y ,1 ,

continuous and bounded in R2 . Note that this class contains many estimator. We can say that Yadav et al. (2014) ratio and product exponential estimators are member of this class. The bias and the MSE of t c are respectively given by Bt c   V02 g 2  YV11 g 3  Y 2V20 g 4 ,

MSEt c   Y 2V20  V02 g12  2YV11g1 .

(18) (19)

where Pak.j.stat.oper.res. Vol.XII No.3 2016 pp429-441

433

Nursel KOYUNCU

g g1  u st

, g2  y st Y ,u st 1

1 2g 2 u st2

, g3  y st Y ,u st 1

By using the optimal value of g1*  

1 2g 2 y st u st

, g4  y st Y ,u st 1

1 2g 2 y st2

. y st Y ,u st 1

Y V11 , the minimum MSE of the estimators in the V02

class t c is found as

 V112  MSE t c min  Y V20 1  (20)   V02V20  which is also the MSE of combined regression type estimators and minimum MSE of Yadav et. al [8]. 2

2.

Proposed Exponential Type Estimators

Motivated by Koyuncu (2012) we propose the following estimator    Ast X  x st    x st   y N   w1 y st  w2    exp    X    Ast X  x st   2 Bst  

(21)

Some new estimators, which are generated from (21) for different combinations of Ah , Bh and  are given in Table 2. To obtain the MSE, we are applying the same procedure as follows:   XAst X 2 Ast2 3    2 (22) y N  w1Y 1  e0   w2 1  e1  1  e1  e  ...  2 1   2 X A  B 8     X A  B st st st st    1 1 3   y N  Y  Y w1  1  w1  Y e0  Y e1  Y e1e0   2Y e12  2 2 8    (23)    1 2 1 1 3 2 2   2  w2 1  e1  e1  e1  e1   e1  2 2 2 8  





where  

XAst . XAst  Bst 

Squaring both sides of (23) and taking expectation we have







 

 

MSE  y N   E Y 2  Y 2 w12 1  e02   2 e12  2e0 e1  w22 1   2   2     1     e12   3 3     Y 2 w1 e0 e1  2   2 e12   Y w2     2     1e12  2  4 4      2 2  Y w1 w2 2  2   e0 e1  2     1  2 e1





 

The MSE of the proposed estimator is given by



MSE y N   Y 2 w12 H  w22 B  Y 2 w1 D  Yw2 G  Y 2  Yw1 w2 F

434

(24)



(25)



where H  1  V20   2V02  2V11  , B  1   2   2     1  2 V02  ,

  3 3    D  V11  2   2V02  , G      2     1V02  2  4 4      2 F  2  2  V11  2     1  2 V02  The optimum values of w1 and w2 , obtained by minimizing (25) respectively, are given by GF  2 DB DF  2GH (26) w1 opt   w2 opt   Y 2  4BH  F   4HB  F 2  Substituting these values in (25) we get the minimum MSE of yN as follows:

 BD 2  DFG  HG 2  MSE  y N   Y 1   4 BH  F 2   2





(27)

Note that the optimum choice of the constants w1 and w2 involve unknown parameters. These quantities can be guessed quite accurately through pilot sample survey or sample data or experience gathered in due course of time (Koyuncu and Kadilar (2009, 2010). Secondly we consider following estimator   *   Ast*  ast*  *  xst  (28) yM   w1 yst  w2    exp  *  *  X    Ast  ast  2 Bst   Some new estimators, which are generated from (28) for different combinations of Ah , Bh and  are given in Table 2. Expressing (28) in terms of e’s, we have

yM   w y  w 1  e1   * 1 st

y

* 2



2   * 3  Ast*  A   * *2 st  1  e  e  ...  1 2 1 *  * 8  Ast  Bst   2  Ast  Bst  

(29)

    1 2 1 * *    Y    w1*Y  Y  w2*  w2* e1   w1 Ye0 e1* 2 2   1 3 3   w2* * e1e1*  w1* *2Ye1*2  w2* *2 e1*2  2 8 8 

M

(30)

Ast* where   * . Ast  Bst *





Squaring both sides of (30) and taking expectation we have  3   MSE  y M   E Y 2  w1Y 2   2   * e0 e1*   *2 e1*2   w12Y 2 1  e02   *2 e1*2  2 * e0 e1* 4    3   (31)  Y w2   2   *e1e1*     1e12   *2 e1*2   w22 1   2     1 e12   *2 e1*2  2e1e1* 4   * * 2 *  w1 w2Y 2  2 e1e1     1e1  2 e0 e1*  2 *2 e1*2  2e0 e1





 










435

Nursel KOYUNCU

The MSE of the proposed estimator is given by MSE  yM   Y 2  w1*2Y 2 H *  w2*2 B*  w1*Y 2 D*  Yw2*G*  w1*w2*YF *  where

* * * H *  1  V200  *2V020  2 *V110 



* * * B*  1   2     1 V002  *2V020  2 *V011

(32)



3  * *  D*   2  *V110   *2V020  4   3  * * *  G*   2  * V011      1V002   *2V020  4   * * * * * * * * F   2  2 V011     1V002  2 V110  2 *2V020  2 V101 

The optimum values of w1 and w2 are given

w1* opt  

G * F *  2 D* B *  4B* A*  F *2 

w2* opt   Y

D* F *  2G* H *  4B* A*  F *2 

(33)

The minimum MSE of yM at optimum values of w1* and w2* , is given by

 B * D *2  D * F *G *  H *G *2  MSE  y M   Y 2 1   4 B * H *  F *2  



3.



(34)

Numerical Example

To analyze the performance of proposed estimator we use the data concerning the number of teachers as the study variable and the number of students as the auxiliary variable in both primary and secondary schools for 923 districts at 6 regions (as 1:Marmara 2:Agean 3:Mediterranean 4:Central Anatolia 5:Black Sea 6:East and Southeast Anatolia) in Turkey in 2007 (Source: Koyuncu and Kadilar (2009)). The summary statistics about the population is given in Table 1. We have generated some new estimators using suitable values for Ah , Bh and  . The MSE values of the k1 , k 2 , y N and y M estimators have been obtained. These values are given in Table 2. When we examine Table 2, we observe that the proposed y M 4 estimator has the smallest. From Table 2 we can conclude that using suitable auxiliary information is very important otherwise we can get worst estimates in each class. 4.

Conclusion

In this paper we have tried to generalize Yadav et al. (2014) estimators and corrected mean square error formula of Singh et al. (2008) and suggested two exponential type estimators in stratified random sampling. To see the performance of estimators we used a real data. We investigate the usage of suitable information of auxiliary variable. We have found that suggested exponential class of estimators is highly efficient than Yadav et al. (2014) and combined regression estimator. 436



References 1.

Diana G. A class of estimators of the population mean in stratified random sampling, Statistica, V.1, 1993, pp.59-66.

2.

Singh H.P., Vishwakarma G.K. A family of estimators of population mean using auxiliary information in stratified sampling, Communications in Statistics-Theory and Methods, V.37, 2008, pp.1038-1050.

3.

Koyuncu N., Kadilar C. Family of estimators of population mean using two auxiliary variables in stratified random sampling, Communications in Statistics: Theory and Methods, V.38, N.14, 2009, pp.2398-2417.

4.

Koyuncu N., Kadilar C. On the family of estimators of population mean in stratified sampling, Pakistan Journal of Statistics, V.26, N.2, 2010, pp.427-443.

5.

Koyuncu N., Kadilar C. On improvement in estimating population mean in stratified random sampling, Journal of Applied Statistics, V.37, N.6, 2010, pp.999-1013.

6.

Tailor R., Chouhan S., Tailor R., Garg N. A ratio-cum-product estimator of population mean in stratified random sampling using two auxiliary variables, Statistica, V.LXXII. N.3, 2012, pp. 287-297.

7.

Koyuncu N. Families of estimators for population mean using information on auxiliary attribute in stratified random sampling, Gazi University Journal of Science, V.26, N.2, 2013, pp.181-193.

8.

Yadav R., Upadhyaya L.N., Singh H.P., Chatterjee S. Improved ratio and product exponential type estimators for finite population mean in stratified random sampling, Communications in Statistics: Theory and Methods, V.43, N.16, 2014, pp.3269-3285.

9.

Singh R., Kumar M., Singh R. D., Chaudhary M.K. Exponential ratio type estimators in stratified random sampling, presented in International Symposium on Optimization and Statistics (I.S.O.S.) at AMU, Aligarh, India, 29–31 Dec., 2008.

10.

Koyuncu N. Efﬁcient estimators of population mean using auxiliary attributes, Applied Mathematics and Computation, V.218, 2012, pp.10900-10905.


437

Nursel KOYUNCU

Table 1:

438

Data Statistics

N 1 =127

N 2 =117

N 3 =103

N 4 =170

N 5 =205

N 6 =201

n1 =31

n2 =21

n3 =29

n4 =38

n5 =22

n6 =39

S y1 =883.835

S y 2 =644.922

S y 3 =1033.467

S y 4 =810.585

S y 5 =403.654

S y 6 =711.723

Y1 =703.74

Y2 =413

Y3 =573.17

Y4 =424.66

Y5 =267.03

Y6 =393.84

S x1 =30486.751

S x 2 =15180.769

S x 3 =27549.697

S x 4 =18218.931

S x 5 =8497.776

S x 6 =23094.141

X 1 =20804.59

X 2 =9211.79

X 3 =14309.30

X 4 =9478.85

X 5 =5569.95

X 6 =12997.59

S xy1 =25237153.52

S xy 2 =9747942.85

S xy 3 =28294397.04

S xy 4 =14523885.53

S xy 5 =3393591.75

S xy 6 =15864573.97

 xy1 =0.936

 xy 2 =0.996

 xy 3 =0.994

 xy 4 =0.983

 xy 5 =0.989

 xy 6 =0.965

 2 x1  =4.593

 2 x2  =18.543

 2 x3  =15.446

 2 x4  =10.162

 2 x5  =21.947

 2 x6  =23.114



Table 2:

Some members of k1 , k 2 , y N and y M and MSE values

Ah

Bh



MSE

t1

C xh

602.5943

t2

 2 h x 

603.8937

t3

 2 h x 

C xh

688.4160

t4

C xh

 2 h x 

589.5579

y N1

 2 h x 

C xh

0

53.1396

yN 2

 2 h x 

C xh

1

556.4491

yN3

 2 h x 

C xh

2

342.4030

yN 4

C xh

 2 h x 

0

52.1180

yN5

C xh

 2 h x 

1

557.3786

yN6

C xh

 2 h x 

2

342.5917

yN7

 h  xy 

C xh

0

52.9641

yN8

 h  xy 

C xh

1

556.6087

yN9

 h  xy 

C xh

2

342.4354

yM1

C xh

 2 h x 

0

51.1939

yM 2

C xh

 2 h x 

1

551.1354

yM 3

C xh

 2 h x 

2

325.882

yM 4

 h  xy 

C xh

0

48.8562

yM 5

 h  xy 

C xh

1

505.5486

yM 6

 h  xy 

C xh

2

331.6634

 

 

b  a  MSEmin t PC  MSEmin t RC


194.2832

439

Nursel KOYUNCU

Appendix

 A*  a * k 2  y st exp  * st * st  Ast  a st  2 Bst

  

(1)

If we take Ah as  2h  x  and Bh as Cxh in Ast* , ast* and Bst respectively we can get L

L

L

h 1

h 1

h 1

Ast*  Wh X h  2 h  x  , ast*  Wh xh  2 h  x  , Bst  WhCxh . Rewriting these values in (1)

we get Singh et al (2008) estimator.  X  x3   t 3  y st exp  3 X  x 3 3  

(2) X 3  Wh X h  2 h x   C xh 

L

L

where x3  Wh xh  2 h x   C xh  h 1

h 1

W A x

 Xh

L

To find the MSE formula let us define e0

y 

L

such that E  e0   E  e1*   0 , E  e02   L

E e e   * 0 1

W h 1

2 h

W h 1

st

Y

 S yh2

2 h h

Y

Y 

2

a *  A* , e  st * st  Ast * 1

L

, E  e1*2  

W h 1

2 h

h 1

h

h

h

Ast*

Ah2 h S xh2

Ast*2

,

Ah h S xyh

YAst*

Expressing k 2 with e term, we have

 Ast*  Ast* 1  e1*    k2  Y 1  e0  exp  *  Ast  Ast* 1  e1*   2 Bst    2 * * 2  A A     e e*2  1 3 1 3 st st * * *2 * * k2  Y  1  A e  e  e  A e e  0 st 0 1 0 1  2  Ast*  Bst  st 1 8  A*  B 2 1  2 8  A*  B 2 st st st st  

* 2 * 2  * A A     e e*2  (3) A 3 1 3 st st * *2 * * st k2  Y  Y   e  e  e  A e e  0 st 0 1 0 1  2  Ast*  Bst  1 8  A*  B 2 1  2 8  A*  B 2 st st st st  

Squaring both sides of (3) and neglecting the terms of e’s having power greater than two, we have

  2 Ast*2 Ast* 2 *2 2 *  k  Y  Y e  e  e e 2  0 1 0 *  4  A*  B 2 1  Ast  Bst   st st  440

(4)



Taking expectation of both sides of (4) we get MSE of estimator k 2

 1 Ast*2 MSE  k2   Y 2  2  4  Ast*  Bst   

L

Wh2 Ah2h S xh2 h 1

*2 st

A

L



Wh2h S yh2 h 1

Y

2

L



* st

A  A  Bst  * st

W h 1

 Ah h S xyh   Ast* Y   

2 h

  YAh 1 Y 2 Ah2  2 2   W   S yh  * S  S  h 1  Ast  Bst  xyh 4  Ast*  Bst 2 xh   L

2 h h

1  2   Wh2h S yh  Rk 2 S xyh  Rk22 S xh2  4   h 1 L

where Rk 2 

YAh .  A  Bst  * st

If we take Ah as  2h  x  and Bh as Cxh in MSE  k2  we get corrected MSE  t3  as     2 2 L Y 2h  x  Y 2h  x  1  2 2 2  MSE  t3   Wh h  S yh  L S xyh  S xh  2 L L 4 L   h 1     Wh X h  2 h  x   WhCxh   Wh X h  2 h  x   WhCxh    h  1 h  1   h 1  h 1    L 1  2   Wh2h S yh  Rt 3 S xyh  Rt23 S xh2  4   h 1

where Rt 3 

Y 2h  x     Wh  X h  2 h  x   Cxh    h 1  L

.


441