Improved Exponential Estimator in Stratified Random Sampling

Improved Exponential Estimator in Stratified Random Sampling Rajesh Singh Department of Statistics B.H.U., Varanasi (U.P.) India [email protected]

Mukesh Kumar Department of Statistics B.H.U., Varanasi (U.P.) India

Manoj K. Chaudhary Department of Statistics B.H.U., Varanasi (U.P.) India

Cem Kadilar Department of Statistics Hacettepe University Beytepe 06800, Ankara Turkey [email protected]

Abstract

 

In this article we have considered the problem of estimating the population mean Y in the stratified random sampling using the information of an auxiliary variable x which is correlated with y and suggested improved exponential ratio estimators in the stratified random sampling. The mean square error (MSE) equations for the proposed estimators have been derived and it is shown that the proposed estimators under optimum condition performs better than estimators suggested by Singh et al. (2008). Theoretical and empirical findings are encouraging and support the soundness of the proposed estimators for mean estimation.

Keywords: Auxiliary variable, exponential ratio-type estimates, mean square error, stratified random sampling. 2000 AMS Classification: 62D05 1. Introduction In survey sampling, it is well established that the use of auxiliary information results in substantial gain in efficiency over the estimators which do not use such information. However, in planning surveys, the stratified sampling has often proved needful in improving the precision of estimates over simple random sampling. In some cases, in addition to mean of auxiliary variable, various parameters related to auxiliary variable such as coefficient of variation, kurtosis, correlation coefficient, etc. may also be known. For these cases, many authors, Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

Rajesh Singh, Mukesh Kumar, Manoj K. Chaudhary, Cem Kadilar

such as Upadhayaya and Singh (1999), Sisodia and Dwivedi (1981), Singh and Tailor (2003), Singh et al. (2007), developed various estimators to improve the ratio estimators in the simple random sampling. Kadilar and Cingi (2003) adapted the estimators in Upadhyaya and Singh (1999) to the stratified random sampling. In this study, under stratified random sampling without replacement scheme, we suggest two improved exponential estimators which are more efficient than estimators proposed by Singh et al. (2008). There are some recent studies proposing the family of estimators without using the exponential function in literature, such as Koyuncu and Kadilar (2009(a,b)), Khoshnevisan et al. (2007), Singh and Vishwakarma (2006, 2008) etc. However, in this article, the proposed family of estimators depends on the exponential function. We assume that the population comprises N units, which can be uniquely L

partitioned into L strata of size N1, N2,…, NL such that  N h  N. The strata h 1

weights W h = Nh / N (h=1, 2,…, L) are assumed known. Let (yhi, xhi) (i=1, 2,…, Nh) denote the values of variates (y, x) respectively for the i-th unit in the h-th stratum and Yh and X h denote stratum means. When the population mean of the auxiliary variable, X , is known, Hansen et al. (1946) suggested a combined ratio estimator for estimating the population mean of the study variable Y  :

y t1  st X , x st

(1.1)

L

L

where y st   w h y h , h 1

Here y h 

nh

1  y hi , n h i 1

L

x st   w h x h , and X   w h X h . h 1

and x h 

h 1

nh

1  x hi . n h i 1

The MSE of t1 to a first degree of approximation is given by





L MSE t1    w 2h  h S 2yh  R 2S 2xh  2RS yxh , h 1

(1.2)

where  h   1  1  , R  Y  st is the population ratio, S2yh is the X X st Nh  nh population variance of a study variable, S 2xh is the population variance of the auxiliary variable and S yxh is the population covariance between study and auxiliary variables in the stratum h. Y

68

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

Improved Exponential Estimator in Stratified Random Sampling

2. Kadilar and Cingi estimator Kadilar and Cingi (2003) introduced an estimator for population mean using known value of some population parameters in stratified random sampling given by y st t2 = X st , a , b (2.1) x st , a , b L

L

h 1

h 1

where x st , a , b   w h a h x h  b h  , X st ,a , b   w h a h X h  b h  and ah, bh are the functions of the known parameters of the auxiliary variable such as coefficient of variation Cxh, coefficient of kurtosis 2h (x) etc., or some constants in the h th stratum . The MSE of the estimator t2 is given by L

MSE( t 2 ) =  W h 1

where R a , b 

Yst X st , a , b

2 h h

S

2 yh

 R

2 a ,b

a 2h S 2xh  2 R

a ,b

a h S yxh



,

(2.2)

.

Bahl and Tuteja (1991) suggested an exponential ratio type estimator for population mean in simple random sampling as

X  x t 3  y exp  . X  x

(2.3)

Motivated by Bahl and Tuteja (1991), Singh et al. (2008) adapted this estimator to the stratified random sampling as

 X  x 4i  t 4i  y st exp 4i , X  x 4i   4i

i = 0, 1, 2, 3, 4.

(2.4)

where L

x 40   w h x h , h 1

L

x 41 



w h x h  C xh  ,

L

X 40   w h X h , h 1 L

X 41 

h 1

w

h

x h   2h (x) ,

L

w L



,

 w X

h

  2h (x )

h

h

,

h 1

x h 2 h ( x )  C xh  ,

h 1

x 44 

 C xh

L

X 42 

h 1

x 43 

h

h

h 1

L

x 42 

 w X

w h x h C xh   2h ( x ) ,

h 1


 w X 

 C xh

 w X

  2h (x )

L

X 43 

h 2h (x )

h

,

h 1 L

X 44 

h

h C xh

.

h 1

69


The bias and MSE of the estimators t4i ( i=0,1,2,3,4) can be obtained respectively by following expressions: 1 L 2 3 1  Bias( t 4i )  (2.5)  w h  h  R 4i a 2hi S 2xh  a hi S yxh  , (i=01,2,3,4) X 4i h 1 2 8    L R2 MSE ( t 4i )   w 2h  h  S 2yh  R 4i a hi S yxh  4i a 2hi S 2xh  , (i=01,2,3,4) (2.6)   4 h 1   where, R 40 

Yst L

h 1

Yst L

 w h ( X h   2h ( x )) Yst L

Yst

R 43 

,

L

,

 w h ( X h  2h ( x )  C xh )

h 1

R 44 

,

L

 w h ( X h  C xh )

 w h Xh

h 1

R 42 

Yst

R 41 

,

h 1 L

,

Yst   w h Yh  Y. h 1

 w h ( X h C xh   2h ( x ))

h 1

We would like to remark that for various values of parameters in (2.4), we get five estimators (for I = 0, 1, 2, 3, 4) as shown in Table 1. The MSE expressions for these estimators are also given in the Table 1. Table 1: Some members of family of estimators t4i The values of ah,bh

Estimator

ah=1, bh=0

 X  x st  t 40  y st exp    X  x st 

ah=1, bh=Cxh

 X  x 41  t 41  y st exp  41   X 41  x 41 

ah=1, bh=2h(x)

 X  x 42  t 42  y st exp  42   X 42  x 42 

ah=2h(x), bh=Cxh

 X  x 43  t 43  y st exp  43   X 43  x 43 

ah=Cxh,, bh=2h(x)

 X  x 44  t 44  y st exp  44   X 44  x 44 

70

MSE of the Estimator   R240 2  2  2 S  whh Syh R40Syxh 4 xh  h1  L

   L R241 2  2  2 w  S  R S  Sxh  h h  yh 41 yxh  4 h1  

 R242 2  2  2 S whhSyh R42Syxh 4 xh h1   L

L



h 1



L

w 2h  h  S 2yh 

 w 2h  h  S2yh  R 43 2h x S yxh 





h 1



 R 243 2  2h ( x )S 2xh   4 

R 244 2 2   R 44 C xh S yxh  C xh S xh  4 



Remark 1 - Here we would like to mention that the choice of the estimator depends on the availability and values of the various parameter(s) used (for choice of the parameters ah and bh refer to Singh et al. (2008) and Singh and Kumar(2009)). Singh et al. (2008) showed that all of these modified estimators (t 4i) have a smaller MSE than the MSE of the stratified version of exponential ratio type estimator t3st under certain conditions:  X  x st  t 3st  y st exp (2.7)   X  x st  Note that t3st is the same estimator with t40. Singh et al. (2008) reported the minimum value of mean square error of the L

estimator t4i as MSE ( t 4i ) min   w 2h  h S 2yh (1   c2 )

(2.8)

i 1

where,  c is combined correlation coefficient in stratified sampling across all  L 2    w h  h  h S yh S xh     i 1 

2

strata. It is calculated as  c2  L

L 2 2 w  S  h h yh  w 2h  h S 2xh 11 i 1

.

3. Suggested modified exponential ratio estimator In stratified random sampling, Kadilar and Cingi (2005) introduced a ratio estimator as t 5  k t1 , (3.1) where the constant k is obtained by minimizing the MSE of t 5. Motivated by Kadilar and Cingi (2005), we propose the following modified estimator t 6i  k t 4i

 X  x 4i   k y st exp 4i ,  X 4i  x 4i 

i = 0, 1, 2, 3, 4.

(3.2)

To obtain the MSE of t6i to the first degree of approximation, let us define y Y a *  A *i . Using these notations we have e 0  st and e1*i  i Y A *i





y st  Y 1  e 0  and a *i  A *i 1  e1*i ,


71


where L

L

A *i   w h a hi X h , h 1

a *i   w h a hi x h , h 1

 

* E e 0   E e1i  0,

 

E e 02 

1 Y2

L

 w 2h  h S 2yh , h 1

 

1 L 2 E e12i   w h  h a hi S 2xh , *2 A i h 1





E e 0 e1*i 

1

L

 w 2h  h a hi S yxh .

Y A *i h 1

Expressing (3.2) in terms of e’s, we have  * *  * *  1   e  e1i   1i 1  t 6i  k i Y 1  e 0  exp   2  2     *2 *2  *e1*i 3θ e1i *e 0 e1*i  .  k i Y 1  e 0    2 8 2    

(3.3)

A* where *i  i . X 4i Bias (t 6i )  E (t 6i  Y )

 Yst ( k i  1)  k i Yst (

 Yst (k i  1) 

3*i E (e1*i2 ) 8

* E (ee1*i )  i ) 2

ki L 2 3 1   w h  h  R 4i a 2hi S 2xh  a hi S yxh  X 4i h 1 2 8 

 Yst ( k i  1)  k i Bias ( t 4i )

(3.4)

From (3.3), we have R 24i 2 2  MSE ( t 6i )  k i  1 Y  R 4i a hi S yxh  a S 4 hi xh   L 1 3   2k i ( k i  1)  w 2h  h  R 24i a 2hi S 2xh  R 4i a hi S yxh  . (3.5) 2 8  h 1 2

72

2

L



 k i  w 2h  h  S 2yh  h 1  2



MSE of the estimator t6i can be rewritten as MSE t 6i   k i  12 Y 2  k i 2 MSE t 4i   2 k i k i  1Yst Bias t 4i 

(3.6)

To obtain the optimum value of k, we partially differentiate the expression (3.6) with respect to k and put it equal to zero as follows:  MSE t 6i   0 , k





 2 k MSEt 4i   (k i  1) 2 Y 2  2k i (k i  1)Yst Bias(t 4i )  0 , k k *i 

Y 2  Ai , Y 2  MSE t 4i   2A i

i = 0, 1, 2, 3, 4.

(3.7)

where L 1 3  A i  Yst Bias (t 4i )   w 2h  h  R 24i a 2hi S 2xh  R 4i a hi S yxh . 2 8  h 1 * Note that 0< k i