A Separate Exponential Ratio-Type Estimator of finite Population Mean under Power Transformation

A Separate Exponential Ratio-Type Estimator of finite Population Mean under Power Transformation

1Ahmad Bandiya Jega and 2Ran Vijay Kumar Singh 1,2Kebbi State University of Science and Technology, Aliero, Nigeria

Abstract: In this paper, a separate ratio-type exponential estimator for estimating the finite population mean has been proposed. Mathematical expressions for the bias and mean square error (MSE) of the proposed estimator have been derived to the first order of approximation. Theoretical conditions have been obtained under which the proposed estimator is more efficient than the estimators under study. Numerical illustration has also been carried out to compare the efficiency of proposed estimator and found that the proposed estimator was more efficient.

Keywords:- Bias, Exponential estimator, Mean square error, Separate estimator, Stratified sampling.

1. INTRODUCTION

   The problem of estimation of population parameters has been an important issue in sample survey and many methods have been used in order to improve the efficiency of the estimators. 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. In some cases, in addition to mean of auxiliary variable, various other parameters related to auxiliary variable such as coefficient variation, correlation coefficient etc are used to estimate the population parameter. Sisodia and Dwivedi (1981), Rao (1991), Upadhayaya and Singh (1999), developed various estimators to improve the ratio estimators in simple random sampling. Kadilar and Cingi (2003) modified the various estimators under stratified random sampling. Bahl and Tuteja (1991) introduced ratio and product- type exponential estimators which perform better than the classical ratio and product estimators respectively. Singh et al. (2008) proposed a ratio and product-type exponential estimators which were more efficient than the Bahl and Tuteja (1991) estimators. Upadhyaya et al. (2011), Singh and Ahmed (2014), Singh and Ahmed (2015a, 2015b), Singh (2016) did remarkable work in this direction. Singh et al. (2018) suggested combined ratio-type exponential estimator of population mean which was equally efficient as combine linear regression estimator.

   Let U U1,…..,UN

2. NOTATIONS

   be a finite population of size N which are partitioned into K distinct strata with ith

   K

   stratum

   containing

   Ni units i 1, 2….., k such that Ni N . Let a sample of size ni units i 1, 2,…., k

   i1

   be drawn from

   k

   k

   the population using simple random sampling preferably without replacement (SRSWOR). Such that ni n .

   i1

   Let y x

   be the observed values of Y , X

   on the

   jth

   unit of the

   ith

   stratum j 1, 2,…, N

   Moreover, the

   ij , ij

   1 Ni

   i

   1 Ni

   population means of the variables Y and X in the ith stratum are Yi

   Yij , Xi

   N

   N

   i j1

   Xij and the corresponding

   N

   N

   i j1

   1 ni

   1 ni

   sample means of the variable Y and X in the ith stratum are

   yi

   yij

   n

   n

   i i1

   , xi

   xij

   n

   n

   i i1

   respectively.

   Estimators of the variable Y and X , in stratified random sampling are given by;

   k k k

   yst Wi yi i1

   k

   and

   xst Wi xi

   i1

   are the unbiased estimators of the population means

   N

   Y WiYi

   i1

   and

   X Wi Xi

   respectively. Where Wi

   i

   N

   denotes the stratum weight.

   i1

   Let be the correlation coefficient between the study variable and the auxiliary variable.

3. EXISTING ESTIMATORS

   The separate ratio estimator for the population mean Y is defined as;

   k yi

   ysr Wi x X i

   (1)

   i1 i

   and its bias and MSE are given as;

   k 1 2

   B ysr Wii

   i1

   Ri Sxi SYXi

   Xi

   MSE y

   k

   W 2 S2 R2S2 2R S

   (2)

   sr i i yi i xi i YXi i1

   The separate product estimator for the population mean Y is define as;

   k yi . xi

   ysp Wi X

   (3)

   i1 i

   and its Bias and MSE are given as;

   k 1

   B ysp Wii

   i1

   SYXi

   X

   X

   i

   MSE y

   k

   W 2 S2 R2S2 2R S

   (4)

   sp i i yi i xi i YXi i1

   The separate linear regression estimator for the population mean Y is defined as;

   k

   k

   y W y

   • b X xi

     (5)

     sl i

     i1

     i i i

     And its MSE is given as;

     MSE( y

     ) W 2 S 2

     1 2

     (6)

     k

     k

     sl

     i1

     i i Yi YXi

4. ADAPTED ESTIMATORS

   Bahl and Tuteja (1991) ratio-type exponential estimator for the population mean Y under stratified random sampling can be defined as:

   k X i xi

   ystBTR Wi yi exp X

   • x

   (7)

   i1

   i i

   its bias and MSE are given as;

   Bais y

   k

   W 1

   3 R S 2

   1 S

   stBTR i i X i 8

   i Xi

   2 XYi

   MSE y

   i1

   k

   W 2 S2 1 R2S2 R S

   (8)

   stBTR i i yi 4 i xi i YXi

   i1

   Bahl and Tuteja (1991) product-type exponential estimator for the population mean Y under stratified random sampling can be defined as:

   k xi X i

   (9)

   ystBTP Wi yi exp x X

   and bias and MSE are given as;

   i1

   i i

   k 1 1 1

   Bais y

   W S R S 2

   i1

   i1

   stBTP i i X i 2 XYi 8 i Xi

   MSE y

   k

   W 2 S2 1 R2S2 R S

   (10)

   stBTP i i yi 4 i xi i YXi

   i1

   Singh et al (2018) estimator

   y

   y

   A combined ratio-type exponential estimator for population mean under stratified random sampling using information on single auxiliary variable has been suggested by Singh et al ( 2018) given as:

   y

   x

   st

   st

   S st

   S st

   exp X x

   st

   where is a constant. (11)

   X X xst

   Bias and MSE of tS are obtained as

   2 1 1

   4 2 8 3 1

   B yS Y

   COV yst xst

   2 V xst

   2 XY

   k

   2 12

   8 X

   4

   4

   MSE y W 2 S 2

   R2 S 2

   2 1 RS

   S

   i1

   i i Yi

   Xi YXi

   For the optimum value of , the Minimum MSE of tS is obtained as

   k 1 1

   2 2 *2

   MSE yS

   i1 ni

   • Wi S Yi 1

     Ni

     YX

     (12)

     That is the MSE of combined linear regression estimator.

5. THE SUGGESTED ESTIMATOR

   In stratified random sampling, we suggest a separate ratio-type exponential estimator as

   K x

   i

   X x

   i

   i

   i

   i

   t W y

   i i

   i i

   exp i

   P

   i1

   X i X i xi

   (13)

   Where i is a constant.

6. BIAS AND MEAN SQUARE ERROR OF THE SUGGESTED ESTIMATOR In order to obtain the Bias and MSE to the first order of approximatin, let us define

   yi Yi (1 e0i ) and

   Therefore

   E(e0i ) E(e1i ) 0

   xi Xi (1 e1i )

   (14)

   1 1

   S 2

   E(e2 )

   Yi

   0 n N Y 2

   (15)

   i i i

   1 1 S 2

   E(e2 )

   • Xi

     1i n N X 2

     Now, by substituting the values of xi and y from (14) into (13) and on solving, the

     i i i

     1 1 S

     i

     bias of proposed estimator is obtained as;

     E(e0ie1i )

   • YXi

     K 1 1 1

     4 2 8

     3

     2

     1

     ni

     Ni Yi Xi

     Bais t

     W

     i

     i

     i i R S 2

     i S

     P

     (16)

     i1

     ni

     Ni Xi

     8 i Xi

     2 YXi

     Now, MSE of the proposed estimator tP can be obtained by squaring, simplify up to the first order of approximation and taking the expectation we will have;

     K 1 1 2 12

     MSE t

     W 2 S 2 i R2 S 2

     Yi

     Yi

     2

     1 R S

     (17)

     P i

     i1 ni Ni

     4 i Xi i i YXi

     To obtain the optimum mean squared error,

     MSEtP 0 gives 1 iCYi

     i

     iopt 2

     CXi

     and minimum MSE of tP is obtained as

     k 1 1

     2 2 2

     min

     min

     MSE tP

     i1 ni

   • Wi SYi 1 i

   Ni

   (18)

   That is the MSE of the separate linear regression estimator.

7. THEORETICAL EFFICIENCY COMPARISON

   In this section, conditions have been found under which the proposed estimator is more efficient than existing estimators under study.

   1. Comparison of the separate ratio estimator with the proposed estimator

      MSE tP MSE ysr if and only if

      3

      C 1 3

      C 1

      min

      2i Yi

      i max

      2i Yi

      (19)

      2

      CXi ,

      2

      2

      CXi ,

      2

      Therefore the proposed estimator is more efficient than

      ysr

      if condition (19) is satisfied.

   2. Comparison of the separate product estimator with the proposed estimator

      MSE tP MSE ysp if

      3 1

      min 2

      CYi

      max 3 1 2

      CYi

      (20)

      2, 2

      i C i

      2, 2

      i C

      Xi

      Xi

      Therefore the proposed estimator is more efficient than

      y sp

      if condition (20) is satisfied.

   3. Comparison of the Bahl and Tuteja (1991) ratio-type exponential estimator under stratified random sampling with the proposed estimator

      MSE tP MSE ystBTR

      min 0,1 2

      CYi

      max 0,1 2

      CYi

      (21)

      i C i i C

      Xi Xi

      Therefore the proposed estimator is more efficient than

      ystBTR if condition (21) is satisfied.

   4. Comparison of the Bahl and Tuteja (1991) product-type exponential estimator under stratified random sampling with the proposed estimator

      MSE tP MSE ystBTP

      min1, 2

      CYi

      max1, 2

      CYi

      (22)

      i C i i C

      Xi Xi

      Therefore the proposed estimator is more efficient than

      ystBTP if condition (22) is satisfied.

8. NUMERICAL EFFICIENCY COMPARISON

   For numerical illustration the data of Kadilar and Cingi (2003) is used which is given in table 1. Y represents apple production amount (as variable of interest) and X represents number of apple trees (as auxiliary variable) in 854 villages of Turkey in 1999 (Source: Institute of Statistics, Republic of Turkey). The data is stratified by regions of Turkey and from each stratum (region) the samples (villages) are selected by using Neyman allocation.

   Table 1: Data Statistics

   	Ni	ni	X i	Y i	SXi	SYi	i	i	W 2 i
   Stratum1	106	9	24375	1536	49189	6425	0.82	0.102	0.015
   Stratum 2	106	17	27421	2212	57461	11552	0.86	0.049	0.015
   Stratum 3	94	38	72409	9384	160757	29907	0.90	0.016	0.012
   Stratum 4	171	67	74365	5588	285603	28643	0.99	0.009	0.04
   Stratum 5	204	7	26441	967	45403	2390	0.71	0,138	0.057
   Stratum 6	173	2	9844	404	18794	946	0.89	0.006	0.041

   Table 2: Theoretical comparison supported numerically using data set provided in Table 1

   MSE tP MSE ysr		1 = -1.1997	-1.8994 < -1.1997 < -0.5	Satisfied
   		2 = -1.6433	-2.7866 < -1.6433 < -0.5	Satisfied
   		3 = -0.7920	-1.0839 < -0.7920 < -0.5	Satisfied
   		4 = -0.8213	-1.1426 < -0.8213 < -0.5	Satisfied
   		5 = -0.5220	-0.5439 < -0.5220 < -0.5	Satisfied
   		6 = -0.5916	-0.6831 < -0.5916< -0.5	Satisfied
   MSE tP MSE ysp		1 = -1.1997	-3.8994 < -1.1997 < -1.5	Satisfied
   		2 = -1.6433	-4.7866 < -1.6433 < -1.5	Satisfied
   		3 = -0.7920	-3.0839 < -0.7920 < -1.5	Satisfied
   		4 = -0.8213	-3.1426 < -0.8213 < -1.5	Satisfied
   		5 = -0.5220	-2.5439 < -0.5220 < -1.5	Satisfied
   		6 = -0.5916	-2.6831 < -0.5916 < -1.5	Satisfied
   MSE tP MSE ystBTR		1 = -1.1997	-2.3994 < -1.1997 < 0	Satisfied
   		2 = -1.6433	-3.2866 < -1.6433 < 0	Satisfied
   		3 = -0.7920	-1.5839 < -0.7920 < 0	Satisfied
   		4 = -0.8213	-1.6426 < -0.8213 < 0	Satisfied
   		5 = -0.5220	-1.0439 < -0.5220 < 0	Satisfied
   		6 = -0.5916	-1.1831 < -0.5916 < 0	Satisfied
   MSE tP MSE ystBTP		1 = -1.1997	-3.3994 < -1.1997 < 1	Satisfied
   		2 = -1.6433	-4.2866 < -1.6433 < 1	Satisfied
   		3 = -0.7920	-2.5839 < -0.7920 < 1	Satsfied
   		4 = -0.8213	-2.6426 < -0.8213 < 1	Satisfied
   		5 = -0.5220	-2.0439 < -0.5220 < 1	Satisfied
   		6 = -0.5916	-2.1831 < -0.5916 < 1	Satisfied

   Table 3: MSE and PRE of the Proposed Estimator with other estimators

   ESTIMATORS	MSE	PRE
   yst	673477.7	100
   ysr	159137.4	423.2
   y sp	1790757	37.6
   ystBTR	340940	197.5
   ystBTP	1156750	58.2
   yS	202122.1	333.2
   tp	107065.8	629

9. CONCLUSION

In the present study, Singh et al (2018) estimator has been improved by modifying it as separate ratio-type exponential estimator and its properties have been studied. Theoretical conditions have been obtained under which the proposed estimator is more efficient than the estimators under study and supported numerically as mentioned in table 2. Table 3 reveals that the proposed

estimator tp

has smallest mean square than the conventional ratio, product and Bahl and tuteja (1991) estimators under stratified

random sampling as well as Singh et al (2018) estimator. Therefore, tp

estimating the finite population mean.

REFERENCES

is more efficient than the other existing estimators for

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