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International Journal of Applied Mathematics & Statistical Sciences (IJAMSS) ISSN(P): 2319-3972; ISSN(E): 2319-3980 Vol. 2, Issue 5, Nov 2013, 33-44 © IASET

EXPONENTIAL TYPE RATIO AND PRODUCT ESTIMATOR FOR RATIO OF TWO POPULATION MEANS SHAILENDRA RAWAL1 & RAJESH TAILOR2 1

Maharaja College, Vikram University, Ujjain, Madhya Pradesh, India

School of Studies in Statistics, Vikram University, Ujjain, Madhya Pradesh, India

ABSTRACT In this paper suggested ratio and product type exponential estimator for ratio of two population mean. Bias and mean squared error of suggested estimator have been obtained suggested estimator have been compared with usual estimator, ratio estimator given by Singh (1965). An empirical study has been carried out to demonstrate the performance of suggested estimator.

KEYWORDS: Ratio and Product Type Estimator, Bias, Mean Square Error 1. INTRODUCTION This paper deals with the problem of estimation of ratio of two population means. Bahl and Tuteja (1991) suggested exponential type estimator of ratio of two population means. In this p we have proposed modified exponential type estimator for ratio of two population means. Suggested estimators have been compared with usual estimator and ratio type estimator. It has been shown that proposed estimators are more efficient than other considered estimators under certain given conditions. Let





U  U1 , U2 ,...,U N be a finite population of size N and y 0 and y1 are two study variates. Let x , is a

auxiliary variate taking values xi (i  1,2,..., N ) . A sample of size n is drawn from N with simple random sampling without replacement an estimator of ratio of two population means.

R

Y0 Y1

to estimate ratio of two population mean R usual estimator is

Rˆ 

(1.1)

y1 Let





y0  Y 1  e0 , y1  Y1 1  e1  x  X 1  e2  Such that

 

E e0   E e1   E e2   0 , E e02   C 02 E e12  C12 E e22  C 22

Shailendra Rawal & Rajesh Tailor

E e0 e1    01C0 C1 , E e0 e2    02C0 C2 , E e1e2   12 C1C2 Now we express R in terms of ei s we have

Rˆ 

y0 y1

Rˆ  R  R(e0  e1  e12  e0e1 )

(1.2)

Taking expectation of both sides of (1.2) we get

 

ˆ  R(C2   C C ) Bias R 1 01 0 1

(1.3)

Taking square and expectation of both sides of (1.3), upto the first degree of approximation mean squared error of

Rˆ is



MSE Rˆ  R 2 (C02  C12  201C0C1 )

(1.4)

where

S 02 

1 N 1 N 1 N 2 2 2 2     xi  X 2 y  Y S  y  Y S  , ,    0i 0 1 1i 1 2 N  1 i 1 N  1 i 1 N  1 i 1

C 02 

S 22 S 02 S12 2 2 C  , , C  2 1 X2 Y02 Y1 2

S 01 

1 N 1 N     y 0i  Y0 xi  X , y  Y y  Y S  ,  0i 0 1i 1 02 N  1  N  1 i 1 i 1

S12 

1 N   y1i  Y1 xi  X  N  1 i 1

 01 

S S S 01  02  02 12  12 S0 S2 S1 S 2 S 0 S1

1 n

  

1 . N

Using information of an auxiliary variate x, ratio estimator of ratio of two population mean is defined as

X Rˆ R  Rˆ   x

(1.5)

Rˆ R  R  R(e0  e1  e2  e12  e22  e1e2  e0 e1 )

(1.6)

Taking expectation of both sides of (1.6) we get

Exponential Type Ratio and Product Estimator for Ratio of Two Population Means



ˆ Bias R R

  R(C

2 1

 C22  12 C1C2  01C0 C1 )

(1.7)

Taking square and expectation of both sides of (1.7), upto the first degree of approximation is mean squared error

ˆ is of R R

 

MSE Rˆ R  R 2 (C02  C12  C22  201C0C1  202C0C2  212C1C2 ) where C 2 

(1.8)

Sx X

C 0 = coefficient of variation y 0

C 1 = coefficient of variation y 1

C 2 = coefficient of variation x Bahl and Tuteja (1991) suggested exponential type ratio and product type estimator for population mean as

X x  YˆRe  y exp 1 1   X 1  x1 

(1.9)

 X  x2  YˆPe  y exp 2   X 2  x2 

(1.10)

2.2 SUGGESTED ESTIMATOR In the line of Bahl and Tuteja (1991), proposed ratio and product type estimator for ratio of two population means are

X x  R Rˆ Re  Rˆ exp 1 1   X 1  x1 

(2.1)

 X  x2  YˆPeR  y exp 2   X 2  x2 

(2.2)

When

x 1 and x 2 are positively and negatively correlated auxiliary variate with ( y 0 , y1 ) respectively.

 ee  e2 3e22 ee R 2 ˆ RRe  R  R e0  e1   e1   e0 e1  1 2  0 2  2 8 2 2  

(2.3)

Taking expectation of both sides of (1.6) we get

 

3C 2  CC  C C   1 1  Bias Rˆ ReR  R  C12  2   01C0C1  12 1 2  02 0 2  8 2 2  n N  

(2.4)

Taking square and expectation of both sides of (1.7), upto the first degree of approximation is mean squared error

Shailendra Rawal & Rajesh Tailor

Rˆ ReR

 

  C2 MSE Rˆ ReR  R 2  C02  C12  2  2  01C0C1   02C0C2  12C1C2  4  

(2.5)

Similarly bias and mean squared error of suggested estimator is obtained are

 

(2.6)

 

(2.7)

 2 3C32  CC  C C  R ˆ Bias RPe  R C1    01C0C1  12 1 3  02 0 3  8 2 2     C2 MSE Rˆ PeR  R 2  C02  C12  2  2  01C0C1   02C0C2  12C1C2  4   2.3 BIAS COMPARISION

Some times bias of an estimator is considered as disadvantageous. So in this section we compare the bias of proposed estimators with the biases of other considered estimators. Bias of usual estimator

  Bias Rˆ   R (C

Rˆ and ratio of estimator Rˆ R are

Bias Rˆ  R (C12  01C0C1 ) R

2 1

(3.1)

 C22  12C1C2  01C0C1 )

(3.2)

Comparing (2.4) and (3.1) it is observed that the bias of the proposed estimator

Rˆ ReR

is less then the bias of usual

estimator Rˆ , i.e.

B( Rˆ ReR )  B( Rˆ )

 3C 22 12 C1C 2  02 C0 C 2       2 2  8   2 3C 22  CC  C C   2C1   2  01C0 C1  12 1 2  02 0 2   0 8 2 2   Comparing (2.4) and (3.1) it is observed that bias of the proposed estimator i.e.

B( Rˆ ReR )  B ( Rˆ R )

Rˆ ReR

is less then Rˆ R

Exponential Type Ratio and Product Estimator for Ratio of Two Population Means

 C C  CC  5   C22  02 0 2  12 1 2  2 2  8  CC  3  2 11 2  2C1  C2  2  01C0 C1  12 C1C2  02 0 2   0 8 2 2   

Similarly condition under the bias estimator of

Rˆ PeR

is less then

Rˆ

B( Rˆ ReR )  B( Rˆ )  3C32 13C1C3  03C0C3       2 2  8  2  2 3C3  CC  C C   2C1   2  01C0 C1  13 1 3  03 0 3   0 8 2 2   Comparing (2.6) and (3.1) it is observed that bias of the proposed estimator

Rˆ Re is less then Rˆ R

i.e.

B ( Rˆ PeR )  B ( Rˆ R )  C C  CC  5   C32  03 0 3  13 1 3  2 2  8  CC  3  2 11 2  2C1  C3  2  01C0C1  13C1C3  03 0 3   0 8 2 2   2.4 EFFICIENCY COMPARISION Mean squared error of usual estimator Rˆ an ratio type estimator Rˆ R

 MSERˆ   R  (C

MSE Rˆ  R 2 (C02  C12  201C0C1 ) 2



2 0

 C12  C22  201C0C1  202C0C2  212C1C2 )

Comparison of (2.5) and (4.1) shows that suggested estimator

 

(4.1)

R is more efficient than usual estimator Rˆ if Rˆ Re



MSE Rˆ ReR  MSE Rˆ

 2  C22 2 R   C0  C1   2  01C0C1   02C0C2  12C1C2   R 2 (C02  C12  2  01C0C1 ) 4   2

C  C 2  2   02 C0  12 C1   0  4 

(4.2)

Shailendra Rawal & Rajesh Tailor

C 2  0 and

C2   02 C 0  12 C1  0 4

either C 2  0 and C 2  4 02 C 0  12 C1  or C 2  0 and

C2   02 C 0  12 C1  0 4

 C 2  0 and C 2  4 02 C 0  12 C1  Thus conditions under which proposed estimator

R would be more efficient then Rˆ is Rˆ Re

either 0  C 2  4 02 C 0  12 C1  or 4 02 C 0  12 C1   C 2  0 

Comparison of (2.5) and (4.2) shows that suggested estimator



R MSE YˆRe

R is more efficient than usual estimator Rˆ if Rˆ Re

  MSE Rˆ  R

 2  C 22 2 R   C0  C1   2  01C0 C1   02 C0 C 2  12 C1C 2   4   2 2 2 2 R  C0  C1  C 2  2  01C0 C1  2  02 C0 C 2  2 12 C1C 2  2

3  C2  C 2   02 C 0  12 C1   0  4 

C 2  0 and

3 C2  02C0  12C1  0 4

either C 2  0 and

or C 2  0 and

C2 

4  02 C0  12 C1  3

C2   02 C 0  12 C1  0 4

 C 2  0 and C 2 

4  02 C0  12 C1  3

Thus conditions under which proposed estimator

R ˆ R is either would be more efficient then R Rˆ Re

0  C 2  4 02 C 0  12 C1  or



4  02 C0  12 C1   C2  0 3

Comparison of (2.7) and (4.1) shows that suggested estimator

Rˆ PeR is more efficient than usual estimator Rˆ

Exponential Type Ratio and Product Estimator for Ratio of Two Population Means

 



MSE Rˆ PeR  MSE Rˆ

  C2 R 2  C02  C12  2  2  01C0 C1   02 C0 C 2  12 C1C 2   R 2 (C02  C12  2  01C0 C1 ) 4  

C  C 2  2   02 C0  12 C1   0  4  C 2  0 and

C2   02 C 0  12 C1  0 4

either C 2  0 and C 2  4 02 C 0  12 C1  or C 2  0 and

C2   02 C 0  12 C1  0 4

 C 2  0 and C 2  4 02 C 0  12 C1  Thus conditions under which proposed estimator

R would be more efficient then Rˆ is either Rˆ Re

0  C 2  4 02 C 0  12 C1  or 4 02 C 0  12 C1   C 2  0 

Comparison of (2.7) and (4.2) shows that suggested estimator

 

Rˆ PeR is more efficient than usual estimator Rˆ R

 

MSE Rˆ PeR  MSE Rˆ R

  C2 R 2  C02  C12  3  2  01C0 C1   03C0 C3  13C1C3   0 4   R 2 C02  C12  C32  2  01C0 C1  2  03C0 C3  2 13C1C3 

3  C3  C3   03C0  13C1   0  4  C3  0

either

and

3 C3   03C0  13C1  0 4

C3  0

C3  0

and

C3 

4  03C0  13C1  3

C3   03C0  13C1  0 4

Shailendra Rawal & Rajesh Tailor

 C3  0

and C3



4  03C0  12C1  3

Thus conditions under which proposed estimator

0  C3 

Rˆ PeR would be more efficient then Rˆ

is either

4  03C0  12C1  3

4  03C0  12C1   C3  0 3

Thus the condition under which suggested estimator

R ˆ R given would be more efficient then ratio estimator R Rˆ Re

by Singh (1965) if either

0  C2  4 02C0  12C1 

4  02 C0  12 C1   C 2  0 3

Similarly the condition under which suggested estimator

Rˆ R

ˆ R would be more efficient then ratio estimator R Pe

given by Singh (1965) if either

0  C2  4 02C0  12C1 

4  02 C0  12 C1   C 2  0 3

2.5 EMPERICAL STUDY To show the performance of the suggested estimator Descriptions of the problem is given below Population I

y 0 : Wing length

y1 : Fourth palp length x : Third palp length

R we are considering two natural data sets. Rˆ Re and Rˆ Re

Exponential Type Ratio and Product Estimator for Ratio of Two Population Means

Population I [Source: Johnson & Wichern (2003)] Table 1

 =0.23333 Y 0 =102 N=10 n=3

S 01 =-0.55556 C 0 =0.055073  01 =-0.11513 C02 =0.003033 S02 =31.55556 S 0 =5.617433

R =1.328767 Y 1 =35.6 S 02 =12.66667 C1 =0.0.040164  02 = 0.668165 C12 =0.001613 S12 =2.044444 S 1 =1.429841

R 2 =1.765622 X =26.5 S12 =-0.33333 C 2 =0.127348784 12 = -0.06908

C22 =0.016218 S 22 =11.38889 S 2 =3.374743

Population II [Source: Johnson & Wichern (2003)] (a) For

Rˆ ReR

Y0 : Male length Y1 : Male width X

: Male height Table 2

 =0.208333 Y 0 =113.375 N=24 N=4

S 01 =79.14674 C 0 =0.103902  01 =0.949785 C02 =0.010796 S02 =138.7663 S 0 =11.77991

(b) For

Rˆ PeR

Y0 : Male length Y1 : Male width X

: Male height

R =1.284096 Y 1 =74 S 02 =37.375 C1 =0.080121  02 =0.945558 C12 =0.006419 S12 =50.04166 S 1 =7.074013

R 2 =1.648903 X =37 S12 =21.65399 C 2 =0.082427 12 =0.912265

C22 =0.006794 S 22 =11.25906 S 2 =3.355452

Shailendra Rawal & Rajesh Tailor

Table 3

 =0.208333 Y 0 =113.375 N=24 N=4

R =1.284096 Y 1 =37 S 03 =21.65399 C1 =0.082427  03 =0.912265 C12 =0.006794 S12 =11.25906 S 1 =3.355452

S 01 =79.14674 C 0 =0.103902  01 =0.949785 C02 =0.010796 S02 =138.7663 S 0 =11.77991

R 2 =1.648903 X =74 S13 =37.375 C 3 =0.080121 13 =0.945558

C32 =0.006419 S 22 =50.04166 S 2 =7.074013

Percent Relative Efficiencies Table 4 Estimator

Population I

Population II

Rˆ Rˆ R Rˆ R

100.00

45.64718

166.0676

219.0716

654.3909

Rˆ PeR

107.1302

601.5625

Table 1 shows that suggested estimators to

Rˆ

Rˆ ReR

and

Rˆ PeR

have higher percentage relative efficiency in comparison

ˆ . Thus suggested estimator recommended for use in practice. and R R Section 2.4 provides the conditions under which suggested exponential type estimator are more efficient than

usual estimator

Rˆ

and ratio estimator

Rˆ R

for ratio of two population means.

7. CONCLUSIONS Estimation is a common problem in various field if agriculture, economics, population etc. where some parameters like population total, population mean population variance ratio of two population means etc need to be estimates. In this article we have considered the problem of estimating the population mean of the study variable when the population mean of an auxiliary variable is known in simple random sampling without replacement (SRSWOR). The class of estimators has been proposed and the bias and mean square error expressions of the proposed class of estimators have been obtained up to first degree of approximation.

REFERENCES 1.

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Cochran, W.G. (1940). The estimation of the yields of cereal experiments by sampling for the ratio gain to total produce. Jour. Agri. Soc., 30, 262-275

Exponential Type Ratio and Product Estimator for Ratio of Two Population Means

Murthy, M.N. (1964). Product method of estimation. Sankhya, 26,A, 284-307.

Pandey, B.N. and Dube, V. (1988). Modified product estimator using coefficient of variation of auxiliary variable. Assam Stat. Rev., 2, 64-66.

Robson, D.S. (1957). Application of multivariate polykays to the theory of unbiased ratio-type estimators. Jour. Amer. Statist. Assoc., 52, 511-522.

Searls, D.T. (1964).The utilization of known coefficient of variation in the estimation procedure. Jour. Amer. Stati. Assoc., 59, 1225-1226.

Singh, G.N. and Rani, R. (2005, 2006). Some linear transformations on auxiliary variable for estimating the ratio of two population means in sample surveys. Model assisted statistics and applications, 1, 3-7.

8. Singh, H. P., Singh, R., Ruiz Espejo, M. and Pineda M. D. (2005). On the efficiency of a dual to ratio-cumproduct estimator in sample surveys. Mathematical Proceedings of the Royal Irish Academy, 105 A (2), 51-56. 9. Singh, H.P. and Tailor, R. (2003). Use of known correlation coefficient in estimating the finite population mean. Stat. Transin. 6,4, 555-560 10. Singh, H. P. Tailor, R. Tailor, R. and Kakran, M.S. (2004). An improved estimator of population mean using power transformation, J. Indian Soc. Agric. Statist., 58, 2, 223-230. 11. Singh, M.P. (1965). On the estimation of ratio and product of the population parameters. Sankhya, B, 27, 321-328. 12. Singh, H.P., Ruiz-Espejo, M., (2003). On linear regression and ratio-product estimation of a finite population mean. The Statistician 52(1), 59-67. http://dx.doi.org/10.1111/1467-9884.00341 13. Singh, H.P., Solanki, R.S., (2011a). A general procedure for estimating the population parameter in the presence of random non-response. Pakistan Journal of Statistics 27(4), 427-465. 14. Sisodia, B.V.S. and Dwivedi, V.K. (1981). A modified ratio estimator using coefficient of an auxiliary variable, J. Ind. Soc. Agri. Stat. 33, 13-18. 15. Tailor, R. (2002). Some estimation problems based on auxiliary information in sample surveys. Ph.D. Thesis submitted to Vikram University, Ujjain, M.P., India. 16. Tailor, R. Kumar, M and Tailor, R. (2006). Estimation of finite population mean using power transformation. International journal of Agricultural and Statistical Science, 2, No. 2, pp. 341-345. 17. Upadhyaya, L.N. and Singh, H.P. and Vos, J.W.E. (1985). On the estimation of population means and ratio using supplementary information. Statistica Neerl., 39, 3, 309-318.