1 a new exponential approach for reducing the mean ...

A NEW EXPONENTIAL APPROACH FOR REDUCING THE MEAN SQUARED ERRORS OF THE ESTIMATORS OF POPULATION MEAN USING CONVENTIONAL AND NON-CONVENTIONAL LOCATION PARAMETERES. Housila P. Singh and Anita Yadav School of studies in Statistics Vikram University Ujjain- 456010, M.P., India Abstract In this paper we have proposed a class of ratio estimators t and a class of ratio-type exponential estimators t e of the population mean with their properties It has been shown that the ratio-type exponential estimator t e provides estimators less biased as well as more efficient than the ratio-type estimator t . Hence we have envisaged the estimators less biased and more efficient than those considered by Kadilar and Cingi (2004, 2006), Yan and Tian (2010), Subramani and Kumarpandiyan (2012a, 2012b, 2012c, 2012d), Jeelani et al (2013) and Abid et al (2016a, b, c). An empirical study is given in support of the present study. Keywords: Study Variable, Auxiliary Variable, Bias, Mean Squared Error. AMS Subject Classification: 62D05. 1. INTRODUCTION It is well established fact that the use of auxiliary information at the estimation stage of a survey improves the precision of the estimate(s) of the parameter(s) under investigation. The problem of estimating the population mean or total using population mean of an auxiliary variable has been extensively discussed. Out of many ratio, product and regression methods of estimation are good examples in this context. The ratio method of estimation is most effective for estimating population mean of the study variable when there is a linear relationship between study variable and auxiliary variable and they have the positive (high) correlation. On the other hand if the correlation between study variable and auxiliary variable is negative (high) the product method of estimation can be employed. Let U  U 1 ,U 2 ,..., U N  be the finite population of size N and the variables under study and auxiliary be denoted by y and x respectively. Let Y , X  be the population means of (y, x) respectively. It is desired to estimate the population mean Y using information on population parameters such as mean X , coefficient of variation

C x  ,

coefficient of skewness 1 ( x ) , kurtosis  2 ( x ), deciles, quartiles, median,

midrange (MR), Walsh average (i.e. Hodges-Lehman estimator) (HL) (and tri mean (TM) etc, associated with auxiliary variable x and the correlation coefficient  between y and x. In this context, the reader is referred to Searls (1964), Das and Tripathi (1980), Sisodia 1

and Dwivedi (1981), Upadhyaya and Singh (1999), Singh and Tailor (2003), Singh et al (2004), Kadilar and Cingi (2004, 2006) , Yan and Tian (2010), Subramani and Kumarapandian (2012a,2012b,2012c), Jeelani et al (2013) and Abid et al (2016a,b,c). In what follows we shall use the following notations throughout the paper: N : Population size. n : Sample size. n f  : Sampling fraction. N 1  f   n . 1 S y2  N  1   yi  Y  : Population Variance of the study variable y. N

2

i 1

S x2

 xi  X  : Population variance of the auxiliary variable x.

 N  1

2

1 N

i 1

C y  S y Y : Coefficient of variation of the study variable y. C x  S x X : Coefficient of variation of the auxiliary variable x.

S xy  N  1

 xi  X  yi  Y  : Covariance between y and x.

1 N

i 1

  S xy S x S y  : Correlation coefficient between x and y. C   C y Cx

, M d : Population median of x.

Qi : i t h population quartile (i=1,2,3).

Tm :

Q1  Q2  Q3  : Tri mean.

2 H l = median X j  X k  2 ,1  j  k  N  Hodges-Lehmann estimator. X 1 : Lowest order statistic in a population of size N, X  N  : Highest order statistic in a population of size N,

Mr =

X 1  X  N 

: Mid range, 2 Qr  Q3  Q1  : Inter-quartile range,

Qd 

Q3  Q1 : Semi-quartile range, 2 Ni  xi  X  N

1  x  

i 1

3

N  1N  2S x3

: Coefficient of skewness of the auxiliary variable x,

2

N   xi  X 4   N N  1  2     3 N  1 i 1  2 x      : Coefficient of kurtosis of 4    N  2 N  3     N  1 N  2 N  3 S x     the auxiliary variable x,

 2 N  12  2     2  x   1  x   N  2N  3  ,

Q3  Q1 : Quartile average, 2 Y R : Population ratio of means, X 4 N 2i  N  1 G X i  : Gini’s Mean Difference,  N  1 i 1 2N

Qa 

2  N  N 1  i  X i  : Downton’s method, N N  1 i 1 2 

D

S pw 

 N  2i  N  1X i  : Probability Weighted Moments, N 2 i 1

 r ( x )  1 N   xi  X  : r being non-negative integer. N

r

i 1

We are interested in estimating the population mean Y of the study variable y (taking value yi for i=1, 2,…, N) from a simple random sample size n drawn without replacement from the population U. We use the notation y and x for the sample means, which are unbiased estimators of the population mean Y and X , respectively. We also denote: s xy  n  1

 xi  x  yi  y  : Sample covariance between y and x.

1 n

s x2

i 1

2

 n  1  xi  x  : Sample variance of x.

ˆ  

1 n

i 1

s xy s x2

S xy S x2

: Sample regression coefficient estimate of the population regression coefficient of y on x.

ˆ  y : Ratio of sample means. R x ˆ X  x  : Regression estimator of the population mean Y . Yˆ  y  





3

The organization of the rest of the paper is as follows. Review of the existing estimators alongwith the newly introduced class of estimators t (say) are given in Section 2. Asymptotic bias and mean squared error (MSE) of the suggested class of estimators t are also obtained in Section 2. Several ratio-type exponential estimators along with a class of ratio-type exponential estimators t e (say) have been proposed in Section 3. The expressions of bias and MSE of the proposed class of ratio-type exponential estimators t e under large sample approximation are also obtained in Section 3. In Section 4, we have compared the proposed classes of estimators t and t e . Bias comparison of the proposed estimators t and t e is given in section 5. It has been shown that the proposed ratio-type exponential estimators t e is less biased as well as more efficient than the existing estimators and the suggested class of ratio-type estimators t . Section 6 consists of an empirical study in support of the present study. Finally, Section 7 summaries the finding of this study. 2. Existing Modified Ratio Estimators and the Suggested Class of Ratio Estimators The usual unbiased estimator for population mean Y is defined by (2.1) t0  y . whose MSE is given by 1  f  S 2 . MSE  y   (2.2) y n The classical ratio estimator for the population mean Y in presence of known population mean X of the auxiliary variable x is defined by X x 0. (2.3) yR  y x To the first degree of approximation, the bias and MSE of the ratio estimator y R are respectively given by 1  f  . 1 RS 2   S S B y R   (2.4) x x y n X and 1  f  S 2  R 2 S 2  2 R  S S . MSE  y R   (2.5) y x y x n We below give in Table 2.1, the modiefied versions of the ratio estimator reported by Kadilar and Cingi (2004), Kadilar and Cingi (2006) –type estimator, Yan and Tian (2010), Subramani and Kumarapandiyan (2012a, 2012b, 2012c, 2012d), Jeelani et al (2013) and Abid et al (2016) alongwith their biases and mean squared errors (MSEs) to the first degree of approximation, as reported in Abid et al (2016).









4

Table 2.1- Known modified ratio estimators for the population mean Y . S.No. Estimator MSE of (.) Population Ratio 1.

t1 

ˆ Y .X x







R1 

Y R X







R2 

X  C x 







R3 

Y X   2 x 







R4 

X2 x  Cx 







R5 

XCx  2 x







R6 

Y X 







R7 

XCx  







R8 

X  C x 







R9 

X 2 x  

 R12 S x2  S y2 1   2

Kadilar and Cingi (2004) 2.

ˆ Y X  Cx x  C x 



t2 



 R22 S x2  S y2 1   2

Y


ˆ Y X   2 x  x   2 x 





t3 



 R32 S x2  S y2 1   2


ˆ Y X 2 x   C x x  2 x   C x





t4 



 R 42 S x2  S y2 1   2

Y2 x 


ˆ Y XC x   2 x  x C x   2 x 





t5 



 R52 S x2  S y2 1   2


ˆ Y X  x   



t6 



 R62 S x2  S y2 1  2

YCx

Kadilar and Cingi (2006) -type 7.

t7 

ˆ Y XC   x C x    x





 R72 S x2  S y2 1  2


t8 

ˆ Y X  C x x   C x 





 R82 S x2  S y2 1   2


t9 

ˆ Y X x    x  2 x     2





 R92 S x2  S y2 1   2

YCx


t10 

ˆ Y X   2 x  x    2 x 





Y







R10 









R11 

X  1 x 







R12 

X1 x   2 x







R13 

X  M d 

2 2  R10 S x  S y2 1   2


t11 

ˆ Y X  1 x  x  1 x 





2 2  R11 S x  S y2 1   2

Yan and Tian (2010) 12.

t12 

ˆ Y X x    2 x  x 1 x    2 x  1





2 2  R12 S x  S y2 1   2

Yan and Tian (2010) t13 

13.

ˆ Y X Md x  M d 

Subramani (2012a)



and



Kumarapandian

Y  2 x 

2 2  R13 S x  S y2 1   2

5

Y X   2  x 



Y

Y1 x 

Y

S.No.

Estimator

14.

ˆ Y t14  XC  M d C x x  M d  x

MSE of (.)



Subramani (2012a) t15 

15.

16.

Kumarapandian

ˆ Y X x   M d 1 x x  M d  1



Subramani (2012b) t16 

and

and

17.

ˆ Y X x   M d  2 x x  M d  2



ˆ Y X  D1 x  D1 



Subramani (2012d) t18 

18.

19.

20.



21.

and



Kumarapandian

ˆ Y X  D3  x  D3 



Kumarapandian

ˆ Y X  D4 x  D 4 



Subramani (2012d) t 21 

and

and



Kumarapandian

ˆ Y X  D5  x  D5 





Subramani Kumarapandian(2012d) t 22 

22.

ˆ Y X  D6 x  D6 



23.

ˆ Y X  D7 x  D7 




24.



t 25 

and

 

Kumarapandian

ˆ Y X  D9  x  D9 



and

Kumarapandian

ˆ Y X  D8 x  D8 

Subramani (2012d) 25.

and

and



Subramani Kumarapandian(2012d) t 23 

and

Kumarapandian




and





ˆ Y X  D2 x  D 2 

Subramani (2012d) t19 



Kumarapandian

Subramani Kumarapandian(2012c) t17 





Population Ratio







R14 







R15 







R16 

X 2 x  M d 







R17 

X  D1 







R18 



Y X  D2







R19 

X  D3 







R 20 



Y X  D4







R21 

X  D5 







R22 

X  D6 







R23 

X  D7 







R24 

Y X  D8









R25 

X  D9 

2 2  R14 S x  S y2 1   2

2 2  R15 S x  S y2 1   2

2 2  R16 S x  S y2 1   2

2 2  R17 S x  S y2 1   2

2 2  R18 S x  S y2 1   2

2 2  R19 S x  S y2 1   2

2 2  R20 S x  S y2 1   2

2 2  R 21 S x  S y2 1   2

2 2  R 22 S x  S y2 1   2

2 2  R23 S x  S y2 1   2

2 2  R 24 S x  S y2 1   2

2 2  R25 S x  S y2 1   2

6

YC x

XC x  M d  1 x Y

X1 x  M d   2 x Y

Y



Y



Y

Y

Y

Y



S.No.

Estimator Subramani (2012d) t 26 

26.

27.

Population Ratio

Kumarapandian

ˆ Y X  D10 x  D10 




MSE of (.) and

and



Kumarapandian

ˆ Y X x   Q d x 1 x   Qd  1











R26 

X  D10 







R27 

X1 x  Qd 







R28 

X  Tm 







R 29 

XC x  Tm 







R30 

X  Tm 

2 2  R26 S x  S y2 1   2

2 2  R27 S x  S y2 1   2

Jeelani et al (2013) 28.

t 28 

ˆ Y X  Tm x  Tm 





2 2  R28 S x  S y2 1   2

Abid et al (2016a) 29.

t 29 

ˆ Y XC  T  x C x  Tm  x m





2 2  R29 S x  S y2 1   2


t 30 

ˆ Y X  Tm  x   Tm 





2 2  R30 S x  S y2 1   2


t 31 

ˆ Y X Mr x  M r 





32.

ˆ Y XC  M x C x  M r  x r





33.

ˆ Y X  M r x   M r 





34.

ˆ Y X  Hl x  H l 



X  M r 







R31 

X  M r 







R33 

X  M r 







R34 

X  H l 







R35 

XC x  H l 







R36 

X  H l 







R37 

X  G 







R38 

X  G 







R39 

2 2  R32 S x  S y2 1   2

2 2  R34 S x  S y2 1   2


t 35 

ˆ Y XC  H x C x  H l  x l





Y

R31 

2 2  R33 S x  S y2 1   2



YC x



Abid et al (2016a) t 34 

Y



Abid et al (2016a) t33 

Y 1 x 



2 2  R31 S x  S y2 1  2

Abid et al (2016a) t32 

Y

2 2  R35 S x  S y2 1   2

Y

YCx

Y

Y

YC x


t 36 

ˆ Y X  H l x   H l 





2 2  R36 S x  S y2 1   2

Y


t37 

ˆ Y X G x  G 





2 2  R37 S x  S y2 1   2

Y

Abid et al (2016b) 38.

t 38 

ˆ Y X  G x   G 






t 39 

ˆ Y

x C x

XC x  G   G

2 2  R38 S x  S y2 1   2

2 2  R39 S x  S y2 1   2

7

Y

YC x

XC x  G 

S.No.

Estimator

MSE of (.)

Population Ratio


t 40 

ˆ Y X D x  D 











R40 

X  D







R41 

X  D







R42 

XC x  D







R 43 

X  S pw 







R44 

X  S pw







R45 

YCx XC x  S pw

2 2  R40 S x  S y2 1   2

Y


t 41 

ˆ Y X  D x   D 





2 2  R 41 S x  S y2 1   2

Y


t42 

ˆ Y

x C x

XCx  D   D

2 2  R 42 S x  S y2 1   2

YC x


t 43 

ˆ Y X  S pw x  S pw







2 2  R43 S x  S y2 1   2

Y


t44 

ˆ Y X  S pw x   S pw







2 2  R44 S x  S y2 1  2

Y


t45 

ˆ Y XC x  S pw x C x  S pw







2 2  R45 S x  S y2 1  2




t 46 

ˆ Y X  D1 x   D1 





Y X  D1



Y X  D2









R 46 







R 47 









R48 

X  D3 







R 49 

X  D4 







R50 

Y X  D5









R51 

X  D6 







R52 

X  D7 







R53 

Y X  D8

2 2  R46 S x  S y2 1   2



Abid et al (2016c) 47.

t 47 

ˆ Y X  D 2 x   D 2 





2 2  R47 S x  S y2 1   2


t 48 

ˆ Y X  D3  x   D3 





2 2  R48 S x  S y2 1   2

Y


t 49 

ˆ Y X  D 4 x   D 4 





2 2  R49 S x  S y2 1   2

Y


t 50 

ˆ Y X  D5  x   D5 





2 2  R50 S x  S y2 1   2




t 51 

ˆ Y X  D6 x   D6 





2 2  R51 S x  S y2 1   2

Y


t 52 

ˆ Y X  D7 x   D7 





2 2  R52 S x  S y2 1   2

Y


t 53 

ˆ Y X  D8 x  D8 





2 2  R53 S x  S y2 1   2

Abid et al (2016c)

8







S.No.

Estimator

54.

ˆ Y t 54  X  D9  x   D9 

MSE of (.)





Population Ratio Y







R54 

X  D9 







R55 

X  D10 







R56 

XC x  D1 







R57 

YC x

XC x  D2 







R58 

YC x

XC x  D3 







R59 

XC x  D4 







R60 

YC x

XC x  D5 







R61 

YC x

XC x  D6 







R62 

YC x

XC x  D7 







R63 

XC x  D8 







R64 

YC x

XC x  D9 







R65 

XC x  D10 

2 2  R54 S x  S y2 1   2


t 55 

ˆ Y X  D10 x   D10 





2 2  R55 S x  S y2 1   2

Y


t 56 

ˆ Y XC  D x C x  D1  x 1





2 2  R56 S x  S y2 1   2


t 57 

ˆ Y XC  D x C x  D 2  x 2





2 2  R57 S x  S y2 1   2


t 58 

ˆ Y XC  D  x C x  D3  x 3





2 2  R58 S x  S y2 1   2


t 59 

ˆ Y XC  D x C x  D 4  x 4





2 2  R59 S x  S y2 1   2


t 60 

ˆ Y XC  D  x C x  D5  x 5





2 2  R60 S x  S y2 1   2


t 61 

ˆ Y XC  D x C x  D6  x 6





2 2  R61 S x  S y2 1   2


t 62 

ˆ Y XC  D x C x  D7  x 7





2 2  R62 S x  S y2 1   2


t 63 

ˆ Y XC  D x C x  D8  x 8





2 2  R63 S x  S y2 1   2

Abid et al (2016) 64.

t 64 

ˆ Y XC  D  x C x  D9  x 9





2 2  R64 S x  S y2 1   2


t65 

ˆ Y XC  D x C x  D10  x 10





2 2  R65 S x  S y2 1   2

Abid et al (2016c)

YC x

YC x

YC x

YC x

We note that the estimators t j  j  1to 65 members of the following class of estimators of the population mean Y defined by

 aX  b   t  Yˆ   ax  b  9





ˆ X  x   aX  b  ,  y   ax  b   

(2.6)

where ˆ is the sample estimate of the population regression coefficient  of y on x,

a 0 and b are real numbers (constants) or the functions of population parameters such

as population total X  NX , population standard deviation S x , variance S x2 , coefficient of variation C x , Coefficient of skewness 1 x  and kurtosis  2 x , correlation coefficient  ,  , quartiles, deciles, median, mode, midrange, trimean and Hodgs-Lehmann (HL) estimator etc. Some unknown members of the suggested class of ratio-type estimators t are given in Table 2.2 Table 2.2- Some unknown members of the class of ratio type estimators t. Values of constants S.No. Estimator a B      x X   ˆ  t1  Y 1 1x 1. 1x x   ˆ  x X  Cx  t2  Y 1 1x  Cx 2. 1x x  Cx  ˆ XC x  1 x  t 3  Y 1x Cx 3. xC x  1 x  4.

ˆ  X 2 x   1 x   t 4  Y    x  2 x   1 x  

 2 x 

1x 

5.

ˆ  X  1 x   t 5  Y    x   1 x  



1x

6.

ˆ  X  M d t6  Y   x  M d



Md

7.

ˆ  M X  Cx t 7  Y  d  xM  C d x 

Md

Cx

8.

ˆ  M X   2 x   t 8  Y  d  x M   x   d 2  

Md

 2 x 

9.

ˆ  XM d  1 x   t 9  Y   x M   x   d 1  

Md

1 x 

10.

ˆ  M X     t10 Y d  xM    d  

Md



11.

ˆ  X  Qd  t11 Y  x Q d 

1

Qd

   

   

   

10

Values of constants a B

S.No.

Estimator

12.

ˆ  XC x  Qd  t12 Y  xC  Q x d 

13.

 X  Qd  t13    x  Qd

   

   

Cx

Qd



Qd

Kumarapandiyan and Subramani (2016)-type

14.

ˆ  XM d  Qd   t14 Y  xM  Q  d d  

Md

Qd

15.

ˆ  Q X  Cx  t15 Y d  Q x C x  d

Qd

Cx

16.

ˆ  Q X   2 x    t16 Y d  Q x   x   2  d 

Qd

 2 x 

17.

ˆ  Q X  1 x    t17 Y d  Q x   x   1  d 

Qd

1 x 

18.

ˆ  Q X     t18  Y  d   Qd x   

Qd



19.

ˆ  Q X  Md  t19  Y  d  Qd x  M d

Qd

Md

1 x 

Qd

20.

   

   

ˆ    x X  Qd  t 20 Y 1   x x  Q d  1

   

Kumarapandiyan and Subramani (2016)-type 21.

ˆ  X 2 x   Tm  t 21 Y  x  x   T m  2

   

 2 x 

Tm

22.

ˆ  X x   Tm  t 22 Y 1  x  x   T m  1

   

1 x 

Tm

23.

ˆ  XM d  Tm   t 23 Y  xM  T  d m  

Md

Tm

24.

ˆ  XQd  Tm  t 24  Y   x Q d  Tm

Qd

Tm

25.

ˆ  XT  C x  t 25 Y m  xT  C x  m

Tm

Cx

26.

ˆ  XT   2 x    t 26  Y  m   x Tm   2  x  

Tm

 2 x

27.

ˆ  XT     t 27 Y m  xT     m 

Tm



   

   

11

Values of constants a b

S.No.

Estimator

28.

ˆ  XT  1 x    t 28 Y m  x T   x   1  m 

29.

ˆ  XT  M d  t 29 Y m  xT  M d  m

30.

ˆ  XT  Qd  t 30 Y m  xT  Q d  m

31.

ˆ  X 2 x   M r  t 31  Y   x  2 x   M r

32.

ˆ  X x   M r  t 32  Y  1  x 1 x   M r

33.

ˆ  XM d  M r  t 33 Y  xM  M d r 

34.

ˆ  XQd  M r   t 34 Y  xQ  M  d r  

Qd

Mr

35.

ˆ  XM r  C x  t 35 Y  xM  C r x 

Mr

Cx

36.

ˆ  XM r   2 x    t 36  Y    x M r   2 x  

Mr

 2 x 

37.

ˆ  XM r     t 37  Y    xM r   

Mr



38.

ˆ  XM r  1 x    t 38  Y    x M r  1 x  

Mr

1x 

39.

ˆ  XM r  M d  t 39 Y  xM  M r d 

Mr

Md

40.

ˆ  XM r  Qd  t 40 Y  xM  Q r d 

   

Mr

Qd

41.

ˆ  XT  M r  t 41 Y m  xT  M r  m

   

Tm

Mr

42.

ˆ  XM r  Tm  t 42 Y  xM  T r m 

   

Mr

Tm

43.

ˆ  X2 x   H l  t43  Y   x 2 x   H l

   

 2 x 

Hl

44.

ˆ  X x   H l  t 44 Y 1  x  x   H l  1

   

1 x 

Hl

45.

ˆ  XM d  H l  t 45 Y  xM  H d l 

Md

Hl

           

   

   

Tm

1 x 

Tm

Md

Tm

Qd

 2 x 

Mr

1 x 

Mr

Md

   

   

   

12

Mr


S.No.

Estimator

46.

ˆ  XQd  H l  t 46  Y   x Qd  H l

47.

ˆ  XH l  C x  t 47 Y  xH  C l x 

48.

   

Qd

Hl

Hl

Cx

ˆ  XH l   2 x    t 48 Y  x H   x   l 2  

Hl

 2 x

49.

ˆ  XH l     t 49 Y  xH    l  

Hl



50.

ˆ  XH l  1 x    t 50 Y  x H   x   l 1  

Hl

1 x 

51.

ˆ  XH l  M d  t 51 Y  xH  M l d 

Hl

Md

52.

ˆ  XH l  Qd  t52  Y   x H l  Qd

Hl

Qd

53.

ˆ  XH l  M r  t 53 Y  xH  M l r 

   

Hl

Mr

54.

ˆ  XT  H l  t 54 Y m  xT  H l  m

   

Tm

Hl

55.

ˆ  XH l  Tm  t 55 Y  xH  T l m 

   

Hl

Tm

56.

ˆ  XM r  H l  t 56 Y  xM  H r l 

   

Mr

Hl

57.

ˆ  X  Qa  t 57 Y  x Q a 

1

Qa

58.

ˆ  XC x  Qa  t 58 Y  xC  Q x a 

Cx

Qa

59.

ˆ  X 2 x   Qa  t 59 Y  x  x   Q a  2

 2 x 

Qa

60.

ˆ  X x   Qa  t 60  Y  1  x 1 x   Qa

1 x 

Qa



Qa

Md

Qa

61.

ˆ  X  Qa  t 61 Y  x  Q a 

   

   

   

   

       

   

   

Kumarapandiyan and Subramani (2016)-type 62.

ˆ  XM d  Qa   t 62 Y  xM  Q  d a  

13


S.No.

Estimator

63.

ˆ  XQd  Qa  t 63 Y  xQ  Q d a 

   

Qd

Qa

64.

ˆ  XQd  C x   t 64 Y  xQ  C  d x  

Qd

Cx

65.

ˆ  XQa   2 x    t 65 Y  x Q   x   a 2  

Qa

 2 x

66.

ˆ  Q X     t 66 Y a  xQ    a  

Qa



67.

ˆ  Q X  1x    t67  Y  a   x Qa  1x  

Qa

1x

68.

ˆ  Q X  Md  t68  Y  a  x Qa  M d

Qa

Md

69.

ˆ  XQa  Qd  t69  Y   x Qa  Qd

   

Qa

Qd

70.

ˆ  XT  Qa  t70  Y  m  x Tm  Qa

   

Tm

Qa

71.

ˆ  XQa  Tm   t71  Y    x Qa  Tm 

Qa

Tm

72.

ˆ  XQa  M r  t72  Y   x Qa  M r

   

Qa

Mr

73.

ˆ  XM r  Qa  t73  Y   x M r  Qa

   

Mr

Qa

ˆ  XH l  Qa  t74  Y   x H l  Qa

   

Hl

Qa

75.

ˆ  XQa  H l  t75  Y   x Qa  H l

   

Qa

Hl

76.

ˆ  XQr  Qa  t76  Y   x Qr  Qa

   

Qr

Qa

77.

ˆ  XQa  Qr  t77  Y   x Qa  Qr

   

Qa

Qr

1

Sx

Cx

Sx

74.

78.

ˆ  X  Sx  t 78 Y  xS x 

   

   

Singh (2003)-type 79.

ˆ  XC x  S x  t79  Y   xCx  S x

   

14


S.No.

Estimator

80.

ˆ  X2 x   S x  t80  Y   x 2 x   S x

   

 2 x 

Sx

   

1 x 

Sx



Sx

Singh (2003)-type 81.

ˆ  X x   S x  t81  Y  1  x 1x   S x

Singh (2003) -type 82.

ˆ  X  S x  t 82 Y  x  S x 

83.

ˆ  XM d  S x   t 83 Y  xM  S  d x  

Md

Sx

84.

ˆ  XQd  Qa   t 84 Y  xQ  Q  d a  

Qd

Qa

85.

ˆ  XS x  C x  t 85 Y  xS  C x  x

Sx

Cx

86.

ˆ  XS x   2 x    t 86 Y  x S   x   2  x 

Sx

 2 x

87.

ˆ  XS x     t 87 Y  xS     x 

Sx



88.

ˆ  XS x  1x    t88  Y    x S x  1x  

Sx

1 x 

89.

ˆ  XS x  M d  t89  Y   xS x  M d

Sx

Md

90.

ˆ  XS x  Qd  t90  Y   x S x  Qd

   

Sx

Qd

91.

ˆ  XT  S x  t 91 Y m  xT  S x  m

   

Tm

Sx

92.

ˆ  XS x  Tm   t92  Y    x S x  Tm 

Sx

Tm

93.

ˆ  XS  M r  t93  Y  x  xS x  M r

   

Sx

Mr

94.

ˆ  XM r  S x  t94  Y   xM r  S x

   

Mr

Sx

95.

ˆ  XH l  S x  t 95 Y  xH  S l x 

   

Hl

Sx

   

   

   

15


S.No.

Estimator

96.

ˆ  XS x  H l  t 96 Y  xS  H l  x

   

Sx

Hl

97.

ˆ  XQr  S x  t 97 Y  xQ  S r x 

   

Qr

Sx

98.

ˆ  XS  Qr  t 98 Y x  xS  Q r  x

   

Sx

Qr

99.

ˆ  XQa  S x  t 99 Y  xQ  S a x 

   

Qa

Sx

100.

ˆ  XS x  Qa  t100 Y  xS  Q a  x

Sx

Qa

101.

ˆ X X  t101  Y   xX

1

X  NX

102.

ˆ  XC x  X  t102 Y  xC  X x 

Cx

X  NX

103.

ˆ  X 2 x   X  t103  Y   x  2 x   X

   

 2 x 

X  NX

104.

ˆ  X x   X  t104  Y  1  x 1 x   X

   

1 x 

X  NX

105.

ˆ  X  X  t105  Y   x  X



X  NX

106.

ˆ  XM d  X   t106 Y  xM  X  d  

Md

X  NX

107.

ˆ  XQd  X   t107 Y  xQ  X  d  

Qd

X  NX

108.

ˆ  XX  C x  t108 Y  xX  C x 

X  NX

109.

ˆ  XX  2 x    t109  Y    x X  2 x  

X  NX

110.

ˆ  XX     t110  Y    xX   

X  NX

111.

ˆ  XX  1 x    t111  Y    x X  1 x  

X  NX

112.

ˆ  XX  M d  t112 Y  xX  M d 

X  NX

113.

ˆ  XX  Qd  t113 Y  xX  Q d 

   

   

   

   

   

   

   































Cx





2 x 











1 x 





Md





Qd

X  NX

16




S.No.

Estimator

114.

ˆ  XT  X  t114 Y m  xT  X  m

   

Tm

115.

ˆ  XX  Tm  t115 Y  xX  T m 

   

X  NX

116.

ˆ  XX  M r  t116  Y   xX  M r

   

X  NX

117.

ˆ  XM r  X  t117  Y   xM r  X

   

Mr

X  NX

118.

ˆ  XH l  X  t118 Y  xH  X l 

   

Hl

X  NX

119.

ˆ  XX  H l  t119 Y  xX  H l 

   

X  NX

120.

ˆ  XQ r  X   t120  Y    xQr  X 

Qr

121.

ˆ  XX  Q r  t121  Y   xX  Qr

   

X  NX

122.

ˆ  XX  S x  t122 Y  xX  S x 

   

X  NX

123.

ˆ  XS x  X  t123 Y  xS  X  x

   

Sx

X  NX

124.

ˆ  XQa  X  t124 Y  xQ  X a 

   

Qa

X  NX

125.

ˆ  XX  Qa  t125  Y   x X  Qa

   

X  NX

126.

ˆ  X     t126  Y    x 

1



127.

ˆ  XC x     t127  Y    xCx   

Cx



128.

ˆ  X x      t128  Y  2   x 2 x    

 2 x 



129.

ˆ  X x      t129  Y \  1   x 1x    

1 x 



130.

ˆ  X     t130  Y    x   





131.

ˆ  XM d     t131  Y    xM d   

Md







Tm





Mr











Hl



X  NX



Qr





Sx











17



X  NX











Qa

S.No.

Estimator


132.

ˆ  XQd     t132  Y    x Qd   

Qd



133.

ˆ  X  C x  t133 Y  x  C x 



Cx

134.

ˆ  X  2 x    t134  Y    x   2 x  



135.

ˆ  X     t135  Y    x   



136.

ˆ  XQa     t136  Y    x Qa   

Qa



137.

ˆ  X  Qa  t137 Y  x  Q a 

   



Qa

138.

ˆ  X  S x  t138 Y  x  S x 

   



Sx

139.

ˆ  XS x     t139 Y  xS     x 

Sx



140.

ˆ  X  1 x    t140  Y    x   1 x  



141.

ˆ  X  M d  t141 Y  x  M d 

   



Md

142.

ˆ  X  Qd  t142 Y  x  Q d 

   



Qd

143.

ˆ  XT     t143 Y m  xT     m 

Tm



144.

ˆ  X  Tm  t144 Y  x  T m 

   



Tm

145.

ˆ  X  M r  t145  Y   x  M r

   



Mr

146.

ˆ  XM r     t146  Y    xM r   

Mr



147.

ˆ  XH l     t147 Y  xH    l  

Hl



148.

ˆ  X  H l  t148 Y  x  H l 



Hl

Qr



149.

   

   

ˆ  XQ r     t149  Y    x Qr   

18

 2 x



1 x 

Values of constants a

b

   



Qr

   



X  NX

S.No.

Estimator

150.

ˆ  X  Qr  t150  Y   x   Qr

151.

ˆ  X  X  t151  Y   x  X

152.

ˆ  XX     t152  Y    xX   

X  NX

153.

ˆ  X  Q1   t153  Y    x  Q1  Al-Omar et al (2009)

1

Q1

1

Q3

1x

Q1

1x

Q2

1 x 

Q3

1 x 

Qr

1 x 

Qd



Q1



Q2



Q3

154.





ˆ  X  Q3   t154  Y    x  Q3 





Al-Omar et al (2009) 155.

ˆ  X x   Q1   t155  Y  1   x 1  Q1 

Kumarapandiyan and Subramani (2016)type 156.

ˆ  X x   Q2   t156  Y  1   x 1  Q2 

Kumarapandiyan and Subramani (2016)type

157.

ˆ  X x   Q3   t157 Y 1  x  Q  1 3  


ˆ  X x   Qr  t158  Y  1  x 1  Qr

   


ˆ  X x   Qd  t159  Y  1  x 1  Qd

   


ˆ  X  Q1   t160  Y    x   Q1 


ˆ  X  Q2   t161  Y    x   Q2 


ˆ  X  Q3   t162  Y    x   Q3 

Kumarapandiyan and Subramani (2016)type

19




S.No.

Estimator

163.

ˆ  X  Qr  t163  Y   x   Qr

   



Qr

To obtain the bias and mean squared error of the proposed class of estimators ‘t’ we write y  Y 1 e0  , x  X 1 e1  , s xy  S xy 1 e2  , s x2  S x2 1  e3 

Such that E ei   0 for all i= 1,2,3; and

 

E e02 

1  f  C 2 y

 

, E e12 

1  f  C 2 x

, E e0 e1  

1  f  CC 2 x

n n n N  n   21 , N  N  n   21 1 E e1e2    N  1N  2 n XS xy nN  2 X11

E e1e3  

N  n  30 N N  n   30 1  2 N  1N  2 n XS x nN  2 X 20

where

 rs  E xi  X   yi  Y  ,

 yx 



r

s



C   yx

Cy Cx

,

,

Cy 

Sy Y

, Cx 

Sy X

and

S xy

S x S y  ,r , s  being non-negative integers.

Expressing‘t’ defined by (2.6) in terms of e’s we have

  X t  Y 1  e0    Y 

  e1 1  e2 1  e3 1 1  e1 1 .  

1 We assume that e1  1 and e3  1 so that we 1  e3  and 1  e1  are expandable. 1

Expanding the right hand side of (2.7), multiplying and neglecting terms of e’s having power greater than two we have





t  Y 1  e0  e1   2 e12  e0 e1  c e1  e1e2  e1e3  e12 Or



t  Y   Y e0  e1  2e12  e0e1  ce1  e1e2  e1e3  e12 ,

where  

(2.8)

aX . aX  b 

Taking expectation of both sides of (2.2) we get the bias of ‘t’ to the the first degree of approximation as

20

Bt  

1  f   R 2 S x2



n

 

J

Y



   N   21  30  N  2  11  20 

1  f   A  B 

(2.9)

n

 S2     N aY , A  R J2  x  and B    21  30  . aX  b  N  2  11  20  Y  The correct biases of the estimators listed in Table 2.1 and 2.2 can be easily obtained from (2.9) just by putting the suitable values of (a, b). The biases of the estimators belonging to the class of estimators ‘t’ is negligible if the sample size n is sufficiently large i.e. n  N  . It should be noted that the biases of the estimators t1 to t 45 listed in

where R J 

Table 2.1 reported in Subramani and Kumarapandian ( 2012a,b,c,d ) and Abid et al (2016a, b,c) are not correct. Squaring both sides of (2.8) and neglecting terms of e’s having power greater than two we have

t  Y   Y 2 e02  2e12  C 2e12  2e0e1  2Ce0e  2Ce12 

(2.10)

Taking expectation of both sides of (2.10) we get the MSE of ‘t’ to the first degree of approximation as 1  f  R 2 S 2  S 2 1  e 2 MSE t   (2.11) J x y n The MSE of the estimators belonging to class of estimators ‘t’ can be obtained from (2.11) just by putting the suitable values of (a, b). The proposed class of estimators ‘t’ is more efficient than the usual unbiased estimator y if







MSE t  < MSE  y 

i.e. if

RJ2 S x2   2

(2.12)

Thus the members of the proposed class of estimators ‘t’ is better than the usual unbiased estimator y as along as the condition (6) is satisfied. Further from (2.5) and (2.11) we have that MSE t  < MSE  y r  i.e. if

R J2  R  2

(2.13)

Thus the members of the proposed class of estimators ‘t’ is more efficient than the usual ratio estimator y R as long as the condition (2.13) is satisfied. 3. The Suggested Class of Ratio-Type Exponential Estimators 21

We define a class of ratio-type exponential estimators for the population mean Y as

 aX  x   t e  Yˆ exp   a X  x   2b 

aX  x   ˆ X  x  exp (3.1)  y   ,  aX  x   2b  where (a, b) are same as defined for the class of estimators ‘t’ at (2.1). A large number of estimators can be identified from the proposed class of estimators t e for suitable values





of (a, b). Some members of the proposed class of estimators t e corresponding to the members of the class of estimator t are listed in Table 3.1. Table 3.1 – Some members of the class of estimators t e corresponding to the estimators listed in Table 2.1. S.No.

Estimators

MSE

1.

X x ˆ  t1e  Y exp   X x

2.

  X x ˆ  t2e  Y exp   X  x  2 C x 

3.

  X x ˆ  t3e  Y exp     X  x  2  x 2  

4.

 X  x 2 x  ˆ t4e  Y exp    x  2  X  x  2C x

   

 R 24 S 2x  S 2y 1   2

5.

  Cx X  x ˆ  t5e  Y exp     C X  x  2  x 2  x 

 R 52 S 2x  S 2y 1   2

6.

 X x  ˆ  t6e  Y exp    X  x  2 

 R62 S x2  S y2 1  2

7.

 Cx X  x  ˆ  t 7e  Y exp   C X  x  2   x 

8.

  X x ˆ t 8e  Y exp    X  x  2C x 

9.

 2 x  X  x  ˆ  t9e  Y exp      x X  x  2   2 

10.

   X x ˆ  t10e  Y exp     X  x  2 2 x  













































   











 S x2  S y2 1  2  4 

  R22 

 S x2  S y2 1  2  4 

1

0

R1 

Y R X

1

Cx

R2 

X  Cx 

1

 2 x 

R3 

Y X  2 x 

Cx

R4 

X2 x  Cx 

Y



















Cx

 2 x 

R5 

XCx  2 x 







1



R6 

Y X 







Cx



R7 

XCx  









Cx

R8 

Y X  C x









R9 

X 2 x  



 

 2 x 

R10 

Y X  2 x 

 R32 S x2  S y2 1  2











  R12 

Values of Population Constants Ratio a b

 R 72 S 2x  S 2y 1   2

 R82 S x2  S y2 1  2

 R 92S2x  S2y 1   2 2 2  R10 S x  S y2 1  2

22

 2 x 

2 x 



Y2 x  YCx

YCx





Y  2 x 





S.No.

Estimators

11.

  X x ˆ  t11e  Y exp    X  x  21 x  

12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.



MSE











  1x  X  x ˆ  t12e  Y exp    1x  X  x  22 x  









 X x ˆ t13e  Y exp   X  x  2M d 




   

   C x X  x  ˆ  t14e  Y exp   C X  x   2M  d   x   1x X  x  ˆ  t15e  Y exp        x X  x  2 M d   1   2 x X  x  ˆ  t16e  Y exp    2 x X  x   2 M d   X  x   ˆ t17e  Y exp   X  x   2D1    X  x   ˆ t18e  Y exp   X  x   2D2    X  x   ˆ t19e  Y exp    X  x   2D3   X  x   ˆ t20e  Y exp   X  x   2D4    X  x   ˆ t21e  Y exp    X  x   2D5   X  x   ˆ t22e  Y exp   X  x   2D6    X  x   ˆ t23e  Y exp    X  x   2D7   X  x   ˆ t24e  Y exp   X  x   2D8    X  x   ˆ t25e  Y exp    X  x   2D9   X  x   ˆ t26e  Y exp   X  x   2D10    1x X  x   ˆ  t27e  Y exp    1x X  x   2Qd 







 







 



 

1 x 



 

 2 x 



 





1 x 

R11 

X  1x

 2 x 

R12 

X1x   2 x 

1

Md

R13 

X  M d 

Cx

Md

R14 



Md

R15 

X1x   Md 

Md

R16 

X2 x   Md 

1

D1

R17 

X  D1

 

1

D2

R18 

X  D2 



 

1

D3

R19 

X  D3 





1

D4

R20 



Y X  D4



 

1

D5

R21 

X  D5 







1

D6

R22 

Y X  D6





 

1

D7

R23 

X  D7 







1

D8

R 24 

Y X  D8









1

D9

R25 

X  D9 







1

D10

R 26 









Qd

R27 

X1x  Qd 

2 2  R11 S x  S y2 1  2

1

1 x 

2 2  R12 S x  S y2 1  2

2 2  R13 S x  S y2 1  2



2 2  R14 S x  S y2 1  2

2 2  R15 S x  S y2 1  2

2 2  R16 S x  S y2 1  2

2 2  R17 S x  S y2 1  2 2 2  R18 S x  S y2 1  2

2 2  R19 S x  S y2 1  2

 R 220S 2x  S 2y 1   2



2 2  R21 S x  S y2 1  2

2 2  R22 S x  S y2 1  2

2 2  R23 S x  S y2 1  2

 R 224S2x  S2y 1  2

 R 225S 2x  S 2y 1   2  R 226S 2x  S 2y 1   2  R 227S 2x  S 2y 1   2

23

1 x 

Y

Y 1x 

Y

YCx XC x  M d



1x Y

2 x Y

Y

Y

Y



Y



Y



Y

Y X  D10 Y1x 



S.No.

Estimators

28.

  X x ˆ  t28e  Y exp    X  x  2Tm 

MSE







35.

  Cx X  x   ˆ  t29e  Y exp    Cx X  x   2Tm    X  x  ˆ  t 30e  Y exp      X  x  2 T m   X  x   ˆ t 31e  Y exp    X  x   2M r    C x X  x  ˆ  t32e  Y exp     C X  x  2 M r  x    X  x  ˆ  t33e  Y exp     X  x   2 M r   X  x   ˆ t34e  Y exp   X  x   2H1    C x X  x   ˆ  t35e  Y exp    C x X  x   2H1 

36.

   X x ˆ  t36e  Y exp     X  x  2H1 

37.

  X x ˆ  t37e  Y exp    X  x  2G 

38.

 Xx  ˆ  t 38e  Y exp    X  x  2 G  

39.

 Cx X  x  ˆ  t39e  Y exp    Cx X  x  2G 

40.

  X x ˆ  t40e  Y exp    X  x  2D 

41.

  X x  ˆ  t41e  Y exp    X  x  2 D  

42.

 Cx X  x  ˆ  t42e  Y exp    Cx X  x  2D 

29. 30. 31. 32. 33. 34.




































































 R



 R 228S 2x  S 2y 1   2  R 229S 2x  S 2y 1   2 2 2  R 30 S x  S 2y 1   2

2 2 31S x



Tm

R28 





Cx

Tm

R29 

XCx  Tm 



Cx

Tm

R 30 



1

Mr

R31 

X  M r 

Cx

Mr

R31 

X  M r 

X  M r 



 S y2 1   2



2 2  R 32 Sx  S2y 1  2













Mr

R33 







1

H1

R34 







Cx

H1









H1







1

G









G







Cx

G







1















Cx

2 2  R 33 S x  S 2y 1   2

2 2  R 34 S x  S 2y 1   2

2 2  R 35 S x  S 2y 1   2

2 2  R 36 S x  S 2y 1   2

2 2  R 37 S x  S 2y 1   2

2 2  R 38 S x  S 2y 1   2

2 2  R 39 S x  S 2y 1   2

 R 240S2x  S2y 1  2

 R 241S2x  S2y 1  2

 R 242S2x  S2y 1  2

24

Y X  Tm

1

D

D

D



YCx

Y X  Tm



Y

YCx

Y

Y X  Hl





YCx XC x  H l

R35 



R 36 

X  H l 

R 37 

X  G 

R 38 

X  G 

R 39 

XCx  G 

Y

Y

Y

YC x

Y XD

R 40 



R 41 

X  D

R42 

XCx  D



Y

YCx



S.No.

Estimators

43.

  X x ˆ  t43e  Y exp   X  x  2 S pw   

44.

   X x ˆ  t44e  Y exp    X  x  2 S pw   

45.

  Cx X  x ˆ  t45e  Y exp   C x X  x  2 S pw   

46.

   X x ˆ  t46e  Y exp     X  x  2D1 

47.

   X x ˆ  t47e  Y exp    X  x  2D 2 

48.

   X x ˆ  t 48e  Y exp    X  x  2D  3 

49.

   X x ˆ  t49e  Y exp    X  x  2D 4 


MSE





























































  X x ˆ t 50e  Y exp    X  x  2D 5 







1









S pw







Cx

S pw









D1

R 46 

X  D1 

D2

R 47 

X  D2 

2 2  R44 S x  S y2 1  2

2 2  R45 S x  S y2 1  2

2 2  R46 S x  S y2 1  2





2 2  R47 S x  S y2 1  2





2 2  R48 S x  S y2 1  2

57.























  Cx X  x ˆ  t57e  Y exp   C X  x  2D 2  x





D4

R 49 

X  D4  Y X  D5

Y



D6

R51 

X  D6 







2 2  R52 S x  S y2 1  2

D7

R52 

X  D7 

R







D8

R53 

Y X  D8











D9

R54 

X  D9 









D10

R55 

X  D10 







Cx

D1

R56 

XC x  D1 

R57 

XC x  D2 

 Cx X  x  ˆ  t56e  Y exp    C x X  x  2D1 





X  D3 



56.



R48 



   X x ˆ  t55e  Y exp    X  x  2D 10  



D3



55.









Y



   X x ˆ  t54e  Y exp     X  x  2D9 







R50 

54.



Y

D5

53.





YCx



   X x ˆ  t53e  Y exp    X  x  2D 8 



XCx  S pw



52.



R45 





   X x ˆ  t52e  Y exp    X  x  2D 7 







2 2  R51 S x  S y2 1  2



Y X  S pw



   X x ˆ  t51e  Y exp     X  x  2D6 



R44 



51.



X  S pw 



2 2  R50 S x  S y2 1  2



R 43 



2 2  R49 S x  S y2 1  2

   

50.

Y

S pw

2 2  R43 S x  S y2 1  2



2 2 2 2 53S x  S y 1  

2 2  R54 S x  S y2 1  2

2 2  R55 S x  S y2 1  2 2 2  R56 S x  S y2 1  2





2 2  R57 S x  S y2 1  2

25



Cx

D2

Y



Y

Y



Y

Y

YC x

YC x

S.No.

Estimators

MSE

58.

  Cx X  x ˆ  t58e  Y exp    C x X  x  2D3 

2 2  R58 S x  S y2 1  2

59.


2 2  R59 S x  S y2 1   2

60.


2 2  R60 S x  S y2 1   2

61.


2 2  R61 S x  S y2 1   2

62.


2 2  R62 S x  S y2 1   2

63.


2 2  R63 S x  S y2 1  2

64.

  Cx X  x ˆ  t64  Y exp   C X  x  2D 9  x

2 2  R64 S x  S y2 1  2

65.

  Cx X  x ˆ  t65e  Y exp   C X  x  2D 10   x

2 2  R65 S x  S y2 1  2







































































Values of Population Constants Ratio a b Cx

D3







Cx

D4







Cx

D5







Cx

D6







Cx

D7







Cx

D8







Cx

D9







Cx

D10

YC x

R58 

XC x  D3 

R59 

XC x  D4 

R60 

XCx  D5 

R61 

XCx  D6 

R62 

XCx  D7 

R63 

XCx  D8 

R64 

XCx  D19 

R65 

XCx  D10 

YC x

YCx

YCx

YCx

YCx

YCx

YCx

Expressing t e in terms of e’s we have



t e  Y 1  e0  Ce1 1  e2 1  e3 

1



 e1    1  exp 1  e1    2  2  

(3.2)

C  X   y C    Cx  Y 

where, . Expanding the right hand side of (3.2), multiplying out and neglecting terms of e’s having power greater than two we have

e e  e   t e  Y 1  e0  1  C e1  e1e2  e1e3   0 1  3  4C e12  2 2 8   or

t e  Y   Y e0  e1  Ce1  C e1e2  e1e3   e0 e1   3  4C e12  .

(3.3) 2 2 8   Taking expectation of both sides of (3.3) we get the bias of ‘ t e ’to the first degree of approximation, we have Bt e  

1  f   3 R 2  S x2   n

8 

J

   N  21  30   Y  N  2  11  20    ,

26



1  f   3 A  B  ,  8

n

 

(3.4)

where RJ 

aY , A and B are same as defined earlier. aX  b 

The bias of t e at (3.4) is negligible if the sample size n is sufficiently large. The bias of the members of the proposed class of estimators can be obtained easily from (3.4) just by putting suitable values of the scalars (a, b). Squaring both sides of (3.3) and neglecting terms of e’s having power greater than two we have 

t e  Y 2  Y 2 e02   

  C 2 e12  e0 e1  2Ce0 e  Ce12  4 

2 2 e1

(3.5)

Taking expectation of both sides of (3.5) we get the MSE of t e to the first degree of approximation as

  1  f   2  S x2  2 2   MSE t e    S 1    RJ  . y 

(3.6)   4   Thus the MSE of the members of the proposed class of estimators t e can be easily

n

obtained from (3.6) just by putting the suitable values of (a, b). Remark-3.1- Motivated Swain (2014) one can define a class of ratio-type estimators for population mean Y as 12

 aX  b   . t s  Yˆ  (3.7) a x  b   Thus the form of the estimators t and t s taking into consideration, we define a class of

ratio-cum-product-type estimators for population mean Y as g

 aX  b   , t g  Yˆ  (3.8)  ax  b  where g is a scalar taking real values. We note that for g(>0) the class of estimators t g generates the ratio-type estimators while for g(