Optimal Sampling Strategies for Estimation of Population Mean using Auxiliary Information

International Journal of Agricultural Science and Research (IJASR) ISSN (P): 2250-0057; ISSN (E): 2321-0087 Vol. 10, Issue 3, Jun 2020, 91-100 © TJPRC Pvt. Ltd.

OPTIMAL SAMPLING STRATEGIES FOR ESTIMATION OF POPULATION MEAN USING AUXILIARY INFORMATION CHANDNI KUMARI & RATAN KUMAR THAKUR Department of Statistics, Bahasaheb Bhimrao Ambedkar University,(A Central University), Lucknow, India ABSTRACT This paper considers the problem of estimating the finite population mean in sample surveys. We have proposed the generalized class of estimators under Midzuno (1950)-Lahiri-Sen (1952) type sampling scheme. We obtain the expression for unbiasness and variance of the proposed sampling strategy. Further, we also obtain the unbiased estimate of the variance of the proposed sampling strategy. We compare the proposed sampling strategy with the various conventional estimators under the simple random sampling without replacement. An empirical study is given in support of the present study.

KEYWORDS: Simple Random Sampling, Ratio Type Estimator & Midzuno-Lahiri-Sen type Sampling Scheme

1 INTRODUCTION The use of probability in sampling theory came to be recognized as a relative tool in drawing inferences about the population, whether finite or not. In sample surveys, experimenter introduces the probability element by adopting the technique of randomization. The main objective of this work is to present the theory and techniques of sample

Original Article

Received: Apr 14, 2020; Accepted: May 04, 2020; Published: Jun 03, 2020; Paper Id.: IJASRJUN202011

surveys with their application in different types of problems in the field. This type of approach has been considered

by Singh (2005), Senapati (2006), Kumari et al. (2018, 2019) in which they improve the existing sampling strategies by using some auxiliary information. The purpose of this paper is to give the class of unbiased estimators under the proposed sampling strategy, but they are theoretically biased in simple random sampling without replacement. Sidodia and Dwivedi (1981), Pandey and Dubey (1988), Singh and Upadhya (1999), Singh (2003), Singh and Taylor (2003), Singh et al (2004), Koshnesevan (2007), Al-Omari (2009), Kumari and Thakur (2019, 2020) had proposed various biased ratio and product type estimators where auxiliary information about the parameters is given in the form of coefficient of variation, correlation coefficient, coefficient of skewness, coefficient of kurtosis, standard deviation, quartiles, etc. In this paper, we make all these estimators unbiased by using Midzuno (1950), Lahiri and Sen (1952) type sampling scheme.

We are usually faced with a collection, often called population

U  U1 ,U 2 ,...,U N 

of which some

characteristic yi is defined for every unit ui (i=1,2,…,N). Sample survey is mainly concerned with ways of obtaining samples, in order to estimate the population parameters such as population total (Y), population mean ( Y ), population variance (  ), population correlation coefficient (  ), etc. The method of sampling must also 2

define the sampling scheme, sampling design and sampling strategy. No sampling procedure is complete in itself rather it require two things (i) estimation procedure (ii) a sampling design. Thus, here we introduce a new family of

www.tjprc.org

editor@tjprc.org

92

Chandni Kumari & Ratan Kumar Thakur

sampling strategies (i.e. the combination and estimator and sampling design) denoted by (t, SRS), (t, PPS), etc.

Let yi be the i

th

characteristics of the population under study. The population mean of the study variable yi is

denoted by Y and it is given as Y  ‘x’. It is given as X 

1 N

N

y i 1

also the prior information about study variable is given and is denoted by

i

N

1 N

 x . Further i 1

i

yi and xi are linearly related so there exists a correlation coefficient between

them which measures the degree or extent of linear relationship between study variable and the auxiliary variable. It is denoted by

 and is given as  

variable and

cov  x, y 

 x y

 x is the standard deviation

, where cov( x, y ) is the covariance between study variable and auxiliary

of auxiliary variable and

 y is the standard deviation of study variable. Also

C y , C x and C yx be the coefficient of variation of ‘y’, ‘x’ and ‘xy’ respectively and it is given as

Cy 

y Y

and C x 

x

and C yx 

X

 y x YX

Now let us take a random sample ‘s’ of size ‘n’ from population ‘N’, then the sample mean of study variable and auxiliary variable is given as y 

1 n 1 n x  xi respectively. y  i and n  n i 1 i 1

2. PROPOSED SAMPLING STRATEGY We propose the following estimator under the Midzuno( 1950)-Sen(1952) type sampling scheme for estimating the unknown population parameter i.e. population mean

yAM = where

Y

ys  X+ 

 X+  +A (x-X)

 and 

(2.1)

are known constants and A is the characterizing scalar.

The units vary considerably in size as in often the case, simple random sampling is not an appropriate procedure as no importance is given to the size of the units. There are various ways of using such auxiliary information for obtaining the most efficient estimators. Thus one such strategy is proposed here i.e. Midzuno-Sen-Type sampling scheme is used for obtaining the unbiased estimator of the population mean. Under this system of selection probabilities, due to Midzuno (1950) and independently by Sen (1952) which consists in selecting the first unit with probability proportional to the value of the variate while the remaining (n-1) units are selected from the (N-1) units with equal probability and without replacement. Then the probability of selecting a specified sample‘s’ of size ‘n’ units is given by n

P(s) =

 P {selecting y at first draw} x P{selecting the remaining (n-1) units given that y i 1

i

i

has been selected at

the first draw} Impact Factor (JCC): 7.9083

NAAS Rating: 4.13

Optimal Sampling Strategies for Estimation of Population Mean using Auxiliary Information

 αX + β  + Aα  x n

=

=

i 1

i



- X

93

1 Cn 1

N 1

    αX + β    i 1  N

 αX + β  + Aα  x - X  C  αX + β  N

n

We propose some class of ratio estimators where the auxiliary information is available in the terms of coefficient of variation, correlation coefficient, coefficient of skewness, coefficient of kurtosis, standard deviation, quartiles, etc. under the Midzuno-Sen type sampling scheme. Now, the proposed estimators under the Midzuno-Sen type sampling scheme is listed below and the corresponding probability of selecting a unit from sample is also given in table 1.

Table 1 Some Class Of Proposed Sampling Strategies

y  X + Cx 

t1 =

 X + C  + Aα  x - X  y X +   t =  X +   + Aα  x - X  y X +   t =  X +   + Aα  x - X  y  X +C  t =   X + C  + Aα  x - X  y C X +   t =  C X +   + Aα  x - X  y X +  t =  X +   + Aα  x - X  y   X +  t =   X +   + Aα  x - X  y   X +  t =   X +   + Aα  x - X  y X + q  t =  X + q  + Aα  x - X  y X + q  t =  X + q  + Aα  x - X  x s

2

s

2x

3

2x

s

2x

x

4

2x

x

s

x

2x

5

x

2x

s

x

6

x

s

1x

x

7

1x

x

s

2x

x

8

2x

x

s

1x

9

1x s

10

2x

2x

Probability Of Selecting A Unit From Sample

 X + C  + Aα  x - X  C X + C   X +   + Aα  x - X  Ps  = C X +    X +   + Aα  x - X  Ps  = C X +     X + C  + Aα  x - X  P s  = C  X +C   C X +   + Aα  x - X  Ps  = C C X +    X +   + Aα  x - X  Ps  = C X +    X +   + Aα  x - X  P s  = C   X +    X +   + Aα  x - X  Ps  = C   X +   X + q  + Aα  x - X  Ps  = C X + q   X + q  + Aα  x - X  Ps  = C X + q  P  s1  =

x

N

n

2

x

N

n

2x

3

N

n

2x

4

2x

x

N

n

2x

x

5

x

2x

N

n

x

2x

x

6

N

n

1x

7

x

x

N

n

2x

8

1x

x

2x

x

x

N

n

1x

9

N

n

1x

2x

10

N

n

2x

where t1, t2,…, t10 are the various estimators uses the auxiliary information about coefficient of variation, www.tjprc.org

editor@tjprc.org

94

Chandni Kumari & Ratan Kumar Thakur

correlation coefficient, coefficient of skewness, coefficient of kurtosis, standard deviation, quartiles, etc.

2.1 Properties of Considered Estimator under Proposed Sampling Strategy Theorem 2.1.1 yAM is an unbiased estimator of population mean under the proposed sampling design.

Theorem 2.1.2 The minimum variance of yAM under the proposed sampling design is equal to the variance of linear regression estimator.

Theorem 2.1.3 The unbiased estimate of the variance of yAM under the proposed sampling design is given as follows

   2  1 1  2   X     y AM 2     y     sy  n N     X     A  x  X    3. DOMINANCE CONDITION OF PROPOSED SAMPLING STRATEGY 3.1 Comparison with mean per unit estimator

MSE  yAM   MSE  y wor i.e. if ,  2 ACxy   A2 2Cx2

(3.1.1)

If A>0 and R>0, the dominance condition (3.1.1) becomes

A

2C



i.e. C 

A 2

(3.1.2)

If A<0 and R>0, the dominance condition (3.1.1) becomes

A

2C



i.e. C 

A 2

(3.1.3)

3.2 Comparison with Ratio Estimator

MSE  yAM   MSE  y ratio iff  A 1 A  1  2C   0

(3.2.1)

If A>1, the dominance condition (3.2.1) becomes

C

A  1 2

(3.2.2)

If A<1, the dominance condition (3.2.1) becomes

C

A  1 2

Impact Factor (JCC): 7.9083

(3.2.3)

NAAS Rating: 4.13

Optimal Sampling Strategies for Estimation of Population Mean using Auxiliary Information

95

3.3 Comparison with product estimator

MSE  y AM   MSE  y  pro iff  A  1 A  1  2C   0

(3.3.1)

If A>-1, the dominance condition (3.3.1) becomes

C

A  1 2

(3.3.2)

If A<-1, the dominance condition (3.3.1) becomes

C

A  1 2

(3.3.3)

3.4 Comparison with linear regression estimator

MSE  y AM opt  MSE  y lin reg Both are equally efficient but strategy whereas

 y lin reg

(3.4.1)

 y AM opt

is unbiased estimator for all values of A under the proposed sampling

is biased estimator for population mean under simple random sampling without replacement.

3.5 Comparison with Generalized Ratio Estimator

MSE  y AM opt  MSE  y A opt

(3.5.1)

Both are equally efficient, but the generalized ratio estimator is almost unbiased at A=C only whereas the estimator under the proposed sampling strategy is unbiased at all values of A. 3.6 Concluding Remarks Using suitable value of C, we find some efficient estimators which have minimum mean square error as follows: We know that sample mean in case of SRSWOR

 y wor

is preferred to ratio and product estimator when

0  C  1/ 2 and 1/ 2  C  0 respectively. From efficiency condition (4.1.2) and (4.1.3), we get the class of estimators which have lesser mean square error than mean per unit estimator even in such circumstances i.e. and

0  C  1/ 2

1/ 2  C  0 . Let the range information about C is known as C  C0 where 0  C0  1/ 2 , such that

or A 

2C0



A  C0 2

, where C0 be the prior information of C. Thus, choose A for satisfying dominance condition (4.1.1) to get a

class of estimators as

www.tjprc.org

editor@tjprc.org

96

Chandni Kumari & Ratan Kumar Thakur

y 2C0  

y  X     X     2C0   x  X 



(3.6.1)

Are better than sample mean in simple random sampling without replacement and they have lesser mean sum of square. Further, it is known that C>C1 where (4.1.3) such that A 

2C1



1/ 2  C1  0 then we choose A for satisfying the dominance condition

  0

So that the class of estimators

y 2C1  

y  X     X     2C1   x  X 



(3.6.2)

are better than mean per unit in the sense of lesser mean square error. More specifically, for prior range information C<1/3.

Choosing A 

2 , satisfying efficiency condition (4.1.2), so we get 3

y2/3 

y  X     X     32  x  X 

(3.6.3)

We get more efficient unbiased estimator and have smaller MSE than mean per unit estimator i.e. ywor .

Let the range information about C be known then from the dominance condition (3.2.2), we have C>C 0 (>1), then

A  1  C0 or A  2C0  1 to get the class of estimators.

2

y 2C0 1  

y  X   

 X     

2C0  1



x  X 

(3.6.4)

Which are better than ratio estimator since it has lesser mean sum of square. Further, it is known that C<C1 (<1) then we choose A such that A  2C1  1 satisfying the efficiency condition (4.2.3) so that the class of estimators

y 2C1 1  

y  X   

 X     

Impact Factor (JCC): 7.9083

2C1  1



x  X 

(3.6.5)

NAAS Rating: 4.13

Optimal Sampling Strategies for Estimation of Population Mean using Auxiliary Information

97

Are better than the ratio estimator in the sense of having lesser mean square error. For example, if it is known that C>3/2 we may choose A 

y2  

2



satisfying the dominance condition (3.2.2) to obtain more efficient unbiased estimator

y  X     X     2  x  X 

(3.6.6)



than ratio estimator y Ratio in the sense of having lesser mean square error.

If the range information about C be known as C>C0 (>-1) then we choose A such that

A  1  C0 or 2

A  2C0  1 satisfying the dominance condition (3.3.2) to get the class of estimators

y 2C0 1  

y  X   

 X     

2C0  1



x  X 

(3.6.7)

Which are better than the product estimator in the sense of having the lesser mean square error. Further, if it is known that C<C1(<-1) then we choose A  2C1  1 satisfying the efficiency condition (3.3.3) so that the class of estimators

y 2C1 1  

y  X     X     2C1  1  x  X 

(3.6.8)



which is more efficient than the product estimator y pro in the sense of having smaller mean square error.

Finally, from the comparative study, it is clear that y AM is the most efficient estimator than the mean per unit estimator, ratio estimator and product estimator. In case, linear regression and Walsh estimator, they are equally efficient but the linear regression and Walsh estimator is unbiased only for the characterizing scalar whereas y AM i.e. the proposed Midzuno-Sen Type sampling strategy is unbiased for all values of characterizing scalar and can be better alternative in various practical situations.

4. EMPIRICAL STUDY An empirical study is performed on various populations. We know that the variance of the proposed sampling strategy is equal to the estimators under without replacement (to the first order of approximation). But the advantage of proposed sampling strategy is that the estimators are unbiased while the estimators are biased in case of sampling without replacement. Thus in the following table, the bias under without replacement and the variance (i.e. mean square error which is same) under Midzuno-Sen -Type sampling strategy are obtained as follows:

www.tjprc.org

editor@tjprc.org

98

Chandni Kumari & Ratan Kumar Thakur

Estimator T1 T2 T3 T4 T5 T6 T7 T8 T9 T10

POPULATION 1 BIAS -63.1041 -54.2168 -107.15 -42.8581 -92.1683 -132.134 -129.094 -109.573 -71.382 -127.122

VAR 202517.4 198471.5 242929.6 194533.2 224019.4 509402.3 293850.1 246795.3 207191.1 543194.7

Table 2 POPULATION 2 BIAS 2.537474 2.532575 2.514122 2.538936 2.437171 1.136236 1.136236 1.908737 -0.0446 -0.27276

VAR 1518.527 1516.616 1509.429 1519.097 1479.616 1027.391 1393.93 1282.637 809.4946 846.5869

POPULATION 3 BIAS 0.15604 0.151128 0.145683 0.157271 0.122961 0.065646 -9.5E-0.5 0.088981 0.028308 0.001753

VAR 1.474065 1.423265 1.557671 1.486827 1.137895 0.601586 0.862671 0.810684 0.31414 0.196705

5. CONCLUSIONS For population 1, T4 is the best estimator because T4 has the minimum variance. Similarly for population 2, T9 is the best estimator and for population 3, T10 is the best estimator.

REFERENCES 1.

Kumari, C and Thakur, R. K. (2020). “Improved Ratio type Estimators Using Auxiliary attribute For Population Variance”, Int. J. Sci. Res., 9(4), 1491-1505

2.

Kumari, C and Thakur, R. K. (2020). “Family of Estimators for Population Variance using Two Auxiliary Information”, Int. J. Res. App. Sci. Eng. Tech., 8(5), 310-315

3.

Kumari, C and Thakur, R. K. (2020). “An Advanced Class of Log type estimators for Population Variance Using an Attribute and a Variable”, Int. J. Indus. Eng. Res. Develop., 11(1), 1-7

4.

Kumari, C and Thakur, R. K. (2020). “An Improved Estimation of Population Variance Using the Coefficient of Kurtosis and Median of an Auxiliary Variable”, Int. J. Eng. Sci. Res. Tech., 9(4), 168-175

5.

Kumari, C., Bhushan, S. And Thakur, R. K. (2019). Optimal Two Parameter Logarithmic Estimators for Estimating the Population Variance, Global Journal of Pure and Applied Mathematics, Volume 15, Issue 5, pg 527-237.

6.

Kumari, C., Bhushan, S. and Thakur, R. K. (2018). Modified Ratio Estimators Using Two Auxiliary Information for Estimating Population Variance in Two-Phase Sampling, International Journal of Scientific and Engineering Research, Volume 9, Issue 8, pg 18841892.

7.

Bhushan, S. and Kumari, C. (2018). An Unbiased Family of Sampling Strategies for Estimation of Population Variance, International Journal of Statistics and Systems, 13 ( 1), 41-52

8.

Bhushan, S. and Kumari, C. (2018). A Class of Double Sampling Log-Type Estimators for Population Variance Using Two Auxiliary Variable, International Journal of Applied Engineering Research, Volume 13, Number (13) 2018, pg 11151-11155.

9.

Lahiri D. B. (1951). A method of sample selection providing unbiased ratio estimates, Bull. Int. Stat. Inst., 3, p. 133-140.

10. Midzuno H. (1952). On the sampling system with probability proportional to the sum of the sizes, Ann. Ints. Stat. Math., 3, p.

99-107.

Impact Factor (JCC): 7.9083

NAAS Rating: 4.13

Optimal Sampling Strategies for Estimation of Population Mean using Auxiliary Information

99

11. Murthy M. N. (1967). Sampling theory and method, Statistical Publishing Society, Calcutta, India.

12. Sen A. R. (1952). Present status of probability sampling and its use in the estimation of characteristics, Econometrika, p. 20103 13. Walsh J. E. (1970). Generalization of ratio estimator for population total, Sankhya, A, 32, p. 99-106.

14. Das, A. K., and Tripathi, T. P (1978). Use of auxiliary information in estimating the finite population variance. Sankhya, 40(C), 139-148.

www.tjprc.org

editor@tjprc.org