Estimation of Population Mean Using Co-Efficient of Variation and ...

International Journal of Probability and Statistics 2012, 1(4): 111-118 DOI: 10.5923/j.ijps.20120104.04

Estimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable J. Subramani * , G. Kumarapandiyan Department of Statistics, Ramanujan School of M athematical Sciences, Pondicherry University, R V Nagar, Kalapet, 605014, Puducherry

Abstract The present paper deals with a class of modified rat io estimators for estimation of population mean of the study

variable using the linear comb ination of the known values of the Co-efficient of Variation and the Median of the auxiliary variable. The biases and the mean squared errors of the proposed estimators are derived and are compared with that of existing modified ratio estimators. Further we have also derived the conditions for which the proposed estimators perform better than the existing modified rat io estimators. The performances of the proposed estimators are also assessed with that of the existing estimators for certain natural populations. From the numerical study it is observed that the proposed modified ratio estimators perform better than the existing modified rat io estimators.

Keywords Mean Squared Error, Modified Rat io Estimators, Natural Populations, Simple Random Sampling

1. Introduction The simp lest estimator of population mean is the sample mean obtained by using simple rando m sampling without replacement, when there is no addit ional information on the auxiliary variable available. So metimes in samp le surveys, along with the study variable Y , information on au xiliary variable X , correlated with Y , is also collected. Th is informat ion on au xiliary variable X, may be utilized to obtain a more efficient estimator of the population mean. Ratio method of estimation is an attempt in this direction. This method of estimat ion may be used when (i) X represents the same character as Y, but measured at some previous date when a complete count of the population was made and (ii) the character X is cheaply, quickly and easily available. Consider a finite population U = { U1 , U2 , … , UN } of N distinct and identifiab le units. Let Y is a study variable with value Yi measured on Ui , i = 1,2,3, … , N giving a vector Y = {Y1 , Y2 , … , YN } and let X is an au xiliary variab le which is readily available. The problem is to estimate the � = 1 ∑Ni=1 Yi with some desirable population mean Y N

properties on the basis of a random samp le selected fro m the population U using auxiliary in formation. When the population parameters of the auxiliary variable X such as Popu lat ion Mean , Co -efficient of Variat ion, Co-efficient of Kurtosis, Co-efficient of Skewness, Median are known, a nu mber of estimators such as ratio, product and linear reg ression estimato rs and their mod ificat ions are

* Corresponding author: [email protected] (J. Subramani) Published online at http://journal.sapub.org/ijps Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved

proposed in the literature. Before d iscussing further about the modified rat io estimators and the proposed modified rat io estimators the notations to be used in this paper are described below: • N − Population size • n − Sample size • f = n/N, Sampling fraction • Y − Study variable • X − Auxiliary variable • X� , Y� − Population means • x� , y � − Samp le means • SX , Sy − Population standard deviations • CX , Cy − Co-efficient of variations • ρ − Co-efficient of correlation • β1 =

( �)3 N ∑N i =1 X i −X (N−1)(N−2)S 3

, Co-efficient of skewness of the

auxiliary variab le • β2 =

( �)4 N (N+1) ∑N i=1 X i −X (N−1)(N−2)(N−3)S 4

−

3 (N−1)2 (N−2)(N−3)

, Co-efficient of

kurtosis of the auxiliary variable • M d −Median of the au xiliary variable • B(. ) − Bias of the estimator • MSE ( . ) − Mean squared error of the estimator �pi ) − Exist ing (proposed) modified rat io estimator •� Yi (Y � of Y The Ratio estimator for estimating the population mean � Y of the study variable Y is defined as � = R� X� �R = y� X Y y� where R� = = x�

y x

x�

is the estimate of R =

Y� X�

=

Y

X

(1)

List of modified ratio estimators together with their biases, mean squared errors and constants available in the literature are classified into two classes namely Class 1, Class 2 and

112

J. Subramani et al.: Estimation of Population M ean Using Co-Efficient of Variation and M edian of an Auxiliary Variable

are given respectively in Tab le 1 and Table 2 respectively. It is to be noted that “the existing mod ified ratio estimators” means the list of modified ratio estimators to be considered in this paper unless otherwise stated. It does not mean to the entire list of mod ified ratio estimators available in the literature. For a mo re detailed discussion on the ratio estimator and its modifications one may refer to Cochran[1], Kadilar and Cingi[2,3], Koyuncu and Kadilar[4], Murthy[5], Prasad[6], Rao[7], Singh and Tailor[9, 11], Singh et.al[10], Sisodia and Dwivedi[12], Subramani and Ku marapandiyan [13,14,15,16,17], Upadhyaya and Singh[18], Yan and Tian[19] and the references cited there in. The modified ratio estimators given in Tab le 1 and Table 2 are b iased but have minimu m mean squared errors

compared to the classical ratio estimator. The list of estimators given in Table 1 and Table 2 uses the known values of the parameters like X� , Cx , β1 , β2 , ρ , M d and their linear co mb inations. However, it seems, no attempt is made to use the linear co mb ination of known values of the Co-efficient of variation and Median of the auxiliary variable to imp rove the ratio estimator. The points discussed above have motivated us to introduce modified ratio estimators using the linear co mbination o f the known values of Co-efficient of variation and Median of the auxiliary variab le. It is observed that the proposed estimators perform better than the existing mod ified rat io estimators listed in Table 1 and Table 2.

Table 1. Existing modified ratio estimators (Class 1) with their biases, mean squared errors and their constants Estimator X� + Cx �Y1 = y� � � x� + Cx

Sisodia and Dwivedi[12]

X� + β 2 � � Y2 = y� � x� + β 2

Singh et.al[10]

Bias - 𝐁𝐁( . )

Mean square d e rror 𝐌𝐌𝐌𝐌𝐌𝐌( . )

(1 − f) 2 � Y (θ1 Cx2 − θ1 Cx Cy ρ) n

( 1 − f) 2 2 2 � Y (Cy + θ1 Cx2 − 2θ1Cx Cy ρ) n

� X θ1 = � X + Cx

( 1 − f) 2 2 2 � Y (Cy + θ2 Cx2 − 2θ2Cx Cy ρ) n

� X θ2 = � X + β2

(1 − f) 2 � Y (θ2Cx2 − θ2 Cx Cy ρ) n

X� + β 1 �3 = y� � � Y x� + β 1

(1 − f) 2 � Y (θ3Cx2 − θ3 Cx Cy ρ) n

X� + ρ �4 = y� � � Y x� + ρ

(1 − f) � (θ24Cx2 − θ4 Cx Cy ρ) Y n

X� Cx + β 2 � � Y5 = y� � x�Cx + β 2

(1 − f) � (θ25Cx2 − θ5 Cx Cy ρ) Y n

X� β + Cx � � Y6 = y� � 2 x�β 2 + Cx

(1 − f) � (θ26Cx2 − θ6 Cx Cy ρ) Y n

Yan and Tian[19]

Singh and Tailor[9]

Upadhyaya and Singh[18]

Upadhyaya and Singh[18]

X� β + β 2 � � Y7 = y� � 1 x�β 1 + β 2

Yan and Tian[19]

X� Cx + β 1 � � Y8 = y� � x�Cx + β 1

Yan and Tian[19]

X� + Md �9 = y� � � Y x� + Md

Subramani and Kumarapandiyan[15]

( 1 − f) 2 2 2 � Y (Cy + θ3 Cx2 − 2θ3Cx Cy ρ) n

Constant 𝛉𝛉 𝐢𝐢

� X θ3 = � X + β1

( 1 − f) 2 2 � (Cy + θ24 Cx2 − 2θ4Cx Cy ρ) Y n

� X θ4 = � X+ρ

( 1 − f) 2 2 2 � Y (Cy + θ5 Cx2 − 2θ5Cx Cy ρ) n

X� Cx θ5 = � X Cx + β 2

( 1 − f) 2 2 2 � Y (Cy + θ6 Cx2 − 2θ6Cx Cy ρ) n

X� β 2 θ6 = � X β 2 + Cx

(1 − f) 2 � Y (θ7Cx2 − θ7 Cx Cy ρ) n

( 1 − f) 2 2 2 � Y (Cy + θ7 Cx2 − 2θ7Cx Cy ρ) n

X� β 1 θ7 = � X β1 + β2

(1 − f) � (θ28Cx2 − θ8 Cx Cy ρ) Y n

( 1 − f) 2 2 2 � Y (Cy + θ8 Cx2 − 2θ8Cx Cy ρ) n

X� Cx θ8 = � X Cx + β 1

(1 − f) � (θ29 Cx2 − θ9 Cx Cy ρ) Y n

( 1 − f) 2 2 � (Cy + θ29 Cx2 − 2θ9Cx Cy ρ) Y n

� X θ9 = � X + Md

International Journal of Probability and Statistics 2012, 1(4): 111-118

113

Table 2. Existing modified ratio estimators (Class 2) with their biases, mean squared errors and their constants Estimator

Bias-𝐁𝐁( . )

y� + b( � X − x�) � � Y1 = X x�

(1 − f) S2x 2 � R1 n Y

y� + b( � X − x�) �2 = � + Cx ) (X Y ( x� + Cx )

(1 − f) S2x 2 R � n Y 2

y� + b( � X − x�) � � + β2) (X Y3 = ( x� + β2) Kadilar and Cingi[2]

(1 − f) S2x 2 R � n Y 3

y� + b( � X − x�) � ( X� β2 + Cx ) Y4 = ( x�β2 + Cx ) Kadilar and Cingi[2]

(1 − f) S2x 2 � R4 n Y

y� + b( � X − x�) �5 = ( X� Cx + β2 ) Y ( x�Cx + β2 )

(1 − f) S2x 2 R � n Y 5

y� + b( � X − x�) � (� Y6 = X + β1) ( x� + β1)

(1 − f) S2x 2 � R6 n Y

y� + b( � X − x�) �7 = � + ρ) (X Y ( x� + ρ)

(1 − f) S2x 2 � R7 n Y

y� + b( � X − x�) � (� Y8 = XCx + ρ) ( x�Cx + ρ)

(1 − f) S2x 2 � R8 n Y

y� + b( � X − x�) � �ρ + Cx ) (X Y9 = ( x�ρ + Cx ) Kadilar and Cingi[3]

(1 − f) S2x 2 � R9 n Y

y� + b( � X − x�) � (� Y10 = Xβ2 + ρ) ( x�β2 + ρ) Kadilar and Cingi[3]

(1 − f) S2x 2 � R10 n Y

y� + b( � X − x�) � �ρ + β2 ) (X Y11 = ( x�ρ + β2)

(1 − f) S2x 2 R � n Y 11

Kadilar and Cingi[2]

Kadilar and Cingi[2]

Kadilar and Cingi[2]

Yan and Tian[19]

Kadilar and Cingi[3]

Kadilar and Cingi[3]

Kadilar and Cingi[3]

2. Proposed Modified Ratio Estimators

In this section, we have suggested a class of modified ratio estimators using the linear co mbination of Co-efficient of variation and Median of the au xiliary variab le. The

Mean square d e rror 𝐌𝐌𝐌𝐌𝐌𝐌( . )

Constant 𝐑𝐑𝐢𝐢

( 1 − f) 2 2 � R1Sx + S2y ( 1− ρ2)� n

� Y R1 = � X

( 1 − f) 2 2 � R2Sx + S2y ( 1− ρ2)� n

� Y R2 = � X + Cx

( 1 − f) 2 2 � R3Sx + S2y ( 1− ρ2)� n

� Y R3 = � X + β2

( 1 − f) 2 2 � R4Sx + S2y ( 1− ρ2)� n

�β2 Y R4 = � Xβ2 + Cx

( 1 − f) 2 2 � R5Sx + S2y ( 1− ρ2)� n

� YCx R5 = � XCx + β2

( 1 − f) 2 2 � R6Sx + S2y ( 1− ρ2)� n

� Y R6 = � X + β1

( 1 − f) 2 2 � R7Sx + S2y ( 1− ρ2)� n

� Y R7 = � X+ρ

( 1 − f) 2 2 � R8Sx + S2y ( 1− ρ2)� n

�Cx Y R8 = � XCx + ρ

( 1 − f) 2 2 � R9Sx + S2y ( 1− ρ2)� n

�ρ Y R9 = � X ρ + Cx

( 1 − f) 2 2 � R10Sx + S2y ( 1− ρ2)� n

�β2 Y R10 = � Xβ2 + ρ

( 1 − f) 2 2 � R11Sx + S2y ( 1− ρ2)� n

�ρ Y R11 = � Xρ + β2

proposed modified ratio estimators for estimating the population mean Y� together with the first degree of approximation, the biases and mean squared errors and the constants are given below:

J. Subramani et al.: Estimation of Population M ean Using Co-Efficient of Variation and M edian of an Auxiliary Variable

114

Table 3. Proposed modified ratio estimators with their biases, mean squared errors and their constants Bias 𝐁𝐁( . )

Estimator X� Cx + Md � � Yp1 = y� � x� Cx + Md

(1 − f) � (θ2p1 Cx2 − θp1 Cx Cy ρ) Y n

( 1 − f) 2 2 � (Cy + θ2p1 Cx2 − 2θp1 Cx Cy ρ) Y n

y� + b( � X − x�) � ( X� Cx + Md ) Yp3 = ( x�Cx + Md )

(1 − f) S2x 2 � Rp3 n Y

( 1 − f) 2 2 � Rp3 Sx + S2y ( 1 − ρ2 )� n

y� + b( � X − x�) � � + Md ) (X Yp2 = ( x� + Md )

(1 − f) S2x 2 � Rp2 n Y

3. Efficiency Comparison

where θ1 =

θ5 =

θ8 =

X� C x

X� C x +β 2 X� C x X� C x +β 1

X�

X�+C x

(1−f) n

n

� 2 �Cy2 + θ2i Cx2 − 2θi Cx Cy ρ�; Y

i = 1, 2, 3, . . . , 9 X�

, θ2 =

X�+β 2 X� β 2

, θ6 = and

, θ3 =

, θ7 X� β 2 +C x � X θ9 = � X +M d

=

X�

X�+β 1 X� β 1

X� β 1 +β 2

, θ4 =

,

(2)

X� , � X +ρ

Class 2: The biases, the mean squared errors and the constants of the 11 modified ratio estimators �Y1 to �Y11 listed in the Table 2 are represented in a single class (say, Class 2), wh ich will be very much useful for comparing with that of proposed modified rat io estimators and are given below: (1−f) S 2x 2 Rj B� � Yj � = � �j � = MSE � Y Y�

R8 =

n

n

Y

� R 2j S2x + S2y (1 − ρ2 )� ;

j = 1, 2, 3, . . . , 11

X

Y� X�+C x Y�C x

(3)

Y� , X�+β 2 � � Yβ2 Y Y� , R5 = � , R6 = � , R7 = � , � X β 2 +C x X C x +β 2 X +β 1 X +ρ Y�C x Y�ρ Y�β 2 Y�ρ , R = , R = and R 9 10 11 = X�ρ +β X�C x +ρ X�β 2 +ρ X�ρ+C x 2

where R 1 = � , R 2 =

R4 =

(1−f)

, R3 =

As derived earlier in section 2, the biases, the mean squared errors and the constants of the proposed modified ratio estimators are given below: (1−f) � Y (θ2p 1 Cx2 − θp 1 Cx Cy ρ) B� � Yp 1 � = MSE � � Yp 1 � =

n (1−f) n

� 2 �Cy2 + θ2p 1 Cx2 − 2θp 1 Cx Cy ρ� Y

where θp 1 =

θp1 = �

� Cx X

Rp3 = �

� Y Cx

X Cx + M d

� Y Rp2 = � X + Md

( 1 − f) 2 2 � Rp2 Sx + S2y ( 1 − ρ2 )� n

X Cx + M d

2

For want of space; for the sake of convenience to the readers and for the ease of comparisons, the modified rat io estimators given in Tab le 1, Table 2 are represented into two classes as given below: Class 1: The biases, the mean squared errors and the constants of the modified ratio type estimators �Y1 to �Y9 listed in the Table 1 are represented in a single class (say, Class 1), wh ich will be very much useful for comparing with that of proposed modified rat io estimators and are given below: (1−f) � Y (θ2i Cx2 − θi Cx Cy ρ) B� � Yi � = MSE � � Yi � =

Constants 𝛉𝛉 𝐢𝐢 𝐨𝐨𝐨𝐨 𝐑𝐑𝐢𝐢

Mean square d e rrors 𝐌𝐌𝐌𝐌𝐌𝐌( . )

X�C x X�C x +M d

(4)

(1−f) S x 2 R B� � Yp 2 � = Y� p 2 n (1−f) � R 2p 2 S2x + S2y (1 − ρ2 ) � MSE � � Yp 2 � = n

(1−f) S 2x

B� � Yp 3 � = n MSE � � Y �= p3

Y� (1−f) n

where R p 2 =

R 2p 3

Y� � X +M d

(5)

� R 2p 3 S2x + S2y (1 − ρ2 ) � where R p 3 =

Y�C x X�C x +M d

(6)

Fro m the exp ressions given in (2) and (4) we have derived the conditions for wh ich the proposed estimator Y�p 1 is mo re efficient than the existing modified ratio estimators given in Class 1, � Y ; i = 1, 2, 3, . . . , 9 and are given below:

i

MSE � � Yp 1 � < MSE � � Yi � if ρ