A Family of Estimators of Population Mean Using Information ... - arXiv

International Journal of Statistics and Economics; [Formerly known as the “Bulletin of Statistics & Economics” (ISSN 0973-7022)]; ISSN 0975-556X; Year: 2013, Volume: 10, Issue Number: 1; Int. j. stat. econ.; Copyright © 2013 by CESER Publications

A Family of Estimators of Population Mean Using Information on Point Bi-Serial and Phi Correlation Coefficient Sachin Malik and Rajesh Singh* Department of Statistics, Banaras Hindu University Varanasi-221005, India *Corresponding Author ([email protected], [email protected])

ABSTRACT This paper deals with the problem of estimating the finite population mean when some information on two auxiliary attributes are available. It is shown that the proposed estimator is more efficient than the usual mean estimator and other existing estimators. The study is also extended to two-phase sampling. The results have been illustrated numerically by taking empirical population considered in the literature.

Key words: Simple random sampling, auxiliary attribute, point bi-serial correlation, phi correlation, efficiency. Mathematics subject classification number: 62 DO5 Journal of Economic Literature (JEL) Classification Number: C83

1. INTRODUCTION There are some situations when in place of one auxiliary attribute, we have information on two qualitative variables. For illustration, to estimate the hourly wages we can use the information on marital status and region of residence (see Gujrati and Sangeetha (2007), page-311). Here we assume that both auxiliary attributes have significant point bi-serial correlation with the study variable and there is significant phi-correlation (see Yule (1912)) between the auxiliary attributes. The use of auxiliary information can increase the precision of an estimator when study variable Y is highly correlated with auxiliary variables X. Naik and Gupta (1996) introduced a ratio estimator when the study variable and the auxiliary attribute are positively correlated. Jhajj et al. (2006) suggested a family of estimators for the population mean in single and two phase sampling when the study variable and auxiliary attribute are positively correlated. Shabbir and Gupta (2007), Singh et al. (2008), Singh et al. (2010) and Abd-Elfattah et al. (2010) have considered the problem of estimating population mean Y taking into consideration the point biserial correlation between auxiliary attribute and study variable.

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International Journal of Statistics and Economics

In order to have an estimate of the study variable y, assuming the knowledge of the population proportion P, Naik and Gupta (1996) and Singh et al. (2007) respectively

proposed following

estimators

t1

t2

§P · y¨¨ 1 ¸¸ © p1 ¹

(1.1)

§p · y¨¨ 2 ¸¸ © P2 ¹

(1.2)

t3

t4

§P p · 1 1¸ y exp¨¨ P1 p1 ¸ © ¹

(1.3)

§ p P2 · ¸¸ y exp¨¨ 2 © p 2 P2 ¹

(1.4)

The bias and MSE expression’s of the estimator’s t i (i=1, 2, 3, 4) up to the first order of approximation are, respectively, given by

>

Bt 1 Yf1C 2p1 1 K pb1 Bt 2 Yf 1 K pb 2

@

(1.5)

2

C

p2

(1.6)

C 2p

º 2 ª1 K pb 2 » 2 «¬ 4 ¼

Bt 3 Yf1

(1.7)

C 2p

º 2 ª1 K pb 2 » 2 «¬ 4 ¼

Bt 4 Yf1

(1.8)

>

@

>

@

t1

Y f1 C 2y C 2p1 1 2K pb1

MSE

t 2

Y f1 C 2y C 2p1 1 2K pb2

MSE

t 3

2 ª §1 ·º Y f1 «C 2y C 2p1 ¨ K pb2 ¸» ©4 ¹¼ ¬

MSE

2

2

76

(1.9)

(1.10)

(1.11)


MSE

where,

U pb j

S yI j S y SI j

U pb1

K pb1

s I1I2

Sy

, Cy

Y Cy

C p1

correlation

SI1I2

Pj

s I1I2 s I1 s I2

; ( j 1,2),

U pb 2

, K pb 2

1 n ¦ I1i p1 I 2i p 2 and UI n 1 i 1

correlation between

SI j

1 N ¦ yi Y I ji Pj , N 1 i 1

S yI j

, Cp j

(1.12)

2

1 N ¦ I ji Pj , N 1 i 1

1 1 , SI2 j n N

f1

2 ª §1 ·º Y f1 «C 2y C 2p2 ¨ K pb2 ¸» 4 © ¹¼ ¬

t 4

Cy Cp2

.

be the sample phi-covariance and phi-

I1 and I 2 respectively, corresponding to the population phi-covariance and phi-

1 N ¦ I1i P1 I 2i P2 N 1 i 1 SI1I2

and U I

SI1 SI2

Following Naik and Gupta (1996) and Singh et al. (2007), we propose the estimators t5 and t6 as

t5

§P · y ¨¨ 1 ¸¸ © p1 ¹

D1

§ P2 ¨¨ © p2

· ¸¸ ¹

D2

E1

t6

where

§ p P2 · § P p1 · ¸¸ ¸¸ exp¨¨ 2 y exp¨¨ 1 P p 1 ¹ © p 2 P2 ¹ © 1

(1.13) E2

(1.14)

D1 , D 2 , E1 and E 2 are real constants. 2. BIAS AND MSE of t 5 and t 6

To obtain the bias and MSE of the estimators t 5 and t 6 to the first degree of approximation, we define

e0

yY , e1 Y

p1 P1 , e2 P1 77

p 2 P2 P2


such that,

E (e i )

0; i

0, 1, 2.

Also,

E (e 02 ) E (e 0 e1 )

f1C 2y , E (e12 )

f1K pb 1 C 2p , E (e 0 e 2 ) 1

U pb1

K pb1

Cy C p1

f1C 2p ,

f1C 2 , E (e 22 ) p1 f1K pb 2 C 2p , 2

U pb 2

, K pb 2

Cy Cp 2

2

f1K I C 2p ,

E (e1e 2 )

, KI

UI

2

C p1 Cp

2

Expressing equation (1.13) in terms of e’s, we have

t5

>

Y 1 e 0 1 e1

D1

1 e 2 D

2

@

D D 1 2 D 2 D 2 1 2 ª º e 2 D 1D 2 e1e 2 D 1e 0 e1 D 2 e 0 e 2 » e1 Y «1 D 1e1 D 2 e 2 1 1 2 2 ¬ ¼

(2.1)

Subtracting Y from both the sides of equation (2.1) and then taking expectation of both sides, we get the bias of the estimator t 5 , up to the first order of approximation, as

B( t 5 )

§ ªD2 D º ªD2 D º· Yf1 ¨ C 2p « 1 1 D1k pb 1 » C 2p « 2 2 D 2 k pb 2 D1D 2 k I » ¸ ¨ 1« 2 ¸ 2 2 2 ¬ ¼» ¬« 2 ¼» ¹ ©

(2.2)

From (2.1), we have

(t 5 Y)

Y>e 0 D1e1 D 2 e 2 @

(2.3)

Squaring both sides of (2.3) and then taking expectation, we get, MSE of the estimator t 5 , up to the first order of approximation, as

MSE ( t 5 )

2

>

Y f 1 C 2y C 2p1 D12 2D1K pb1 C 2p 2 D 22 2D 2 K pb 2 2D1D 2 K I

@

(2.4)

To obtain the bias and MSE of t 6 , to the first order of approximation, we express equation (1.14) in term of e’s, as

78


t6

E1 E2 § § e · · § e1 · ¸¸ exp¨¨ 2 ¸¸ ¸ Y ¨1 e 0 exp¨¨ ¨ © 2 e 2 ¹ ¸¹ © 2 e1 ¹ ©

ª Ee e º E e E2e2 E e EE ee E2e 2 E e e Y «1 e 0 1 1 1 1 2 2 1 2 1 2 2 2 2 0 2 1 0 1 » 2 4 2 4 4 2 2 ¼ ¬

(2.5)

Subtracting Y from both sides of equation (2.5) and then taking expectation of both sides, we get the bias of the estimator t 6 up to the first order of approximation, as

B( t 6 )

ª § E2 E Yf1 «C 2p ¨ 1 1 K pb 1 2 «¬ 1 ¨© 4

· § 2 ·º EE ¸ C2 ¨ E2 E2 K 1 2 K I ¸» pb p 2 ¸ ¸» 2 ¨ 4 2 4 ¹ © ¹¼

(2.6)

From (2.5), we have

(t 6 Y)

E e E e º ª Y «e 0 1 1 2 2 » 2 2 ¼ ¬

(2.7)

Squaring both sides of (2.7) and then taking expectation, we get the MSE of the estimator t 6 up to the first order of approximation, as

ª ·º § E2 E E · § E2 2 MSE ( t 6 ) Y f 1 «C 2y C 2p1 ¨¨ 1 E1 K pb1 ¸¸ C 2p 2 ¨¨ 2 1 2 K I E 2 K pb1 ¸¸» 2 ¹¼» © 4 ¹ © 4 ¬«

(2.8)

3. ANOTHER ESTIMATOR

Following Naik and Gupta (1996) and Singh et al. (2007), we propose another improved estimator tp as-

79


tp

§P · w 0 y w 1 y¨¨ 1 ¸¸ © p1 ¹

where D1 , D 2 ,

D1

§ P2 · ¨¨ ¸¸ © p2 ¹

D2

E1

§ P p · § p P2 · ¸¸ w 2 y exp¨¨ 1 1 ¸¸ exp¨¨ 2 © P1 p1 ¹ © p 2 P2 ¹

E1 and E 2 are real constants and w i (i

E2

(3.1)

0,1,2) are suitably chosen constants

whose values are to be determined later. Expressing (3.1) in terms of e’s, we have

tp

ª § E e · § E e ·º D D Y1 e 0 «w 0 w 1 1 e1 1 1 e 2 2 w 2 exp¨ 1 1 ¸ exp¨ 2 2 ¸» © 2 ¹ © 2 ¹¼ ¬

(3.2)

Expanding the right hand side of equation (3.2) and retaining terms, up to second power of e’s, we have

tp

ª § D 1 D 1 2 D 2 1 D 2 2 · Y «1 e 0 w 1 ¨ 1 e1 e 2 D 1 e1 D 2 e 2 D 1 D 2 e 1 e 2 D 1 e 0 e 1 D 2 e 0 e 2 ¸ 2 2 © ¹ ¬

§ E2e2 E e E e e E E e e E e e E2e2 E e w 2 ¨¨ 1 1 1 1 1 0 1 1 2 1 2 2 2 2 2 2 0 2 2 2 4 4 2 2 © 4

·º ¸¸» ¹¼»

(3.3)

Subtracting Y from both sides of (3.3) and then taking expectation of both sides, we get the bias of the estimator t p , up to the first order of approximation as

B tp

° § ªD2 D º Yf1 ®w 1 ¨ C 2p « 1 1 D1k pb 1 » C 2p ¨ 2 2 »¼ °¯ © 1 «¬ 2

§ ªE2 E º w 2 ¨ C 2p « 1 1 k pb1 » C 2p ¨ 1« 4 2 2 ¬ ¼» ©

ª D 22 D 2 º· D 2 k pb 2 D1D 2 k I » ¸ « 2 «¬ 2 »¼ ¸¹

ª E 22 E 2 º ·½° EE k pb 2 1 2 k I » ¸¾ « 2 4 »¼ ¸¹°¿ ¬« 4 (3.4)

From (3.3), we have

(t p Y)

ª E e ·º § Ee Y «e 0 w 1 D 1e1 D 2 e 2 w 2 ¨ 1 1 2 2 ¸» 2 2 ¹¼ © ¬ 80

(3.5)


Squaring both sides of (3.6) and then taking expectation, we get MSE of the estimator

t p up to the

first order of approximation, as

>

MSE t p Y f 1 C 2y w 12 A 1 w 22 A 2 2 w 1 A 3 w 2 A 4 w 1 w 2 A 5 2

@

(3.6)

where,

4A 2 A 3 A 4 A 5

w1

4A1A 2 A 52 4A 2 A 3 A 4 A 5

w2

4A1A 2 A 52

½ ° ° ¾ ° ° ¿

(3.7)

and

A1 A2 A3 A4 A5

½ ° ° 1 2 2 2 2 2 E1 c p E 2 C p 2E1E 2 k I C p ° 1 2 2 4 ° ° D1k pb1 C 2p D 2 k pb 2 C 2p ¾ 1 2 ° ° E1k pb1 C 2p E 2 k pb 2 C 2p 1 2 ° 2 2 2 2 ° D1E1C p D 2E 2 C p D 2E1k I C p D1E 2 k I C p ° 1 2 2 2 ¿

D12 C 2p D 22 C 2p 2D1D 2 k I C 2p

>

1

2

2

@

(3.8)

4. EMPIRICAL STUDY Data: (Source: Government of Pakistan (2004))

The population consists rice cultivation areas in 73 districts of Pakistan. The variables are defined as: Y= rice production (in 000’ tonnes, with one tonne = 0.984 ton) during 2003,

P1 = production of farms where rice production is more than 20 tonnes during the year 2002, and P2 = proportion of farms with rice cultivation area more than 20 hectares during the year 2003. For this data, we have N=73, Y =61.3,

P1 =0.4247, P2 =0.3425, S 2y =12371.4, S I21 =0.225490, SI22 =0.228311,

U pb1 =0.621, U pb 2 =0.673, U I =0.889. 81


Table 4.1: PRE of different estimators of Y with respect to y Choice of scalars

w0

w1

w2

1

0

0

0

1

0

0

w0

0

w1

D1

D2

E1

Estimator

MSE

PRE’S

y

655.28

100.00

1

0

t1

402.80

162.68

0

1

t2

1392.16

47.66

-1

1

t5

580.01

112.97

1

w2

E2

1

1

1

0

t3

462.07

141.80

0

1

t4

1091.20

60.05

1

-1

t6

363.03

180.50

1

1

tp

356.87

183.60

5. DOUBLE SAMPLING It is assumed that the population proportion P1 for the first auxiliary attribute I1 is unknown but the same is known for the second auxiliary attribute I 2 . When P1 is unknown, it is some times estimated from a preliminary large sample of size n c on which only the attribute I1 is measured. Then a second phase sample of size n (n< n c ) is drawn and Y is observed. Let p cj

1 nc ¦ I ji , ( j 1,2). n i1

The estimator’s t1, t2, t3 and t4 in two-phase sampling take the following form

t d1

§ p' · y¨¨ 1 ¸¸ © p1 ¹

82

(5.1)


t d2

td3

td4

§P · y¨¨ 2' ¸¸ © p2 ¹

(5.2)

§ p ' p1 · ¸¸ y exp¨¨ 1' © p1 p1 ¹

(5.3)

§ p' P · y exp¨¨ '2 2 ¸¸ © p 2 P2 ¹

(5.4)

The bias and MSE expressions of the estimators td1, td2, td3 and td4 up to first order of approximation, are respectively given as

>

Bt d 1 Yf 3 C 2p1 1 k pb1

>

@

Bt d 2 Yf 2 C 2p 2 1 K pb 2 C 2p 2

Bt d 3

Yf 3

Bt d 4

Yf 3

4

@

(5.6)

>1 K @

C 2p 2 4

(5.5)

pb 2

(5.7)

>1 K pb @ 2

(5.8)

>

@

>

@

MSE

t d1

Y f1C 2y f 3C 2P1 1 2K pb1

MSE

t d 2

Y f1C 2y f 2 C 2p2 1 2K kp2

MSE

t d3

C 2p 2ª Y «f1C 2y f 3 1 1 4K pb1 4 «¬

MSE t d 4

2

2

(5.9)

(5.10)

º

» »¼

ª C 2p 2 Y «f1C 2y f 3 1 1 4K pb1 « 4 ¬«

where,

83

(5.11)

º

»» ¼»

(5.12)


2

S

1 n I ji p j , S'I j 2 ¦ n 1 i 1

2 IJ

1

f2

n

'

1 , f3 N

n!

1

¦

2

n' 1i 1

I ji p 'j

,

1 1 . n n'

The estimator’s t5 and t6, in two phase sampling, takes the following form

§ p' y¨ 1 ¨p © 1

t d5

t d6

Where m1 , m 2 , n 1

· ¸ ¸ ¹

m1

§ P2 ¨ ¨ p' © 2

§ p' p · y exp¨ 1 1 ¸ ¨ p' p ¸ © 1 1¹

m2

· ¸ ¸ ¹

(5.13)

n1

§ p ' P2 · ¸ exp¨ 2 ¨ p' P ¸ 2¹ © 2

n2 (5.14)

and n 2 are real constants. 6. BIAS AND MSE OF t d 5 and t d 6

To obtain the bias and MSE of t d 5 and t d 6 up to first degree of approximation, we define

eo

Such that,

Ee o

E e '2

f1C 2y , E e1'

E e1'

p1' P1 ' , e2 P1

yY ' , e1 Y

p '2 P2 P2

o.

Also,

E e o2

E e1' e '2

f 2 C 2p 2 , E e o e1'

2

§ 2· f 2 C 2p1 , E¨ e '2 ¸ ¹ ©

f 2 K pb1 C 2p1 , E e o e '2

f 2 C 2p , 2

f 2 K pb 2 C 2p . 2

Expressing equation (5.13) in terms of e’s, we have

t d6

Y 1 e o 1 e1'

m 1 e1 m 1 e '2 m 1

1

2

Expanding the right hand side of above equation and retaining terms up to second power of e’s, we have

84


t d5

m m 1 2 m m 1 ' 2 ª Y «1 e o m1e1 1 1 e1 m12 e1e '2 1 1 e1 2 2 ¬ m m 1 ' 2 º m1e1' m 2 e '2 2 2 e 2 m1e1e 0 m1e 0 e1' m 2 e 0 e '2 » 2 ¼

(6.1)

Subtracting Y from both sides of (6.1) and then taking expectation, we get the bias of the estimator

t d5 , up to the first order of approximation, as ª ·º § m2 m · § m2 m Bt d 5 Y «f 3C 2p1 ¨¨ 1 1 m1K pb1 ¸¸ f 2 C 2P2 ¨¨ 2 2 m 2 k pb 2 ¸¸» 2 2 2 2 ¹¼ © ¹ © ¬

(6.2)

From (5.1), we have

( t d 5 Y)

>

Y e 0 e1m1 m1e1' m 2 e '2

@

(6.3)

Squaring both sides of (6.3) and then taking expectations, we get MSE of the estimator td5, up to the first order of approximation, as

>

MSEt d 5 Y f1C 2y f 3C 2p1 m12 2m1K pb1 f 2 C 2p2 2 m 22 2m 2 K pb2

@

(6.4)

Now to obtain the bias and MSE of td6 to the first order of approximation, we express equation (5.14) in terms of e’s

t d6

§ n e' · § n e' · §n e · Y1 e 0 exp¨¨ 1 1 ¸¸ exp¨ 1 1 ¸ exp¨¨ 2 2 ¸¸ © 2 ¹ © 2 ¹ © 2 ¹

Expanding the right hand side of above equation and retaining terms up to second power of e’s, we have

t d6

§ n e ' n e n 2 e 2 n e ' n ec 2 n e e c n e e n e e c · Y¨¨1 e 0 1 1 1 1 1 1 2 2 2 2 1 0 1 1 0 1 2 0 2 ¸¸ 2 2 4 2 4 2 2 2 ¹ ©

(6.5)

Subtracting Y from both sides of (6.5) and then taking expectations, we get the bias of the estimator

t d 6 up to the first order of approximation, as ª § n2 n n ·º § n2 n · n Bt d 6 Y «f 3 ¨¨ 1 1 1 K pb1 ¸¸C 2p1 f 2 ¨¨ 2 2 2 K pb2 ¸¸» 8 2 8 2 ¹¼ © 8 ¹ ¬ © 8

85

(6.6)


From (6.5), we have

( t d 6 Y)

n ec n e n ec · § Y¨ e 0 1 1 1 1 2 2 ¸ 2 2 2 ¹ ©

(6.7)

Squaring both sides of (6.7) and then taking expectations, we get the MSE of td6 up to the first order of approximation, as 2ª § n2 · § n2 · º MSE t d 6 Y «f1C 2y f 3 ¨¨ 1 n1K pb1 ¸¸C 2p1 f 2 ¨¨ 2 n 2 K pb2 ¸¸C 2p2 » 4 4 © ¹ © ¹ ¼ ¬

(6.8)

7. IMPROVED ESTIMATOR tp IN TWO-PHASE SAMPLING The estimator tp in double sampling is written as

§ pc · h 0 y h 1 y¨¨ 1 ¸¸ © p1 ¹

t pd

where,

m1

§ P2 · ¨¨ ' ¸¸ © p2 ¹

m2

n1

§ p' P · § p ' p1 · ¸¸ exp¨¨ '2 2 ¸¸ h 2 exp¨¨ 1' © p 2 P2 ¹ © p1 p 1 ¹

m1 , m 2 , n1 and n 2 are real constants and h i (i

n2

(7.1)

0,1,2) are suitably chosen constants

whose values are to be determined later. Expressing (7.1) in terms of e’s, we have

t pd

h 0 y h1 y1 e1c m1 1 e1 m 2 1 ec2 m 2 § n ec · §n e · § n ec · h 2 y exp¨ 1 1 ¸ exp¨ 1 1 ¸ exp¨ 2 2 ¸ © 2 ¹ © 2 ¹ © 2 ¹

(7.2)

Expanding the right hand side of (7.2) and retaining terms up to second power of e’s as

t pd

ª m m 1 2 '2 § m m 1 2 Y «1 e 0 h1 ¨ 1 1 e1 m1e1 m1e 0 e1 m 22 e1e1' 1 1 e1 m1e1' 2 2 © ¬ m1e 0 e1' m 2 e '2 m 2 e 0 e '2

m 2 m 2 1 '2 · e2 ¸ 2 ¹

§ n e' n e n e ec ·º n 2e 2 n e' n ec 2 n e ec n e e w 2 ¨ 1 1 1 1 1 1 2 2 2 2 1 0 1 1 0 1 2 0 2 ¸» ¨ 2 2 4 2 4 2 2 2 ¸¹» © ¼ (7.3)

86


Subtracting Y from both the sides of (7.3) then taking expectations on both the sides, we get the bias of the estimator

t d up to the first order of approximation as ª §m 2 m Y «h1f 3 C 2p ¨ 1 1 m1 K pb 1 1¨ 2 2 «¬ ©

B( t pd )

§ n2 n n h 3 f 3 C 2p ¨ 1 1 1 K pb1 1¨ 8 8 2 ©

· § 2 ¸ h f C2 ¨ m 2 m 2 m k 2 pb 2 1 2 p ¸ 2 ¨ 2 2 © ¹

· § 2 ·º ¸ h f C2 ¨ n 2 n 2 n 2 k ¸» pb 3 2 p 2 ¸ ¸» 2 ¨ 8 8 2 ¹ © ¹¼

· ¸ ¸ ¹

(7.4)

From (7.3), we have

t pd Y

ª § n e' n e n e ' ·º Y «e 0 h1 m1e1 m1e1' m1 e '2 h 2 ¨ 1 1 1 1 2 2 ¸» ¨ 2 2 2 ¸¹» «¬ © ¼

(7.5)

Squaring both sides of (7.5) and then taking expectations, we get MSE of the estimator t pd up to the first order of approximation as

MSE t pd

2

>

Y f1 C 2y h12 B1 h 22 B 2 2h1B3 h 2 B 4 h1h 2 B5

@

(7.6)

Where,

h1 h2

4 B 2 B3 B 4 B5 ½ 4B1B 2 B52 °° ¾ 2B1B 4 2B3 B5 ° 4B1B 2 B52 °¿

(7.7)

and

B1 B2 B3 B4 B5

½ ° ° 1 2 2 2 2 f 2 n 2 C p f 3 n1 C p ° 2 1 4 ° ° f 3 m1k pb1 C 2p f 2 m 2 k pb 2 C 2p ¾ 1 2 ° f 3 n1k pb1 C 2p f 2 n 2 k pb 2 C 2p ° 1 2 ° ° 2 2 f 3 n1m1C p f 2 n 2 m 2 C p ° 1 2 ¿

f 2 m 22 C 2p f 3 m12 C 2p 2 1

>

@

87

(7.8)


8. EMPIRICAL STUDY Data: (Source: Singh and Chaudhary (1986), P. 177). The population consists of 34 wheat farms in 34 villages in certain region of India. The variables are defined as: y = area under wheat crop (in acres) during 1974.

p1 = proportion of farms under wheat crop which have more than 500 acres land during 1971. and p 2 = proportion of farms under wheat crop which have more than 100 acres land during 1973. For this data, we have N=34, Y =199.4,

P1 =0.6765, P2 =0.7353, S 2y =22564.6, S I21 =0.225490, SI22 =0.200535,

U pb1 =0599, U pb 2 =0.559, U I =0.725. Table 7.1: PRE of different estimators of

Y with respect to y

Choice of scalars

h0

h1

h2

1

0

0

0

1

0

0

h0

0

h1

m1

m2

n2

Estimator

MSE

PRE’S

y

1592.79

100

1

0

t d1

1256.94

126.71

0

1

t d2

1538.00

103.90

1

1

t d5

1197.15

133.04

1

h2

n1

1

1

1

0

t d3

1131.00

140.82

0

1

td4

2425.83

65.65

1

1

t d6

1278.00

124.62

1

1

t pd

1032.36

154.28

88


9. Conclusion In this paper, we have suggested a class of estimators in single and double sampling by using point bi serial correlation and phi correlation coefficient. From Table 4.1 and Table 7.1, we observe that the proposed estimator tp and its double sampling version tpd, performs better than other estimators considered in this paper.

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