Enhancing the efficiency of the ratio-type estimators of ...scientiairanica.sharif.edu/article_21219_39bd2fed4... · Kadilar and Cingi [11] suggested some modified ratio-type estimators

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Enhancing the efficiency of the ratio-type estimators of population variance with a blend of information on robust location measures

Farah Naza,*, Muhammad Abidb, Tahir Nawazb,c and Tianxiao Panga

a. Department of Mathematics, Institute of Statistics, Zhejiang University, Hangzhou 310027, People’s Republic of China.

b. Faculty of Physical Sciences, Department of Statistics, Government College University Faisalabad, Pakistan.

c. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China.

Abstract

Numerous ratio-type estimators of the population variance are proposed in the existing literature based on different characteristics of the study as well as the auxiliary variable. However, mostly the existing estimators are based on the conventional measures of the population characteristics and their efficiency is dubious in the presence of outliers in the data. This study presents improved families of variance estimators under simple random sampling without replacement assuming that the information on some robust non-conventional location parameters of the auxiliary variable is known besides the usual conventional parameters. The bias and mean square error of the proposed families of estimators are obtained and the efficiency conditions are derived mathematically. The theoretical results are supplemented with the numerical illustrations by using real datasets which indicates the supremacy of the suggested families of estimators.

Keywords: Auxiliary variable; Bias; Efficiency; Mean square error; Outliers; Ratio estimators.

* Corresponding Author

E-mail addresses: [email protected] (Naz, F.); [email protected], [email protected] (Abid, M.); [email protected], [email protected] (Nawaz, T.); [email protected] (Pang, T.). Tel: +86-13093790613 (Naz, F.)

mailto:[email protected]





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1. Introduction

The Estimation of population variance along with the estimation of population mean is of interest in many practical situations such as business, manufacturing industry, Services industry, the pharmaceutical industry, medical sciences, biological sciences and agriculture (cf. Solanki et al., [1]). Therefore, the development of efficient estimators of population variance constantly attracts the researchers. Commonly in survey sampling, the information on an auxiliary variable which is closely related to the study variable is utilized to enhance the efficiency of the estimators of population characteristic under investigation. When auxiliary information is available and the study variable is positively correlated with the auxiliary variable, the ratio estimation method is frequently used to improve the efficiency of the estimators of population characteristics [2]. The ratio-type estimators are now being frequently used in different fields such as process monitoring, designing of acceptance sampling plan for lot sentencing, environmental studies, and forestry are few to mention [3–8]. Generally, the information of conventional auxiliary parameters such as the coefficient of kurtosis, the coefficient of skewness, the coefficient of variation, and the coefficient of correlation is used in a linear combination with the sample information of the study and the auxiliary variable to design the estimators of population variance. For instance, see Isaki [9]; Upadhyaya and Singh [10]; Kadilar and Cingi [11]; Subramani and Kumarapandiyan [12–14]; Khan and Shabbir [15]; Solanki et al., [1]; Yaqub and Shabbir [16]; Maqbool and Javaid [17]; Adichwal, Sharma and Singh [18]; Maji, Singh and Bandyopadhyay [19]; Abid et al., [20]; Singh, Pal and Yadav[21]; Muneer et al., [22] and the references cited therein. The development of ratio-type estimator that deals with the presence of outliers in the data is a neglected area in survey sampling. Since the presence of outliers in the data can negatively affect the efficiency of the ratio-type estimators. Therefore, there is a need for ratio-type estimators that can accommodate the outliers without loss of efficiency.

The present study incorporates the information on robust non-conventional location parameters of the auxiliary variables coupled with the conventional parameters under the ratio-type structure of estimation to propose new ratio-type estimators of population variance. The non-conventional measures used in this study are tri-mean (TM), mid-range (MR), Hodges-Lehmann estimator of the mean (HL) and deciles mean (DM). Although MR is not a robust measure against outliers, however, it is an effective estimator of the mean

when the underlying distribution has excess kurtosis and the sample size n is small i.e.

4 20n (cf. [23–25]). Since the objective of the current study is to use non-conventional and robust measures of location for variance estimation, therefore MR is included in this study as a non-conventional measure of the location. The advantage of using TM, HL and DM lies in their ability to be stable in the presence of outliers. These measures are used in a linear combination with the other conventional measures to improve the efficiency of the variance estimator in simple random sampling without replacement (SRSWOR) scheme. The properties of these non-conventional measures can be found in Wang et al., [26]; Ferrell [25]; Hettmansperger and McKean [27]; and Rana et al., [28].

The rest of the manuscript is structured as follows. The existing estimators of population variance that utilizes the known information on conventional parameters of the auxiliary

3

variable are described in Section 2. The proposed improved families of estimators of population variance which combines the information on known conventional and non-conventional parameters of the auxiliary variable are described in Section 3. In Section 4, the performance comparison of the proposed estimators is given with some of the existing estimators of population variance. Finally, the conclusion and some recommendations are presented in Section 5.

2. Existing Estimators of Variance

This section reviews some of the existing estimators of population variance under SRS which are based on known auxiliary information of conventional parameters.

Suppose a finite population 1 2, , , NV V V V has N units, and Let Y be a variable of

interest with measurements iY taken from each population unit for constituting a set of

observations 1 2, , , NY Y Y Y The purpose of the measurement process is to estimate the

population variance 22 1

1

N

Y i

i

S N Y Y

by drawing a random sample from the population.

Assume that the information on auxiliary variable parameters (both the conventional and non-conventional) is readily available or can be easily obtained without involving much time and cost. The additional information can be advantageously used in the ratio, the regression, the product, and the ratio-cum regression methods of estimation to define the estimators of population variance or other population characteristics such as mean. One such estimator is defined by Isaki [9] that utilizes the information on the known variance of

the auxiliary variable 2

XS to estimate the population variance 2 YS . It is also known as the

traditional ratio-type estimator. The estimator along with its approximate bias and MSE is given as:

2

2 2

2ˆ y

R X

x

sS S

s ; 2 2

222ˆ 1 1R Y X

B S S

2 4

222 2ˆ 1 1 2 1R Y Y X

MSE S S (1)

Upadhyaya and Singh [10] proposed a modified ratio-type estimator of population variance by incorporating information on the coefficient of kurtosis of the auxiliary variable and is given as:

2

22 2

1 2

2

ˆ X X

y

x X

SS s

s

Kadilar and Cingi [11] used the coefficient of variation of the auxiliary variable as a linear combination with the variance of the auxiliary variable to define a ratio-type estimator as:

22 2

2 2ˆ X X

y

x X

S CS s

s C

4

Subramani and Kumarapandiyan [14] used the median of the auxiliary variable to propose a modified ratio-type estimator of the population variance which is defined as.

2

2 2

3 2ˆ X d X

y

x d X

S MS s

s M

Subramani and Kumarapandiyan [13] suggested the following quartiles based estimators of the population variance.

2

2 2

2ˆ X j X

i y

x j X

S QS s

s Q

for 4,5, ,8i and 1 3, , , ,

j X X X r X d X a XQ Q Q Q Q Q , respectively.

Subramani and Kumarapandiyan [13] established that these estimators outperforms some of the existing ratio-type estimators of population variance in terms of smaller MSEs.

Subramani and Kumarapandiyan [12] used deciles of the auxiliary variable to suggest new ratio-type estimators of the population variance. The deciles based estimators showed superior efficiency as compared to the existing estimators. These estimators are defined as:

2

2 2

2ˆ X j X

i y

x j X

S DS s

s D

for 9,10, ,18i and 1 2 10, , ,

j X X X XD D D D , respectively.

Kadilar and Cingi [11] suggested some modified ratio-type estimators of population variance by integrating information on the population coefficient of variation and the population coefficient of kurtosis of the auxiliary variable together with the variance of the auxiliary variable as:

2

22 2

19 2

2

ˆ X X X

y

X x X

C SS s

C s

;

2

22 2

20 2

2

ˆ X XX

y

x XX

S CS s

s C

Taking motivation from Kadilar and Cingi [11], and Subramani and Kumarapandiyan [14] a new ratio-type estimator of the population variance was introduced by Subramani and Kumarapandiyan [29] that utilizes the population information on the coefficient of variation and the median of the auxiliary variable and is given as:

2

2 2

21 2ˆ X X d X

y

X x d X

C S MS s

C s M

Khan and Shabbir [15] have suggested a similar estimator of population variance, which utilizes the information on the coefficient of correlation between the study and auxiliary variable in addition to the information on upper quartile of the auxiliary variable as:

2

32 2

22 2

3

ˆ X X

y

x X

S QS s

s Q

5

In general, the bias and MSEs of the estimators 2ˆ 1, ,22iS i up to the first degree of

approximation are given as:

2 2

222ˆ 1 1i Y i i X

B S S

42 2

222 2 1 1 2 1ˆ

Y Y Xi i iM SSE S (2)

where

2

1 2

2

X

X X

S

S

,

2

2 2

X

X X

S

S C

,

2

3 2

X

X d X

S

S M

,

2

4 2

1

X

X X

S

S Q

,

2

5 2

3

X

X X

S

S Q

,

2

6 2

X

X r X

S

S Q

,

2

7 2

X

X d X

S

S Q

,

2

8 2

X

X a X

S

S Q

,

2

9 2

1

X

X X

S

S D

,

2

10 2

2

X

X X

S

S D

,

2

11 2

3

X

X X

S

S D

,

2

12 2

4

X

X X

S

S D

,

2

13 2

5

X

X X

S

S D

,

2

14 2

6

X

X X

S

S D

,

2

15 2

7

X

X X

S

S D

,

2

16 2

8

X

X X

S

S D

,

2

17 2

9

X

X X

S

S D

,

2

18 2

10

X

X X

S

S D

,

2

19 2

2

X X

X X X

C S

C S

,

2

2

20 2

2

XX

X XX

S

S C

,

2

21 2

X X

X X d X

C S

C S M

, and

2

22 2

3

X

X X

S

S Q

.

Upadhyaya and Singh [30] used the information on the population mean of the auxiliary variable as a ratio to define a new ratio-type estimators of variance. The estimator, its bias and MSE are defined as:

22

23ˆ

y

XS

xs ; 2 2

123

2

2ˆ

Y X XB S CS C ;

4 2

212

2

23 1ˆ 2 Y X XYMSE S CS C (3)

Recently, Subramani and Kumarapandiyan [31] proposed a new class of modified ratio-type estimators of population variance by modifying the estimator proposed by Upadhyaya and Singh [30]. They showed that their new class of estimators outperforms the estimators proposed by Upadhyaya and Singh [10], Upadhyaya and Singh [30], Kadilar and Cingi [11], and Subramani and Kumarapandiyan [12–14,29] under certain efficiency conditions.

The general structure of Subramani and Kumarapandiyan [31] class of estimators with its bias and MSE is given as:

2 2ˆ i

ySK i

i

sX

Sx

; for 1,2, ,51i ,

2

2 2 2 2

1ˆ

Y X XSK i SK i SK iB S S C C ;

6

4 2 2

212

2 1 ˆ 2Y X XSK i Y SK i SK iM SSE S C C (4)

where SK i

i

X

X

and

1 XC , 2 2 X

, 3 1 X

, 4 , 5 XS , 6 d X

M , 7 1 X

Q , 8 3 X

Q , 9 r X

Q ,

10 d XQ ,

11 a XQ ,

12 1

XD ,

13 2 XD ,

14 3 XD ,

15 4 XD ,

16 5 XD ,

17 6 XD ,

18 7 XD ,

19 8 XD ,

20 9 XD ,

21 10 XD ,

22

2

X

X

C

,

2

23

X

XC

,

24

1

X

X

C

,

1

25

X

XC

,

26XC

, 27

XC

,

28X

X

C

S ,

29X

X

S

C ,

30X

d X

C

M ,

31

d X

X

M

C ,

2

32

1

X

X

,

1

33

2

X

X

,

2

34

X

,

35

2 X

,

2

36

X

XS

,

37

2

X

X

S

,

2

38

X

d XM

,

39

2

d X

X

M

,

1

40

X

,

41

1 X

,

1

42

X

XS

,

43

1

X

X

S

,

1

44

X

d XM

,

45

1

d X

X

M

,

46

XS

,

47XS

,

48

d XM

,

49

d XM

,

50X

d X

S

M , and

51

d X

X

M

S .

All of the above-modified estimators of variance are biased. However, under certain conditions, these estimators have superior efficiency in terms of lesser MSEs as compared to the traditional ratio estimator of variance suggested by Isaki [9] as they utilize more auxiliary information as compared to the traditional ratio estimator.

3. Proposed Estimators of Variance

This section presents three different families of the ratio-type estimator of population variance assuming that the information on some robust non-conventional measures of the

location of the auxiliary variable is readily available. Firstly, we use tri-mean XTM

which is the weighted average of the population median, upper and lower quartiles of the

auxiliary variable and is defined as

2 X 1 X 3 X / 2

2X

Q Q QTM

. It is a robust measure of location and remains stable in the presence of outliers with a breakdown point of 0.25. It means it remains bounded for a data having nearly 25% contamination. The properties of

XTM can be found in Wang et al., [26] and Abid et al., [32]. Secondly, this study includes

estimators based on mid-range XMR

which is the average of the minimum and the maximum values occurring in a data set. Although it is highly sensitive to outliers, but for small samples sizes and excess kurtosis it is a useful measure of location (see [23–25] for

7

more details). Thirdly, we employ Hodges Lehmann XHL

estimator, discussed in Hettmansperger and McKean [27], which is based on the median of the pair-wise Walsh averages. It is robust against outliers and has breakdown point of 0.29. It is defined as

;1

2X

X j X kHL median j k N

. Lastly, this study utilizes another robust

measure of location suggested by Rana et al., [28] namely deciles mean XDM . It is simply

the average of deciles and is defined as

9

1

9

j Xj

X

DDM

. Rana et al., [28] have compared

its performance with the mean, median and the XTM and observed that it performs better

than the other three measures of location as its MSE is lower.

All these non-conventional or robust measures are integrated with other conventional measures such as the coefficient of skewness, the coefficient of variation, the coefficient of correlation between the study and the auxiliary variable and the coefficient of kurtosis in the context of ratio method of estimation under SRS to estimate the population variance. Since, the non-conventional and robust measures are being incorporated it is, therefore, expected that the use of these measures will result in enhancing the efficiency of the ratio-type estimators of population variance.

3.1. The suggested estimators of class-I

Taking motivation from the Searls estimator [33], the proposed class of estimators for population variance is defined as:

2

1

2ˆ Φ

Φyp j

XS Ks

x

(5)

where K is the Searls [33] constant and its value will be determined at a later stage, 2

ys is

the sample mean of the study variable, X and x are the population and sample means of

the auxiliary variable respectively. It is worth mentioning that Φ 0X , Φ 0x

and Φ can either be a known real number or known conventional parameter of the auxiliary variable X , whereas is considered as the known non-conventional parameter of

the auxiliary variable X .

To obtain the bias and MSE of 2

1ˆ

p jS , in terms of relative errors, we can express

2 2

0 2

y y

y

s Se

S

and 1ex X

X

such that 2 2

01y ys S e , 11x eX and 0 1 0E e E e ,

2

0 21

yE e , 2 2

1 xE e C , 0 1 21 xE e e C .

Expressing, 2

1ˆ

p jS in terms of e ’s we have

8

122

1 0 1ˆ 1 1p j yKS S e e

, (6)

where Φ

Φ

X

X

.

Assuming 1 1e so that 1

11 e

is expandable, expanding the right hand side of

Equation (6) and neglecting the terms of e ’s having power greater than two, we have

2 2 2 2 2 2 2

01 1 0

2

1 1ˆ

y y y y yp j K S S e S e S e e S eS OR

22 2 2 2

0 1 0 1 11ˆ 1j yp yS S K K e e e eS e (7)

The bias of 2

1ˆ

p jS up to the first degree of approximation is obtained by applying

expectation on both sides of Equation (7) as:

2 2 2

11

2

21ˆj y xp xB S S K K C C (8)

Squaring both sides of Equation (7), keeping terms of 𝑒’s only up to 2nd order and applying

expectation we get the MSE of 2

1ˆ

p jS up to the first degree of approximation as

2 2 2 2 2 2

24

2

2

21

1

1 1 3ˆ

2

2 4p

x

yj

y

x

K K C K KMSE S

C K KS

(9)

Differentiating Equation (9) with respect to K , equating it to zero and after simplification, we get the optimum value of K as:

21

212

1

1 1 3 4

x x

opt

x xy

C CK K

C C

. (10)

Let 1 211 x xK C C and 2 212

1 1 3 4x xyK C C , so that

1

2

opt

KK

K .

Substituting the above result in Equation (9) and simplifying, the minimum MSE of 2

1ˆ

p jS is

given as:

22

4

11

2

ˆ 1yjmin

p

KMSE S

KS

(11)

Similarly, substituting the optimal value of K i.e. opt

K in Equation (8) we have the

minimum bias of 2

1ˆ

p jS as:

2 2 22

1 211ˆy x xopt oj

minp pt

B S K KS C C (12)

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Many different kinds of estimators can be generated from 2

1ˆ

p jS by setting different values

of K ,Φ , and . For example; If Φ 1K and 0 then the estimator suggested by

Upadhyaya and Singh [30] becomes a member of the proposed class-I of estimators. Similarly, if Φ 1K and i then the modified class of estimators proposed by

Subramani and Kumarapandiyan [31] also becomes a member of the class 2

1ˆ

p jS . Some new

members of the class 2

1ˆ

p jS are summarized in Table 1.

3.1.1. Efficiency conditions for class-I estimators

The proposed estimators of class-I perform better than the Isaki’s [9] traditional estimator

of population variance if 2 2

1ˆ ˆ

mp j R

inMSE MSES S i.e.

2

4 41222 2

2

1 1 1 2 1y Y Y X

KS S

K

OR

2

1 2 222 21 2

Y XK K

.

Similarly, the estimators contained in 2

1ˆ

p jS will achieve superior efficiency as compared to

the existing estimators 2( 1,...,22)ˆi iS detailed in Section 2 if:

2

4 4 21222 2

2

1 1 1 2 1y Y i iY X

KS S

K

OR

2 2

1 2 222 21 1 1 2 1i iY X

K K

.

The suggested class-I estimators will be more efficient as compared to the Upadhyaya and Singh [30] if:

2

4 4 21212

2

1 1 2 y Y X XY

KS S C C

K

OR

2 2

1 2 2121 1 2 X XY

K K C C

.

The estimators envisaged in the proposed class 2

1ˆ

p jS will exhibit superior performance as

compared to Subramani and Kumarapandiyan [31] modified class of estimators if

10

2

4 4 2 21212

2

1 1 2y Y X XY SK i SK i

KS S C C

K

OR

2 2 2

1 2 2121 1 2X XY SK i SK i

K K C C

.

3.2. The suggested estimators of class-II

This section deals with another proposed class of estimators of population variance under SRS which is based on power transformation. The general structure of the estimators of class-II is defined as

2

2

2

φˆ φ

X

X

p l yLsX

Sx

(13)

where L is the Searls [33] constants and its value will be determined later,φ and can be

real numbers or functions of a known conventional parameter of the auxiliary variable X , whereas and are the known functions of the non-conventional parameters of the

auxiliary variable X .

Following the procedure described in section 3.1, the minimum bias and minimum MSE of the proposed class-II estimators are given as

2

2 22

1

2

2 1 2 212

ˆ 1y X Xoptl op ptmin

B S S L L C C

(14)

22

4

21

2

ˆ 1ylmin

p

LMSE S

LS

(15)

where, 1

φ

φ

X

X

,

2

X

X

,

1

2

opt

LL

L , 2

1 1 2 2 1 2 211 22

X XL C C , and

2 2 2

2 2 2 1 1 2 2121 1 2 4X XY

L C C .

Various ratio-type estimators can be obtained by assuming different values of ,φ, , ,L

and in the proposed class 2

2ˆ

p lS . For instance, if we set φ 1L and 0 then the

estimator suggested by Upadhyaya and Singh [30] is a member of the 2

2ˆ

p lS class of estimators. On the other hand, if we set φ 1L , i and 0 and then Subramani

and Kumarapandiyan [31] class of estimators is the member of the proposed class-II estimators. Similarly, if we set L K , 1 and 0 the class of estimators defined in section 3.1 becomes a member of class-II estimators.

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Some new members of the proposed class-II are defined in Table 2, which are based on a mixture of conventional and non-conventional parameters of the auxiliary variable.

3.2.1. Efficiency conditions for class-II estimators

The estimators 2

2ˆ

p lS perform better than the Isaki [9] traditional estimator of population

variance if 2 2

2ˆ ˆ

mp l R

inMSE MSES S i.e.

2

4 41222 2

2

1 1 1 2 1y Y Y X

LS S

L

OR

2

1 2 222 21 2

Y XL L

The estimators defined in class-II will achieve greater efficiency as compared to the

existing estimators 2 2ˆ 1, ,2i iS defined in Section 2 if the following condition is

satisfied.

2

4 4 21222 2

2

1 1 1 2 1y Y i iY X

LS S

L

OR

2 2

1 2 222 21 1 1 2 1i iY X

L L

The suggested class-II estimators will outperform Upadhyaya and Singh [30] modified ratio-type estimator of population variance in terms of efficiency if

2

4 4 21212

2

1 1 2 y Y X XY

LS S C C

L

OR

2 2

1 2 2121 1 2 X XY

L L C C

The estimators envisaged in the proposed class 2

2ˆ

p lS will exhibit superior performance as compared to Subramani and Kumarapandiyan [31] modified class of estimators if

2

4 4 2 21212

2

1 1 2y Y X XY SK i SK i

LS S C C

L

OR

12

2 2 2

1 2 2121 1 2X XY SK i SK i

L L C C

3.3. The suggested estimators of class-III

In this Section, we have presented a new class of ratio-cum regression estimators of

population variance. The proposed class of estimators 2

3ˆ

p mS is defined as:

2

3

2ˆ α

α

X

X

p m y

XM s bS X x

x

(16)

where, M is the Searls [33] constants and its value will be determined later,α and can be real numbers or functions of known conventional parameter of the auxiliary variable X ,

and are the known functions of the non-conventional parameters of the auxiliary variable X , whereas b is the regression coefficient between the study and auxiliary variable.

The minimized bias and minimized MSE of the class-III estimators are obtained by adapting the procedure detailed in section 3.1 as follows:

2

2 22

1

2

3 1 2 212

ˆ 1y X Xoptm op ptmin

B S S M M C C

(17)

2

4 12

2

3ˆ

yp mmin

MMSE S

MS

(18)

where, 1

α

α

X

X

,

2

X

X

,

1

2

opt

MM

M , 4 2

1 1 2 2 1 2 211 22

y X XM S C C

,

and

4 4 2 2 2 2 2 2 2

2 2 2 1 1 2 21 21 1 221 2 4 2y y X X y X X XY

M S S C C bS X C C bX C .

Numerous ratio and regression type estimators can easily be obtained by choosing

different values of the ,α, , ,M and b in the proposed class 2

3ˆ

p mS such as: if we set

α 1M and 0b then the estimator suggested by Upadhyaya and Singh [30]

is a member of the 2

3ˆ

p mS class of estimators. Similarly, if we set α 1M , i and

0b then Subramani and Kumarapandiyan [31] class of estimators becomes a member

of the proposed 2

3ˆ

p mS class of estimators. If we set M K , 1 and 0b the class of

estimators defined in section 3.1 becomes a member of class 2

3ˆ

p mS estimators. Moreover, if

the regression coefficient 0b , the class of estimators defined in section 3.2 is also a

member of the proposed class 2

3ˆ

p mS .

13

Table 3 contains some new members of the proposed class-III to estimate the population variance based on the auxiliary information.

3.3.1. Efficiency conditions for class-III estimators

The estimators in class-III 2

3ˆ

p mS perform better than the traditional ratio estimator of

population variance proposed by Isaki [9] if 2 2

3ˆ ˆ

mp m R

inMSE MSES S i.e.

2

4 41222 2

2

1 1 2 1y Y Y X

MS S

M

OR

2 4

1 2 222 21 2Y Y X

M M S

The estimators defined in class-III have superior in efficiency as compared to the

estimators defined in section 2 i.e. 2 2ˆ 1, ,2i iS if

2

4 4 21222 2

2

1 1 2 1y Y i iY X

MS S

M

OR

2 4 2

1 2 222 21 1 1 2 1y i iY X

M M S

The estimators suggested in class-III outperform Upadhyaya and Singh [30] modified ratio-type estimator of population variance in terms of efficiency if

2

4 4 21212

2

1 2 y Y X XY

MS S C C

M

OR

2 4 2

1 2 2121 1 2 y X XY

M M S C C

The estimators envisaged in the proposed class-III 2

3ˆ

p mS will exhibit superior performance

as compared to Subramani and Kumarapandiyan [31] modified class of estimators if

2

4 4 2 21212

2

1 2y Y X XY SK i SK i

MS S C C

M

OR

2 4 2 2

1 2 2121 1 2y X XY SK i SK i

M M S C C

14

From the theoretical results, it is established that the proposed families of the estimators have superior efficiency under certain conditions. To further support these theoretical findings an empirical study is conducted in the following Section.

4. Empirical Study

To gauge the performance of the proposed classes of estimators against their competing estimators of variance, two real datasets (population-I and population-II) are taken from the Singh and Chaudhary [34] page 177. These datasets are also considered in Subramani

and Kumarapandiyan [31] and some other studies [35–38]. In population-I, Y denotes the

area under wheat in 1974 and X denotes the area under wheat in 1971; whereas, in

population-II, Y represents the area under wheat in 1974 and X is the area under wheat in 1973. The original populations do not contain many outlier observations. To compare the performance of the proposed and existing estimators using a population crippled with outlier observations, some outliers are introduced in population-I and it is named as population-III. It is expected that the performance of the proposed classes of estimators remains relatively stable in the presence of outliers in the data. Figures 1-3 depicts the nature of populations data considered in this study.

To compare the performance of the proposed and the existing estimators the percentage relative efficiencies (PREs) of the proposed classes of estimators with respect to the traditional ratio estimator suggested by Isaki [9] are obtained as:

100TR

p

p

MSEPRE

MSE

where, pPRE

denotes the percentage relative efficiency of the proposed estimators as

compared to the traditional ratio estimator, TRMSE

is the mean square error of the

traditional ratio estimator, and pMSE

is the mean square error of the proposed estimator. It is worth mentioning that due to lack of space, we have taken only the most efficient estimator from the class of estimators proposed by Subramani and Kumarapandiyan [31] which is based on the

10 XD for comparison purpose. The population characteristics are

summarized in Table 4.

The computed PREs of the existing estimators are compared with the traditional ratio estimator of Isaki [9] as shown in Tables 5-7, while the PREs of the proposed estimators as compared to the traditional ratio estimator are given in Table 8-10 respectively.

From the results reported in Tables 5-10 the main findings are summarized as:

1. It is observed that, in general, the estimators proposed in this study have greater PREs as compared to the existing estimators of Isaki [9], Upadhyaya and Singh [10,30], Kadilar and Cingi [11], Khan and Shabbir [15], and Subramani and Kumarapandiyan [12–14,29,31] for all the populations considered in this study(cf.

15

Tables 5-10). For example; Subramani and Kumarapandiyan [31] estimator has the maximum PRE among the existing estimators i.e. 111.56, 117.10 and 178.49 for population-I, II and III, respectively. On the other hand the maximum level of PRE of the proposed estimators is 141.59, 148.42 and 202.40 for population-I, II and III, respectively. It indicates the better performance of the proposed classes of estimators in terms of PREs.

2. It is observed that the PREs of the existing and the proposed estimators increases as the number of outliers in the data increases (cf. Tables 5-6 and Tables 9-10). For instance; the PRE of Subramani and Kumarapandiyan [31] (best among the existing estimators considered in this study) has increased from 111.56 to 178.49, whereas, the maximum increase in PRE of the proposed estimators is observed to be 148.42 to 202.40.

3. The results indicates that when the number of outlier in the data are increased, the majority of the proposed estimators still remain more efficient as compared to all the existing estimators considered in this study (cf. Tables 9-10). However, some of the proposed estimators have lesser PREs as compared to the Subramani and Kumarapandiyan [31] estimator. These estimators still remained more efficient as compared to the rest of the existing estimators.

4. Among the proposed classes of estimators in this study, the estimators proposed in class-II exhibits the best performance in terms of PREs (cf. Tables 6, 8 and10).

5. In each of the proposed classes of estimators, the estimators which integrate the information of the correlation coefficient between the study and the auxiliary and the non-conventional parameter of an auxiliary variable are superior in terms of PREs as compared to other proposed estimators (cf. Tables 6, 8 and 10).

6. The proposed estimator of population variance that combines the mid-range (𝑀𝑅𝑋) and the correlation coefficient between the study and the auxiliary variable turns out to be the most efficient estimator in all the proposed classes (cf. Tables 6, 8 and 10).

7. It is also observed that in presence of outliers the performance of the majority of the existing estimators is almost alike as of the traditional ratio estimator (cf. Tables 5, 7 and 9), whereas, the suggested estimators perform quite well as compared to the existing and the traditional ratio-type estimators (cf. Tables 6, 8 and 10).

These findings highlight the significance of the incorporation of the non-conventional and robust measures of the auxiliary variable in the ratio method of estimation for estimating the population variance in the presence of outliers. The inclusion of these measures resulted in higher PREs of the proposed estimators as compared to the existing estimators which are only based on conventional measures.

5. Conclusion and recommendations

The present study introduces three new classes of ratio-type estimators of population variance under the SRS by integrating the information of non-conventional measures of an auxiliary variable with the conventional measures. The algebraic expressions for the bias and MSEs are obtained theoretically; moreover, the efficiency conditions under which the proposed estimators perform better than the existing estimators are also derived. An empirical study is conducted to supplement the theoretical results. The empirical results

16

reveals that the proposed estimators outperform the existing estimators of Isaki [9], Upadhyaya and Singh [10,30], Kadilar and Cingi [11], Khan and Shabbir [15]and Subramani and Kumarapandiyan [12–14,29,31] in terms of MSEs and PREs when outliers are present in the data. Based on the findings of the present study, it is strongly recommended that if the information on robust non-conventional measures of the auxiliary variable is available together with the information on conventional measures of the auxiliary variable, the proposed estimators should be used to efficiently estimate the population variance under SRS after making necessary adjustment for bias. This study is limited to the use of non-conventional location or robust measures of location and the proposed estimators are based on only a single auxiliary variable in the context of ratio method of estimation. The scope of the present study can be further extended by using non-conventional dispersion measures and inclusion of additional auxiliary variables. Moreover, the robust measures can be employed to enhance the efficiency of the variance estimators under different sampling schemes such stratified random sampling, two phase sampling and ranked set sampling.

Acknowledgments

The authors are thankful to the reviewers and the editor for their valuable comments and suggestion that led to improving the article.

References

1. Solanki, R.S., Singh, H.P., and Pal, S.K."Improved ratio-type estimators of finite population variance using quartiles", Hacettepe Journal of Mathematics and Statistics, 44(3), pp. 747–754 (2015).

2. Cochran, W.G."Sampling techniques", 3rd ed., John Wiley and Sons, New York(1977).

3. Sanusi, R.A., Abujiya, M.R., Riaz, M., and Abbas, N."Combined Shewhart CUSUM charts using auxiliary variable", Computers and Industrial Engineering, 105, pp. 329–337 (2017).

4. Sanusi, R.A., Abbas, N., and Riaz, M."On efficient CUSUM-type location control charts using auxiliary information", Quality Technology and Quantitative Management, 15(1), pp. 87–105 (2018).

5. Temesgen, H., Monleon, V., Weiskittel, A., and Wilson, D."Sampling strategies for efficient estimation of tree foliage biomass", Forest Science, 57(2), pp. 153–163 (2011).

6. Stehman, S. V."Use of auxiliary data to improve the precision of estimators of thematic map accuracy", Remote Sensing of Environment, 58(2), pp. 169–176 (1996).

7. Hussain, S., Song, L., Ahmad, S., and Riaz, M."On auxiliary information based improved EWMA median control charts", Scientia Iranica, 25(2), pp. 954–982 (2018).

8. Raza, M.A., Nawaz, T., and Aslam, M."On designing CUSUM charts using ratio-type estimators for monitoring the location of normal processes", Scientia Iranica, In Press,

17

(2018).

9. Isaki, C.T."Variance estimation using auxiliary information", Journal of the American Statistical Association, 78, pp. 117–123 (1983).

10. Upadhyaya, L.N. and Singh., H.P."An estimator for population variance that utilizes the kurtosis of an auxiliary variable in sample surveys", Vikram Mathematical Journal, 19, pp. 14–17 (1999).

11. Kadilar, C. and Cingi, H."Ratio estimators for population variance in simple and stratified sampling", Applied Mathematics and Computation, 173, pp. 1047–1058 (2006).

12. Subramani, J. and Kumarapandiyan, G."Estimation of variance using deciles of an auxiliary variable"International Conference on Frontiers of Statistics and Its Applications, Bonfring Publisher: 143–149(2012).

13. Subramani, J. and Kumarapandiyan, G."Variance estimation using quartiles and their functions of an auxiliary variable", International Journal of Statistics and Applications, 2(5), pp. 67–72 (2012).

14. Subramani, J. and Kumarapandiyan, G."Variance estimation using median of the auxiliary variable", International Journal of Probability and Statistics, 1(3), pp. 36–40 (2012).

15. Khan, M. and Shabbir, J."A ratio type estimator for the estimation of population variance using quartiles of an auxiliary variable", Journal of Statistics Application and Probability, 2(3), pp. 319–325 (2013).

16. Yaqub, M. and Shabbir, J."An improved class of estimators for finite population variance", Hacettepe Journal of Mathematics and Statistics, 45(5), pp. 1641–1660 (2016).

17. Maqbool, S. and Javaid, S."Variance Estimation Using Linear Combination of Tri-mean and Quartile Average", American Journal of Biological and Environmental Statistics, 3(1), pp. 5 (2017).

18. Adichwal, N.K., Sharma, P., and Singh, R."Generalized Class of Estimators for Population Variance Using Information on Two Auxiliary Variables", International Journal of Applied and Computational Mathematics, 3(2), pp. 651–661 (2017).

19. Maji, R., Singh, G.N., and Bandyopadhyay, A."Effective estimation strategy of finite population variance using multi-auxiliary variables in double sampling", Journal of Modern Applied Statistical Methods, 16(1), pp. 158–178 (2017).

20. Abid, M., Ahmed, S., Tahir, M., Nazir, H.Z., and Riaz, M."Improved Ratio Estimators of Variance Based on Robust Measures", Scientia Iranica, In Press, (2018).

21. Singh, H.P., Pal, S.K., and Yadav, A."A study on the chain ratio-ratio-type exponential estimator for finite population variance", Communications in Statistics - Theory and Methods, 47(6), pp. 1442–1458 (2018).

18

22. Muneer, S., Khalil, A., Shabbir, J., and Narjis, G."A new improved ratio-product type exponential estimator of finite population variance using auxiliary information", Journal of Statistical Computation and Simulation, 88(16), pp. 3179–3192 (2018).

23. Vinson, W.D."An Investigation of Measures of Central Tendency Used in Quality Control", The University of North Carolina, Chapel Hill(1951).

24. Johnstone, C.D."Statistical methods in quality control", Prentice-Hall(1957).

25. Ferrell, E."Control charts using midranges and medians", Industrial Quality Control, 9(5), pp. 30–34 (1953).

26. Wang, T., Li, Y., and Cui, H."On weighted randomly trimmed means", Journal of Systems Science and Complexity, 20(1), pp. 47–65 (2007).

27. Hettmansperger, T. and McKean, J.W."Robust nonparametric statistical methods", 2nd ed., Chapman & Hall /CRC Press, Boca Raton(2011).

28. Rana, S., Siraj-Ud-Doulah, M., Midi, H., and Imon, A.H.M.R."Decile mean: A new robust measure of central tendency", Chiang Mai Journal of Science, 39(3), pp. 478–485 (2012).

29. Subramani, J. and Kumarapandiyan, G."Estimation of variance using known coefficient of variation and median of an auxiliary variable", Journal of Modern Applied Statistical Methods, 12(1), pp. 58–64 (2013).

30. Upadhyaya, L.N. and Singh, H.P."Estimation of population standard deviation using auxiliary information", American Journal of Mathematics and Management Sciences, 21(3–4), pp. 345–358 (2001).

31. Subramani, J. and Kumarapandiyan, G."A class of modified ratio estimators for estimation of population variance", Journal of Applied Mathematics, Statistics and Informatics, 11(1), pp. 91–114 (2015).

32. Abid, M., Abbas, N., Nazir, H.Z., and Lin, Z."Enhancing the mean ratio estimators for estimating population mean using non-conventional location parameters", Revista Colombiana de Estadística, 39(1), pp. 63–79 (2016).

33. Searls, D.T."Utilization of known coefficient of kurtosis in the estimation procedure of variance", Journal of American Statistical Association, 59, pp. 1225–1226 (1964).

34. Singh, D. and Chaudhary, F.S."Theory and analysis of sample survey designs", New Age International Publisher, New Dehli(1986).

35. Shabbir, J. and Gupta, S."Estimation of the finite population mean in two phase sampling when auxiliary variables are attributes", Hacettepe Journal of Mathematics and Statistics, 39(1), pp. 121–129 (2010).

36. Yadav, S.K., Mishra, S.S., and Shukla, K.A."Improved ratio estimators for population mean based on median using linear combination of population mean and median of an auxiliary variable", American Journal of Operational Research, 4(2), pp. 21–27 (2014).

19

37. Abid, M., Abbas, N., and Riaz, M."Improved modified ratio estimators of population mean based on deciles", Chiang Mai Journal of Science, 43(1), pp. 11311–1323 (2016).

38. Shabbir, J. and Gupta, S."Estimation of finite population mean in simple and stratified random sampling using two auxiliary variables", Communications in Statistics - Theory and Methods, 46(20), pp. 10135–10148 (2017).

Appendix

The symbols and notations used in the study are summarized as follows:

N Population size; n Sample size

nfN

Sampling fraction; 1 f

n

Y Study variable; X Auxiliary variable

, Y X Population means; , y x Sample means

, Y XS S Population standard deviations; , y xs s Sample standard deviations

, Y XC C Population coefficient of variations; Population correlation coefficient

40

2 2

20

Y

Population coefficient of kurtosis of the study variable

04

2 2

02

X

Population coefficient of kurtosis of the auxiliary variable

2

03

1 3

02

X

Population coefficient of skewness of the auxiliary variable

d XM Population median of the auxiliary variable

1 XQ Population lower quartile of the auxiliary variable

3 XQ Population upper quartile of the auxiliary variable

r XQ Population inter quartile range of the auxiliary variable

a XQ Population inter quartile average of the auxiliary variable

d XQ Population semi inter quartile range of the auxiliary variable

i XD ith ( 1,2, ,10)i population decile of the auxiliary variable

20

XTM Population tri-mean of the auxiliary variable X

XMR Population mid-range of the auxiliary variable X

XHL Hodges-Lehmann Mean of the auxiliary variable X

XDM Population deciles mean of the auxiliary variable X

.B Bias of the estimator; .MSE Mean square error of the estimator

2ˆRS Traditional ratio estimator; 2ˆ

iS Existing ratio estimator

2

( )ˆ

SK iS Subramani and Kumarapandiyan (2015) class of estimators

2

1ˆ

p jS Proposed class-I estimators; 2

2ˆ

p lS Proposed class-II estimators

2

3ˆ

p mS Proposed class-III estimators

1

1 Nr s

rs i i

i

Y Y X XN

; /2 /2

20 02

rsrs r s

; 22

22

20 02

; 21

21

20 02

21

Figures Captions Figure 1: Scatter plot of Population-I

Figure 2: Scatter plot of Population-II

Figure 3: Scatter plot of Population-III

Table Captions

Table 1: Some new members of proposed class-I estimators

Table 2: Some new members of proposed class-II estimators

Table 3: Some new members of proposed class-III estimators

Table 4: Values of the population characteristics

Table 5: The PREs of the existing estimators vs. traditional ratio estimator for population-I

Table 6: The PREs of the proposed classes of estimators vs. traditional ratio estimator for

population-I

Table 7: The PREs of the existing estimators vs. traditional ratio estimator for population-II


population-II

Table 9: The PREs of the existing estimators vs. traditional ratio estimator for population-

III


population-III

22

Figure 1

Figure 2

23

Figure 3

24

Table 1

Estimator Value of Constant

Φ

2 12

1

1

1ˆ X

y

XX

p

XX TM

S KsTMx

1 X

XTM

2 12

1

1

2ˆ X

y

XX

p

XX MR

S KsMRx

1 X

XMR

2 12

1

1

3ˆ X

y

XX

p

XX HL

S KsHLx

1 X

XHL

2 12

1

1

4ˆ X

y

XX

p

XX DM

S KsDMx

1 X

XDM

2 2

1 5ˆ X

X

pX

y

X

C X TMKs

C TS

Mx

XC XTM

2 2

1 6ˆ X

X

pX

y

X

C X MRKs

C MS

Rx

XC XMR

2 2

1 7ˆ X

X

pX

y

X

C X HLKs

C HS

Lx

XC XHL

2 2

1 8ˆ X

X

pX

y

X

C X DMKs

C DS

Mx

XC XDM

9

22

1ˆ X

p y

X

X T

x

MKs

TS

M

XTM

22

1 10ˆ

X

pX

y

X MRKs

MS

x R

XMR

22

1 11ˆ

X

pX

y

X HLKs

HS

x L

XHL

22

1 12ˆ

X

pX

y

X DMKs

DS

x M

XDM

22

2

2

1 13ˆ XX

y

X

p

X

X T

TS

MKs

Mx

2 X

XTM

22

2

2

1 14ˆ XX

y

X

p

X

X M

MS

RKs

Rx

2 X

XMR

25


Φ

22

2

2

1 15ˆ XX

y

X

p

X

X H

HS

LKs

Lx

2 X

XHL

22

2

2

1 16ˆ XX

y

X

p

X

X D

DS

MKs

Mx

2 X

XDM

26

Table 2


φ

1

1

22

2 1

1

1

ˆ

X

XX

X

X TMXX

y

X

p

X

X TMLs

TMS

x

1 X

XTM 1 X

XTM

1

1

22

2 2

1

1

ˆ

X

XX

X

X MRXX

y

X

p

X

X MRLs

MRS

x

1 X

XMR 1 X

XMR

1

1

22

2 3

1

1

ˆ

X

XX

X

X HLXX

y

X

p

X

X HLLs

HLS

x

1 X

XHL 1 X

XHL

1

1

22

2 4

1

1

ˆ

X

XX

X

X DMXX

y

X

p

X

X DMLs

DMS

x

1 X

XDM 1 X

XDM

22

2 5ˆ

X

X X

C X

C X TMX X

y

X X

pSx

C X TMLs

C TM

XC XTM XC XTM

22

2 6ˆ

X

X X

C X

C X MRX X

y

X X

pSx

C X MRLs

C MR

XC XMR XC XMR

22

2 7ˆ

X

X X

C X

C X HLX X

y

X X

pSx

C X HLLs

C HL

XC XHL XC XHL

22

2 8ˆ

X

X X

C X

C X DMX X

y

X X

pSx

C X DMLs

C DM

XC XDM XC XDM

2

9

2

2ˆ

X

X

X TMX

y

X

p

X TMLs

TMxS

XTM XTM

10

22

2ˆ

X

X

X MX

X

p

R

y

X MRLs

MRS

x

XMR XMR

11

22

2ˆ

X

X

X HX

X

p

L

y

X HLLs

HLS

x

XHL XHL

12

22

2ˆ

X

X

X DX

X

p

M

y

X DMLs

DMS

x

XDM XDM

27


φ

2

2

22

2 13

2

2

ˆ

X

XX

X

X TMXX

y

X

p

X

X TMLs

xS

TM

2 X

XTM 2 X

XTM

2

2

22

2 14

2

2

ˆ

X

XX

X

X MRXX

y

X

p

X

X MRLs

xS

MR

2 X

XMR 2 X

XMR

2

2

22

2 15

2

2

ˆ

X

XX

X

X HLXX

y

X

p

X

X HLLs

xS

HL

2 X

XHL 2 X

XHL

2

2

22

2 16

2

2

ˆ

X

XX

X

X DMXX

y

X

p

X

X DMLs

xS

DM

2 X

XDM 2 X

XDM

28

Table 3


α

1

1122

3 1

1

ˆ

X

XX

X

X TMXX

y

XX

p

X TMMs b X x

x TMS

1 X

XTM 1 X

XTM

1

1122

3 2

1

ˆ

X

XX

X

X MRXX

y

XX

p

X MRMs b X x

x MRS

1 X

XMR 1 X

XMR

1

1122

3 3

1

ˆ

X

XX

X

X HLXX

y

XX

p

X HLMs b X x

x HLS

1 X

XHL 1 X

XHL

1

1122

3 4

1

ˆ

X

XX

X

X DMXX

y

XX

p

X DMMs b X x

x DMS

1 X

XDM 1 X

XDM

22

3 5ˆ

X

X X

C X

C X TM

pX X

y

X X

C X TMMs b X x

TS

C x M

XC XTM XC XTM

22

3 6ˆ

X

X X

C X

C X MR

pX X

y

X X

C X MRMs b X x

MS

C x R

XC XMR XC XMR

22

3 7ˆ

X

X X

C X

C X HL

pX X

y

X X

C X HLMs b X x

HS

C x L

XC XHL XC XHL

22

3 8ˆ

X

X X

C X

C X DM

pX X

y

X X

C X DMMs b X x

DS

C x M

XC XDM XC XDM

22

3 9ˆ

X

X

X TMX

y

X

p

X TMMs b X x

x TMS

XTM XTM

22

3 10ˆ

X

X

X MRX

y

X

p

X MRMs b X x

x MRS

XMR XMR

22

3 11ˆ

X

X

X HLX

y

X

p

X HLMs b X x

x HLS

XHL XHL

22

3 12ˆ

X

X

X DMX

y

X

p

X DMMs b X x

x DMS

XDM XDM

29


α

2

2222

3 13

2

ˆ

X

XX

X

X TMXX

y

XX

p

X TMMs b X x

x TMS

2 X

XTM 2 X

XTM

2

2222

3 14

2

ˆ

X

XX

X

X MRXX

y

XX

p

X MRMs b X x

x MRS

2 X

XMR 2 X

XMR

2

2222

3 15

2

ˆ

X

XX

X

X HLXX

y

XX

p

X HLMs b X x

x HLS

2 X

XHL 2 X

XHL

2

2222

3 16

2

ˆ

X

XX

X

X DMXX

y

XX

p

X DMMs b X x

x DMS

2 X

XDM 2 X

XDM

30

Table 4

Characteristic POP-I POP-II Pop-III Characteristic POP-I POP-II POP-III

N 34 34 34 d XQ 8.03 8.94 8.03

n 20 20 20 a XQ 17.45 18.86 17.45

Y 85.64 85.64 129.76 1 XD 7.03 6.06 7.03

X 20.89 19.94 20.89 2 XD 7.68 8.30 7.68

YS 73.31 73.31 120.76 3 XD 10.82 10.27 10.82

YC 0.86 0.86 0.93 4 XD 12.94 11.12 12.94

XS 15.05 15.02 15.05 5 XD 15.00 14.25 15.00

XC 0.72 0.75 0.72 6 XD 22.72 21.02 22.72

1 X 0.87 1.28 0.87 7 X

D 25.04 26.45 25.04

2 X 2.91 3.73 2.91 8 X

D 33.56 30.44 33.56

2 Y 13.37 13.37 3.54 9 X

D 43.61 37.32 43.61

22 1.15 1.22 0.78 10 XD 56.40 63.40 56.40

21 -0.31 -0.29 -0.43 XTM 16.23 16.56 16.23

d XM 15.00 14.25 15.00 XMR

28.45 32.00 28.45

1 XQ 9.43 9.93 9.43 XHL

18.95 18.25 18.95

3 XQ 25.48 27.80 25.48 XDM

19.82 18.36 19.82

r XQ 16.05 17.88 16.05

31

Table 5 Estimator PRE Estimator PRE Estimator PRE Estimator PRE

2

1S 100.32 2

2S 100.08 2

3S 101.53 2

4S 100.99 2

5S 102.47 2

6S 101.63 2

7S 100.85 2

8S 101.76 2

9S 100.75 2

10S 100.82 2

11S 101.13 2

12S 101.34 2

13S 101.53 2

14S 102.23 2

15S 102.43 2

16S 103.12 2

17S 103.85 2

18S 104.69 2

19S 100.44 2

20S 100.03 2

21S 102.06 2

22S 104.71 2

23S 104.81 2ˆSKS 111.56

Note: Bold value indicates maximum PRE

Table 6 Proposed Class-I Proposed Class-II Proposed Class-III

Estimator PRE Estimator PRE Estimator PRE 2

1 1ˆ

pS 139.59 2

2 1ˆ

pS 140.83 2

3 1ˆ

pS 140.78 2

1 2ˆ

pS 140.26 2

2 2ˆ

pS 141.30 2

3 2ˆ

pS 141.25 2

1 3ˆ

pS 139.78 2

2 3ˆ

pS 140.98 2

3 3ˆ

pS 140.93 2

1 4ˆ

pS 139.84 2

2 4ˆ

pS 141.02 2

3 4ˆ

pS 140.97 2

1 5ˆ

pS 139.83 2

2 5ˆ

pS 141.02 2

3 5ˆ

pS 140.96 2

1 6ˆ

pS 140.46 2

2 6ˆ

pS 141.40 2

3 6ˆ

pS 141.36 2

1 7ˆ

pS 140.01 2

2 7ˆ

pS 141.15 2

3 7ˆ

pS 141.10 2

1 8ˆ

pS 140.07 2

2 8ˆ

pS 141.18 2

3 8ˆ

pS 141.13 2

1 9ˆ

pS 140.37 2

2 9ˆ

pS 141.36 2

3 9ˆ

pS 141.31 2

1 10ˆ

pS 140.88 2

2 10ˆ

pS 141.59 2

3 10ˆ

pS 141.55 2

1 11ˆ

pS 140.53 2

2 11ˆ

pS 141.44 2

3 11ˆ

pS 141.39 2

1 12ˆ

pS 140.57 2

2 12ˆ

pS 141.46 2

3 12ˆ

pS 141.41 2

1 13ˆ

pS 138.17 2

2 13ˆ

pS 139.22 2

3 13ˆ

pS 139.14 2

1 14ˆ

pS 138.78 2

2 14ˆ

pS 140.03 2

3 14ˆ

pS 139.96 2

1 15ˆ

pS 138.32 2

2 15ˆ

pS 139.44 2

3 15ˆ

pS 139.37 2

1 16ˆ

pS 138.37 2

2 16ˆ

pS 139.51 2

3 16ˆ

pS 139.43 Note: Bold value indicates maximum PRE

32


2

1S 100.55 2

2S 100.11 2

3S 102.00 2

4S 101.43 2

5S 103.65 2

6S 102.47 2

7S 101.29 2

8S 102.59 2

9S 100.89 2

10S 101.20 2

11S 101.47 2

12S 101.59 2

13S 102.00 2

14S 102.86 2

15S 103.50 2

16S 103.95 2

17S 104.68 2

18S 107.07 2

19S 100.73 2

20S 100.03 2

21S 102.60 2

22S 106.99 2

23S 109.46 2ˆSKS 117.10


Table 8 Proposed Class-I Proposed Class-II Proposed Class-III

Estimator PRE Estimator PRE Estimator PRE 2

1 1ˆ

pS 145.80 2

2 1ˆ

pS 147.19 2

3 1ˆ

pS 147.13 2

1 2ˆ

pS 146.69 2

2 2ˆ

pS 147.91 2

3 2ˆ

pS 147.86 2

1 3ˆ

pS 145.93 2

2 3ˆ

pS 147.32 2

3 3ˆ

pS 147.26 2

1 4ˆ

pS 145.94 2

2 4ˆ

pS 147.33 2

3 4ˆ

pS 147.27 2

1 5ˆ

pS 146.52 2

2 5ˆ

pS 147.79 2

3 5ˆ

pS 147.74 2

1 6ˆ

pS 147.30 2

2 6ˆ

pS 148.24 2

3 6ˆ

pS 148.20 2

1 7ˆ

pS 146.65 2

2 7ˆ

pS 147.88 2

3 7ˆ

pS 147.83 2

1 8ˆ

pS 146.65 2

2 8ˆ

pS 147.89 2

3 8ˆ

pS 147.83 2

1 9ˆ

pS 147.16 2

2 9ˆ

pS 148.17 2

3 9ˆ

pS 148.13 2

1 10ˆ

pS 147.76 2

2 10ˆ

pS 148.42 2

3 10ˆ

pS 148.38 2

1 11ˆ

pS 147.26 2

2 11ˆ

pS 148.22 2

3 11ˆ

pS 148.18 2

1 12ˆ

pS 147.27 2

2 12ˆ

pS 148.23 2

3 12ˆ

pS 148.18 2

1 13ˆ

pS 144.51 2

2 13ˆ

pS 145.57 2

3 13ˆ

pS 145.48 2

1 14ˆ

pS 145.25 2

2 14ˆ

pS 146.59 2

3 14ˆ

pS 146.52 2

1 15ˆ

pS 144.60 2

2 15ˆ

pS 145.71 2

3 15ˆ

pS 145.63 2

1 16ˆ

pS 144.61 2

2 16ˆ

pS 145.72 2

3 16ˆ


33


2

1S 101.11 2

2S 100.28 2

3S 105.56 2

4S 103.54 2

5S 109.19 2

6S 105.93 2

7S 106.43 2

8S 103.03 2

9S 102.66 2

10S 102.90 2

11S 104.05 2

12S 104.82 2

13S 105.56 2

14S 108.26 2

15S 109.05 2

16S 111.86 2

17S 115.03 2

18S 118.82 2

19S 101.54 2

20S 100.10 2

21S 107.60 2

22S 140.23 2

23S 133.23 2ˆSKS 178.49


Table 10

Proposed Class-I Proposed Class-II Proposed Class-III Estimator PRE Estimator PRE Estimator PRE

2

1 1ˆ

pS 172.96 2

2 1ˆ

pS 187.08 2

3 1ˆ

pS 186.97 2

1 2ˆ

pS 180.71 2

2 2ˆ

pS 194.20 2

3 2ˆ

pS 194.09 2

1 3ˆ

pS 175.17 2

2 3ˆ

pS 189.36 2

3 3ˆ

pS 189.24 2

1 4ˆ

pS 175.82 2

2 4ˆ

pS 189.98 2

3 4ˆ

pS 189.87 2

1 5ˆ

pS 177.65 2

2 5ˆ

pS 191.67 2

3 5ˆ

pS 191.56 2

1 6ˆ

pS 185.08 2

2 6ˆ

pS 197.23 2

3 6ˆ

pS 197.13 2

1 7ˆ

pS 179.85 2

2 7ˆ

pS 193.53 2

3 7ˆ

pS 193.42 2

1 8ˆ

pS 180.48 2

2 8ˆ

pS 194.02 2

3 8ˆ

pS 193.92 2

1 9ˆ

pS 194.36 2

2 9ˆ

pS 201.53 2

3 9ˆ

pS 201.44 2

1 10ˆ

pS 197.71 2

2 10ˆ

pS 202.40 2

3 10ˆ

pS 202.31 2

1 11ˆ

pS 195.47 2

2 11ˆ

pS 201.86 2

3 11ˆ

pS 201.77 2

1 12ˆ

pS 195.77 2

2 12ˆ

pS 201.94 2

3 12ˆ

pS 201.85 2

1 13ˆ

pS 159.99 2

2 13ˆ

pS 169.68 2

3 13ˆ

pS 169.55 2

1 14ˆ

pS 165.92 2

2 14ˆ

pS 178.52 2

3 14ˆ

pS 178.40 2

1 15ˆ

pS 161.51 2

2 15ˆ

pS 172.09 2

3 15ˆ

pS 171.97 2

1 16ˆ

pS 161.98 2

2 16ˆ

pS 172.81 2

3 16ˆ


34

Farah Naz earned her M.Sc and Mphil degrees in Statistics from the Islamia University

Bahawalpur, Pakistan and Government College University Faisalabad, Pakistan respectively.

Currently, she is pursuing PhD in Statistics from the School of Mathematical Sciences,

Institute of Statistics, Zhejiang University Hangzhou People’s Republic of China under the

Chinese Government Scholarship Program (2017). Her research interest is in Survey

Sampling and Distribution Theory.

Muhammad Abid obtained his MSc and MPhil degrees in Statistics from Quaid-i-Azam University, Islamabad, Pakistan, in 2008 and 2010, respectively. He did his PhD in Statistics from the School of Mathematical Sciences, Institute of Statistics, Zhejiang University, Hangzhou, People’s Republic of China, in 2017. He served as a Statistical Officer in National Accounts Wing, Pakistan Bureau of Statistics (PBS) during 2010-2011. He is now serving as an Assistant Professor in the Department of Statistics, Government College University Faisalabad, Pakistan, from 2017 to present. He has published more than 15 research papers in research journals. His research interests include Statistical Quality Control, Bayesian Statistics, Non-parametric Techniques, and Survey Sampling.

Tahir Nawaz obtained his MSc and MPhil degrees in Statistics from the Islamia University

Bahawalpur, Pakistan and Government College University Lahore, Pakistan respectively. He

served as a Statistical Officer in Punjab Bureau of Statistics, Pakistan during May 2007 to

November 2009. He served as a Lecturer in Statistics in the Islamia University Bahawalpur,

Pakistan during November 2009 to October 2013. He is working as a Lecturer in the

Department of Statistics, Government College University Faisalabad, Pakistan since

November 2013. Currently, he is pursuing his PhD in Statistics from the School of

Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, People’s Republic of China

under the Chinese Government Scholarship Program (2016). His research interest includes

Statistical Process Control, Distribution Theory, and Survey Sampling.

Tianxiao Pang earned his PhD in Probability and Mathematical Statistics from the

Department of Mathematics, Zhejiang University, Hangzhou, People’s Republic of China in

2005. He is an Associate Professor of Statistics at Zhejiang University, Hangzhou, People’s

Republic of China. More than 30 refereed publications in various reputed international

journals are at his credit. His research interest is in Probability Limit Theory, Large Sample

Theory in Statistics, and Econometrics.

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