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Estimation of general parameters using auxiliary information in simple random sampling without replacement

Department of General Management, University of Petroleum and Energy Studies, Dehradun, India
Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia
Department of Statistics, Banaras Hindu University, India
Department of Statistics & Operations Research, Aligarh Muslim University, India

⁎Corresponding author. irfii.st@amu.ac.in (Irfan Ali)

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.
https://orcid.org/0000-0002-1790-5450

Abstract

Estimation of population parameters plays a vital role in the area of sampling. Many authors have proposed several estimators for estimating population parameter(s) using auxiliary information. This paper has attempted to suggest a new estimator for estimating the general parameter ta,b using auxiliary information in simple random sampling without replacement (SRSWOR). A conventional estimator ta,b is used to define the population constants: coefficient of variation, population mean, standard deviation, and population mean square. The expression for the minimum mean squared error has been derived. The efficiency of the suggested estimator and the existing estimators has been analyzed using a simulation study. Theoretical and empirical studies reveal the effectiveness of the proposed estimator over other existing estimators.

Keywords

Auxiliary variable
Conventional estimator
MSE
Efficiency
Simulation study
1

1 Introduction

Auxiliary information, when suitably used, improves the efficiency of the population parameters estimators. There are various techniques for applying auxiliary information, which helps improve the estimator’s performance. These techniques include product methods, ratio, and regression, among others. Auxiliary information can be available in various forms, such as in the form of variables or attributes. Many researchers have widely used it to formulate various estimators for estimating population parameters in different sampling schemes. Srivastava (1971), Kadilar and Cingi (2004) and Gupta and Shabbir (2008) have proposed estimators for estimating unknown population values. Another estimator’s class that uses correlation coefficient value and minimum MSE have been used in population mean estimation as documented in Srivastava and Jhajj (1980), Srivastava and Jhajj (1981), Srivastava and Jhajj (1986). Shabbir and Gupta (2007) proposed an estimator using the information in the form of attributes. Tracy et al. (1996) and Gupta and Shabbir (2007) used information on two auxiliary variables and proposed estimators to estimate the population constants. Some other related work can be found in Singh and Singh (2001), Tripathi et al. (2002), Khoshnevisan et al. (2007), Singh et al. (2008a), Singh et al. (2008b), Singh and Kumar (2011), Singh and Solanki (2011), Singh and Malik (2014), Sharma et al. (2017), Adichwal et al. (2016), Adichwal et al. (2019), Adichwal et al. (2017), Singh et al. (2018) and Mishra and Singh (2017).

Motivated by the work of these authors, we have proposed an improved estimator in the SRSWOR scheme for the general parameter of the population.

2

2 Notations

Let us consider a sample of size n is drawn from a population W=W1,W2,...,WN using SRSWOR scheme. Let Yi and Xi be the study and auxiliary population variables for the ith units (i = 1,2,3,….…, N), and let yi and xi, be ith units, respectively, in the sample for i = 1,2,3,....,n.

In general, consider the following population parameters μrs=1Ni=1Nyi-y¯rxi-X¯s

δrs=μrs/μ20r/2μ02s/2 and (r, s) is non-negative integers.

Note that

μ20=SY2 , μ02=SX2 and μ11=SXY , So that CY2=SY2/Y¯2=μ20/Y¯2 , CX2=SX2/X¯2=μ02/X¯2 and ρXY=SXY/SXSY=μ11/μ20μ02

Let us define,

ε0=y¯Y¯-1 , ε1=sy2SY2-1 , ε2=x¯X¯-1 and ε3=sx2SX2-1

Subject to the condition E(εi)=0,i=0,1,2,3.

Ignoring fpc, we have E(ε02)=n-1CY2E(ε12)=n-1δ40-1E(ε22)=n-1CX2E(ε32)=n-1δ04-1 E(ε0ε1)=n-1δ30CYE(ε0ε2)=n-1ρyxCXCYE(ε0ε3)=n-1δ12CY E(ε1ε2)=n-1δ21CXE(ε1ε3)=n-1δ22-1E(ε2ε3)=n-1δ03CX

3

3 Conventional estimator

The general form of the parameter under consideration can be stated as

(3.1)
ta,b=Y¯aSyb

In Eq. (3.1), a and b are scalars, suitably chosen and y¯=1ni=1nyi and sy2=1n-1i=1nyi-y¯2 are unbiased estimators of Y¯ and Sy2 , respectively.

For choices of a, b the general parameter ta,b will take form of

  1. If a=1,b=0 , ta,b reduces to t1,0=Y¯

  2. If a=0,b=2 , ta,b reduces to t0,2=SY2

  3. If a=-1,1 , ta,b reduces to t-1,1=CY

  4. If a=0,b=1 , ta,b reduces to t0,1=SY

The general parameter ta,b conventional estimator is defined as-

(3.2)
t̂a,b=y¯asyb where y¯=1ni=1nyi and sy2=1n-1i=1nyi-y¯2 .

Expressing Eq. (3.2) in terms of ε ’s, we have

(3.3)
t̂a,b=Y¯aSyb1+ε0a1+ε1b2

Eq. (3.3) can be written as t̂a,b=ta,b1+aε0+b2ε1+aa-12ε02+ab2ε0ε1+bb-28ε12

or equivalently,

(3.4)
t̂a,b-ta,b=ta,baε0+b2ε1+aa-12ε02+ab2ε0ε1+bb-28ε12

Squaring Eq. (3.4) and neglecting the terms of ε with power three or more, we have

(3.5)
t̂a,b-ta,b2=ta,b2a2ε02+abε0ε1+b24ε12

Take the expectations on both sides of Eq. (3.5), we have MSE’s estimators of t̂a,b to as given by

(3.6)
MSEt̂a,b=ta,b2na2CY2+abδ30CY+b24δ40-1 or
(3.7)
MSEt̂a,b=ta,b2nf1a,b
where, f1a,b=a2CY2+abδ30CY+b24δ40-1.

4

4 Proposed estimator

We propose a class of difference-cum exponential ratio type estimators to estimate the parameter of the population ta,b as

(4.1)
t=t̂a,b+kX¯-x¯expw1(X¯-x¯)X¯+(α-1)x¯expw2(SX2-sx2)SX2+(β-1)sx2 where w1 , w2 being scalar having the values (0,-1,1), and k, α and β are constants and can be defined suitably.

We express Eq. (4.1) in terms of ε ’s to obtain t=ta,b1+ε0a1+ε1b2-kX¯ε2exp-w1ε2α1+α-1αε2-1exp-w2ε3β1+β-1βε3-1t=ta,b1+aε0+b2ε1+aa-12ε02+ab2ε0ε1+bb-28ε12+ ×1-w2βε3-w1αε2+w22β-1+w22β2ε32+w12α-1+w12α2ε22+w1αw2βε2ε3+

(4.2)
-kX¯ε21-w2βε3-w1αε2+w22β-1+w22β2ε32+w12α-1+w12α2ε22+w1αw2βε2ε3+

Multiplying out and neglecting the higher-order terms of ε ’s which are greater than two powers in Eq. (4.2), we have t=ta,b1+aε0+b2ε1-w1αε2-w2βε3+aa-12ε02+ab2ε0ε1+bb-28ε12-w1αaε0ε2+b2ε1ε2

(4.3)
-w2βaε0ε3+b2ε1ε3+w1αw2βε2ε3+w12α-1+w12α2ε22+w22β-1+w22β2ε32-kX¯ε2+w2βkX¯ε2ε3+w1αkX¯ε22

or

(4.4)
t-ta,b=ta,baε0+b2ε1-w1αε2-w2βε3+aa-12ε02+ab2ε0ε1+bb-28ε12-w1αaε0ε2+b2ε1ε2-w2βaε0ε3+b2ε1ε3+w1αw2βε2ε3+w12α-1+w12α2ε22+w22β-1+w22β2ε32-kX¯ε2+w2βkX¯ε2ε3+w1αkX¯ε2ε3

Squaring Eq. (4.4) both sides and neglect the higher order of ε ’s which are greater than two powers, the result is as follows t-ta,b2=ta,baε0+ta,bb2ε1-ta,bw1α+kX¯ε2-ta,bw2βε32

or

(4.5)
t-ta,b2=ta,baε0+b2ε1-w1αε2-w2βε2-kX¯ε22

From Eq. (4.5), we have t-ta,b2=ta,b2a2ε02+b24ε12+w1α2ε22+w2β2ε32+2ab2ε0ε1-2aw1αε0ε2 -2aw2βε0ε3-2b2w1αε1ε2-2b2w2βε1ε3+2w1αw2βε2ε3 -2ta,bakX¯ε0ε2-2ta,bb2kX¯ε1ε2+2ta,bw1αkX¯ε22+2ta,bw2βkX¯ε2ε3

(4.6)
+k2X¯2ε22 or t-ta,b2=ta,b2a2ε02+b24ε12+2ab2ε0ε1+ta,b2w1α2ε22+w2β2ε32-2w1αaε0ε2+b2ε1ε2 -2w2βaε0ε3+b2ε1ε3+2w1βw2βε2ε3+2ta,bkX¯w1αε22+w2βε2ε3
(4.7)
-2ta,bkX¯aε0ε2+b2ε1ε2+k2X¯2ε22

Take the expectation of Eq. (4.7), we have MSEt=MSEt̂a,b+ta,b2nw1α2CX2+w2β2δ04-1-2w1αaρXYCY+b2δ21CX -2w2βaδ12CY+b2δ22-1+2w1βw2βδ03CX+2ta,bnkX¯w1αCX2+w2βδ03CX

(4.8)
-2ta,bnkX¯aρXYCY+b2δ21CX+k2X¯2nCX2 or
(4.9)
MSEt=MSEt̂a,b+ta,b2nA12CX2+A22δ04-1-2A1f2a,bCX-2A2f3a,b+2A1A2δ03CX+2ta,bnkX¯A1CX2+A2δ03CX-2ta,bnkX¯f2a,bCX+k2X¯2nCX2
where, A1=w1α A2=w2β f2a,b=aρXYCY+b2δ21 f3a,b=aδ12CY+b2δ22-1.

Differentiate partially Eq. (4.9) w.r.to A1 and A2 and equate to zero, we have

ta,bCXta,bδ03ta,bδ03CXta,b4δ03-1 A1A2 = ta,baρXYCY+b2δ21-kX¯CXta,baδ12CY+b2δ22-1-kX¯CXδ03

After simplifying, we obtain the optimum value of A1 and A2 , that is,

(4.10)
A1opt=δ04-1f2a,b-δ03f3a,bδ04-δ032-1CX-KX¯ta,bA2opt=f3a,b-δ03f2a,bδ04-δ032-1

Substituting A1opt and A2opt from Eq. (4.10) in Eq. (4.9), we have

(4.11)
MSEtmin=MSEt̂a,b-ta,b2nf3a,b2-2f2a,bf3a,bδ03+δ04-1f2a,b2δ04-δ032-1
(4.12)
=MSEt̂a,b-ta,b2nf2a,b2-ta,b2nf2a,bδ03-f3a,b2δ04-δ032-1
(4.13)
=MSEt̂a,b-ta,b2nf3a,b2δ04-1-ta,b2nδ04-1f2a,b-δ03f3a,b2δ04-δ032-1δ04-1

Remark 4.1: The optimum values of constants A1 and A2 at (4.10) involve unknown population parameters. The values of these quantities can be guessed accurately through a pilot sample survey or sample data at hand or experience gathered in due course of time, see Srivastava and Jhajj (1980), Singh et al. (2003) and Singh and Solanki (2013).

When X¯ is known, the following Difference-cum exponential ratio type estimator (DCERTE) to obtain population parameter ta,b on putting w2=0 in Eq. (4.1) is defined as:

(4.14)
t1=t̂a,b+kX¯-x¯expw1(X¯-x¯)X¯+(α-1)x¯ where w1 being scalar has real values (0,-1,1) and k and α is arbitrary constants.

MSE of the estimator t1 is given by

(4.15)
MSEt1=MSEt̂a,b+ta,b2nw1α2CX2-2w1αf2a,bCX

MSEt1 defined in Eq. (4.15) is minimized for w1αopt=f2a,bCX

(4.16)
MSEt1min=MSEt̂a,b-ta,b2nf2a,b2

Table 4.1 presents the existing estimators obtained from Eq. (4.14) on taking appropriate values of a, b, k, w1 and α accordingly.

Table 4.1 Particular cases of the estimator t1 .
Subset of the proposed estimator a b K w1 α
t11=t̂a,bexp(X¯-x¯)X¯+(α-1)x¯ (Singh and Pal, 2017) a b 0 1 α
t12=y¯exp(X¯-x¯)X¯+(α-1)x¯ (Upadhyaya et al., 2011) 1 0 0 1 α
t12=y¯exp(X¯-x¯)X¯+x¯ (Bahl and Tuteja, 1991) 1 0 0 1 2
t13=y¯+kX¯-x¯ (Difference Estimator) 1 0 0 0 α

Using known Sx2 , following exponential ratio type estimator to estimate the parameter ta,b of the population by putting k = 0 and w1=0 in Eq. (4.1) is defined as

(4.17)
t2=t̂a,bexpw2(SX2-sx2)SX2+(β-1)sx2 where w2 being scalar has real values (0,-1,1) and β is suitably chosen constants, MSE of the estimator t2 is defined as:
(4.18)
MSEt2=MSEt̂a,b+ta,b2nw2β2δ04-1-2w2βf3a,b

MSEt2 in (4.18) is minimized form for w2βopt=f3a,bδ04-1

(4.19)
MSEt2min=MSEt̂a,b-ta,b2nf3a,b2δ04-1

Table 4.2 presents a set of existing estimators obtained from (4.17) by suitable a, b, w2 and β .

Table 4.2 Particular cases of the estimator t2 .
Subset of the proposed estimator A b w2 β
t21=t̂a,bexp(SX2-sx2)SX2+(β-1)sx2 (Singh and Pal, 2017) a b 1 β
t21=sy2exp(SX2-sx2)SX2+(β-1)sx2 (Yadav and Kadilar, 2013) 0 2 1 β
t21=sy2exp(SX2-sx2)SX2+sx2 (Singh et al., 2011) 0 2 1 2

5

5 Efficiency comparison

(5.1)
MSEt̂a,b-MSEtmin=ta,b2nf3a,b2-2f2a,bf3a,bδ03+δ04-1f2a,b2δ04-δ032-10
(5.2)
MSEt1min-MSEtmin=ta,b2nf2a,bδ03-f3a,b2δ04-δ032-10
(5.3)
MSEt2min-MSEtmin=ta,b2nδ04-1f2a,b-δ03f3a,b2δ04-δ032-1δ04-10

We note that δ04-δ032-10 always.

It can be observed from (5.1), (5.2) and (5.3), the proposed difference-cum exponential ratio type conventional estimator t performs efficiently than the estimators t̂a,b , t1 and t2 .

6

6 Estimation of the population mean of the study variable Y

For a,b,k,w1,w2,α,β=1,0,k,w1,w2,α,β , class of estimator ‘t’ written in Eq. (4.1), will take the following form

(6.1)
t=y¯+kX¯-x¯expw1(X¯-x¯)X¯+(α-1)x¯expw2(SX2-sx2)SX2+(β-1)sx2

MSE’s expressions of the estimator t up to O(n−1) is given by MSEt=MSEy¯+Y¯2nA12Cx2+A22δ04-1-2A1ρXYCXCY-2A2δ12CY+2A1A2δ03CX

(6.2)
+2nkX¯Y¯A1CX2+A2δ03CX-2nkX¯Y¯ρXYCXCY+k2X¯2nCX2 where, A1=w1α and A2=w2β

MSE of t at (6.2) is minimized for values

(6.3)
A1opt=δ04-1ρXY-δ03δ12CYδ04-δ032-1Cx-KX¯Y¯A2opt=δ12-δ03ρXYCYδ04-δ032-1

Putting the value of A1opt and A2opt from Eq. (6.3) in Eq. (6.2) to get the minimum MSE of the estimator t as (Table 6.1)

(6.4)
MSEtmin=Vy¯-Y¯2nδ122-2ρXYδ03δ12+δ04-1ρXY2CY2δ04-δ032-1
Table 6.1 Particular cases of the estimator t .
Subset of the proposed estimator k w1 w2 α
t12=y¯exp(X¯-x¯)X¯+(α-1)x¯ (Upadhyaya et al., 2011) 0 1 0 α
t12=y¯exp(X¯-x¯)X¯+x¯ (Bahl and Tuteja, 1991) 0 1 0 2
t13=y¯+kX¯-x¯ (Difference Estimator) 0 0 0 α

The minimum MSE’s of the estimator t12 , t12 and t13 is given by

(6.5)
MSEt12min=Vy¯-Y¯2nρXY2CY2
(6.6)
MSEt12min=Vy¯-Y¯2n14CX2-ρXYCXCY
(6.7)
MSEt13min=Vy¯-Y¯2nρXY2CY2

7

7 Efficiency comparison

(7.1)
Vy¯-MSEtmin=Y¯2nδ122-2ρXYδ03δ12+δ04-1ρXY2CY2δ04-δ032-10
(7.2)
MSEt12-MSEtmin=Y¯2nδ122-2ρXYδ03δ12+δ04-1ρXY2CY2δ04-δ032-1-Y¯2nρXY2CY20
(7.3)
MSEt12min-MSEtmin=Y¯2nδ122-2ρXYδ03δ12+δ04-1ρXY2CY2δ04-δ032-1-Y¯2n14CX2-ρXYCXCY0
(7.4)
MSEt13min-MSEtmin=Y¯2nδ122-2ρXYδ03δ12+δ04-1ρXY2CY2δ04-δ032-1-Y¯2n14CX2-ρXYCXCY0

8

8 Empirical study

In this section, we compare the performance of the proposed estimators using a known population data set.

The description of population data sets is as follows.

Population

Y = Number of person per block

X = Number of rooms per block

The values of the parameters are

n = 10, X-=58.8 , Y-=101.1 , CX=0.1281 , CY=0.1450 , ρXY=0.6500 , δ12=0.5714 , δ21=0.4537 , δ03=0.4861 , δ30=0.3248 , δ04=2.2387 , δ40=2.3523 , δ13=1.5041 , δ31=1.6923 , δ22=1.5432 .

Tables 8.1 is concluded that the performance of the proposed estimator t is more efficient in comparison to the usual mean estimator and other existing estimators t12 , t12 and t13 as the PRE of the proposed estimator t is greater than the existing estimators.

Table 8.1 PRE’s of the proposed and existing estimators.
Estimators PRE
y¯ 100.0000
t 195.1564
t12 173.1602
t12 72.5100
t13 173.1602

9

9 Simulation study

This section shows the procedure for comparing estimators t12 , t12 and t13 with the estimator t computationally based on the Reddy et al. (2010) algorithm. The following stepwise simulation algorithm is used to evaluate the efficiency of Y¯ :

Step 1: Simulate XÑμ,σ2 and X1Ñμ1,σ12 independently and randomly using the method of box-Muller.

Step 2: Let Y=ρX+1-ρ2X1 such that 0<ρ=0.4,0.6,0.8<1 .

Step 3: Gives the pair (Y, X).

Step-4: Let define the parameters μ=5 , σ=3 , μ1=5 and σ1=3 for population-I in step 1, repeat steps 1 to 3 for 1000 times. Variable Y and X will have the same variances in the population.

Step-5: Similarly, use parameters μ=3 , σ=2 , μ1=5 and σ1=3 in step-1 to generate the population-II, and then repeat the process from steps 1 to 3 for 1000 times. Variable Y and X will have different variances in the population.

Step-6: Use SRS to draw 500 samples yi,xi, for i = 1, 2,…,n from the population of size N = 1000, WOR of size n = 40, 50 and 60.

Step-7: Calculate AMSEt=1500k=1500Etk-y¯2

and PRE of an estimator t with respect to the usual estimator y¯ is PREt=Vary¯×100MSEt

Tables 9.1 and 9.2 shows simulation results that have been obtained following the stepwise procedures.

Table 9.1 Average PRE of the estimators for Population-I.
ρ N Average PRE’s
y¯ t t12 t12 t13
0.5 40 100.0000 140.2242 138.7793 80.36167 138.7793
50 100.0000 139.6806 138.6224 80.1948 138.6224
60 100.0000 139.5516 138.5826 80.29106 138.5826
0.6 40 100.0000 165.8203 164.1184 72.5535 164.1184
50 100.0000 165.1557 163.9098 72.47781 163.9098
60 100.0000 164.9832 163.841 72.51843 163.841
0.8 40 100.0000 299.7129 296.6709 60.68814 296.6709
50 100.0000 298.4635 296.2357 60.68819 296.2357
60 100.0000 298.0521 296.0032 60.66814 296.0032
Table 9.2 Average PRE of the estimators for Population-II.
ρ N Average PRE’s
y¯ t t12 t12 t13
0.5 40 100.0000 119.2466 118.0128 97.4543 118.0128
50 100.0000 118.8099 117.9053 96.43723 117.9053
60 100.0000 118.7146 117.8876 96.7558 117.8876
0.6 40 100.0000 130.8356 129.4851 87.2013 129.4851
50 100.0000 130.3386 129.3491 86.7238 129.3491
60 100.0000 130.2249 129.3196 86.9226 129.3196
0.8 40 100.0000 191.0631 189.1082 68.4768 189.1082
50 100.0000 190.2839 188.8529 68.3965 188.8529
60 100.0000 190.0674 188.7543 68.4277 188.7543

From Tables 9.1 and 9.2, we observe that from the various values of correlation coefficient ρ=0.4,0.6,0.8 and the sample size n = 40,50,60, it can be concluded that the performance of the proposed estimator t is more efficient in comparison to the usual mean estimator and other existing estimators t12 , t12 and t13 . Table 9.1 and 9.2 indicates that the average PRE of the proposed t estimator is higher in all the cases (for variations in ρ and n) and for both the simulated data sets discussed above.

10

10 Conclusion

The present study proposed an improved estimator for a general parameter estimation using the information on “additional value X”. Differential cum exponential ratio type estimator ‘t’ of a general parameter is suggested in SRSWOR. The estimator ‘t’ proposed in Eq. (4.1) can be used to estimate different population parameters. We have verified the performance of the proposed estimator using simulated data set for the case of estimation of population mean only. The result of this study shows that the usual mean estimator, Singh and Pal (2017), Upadhyaya et al. (2011), Bahl and Tuteja (1991), Yadav and Kadilar (2013) and Singh et al. (2011) can be shown as the members of the proposed class. An empirical study has been carried out using real data set (presented in Table 8.1) and simulated data sets (presented in Table 9.1 and Table 9.2 by taking two different simulated population data sets) to illustrate the efficiency and effectiveness of the proposed estimator. The result of simulation studies shows that the average PRE of the suggested estimator t is higher as compared to the existing estimators Upadhyaya et al. (2011) t_{1\left(2 \right)}, Tuteja and Bahl (1991) t12 t12 and difference estimator t13 for different choices of correlation coefficient ρ and sample size n in all the cases and for both the simulated data sets. The same pattern can also be obtained in case of real data set i.e. the proposed estimator t performing better as comparison to the usual mean estimator and other existing estimators t12 , t12 and t13 as the PRE of the proposed estimator t is greater than the existing estimators. Through a literature survey, it can be found that the best estimator is one having the minimum mean square error. Hence, by the result of the simulation study, we can conclude that the proposed estimator t performs better than the existing estimators t12 , t12 and t13 . Hence, this method is recommended for practical application.

Acknowledgement

The authors wish to acknowledge the editor in chief and the anonymous reviewers for their constructive comments, which improved the quality of the work.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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