LEVERAGING AUXILIARY VARIABLES: ADVANCING MEAN ESTIMATION THROUGH CONDITIONAL AND UNCONDITIONAL POST-STRATIFICATION Текст научной статьи по специальности «Математика»

LEVERAGING AUXILIARY VARIABLES: ADVANCING MEAN ESTIMATION THROUGH CONDITIONAL AND UNCONDITIONAL POST-STRATIFICATION

G.R.V. Triveni1 and Faizan Danish2

^Department of Mathematics, School of Advanced Sciences, VIT-AP University, Inavolu, Beside

AP Secretariat, Amaravati AP-522237, India 1trivenigullinkala@gmail.com ,2danishstat@gmail.com

Abstract

This article presents a novel class of estimators designed for post-stratification to estimate the mean of a study variable using information from auxiliary variables. Through a rigorous examination of bias and Mean Square Error (MSE), we demonstrate the potential to improve estimation accuracy up to the first order of approximation. We also thoroughly explore both Conditional and Unconditional post-stratification properties, enhancing our understanding of the estimator's performance. To assess the effectiveness of our proposed estimator, we conduct a comprehensive numerical illustration. The results affirm its superiority over existing estimators in both Conditional and Unconditional Poststratification scenarios, exhibiting the highest Percentage Relative Efficiency. Additionally, graphical analysis reveals that Conditional post- stratification outperforms Unconditional post-stratification. These findings underscore the significant practical value of our proposed estimator in enhancing the accuracy of mean estimation in post-stratification studies. By accurately estimating population parameters, our novel class of estimators contributes to more informed decision-making in various fields of study. The utilization of auxiliary variables allows for better utilization of available information and leads to more reliable and robust conclusions. Overall, the novel class of estimators introduced in this article represents a valuable contribution to the field of post-stratification. As researchers continue to explore and apply these estimators, they have the potential to revolutionize data analysis methods, becoming indispensable tools for survey and research design. The improvements in estimation accuracy brought about by these estimators are particularly crucial in situations where reliable data is scarce or challenging to obtain, making them invaluable for decision-makers and researchers alike. With the increased accuracy and efficiency of our proposed estimators, they provide a pathway for better resource allocation, cost-effective decision-making, and improved policy formulation. Policymakers and researchers can confidently rely on these estimators to produce more accurate results and achieve better outcomes in various domains. In conclusion, the novel class of estimators for post-stratification presented in this article opens up new avenues for advancing statistical estimation methods. The fusion of auxiliary variables with traditional poststratification techniques represents a powerful approach to enhance estimation accuracy. Embracing and incorporating these estimators into research practices will undoubtedly bring us closer to making data-driven decisions that have a meaningful impact on society.

Keywords: Conditional post stratification, Unconditional post stratification, Mean square error.

I. Introduction

A common statistical method used in research studies to increase the precision and representativeness of survey data is post-stratification. It entails breaking down the sample population into discrete subgroups according to certain traits or factors, such age, gender, income level, or geography. Researchers can reduce potential biases and improve the generalizability of the results by stratifying the population to make sure that each subgroup is appropriately represented in the sample. After stratifying the sample, researchers can determine the population parameters by giving each subgroup the proper weights depending on its relative size. This method enables researchers to account for differences and generate more accurate and trustworthy estimates when the sample does not properly reflect the makeup of the target population. By

G.R.V. Triveni and Faizan Danish RT&A, No 4 (76) LEVERAGING AUXILIARY VARIABLES: ADVANCING MEAN_Volume 18, December 2023

taking into account population changes and enhancing the precision of inferential statistics, poststratification is a useful tool that improves the validity and robustness of study findings.

Auxiliary variable information is used in various types of literature to estimate population mean or variance. The significance of post-stratification and the proper framework for statistical inference were covered in [1]. By utilizing auxiliary data and empirical research, [2] proposed estimators for population mean and shown that the proposed estimator outperformed the alternative. For judgement post-stratification, [3] offered an alternative estimate that regularly beats the usual non-parametric mean estimator, and they noted a decrease in Mean Square Error (MSE) in their proposed estimator compared to the standard estimator. [4] proposed a class of estimators and compared them with a few already in use. They came to the conclusion that their suggested class of estimators performed well based on a numerical investigation. In post-stratified sampling, [5] created a new family of combined estimators of the population mean, and the outcomes are empirically demonstrated. Exponential estimators are later proposed in poststratification by [6], and its bias and MSE equations are obtained. The theoretical findings are further supported by a numerical analysis. [7] suggested an estimator of the population mean utilizing information from an auxiliary variable and demonstrated the superiority of the proposed estimator over others through comparison analysis. Additionally, [8] constructed a generalized class of estimators for population variance and demonstrated the effectiveness of the suggested estimator through a numerical investigation. Through empirical research, [9] demonstrated the superiority of the suggested estimator and developed a new family of exponential estimators. The ratio and product type exponential estimators were improved in the case of post-stratification by [10]. They demonstrated that the suggested estimator performed more effectively after stratification than unbiased, ratio, and product estimators. Additionally, [11] raises the issue of estimating a population proportion in a decision following stratification. They conducted Monte Carlo simulation research to evaluate the performance of proportion estimators. According to [12], a family of Ratio estimators and the formulations for bias and an MSE are constructed in the case of the non-response issue. It is demonstrated by numerical analysis that the suggested estimator has reduced MSE values. A novel class of estimators was recently developed [13], and by numerical analysis, under ideal circumstances, the proposed class of estimators outperformed the previously taken into consideration existing estimators.[14] suggested some post-stratification enhanced estimators. They demonstrated the effectiveness of the proposed estimator using two real data sets. In this paper, a class of estimators for estimating population mean under poststratification has been developed.

II. Terminology

Consider a finite population x = (1-2, ■■■ N} of size N is stratified into K strata with stratum each of which has Nhunits such that Eh=i Nh = N. With the use of Simple random sampling without replacement, a sample with the dimension nh is taken from stratum. Let d serve as study(dependent) variable and i serve as auxiliary (independent) variable. We have used the following notations:

1 WhDh is the population mean of study variable 1 Wh^h is the population mean of auxiliary variable Dh = dhi and Th = ~ EN=hi ihi are the stratum means of study and auxiliary variables

Sdh = (N 1 ^ 2Nhi(dhi - Dh)2 is the variance of study variable at stratum

S2h ^N\)IN^1(ihi - Ih)2 is the variance of auxiliary variable at stratum

Sdih = —1—ENhi(dhi - Dh)((ihi - Ih) is the covariance at stratum

(Nh-1)

Cdh = —^ENhi(dhi - Dh)2 be the square of coefficient of variation of d

• Cih = -2 1 X^iOhi - 11)2 be the square of coefficient of variation of i

N^-1S[=h1(dhi-Dh)(ihi-îh)

• p= -s—=---be the correlation coefficient of d and i.

dih (Dh*Cdh)(Ih*Idh)

—h

• Wh = — represents stratum weight.

I. Properties of Estimators in Unconditional Post-Stratification

To derive bias and mean squared error (MSE), we write

dh = Dh(1 +_e0h), ïh = Th(1 + e1h),

_ Xh=1WhDheoh _, _ Xh=1Whïhe1h eo = = and e1 = y

Where

e = dh~Dh , e =îh-îh

e0h = „ and e1h = t

Dh !h

E (e0h) = E (e1h) = 0 1 1 .nWh--h

E(e2h)= [nwWh- —h] C

E(e0h) =

dh

E (e0he1h) = - pdihCihCdh.

E (eo) = Erh=1 »J^ 0h)= — ( > WhFh(y)E(eoh)) = 0

we will find the expected values of error terms as

^h=iWhFh(y)eoA 1 iy, F(y) ) F(y)(Y

Similarly, E (eo) = E (ei) = 0

E (eg) = E (^yhe°h)2 = -l2ShK=iWh2DhE(e2oh)

= "52£h=i Wh2üh - ¿I Cgh = D £h=i Wh2 Dh - Cgh

= ^[n-N]^WhS2h= vd (say) Similarly,

K

E (e2) = -^i1-1! Y Y WhSi2h = vi

h=1

E (eoei)= ¡^ [1 -1] Zh=i WhSdih = Vm (1)

II. Properties of Estimators in Conditional Post-Stratification

K

e i(e2) = (- - -1) S2h = V1D(say)

D2 f-< Vnh N^

h=1 K

E i(e2) = ^rYwh (- - S?h = V I2 ^ Vnh

h=1

G.R.V. Triveni and Faizan Danish RT&A, No 4 (76) LEVERAGING AUXILIARY VARIABLES: ADVANCING MEAN_Volume 18, December 2023

E 1(eoe1) = ¿SU Wh - Sdih = V1D1 (2)

III. Estimators in Literature We write the following estimators in terms of Unconditional case in post-stratification as a. The usual unbiased estimator of population mean D = 2h=1 Wh Dh is given by

Ui = dpS = £K=iWhdh (3)

Using the results from Stephen (1945), the variances of dps to the first degree of approximation is given by

For Unconditional post-stratification,

Var(uia) = [1 -1] 2h=i 2 WhS2h + £2h=i(1 - Wh) S2h (4)

ps-

Where Var(u1a) is the Unconditional variance of post stratified estimator dp and S2h = i^S1N=hi(dhi - Dh)2 For Conditional post- stratification,

Var(uib) = ^^=iWh2^^-^]s2h (5)

Var(u1b) is the Conditional variance of post stratified estimator dps.

b. The Ratio estimator for population mean according to Naik and Gupta [2] is given by

U2 = dPS ("i Where TpS = Eh=i Wh"h

Up to the first degree of approximation, the MSE of estimator u2 is given by

MSE (u2) = D2

Vd + V[ [1 - 2 ("V"1)]

■V, .

c. The usual product estimator is given by

u3 = dps (y

(6)

Up to the first degree of approximation, the MSE of estimator u3 is given by

MSE (u3) = D2

Vd + V, [1 + 2 (VD,"-)]

(7)

d. The Usual regression estimator for D is given by

u4 = dps +bps(^-Tps)

The MSE of estimator u4 is given by

MSE K) = D2VD(1-?D,) (8)

„2

Where & =

D[ vdv,

e. Koyuncu [7] proposed a class of estimators as

u5 = [r1dps + r2 0-ips)](-7

\aps"p

Its MSE is given by

MSE (U5) = D2[1 + rlAi + r2A2 + 2rir2A3 - 2r^ - 2r2A5] (9)

Where Ai = 1 + vd + 9pSVi ^ - 4 (VD1)] V,

A2 d2

R

V, / VDi

A3 R (2^ps V,

A4 = 1 + 9pSVi(9pS-VDi

A fM D V

A = (—) ^ , R=- , ^ = ■— 5 VR/ i ^ps

A2A4-A3A5 , AiA5-A3A4

ri = 2 4 V and r2 = ——^r4.

1 AiAr-Aij 2 AiAr-A3

f. Sharma and Singh [9] proposed exponential type estimators as Ratio type exponential estimator

u6a = dps exp I

V + "ps

Product type exponential estimator as

j Aps

u6b = dps exp l -

, "ps + 1

/a(1 - "ps)

u6c = dps exp I -f—^-

\ 1 + "ps

Where a being a suitable constant The MSEs of the above estimators as

MSE (U6a)= d2{vd+V1[i-4(Vd1)]} MSE (u6b) = D 2 {vd + -4- [l + 4 ("VJ1)]}

MSE (U6c) = D2{VD+aVi[a-4(VDi)]} (10)

Where a = 2 * (V^) g. Sharma and Singh [9] suggested a class of estimators as

U' = [ridps + r2(1"Tp-)]eXp(ap!ff+(Tp^i)2bp!)

Where aps, bpsare either real numbers or the functions of the auxiliary variable. Its MSE is given by

MSE (u7) = D2 [1 + rlB1 + r^B2 + 2r1r2B3 - 2r1B4 - 2r2B5] (11)

Where Bi = 1 + vd + 9pSV: [9pS - 2 (VDl)]

B2 =R?

B =V±( -VD"

3 R (9ps V

V, / Vd,

B5 —(*)<p

5 V2R/ ps

_ B2B4-B3B5 ^ _ B1B5-B3B4

r — 9 and r? — 9 .

1 b1b2-b2 2 b1b2-b3

h. Singh et al. [13] suggested another class of estimators for population mean as

Uo =

5apS(I-ipS)

ridpS + Г2 exp '

aps1 + bps 4

apsips + bps

^apS(I + ipS) + 2bpS/ Where (S,n)are constants belongs to real numbers like (-1,0,1).

MSE (u8) — D2 [1 + r2C1 + r|C2 + 2r1r2C3 - 2r1C4 - 2r2C5] (12)

Where, C1 — 1 + Vd - 4WpSVM + n(2n + 1)vpsV,

1

C2 — —j[ 1+0(20 + 1)^]

1

C3 = — 3 IR

2 L J-+0(20 + i)Tps

(n + 0)(n + 0 + 1)

2 Tp^I v"' ' "^ps

1 + „ . . . . _^PsV - (n + 0)^psVci

n / (n +1) C4 = 1 + 9ps-i—9 V,-2Vd,

5 IR L 2 ^ps J

Q /TC -l\/o - C2C4-C3C5__C1C5-C3C4

9 = (2^dih - 1)/2 , r1 = r 4 r2 and r2 = r 5 r2 .

C1C2-C3 c1c2-c3

We have written the above considered pre-existing estimators in Unconditional case. If we change the expectations of error terms like in equation (2), we get the estimators in Conditional case. IV. Suggested class of estimators in post stratification

We propose a class of estimators for population mean D as

Uprop

r1dps + r2 (l - ips)

r3dps. -

3 ps \ я 1 _|_h

I-ip exp ( —

1 I+ip

(13)

Where (r1, r2, n) are suitable constants and (aps, bps) are either constants or functions of auxiliary variable.

Expressing the equation (13) in terms of e0h and e1h, we have

n

uprop = [ri WhDh (1 + eoh)

+ JV Wh1h-y Wh1h(l + eih))- r-Y WhDh(l + eoh)(1 <h=i <h=i / <h=i

/ ¿-h=i vvh'h - Z,h=i "^'h^ + eih

. e< """ -

ps

+ 9psei)" | exp(EK=iWh1h + EK=iWh1h(i + eih)

Where Vps = rr+r

ps ps ~ ps

=[riD(i + eo) + rr (I- 1(i + ei)) - r-D(i + eo)(i + ^i)-1] exp (1-l(i+ei))n

riD(i + eo) - rrTei - r-D(i + eo)(i - + <e2] exp

riD(i + eo) - r21 ei - r3D(i + eo - 9psei - 9pse0ei + Vpsei)] [i - n ei + 3^e2]

Uprop ~ D =D {[(ri - i) + rieo - r2mei - r3 (i + eo - 9psei - Vpseoei + Vpse2)] [i - iei + 3^e2]}

(14)

Where m = ^

- D = D {(ri - i) + eo(ri - r3) - r3 + ei (-r2m + ----

Uprop - D = D{(ri - i) + eo(ri - r3) - r3 + ei (-12'

n(ri - i) n'-

ps 2 2

. 2 i3n(ri- ^ , /mm 2 n^ps 3n\

+ e (—8— + r2(_2) r3^ps r3 2 r3 "8/

n n

+ (-r^ + r-^ps + r3:

soei (~ri 2 + r3^ps + r3 3)} By taking expectation on both sides of equation (14), we get bias as

Bias (Up„p)- D {(r, - i) - r3 + V, p^ + rr (=) - r3 (< + ^ + -?)] + V„ (~r. 2 + r-^ +

'3-3)}

By taking square on both sides of equation (14), we have

(uprop - D)2 = D2 {(('i - i) - r-)2 + e2(ri - r-)2 + ei (^m - r-^ + ^^ l-)2 ~

2eoei [('i - r-) (r2m - '-9ps + + 3'-)] + 2(('i - i) + '-) [eo ('i - '-) - ei (r2m - '-9ps +

^ _ ?)] + e2 K-^ + '2 (=,) _ ^ _ '3 _ '- -3)] + eoei (_ri r + ^ + '- -)}

(15)

By considering expectation on both sides of equation (15), we get MSE as

MSE (Uprop)- D2 {(('i -i) - r-)2 + vd('I - r-)2 + V, ('rm - r-^ + ^ - f)2 - 2VM |(ri -

. ( , n('i-i) nr-\l . „ rz-ncri-i) /mn\ 2 n^ps , „ ( 3 ,

r-) (r2m - r-9ps + -2--T-)] + V,[(—¡T" + '2 ("2") - r-^ps -'-— ~'- -3)] + vd, (-'i 2 +

'-^ps + '- -

- D2 { ('i - i)2 + ( r-)2 - 2r-('i - i) + vd('i - r-)2 + V, ('rm - r-^ + ^ - f)2 -

n,r , / , 3(ri~i) nr-\l , „ lY-n(ri-i) , /m^ 2 n^ps -nM ,

2vd, |(ri ~ r-) (r2m ~ r-^ps +——T)] + [+ '2 (mrj ~ r-^ps~r-— ~'- -n)] +

Vd., 12

(-ri2 + r3Vps+r3 3)} We rewrite the above equation as

MSE (Uprop)= D2 [l - 3П V. + Sir! + -02r? + -ЭзГ2 + «4^ + «5^ + ^2 + «7^2 + Vi^ + 89^3]

(16)

Where «1 — -2 + ^V, + nVM - 7 V, - ^d, n2

«2 —1+Vd + 7 V, - nVD, mn

«3 — -m nV,+—V, «4 — m2V,

«5 — 2 + n2 V, + 9pS nV, - ^V, - n^pSV, - ^V, + nVD, + 9pVM + n Vdi «6 —1+Vd + 9psV, + n2V, + 9psV, - 29pVra - nVD, «7 — m nV, - 2mVDI

n2

«8 — -2 - Vd - y V, - 9ps nV, + 2^psVDi + 2nVM «9 — -2mm V, - mnV, + 2mVDi

To get the values of r1, r2 and r3, differentiate equation (16) with respect to r1, r2 and r3 and equate them to zero. We get,

_ (2«2«3-«1«7)(2«4«8-«7«9)-(«5«7-«3«8)(«72-4«2«4) Q , . 3 («7«8-2«2«9)(2«4«8-«7«9)-(«72-4«2«4)(«8«9-2«6«7) 1( y) _ (2«2«3-«1«7)-01(«7«8-2«2«9) Q / , r2 («72-4«2«4) 2 (say)

1 2fl2

V. Efficiency comparison

Theoretically, we establish the following criteria to assess the effectiveness of suggested estimator and those estimators taken into consideration in the literature. By comparing equations (4) and (16), we have MSE (Upr0p) - MSE(uia)< 0

D2 [1 - ^V, + «1r1 + «2r2 + «3r2 + «4r2 + «5r3 + «6r3 + «7^2 + «8r1r3 + «9r2r3] < [± 1] ZK=i Z WhS2h + ¿ZK=i(1 - Wh) s2h

By comparing equations (5) and (16), MSE (upr0p) - MSE(uXb)< 0

D2 [l - 3nV. + 0iri + «2^ + 8ЗГ2 + 04r2 + 05ГЗ + «6^ + «7^2 + 8^3 + 89Г2ГЗ] < ^W^-NJ^ By comparing equations (6) and (16), MSE (uprop)- MSE (u2)< 0

[i - 3?Vi + «Л + 02r? + 8зГ2 + 04r22 + flsrs + flerl + 0yrir2 + 8^3 + 89r2r3] <

V +

By comparing equations (7) and (16), MSE (Uprop)- MSE (u-) < 0

[i - -2 V, + Vi + Sr'2 + S-'r + ^'2 + -05'- + V2 + Syri'r + Ss'i'- + Vr'-^

V+

v.[i+r(VD1)]

By comparing equations (8) and (16), MSE (uprop) - MSE (u4) < 0

[i — —I + Si'i + 02'2 + S3'2 + 04'2 + S5'3 + S6'2 + S7'i'2 + S8'1'3 + S9'2'3]<Vd(1-?d,) By comparing equations (9) and (16), MSE (Uprop)- MSE (U5) < 0

[i ~3tVi + Si'i + 02'? + 03'2 + 04'! + 05'3 + 06'l + 07'i'2 + 08'i'3 + S^^U + r2Ai + ^2 + 2'i'2A- - 2riA4 - 2^5]

By comparing equations (10) and (16), we have MSE (Uprop)- MSE (U6a)< 0

[i - -n V, + 0i'i + 02'2 + 0-'2 + 04'2 + 05'- + 06'- + 07ri'2 + 08'ir- + 09'2r-]<{vD +

^4®]}

MSE (Uprop) < MSE (U6b)

[i ~3rVl + 0i'i + 02'2 + 03'2 + 04'2 + 05'3 + 06'2 + 07'i'2 + 08'i'3 + 09'2'3]<{vd + MSE (Uprop) - MSE (U6c) < o

[i — -n V, + 0i'i + 02'2 + 0-'2 + 04'2 + 05'- + 06'- + 07'i'2 + 08'ir- + 09'2r-]<{vD +

2?[« —4(^)]}

By comparing equations (11) and (16), MSE (Uprop) - MSE (u7) < o

[i ~3rVl + 0i'i + 02'l + 03'2 + 04'2 + 05'3 + 06'2 + 07'i'2 + 08'i'3 + 09'2'3]< [i + + '2^2 + 2'i'2B3 ~ 2riB4 - 2r2Bs]

By comparing equations (12) and (16), MSE (Uprop) - MSE (u8) < o

[i - -n V, + 0i'i + 02'i + 0-'2 + 04'2 + 05'- + 06'2 + 07ri'2 + 08'ir- + 09r2r-]<[i + '2C + 'fo + r'i'rC- - 2riC4 - 2rrC5]

VI. Empirical study

We use information from the Ministry of Education of the Turkish Republic from 2007 on the number of teachers as the study variable (d) and the number of students classifying more or less than 750 in primary and secondary schools as the auxiliary attribute (i) for 923 districts across 6 regions (as 1: Marmara) 2, Atlantic, 3, Mediterranean, and 4, Central Anatolia Black Sea 5 and 6: East and Southeast Anatolia). Table 1 provides the data's summary statistics. We used Neyman allocation to place the samples in different strata.

The functions of auxiliary variable which we used in numerical calculation are: £ Wh Cih = o.266448, XWh Sih =0.246447 and pdih = o.i458—

In the case of unconditional post stratification, Table.2 shows the MSE values for our suggested estimator and the other estimators that were taken into consideration, together with the PRE values. It has been noted that the proposed estimator exhibits the maximum relative efficiency. It is also same in case of Conditional post stratification by observing Table.3.

Stratum no. Nh nh Dh rh ^dh ^ih ^dih P2(ih)

1 127 31 703.74 0.952 883.835 0.213 25.267 16.922

2 117 21 413 0.974 644.922 0.159 9.982 35.579

3 103 29 573.17 0.932 1033.467 0.253 37.453 10.34

4 170 38 424.66 0.888 810.585 0.316 44.625 4.231

5 205 22 267.03 0.912 403.654 0.284 21.04 6.675

6 201 39 393.84 0.95 711.723 0.218 18.66 15.56

Table 2. Unconditional case: MSE and PRE values of existing estimators and proposed estimator.

S. No. Estimator MSE value Percentage Relative Efficiency

1. Ula 2539.82 100

2. U2 2400.58 105.80

3. U3 2629.67 96.58

4. U4 2398.50 105.89

5. U5 2397.75 105.92

6. U6a 2405.89 105.57

7. U66 2520.44 100.77

8. U6c 2398.50 105.89

9. U7 2397.75 105.93

10. U8 1931.21 131.51

11. Uprop 1714.78 148.11

Table 3. Conditional case: MSE and PRE values of existing estimators and proposed estimator.

S. No. Estimator MSE value Percentage Relative Efficiency

1. Uib 2229.27 100

2. U2 1638.65 136.10

3. U3 2983.40 77.32

4. U4 846.88 263.23

5. U5 846.78 263.26

6. U6a 1913.52 116.50

7. U66 2585.89 86.21

8. U6c 2204.72 101.11

9. U7 846.78 263.26

10. U8 519.11 429.44

11. Uprop 68.54 3252.51

We may conclude that conditional post stratification outperformed unconditional post stratification by comparing MSE and PRE values in Tables.2 and 3.

MSE values of unconditional and conditional post-stratification

for different estimators

3500 3000 2500 2000 1500 1000 500 0

6

Estimators

10

11

Unconditional post-stratification > Conditional post-stratification

Figure 1. MSE values for both Unconditional and Conditional Post-stratified estimators.

We have presented the estimators in Table 2 and 3 in the above figure graphically. A line that represents the MSE values of Unconditional post stratified estimators can be seen in the picture together with a line with markers that represents the MSE values of conditional post stratified estimators. According to the graphic, conditional post stratified estimators have lower MSE values than unconditional post stratified estimators.

VII. Conclusion

1

2

3

4

5

7

8

9

In this research paper, we introduced a novel class of estimators and derived their Mean Square Error (MSE). We also investigated existing estimators and considered two cases in poststratification: conditional and unconditional. Through a real data analysis, we computed the MSE and Percentage Relative Efficiency (PRE) values for all estimators presented in this study. The results, as shown in Table 2 and 3, clearly demonstrate that our proposed estimator exhibits the highest relative efficiency compared to the other estimators considered. Furthermore, we observed from the figure 1 that conditional post-stratification outperformed unconditional post-stratification in our analysis. These findings highlight the potential of our proposed estimators for enhancing mean estimation accuracy in post-stratification studies.

References

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