IMPROVING VARIANCE PRECISION IN POPULATION STUDIES: THE ROLE OF POST-STRATIFICATION AND AUXILIARY DATA Текст научной статьи по специальности «Математика»

IMPROVING VARIANCE PRECISION IN POPULATION STUDIES: THE ROLE OF POST-STRATIFICATION AND

AUXILIARY DATA

M. I. Khan1, S. Qurat U1 Ain2, M. Younis Shah3*

!Department of Mathematics, Faculty of Science, Islamic University of Madinah,

Madinah 42351, Saudi Arabia 2Department of data analytics, Harrisburg University of science and technology 3* Division of Statistics & Computer Science, 180009 Jammu, Jammu and Kashmir,

India.

1 [email protected], [email protected], [email protected]

Abstract

In this study, we propose an enhanced estimator for the finite population variance in the context of post-stratified sampling, incorporating an auxiliary variable to improve accuracy. We derive expressions for the bias and mean square error (MSE) of the proposed estimator, providing an approximation accurate up to the first order. The theoretical analysis highlights the conditions under which the proposed estimator yields lower bias and reduced MSE, making it a more efficient alternative to traditional methods. To evaluate the practical performance of this estimator, we apply it to two real-world data sets, where our results demonstrate a marked improvement in efficiency over existing estimators. The numerical findings confirm that, in post-stratified sampling, the proposed estimator significantly enhances the precision of variance estimation, especially when the auxiliary variable is highly correlated with the study variable. This work not only contributes a more efficient estimator but also provides valuable insights into the application of auxiliary information in post-stratified sampling designs.

Keywords: Post stratification, Auxiliary variable, Estimation, Population variance and Mean squared error.

I. Introduction

This paper presents an enhanced estimator for population variance under post-stratification by utilizing auxiliary information. The use of auxiliary information in survey sampling has long been recognized for its ability to improve the efficiency of estimators for various population parameters, such as the mean, variance, median, mode, quartiles, interquartile range, percentiles, coefficient of variation, and proportions. Numerous methods for effectively incorporating auxiliary information have been extensively documented in survey sampling literature. Estimators of the ratio, product, and regression types leverage the correlation between the study variable and the auxiliary variable to achieve better precision. In this study, the mean square error (MSE) and bias of the proposed estimator are derived under large sample conditions, accurate to the first order of approximation. Theoretical comparisons with existing estimators are made, and conditions are established under which the proposed estimator is more efficient than those previously developed.

M. I. Khan, S. Qurat U1 Ain, M. Younis Shah IMPROVING VARIANCE PRECISION IN POPULATION

The application of stratified random sampling (STRS) presumes that the sizes and structure of sampling frames for each stratum are already available. Whereas the total population size and the percentage of the unit that belongs to each stratum may be known in many existing system, it is possible that the sample frame for every stratum is neither available or would be costly and difficult to construct. In social surveys relevant census information, where it is necessary to partition the heterogeneous population into different sub-groups, the sampling frame may not be available. In such types of situations, STRS is not applicable as such. In order to resolve these difficulties, post stratification technique is applied, in which a sample of necessary size is first selected from the population employing simple random sampling with or without replacement, and it is then stratified using the stratification variable

The procedure is identical to the one of stratified sampling and the only difference is that the allocation into strata is made ex-post. The gain in precision is related to the sample size in each stratum and (inversely) to the difference between the sample weights and the population ones. The standard error for the post-stratified mean estimator is larger than the stratified sampling one, because additional variability is given by the fact that the sample stratum sizes are themselves the outcome of a random process.

Initially introduced the concept of post-stratification [1]. Later extended this work by [2] investigating [3] classic ratio estimator in the context of post-stratification. They first considered the sample sizes within each stratum as fixed and then accounted for variations across possible stratum sample sizes, drawing on a result from [4]. Several researchers have made notable contributions to the development of post-stratification techniques. Important groundwork in the field laid by [5], followed by [6], who further advanced the methodology. Significant strides in [7] refining the theoretical foundations, while key modifications that improved estimator efficiency introduced by [8]. The scope of post-stratification by applying it to new contexts expanded by [9], and [10] provided valuable insights, enhancing the understanding and application of these methods in finite population estimation. More recently, characterized post-stratification product and ratio-type exponential estimators by [11]. While the regression estimator has generally been shown to outperform ratio and product estimators, this is not the case when the regression line of the primary variable on the auxiliary variable passes through a region near the origin [12].

In the literature, seldom is known about estimation of population variance under post-stratified sampling. A number of estimators for the limited population variance of the post-stratified sample mean utilizing data from the auxiliary variable developed by [13], a new ratio estimators in stratified random sampling using the information of an auxiliary attributes suggested by [14], An exponential estimator in the stratified random sampling taking an auxiliary attribute proposed by [15], An efficient exponential ratio estimator allows estimating the population mean in stratified random sampling using an auxiliary variables developed by [16], memory-type ratio and product estimators for the estimation of population variance based on exponentially weighted moving averages (EWMA) statistic proposed by [17], the generalized estimator of population mean using auxiliary attributes in stratified two- phase sampling introduced by [18], the estimation of rare and clustered population mean using stratified adaptive clustered sampling proposed by [19]. The difficulty of estimating the population mean in the situation of post-stratification discussed by [20].

II. Methods

Let the population of size N that is finite and partitioned into L strata of sizes N1,N2, ■■■,Nl such that Nh = N. Simple random sampling without replacement (SRSWOR) is used to select a sample of size n from the whole population. Following the method of selection from the population, the number of units falling under the hth stratum is indicated. Let nh be the size of the

M. I. Khan, S. Qurat U1 Ain, M. Younis Shah v . ^^M^Vnrs IMPROVING VARIANCE PRECISION IN POPULATION..._Volume 20, March 2025

sample falling in the hth stratum such that "h=i nh = n, here, it is expected that n is large enough

so that the probability of nh being zero is too low. Let yhi and xhi are the observed values of y and x

respectively on the ith unit of the hth stratum. Let yh = — "^yhi and xh = — xhi represent the

sample means corresponding to the population means Yh = -^-"¡^yhi and Xh = °f the

study variable (y) and auxiliary variable (x) respectively in the hth stratum.

Let syh = r^T^iC^i -yh)2 andsxh = -^—I^^iXhi - *h)2represent the sample variances

nh 1 nh 1

1 VNh ____ 1 v^fr

corresponding to the population variances = —— X^Oju - Yh)2andS%h = ——'Zi=1(.xhi - Xh)2 of the study variable (x) and auxiliary variable (x) respectively

—, and cxh = — represent the sample coefficient of variation corresponding to the population

yh xh

coefficient of variation Cyh = and Cxh = respectively. Let ryxh = Syxh represent the

Yh xh syhsxh

sample correlation coefficient corresponding to the population correlation coefficient pyxh = s between their respective subscripts in the hth stratum.

Further syxh = - Vh)(xhi - xh) represents the sample

nh-l

covariance

corresponding to population variance Syxh = ^ 1 ^ X^Cyhi — ^h)(xhi — ) , the study variable (y) and auxiliary variable (x).

s2

' y

nh r— — - - nh

sample variances of y, where wh = -jp is the hth stratum weight respectively. Ignoring the finite population correction factor to make calculations easier in post-stratified sampling

5pst -'Wtf

nh

h=l n

Nh

№rsh 1 - -

where Xrsth = ——— and ^rsth = Y~—ll.l(yhi — Yh^ (Xhi — Xfl)S ^200h^020h h i = 1

A

post-stratified regression estimator @pSt(reg) ^^^ the finite population variance of the post-stratified sample mean utilizing data from the study's auxiliary variables developed [13] as:

Gpst(reg) = ^ ~~ [(.Syh + "^220h($xh — sxh)] h=i h

Where W22oh is the sample regression coefficient of y on x with corresponding population

Let OpSt = Vh=iwh~ and êpst = Vh=iwh~ represent the post-stratified population and

The variance of the estimator ôpSt is provided as

ZL Wh

ttwoh- 1) (1)

regression coefficient W22oh = v2h( 220h—) in hth stratum. The variance of ôpSt(reg) is given by

L 4

1/ (-2 \ = Wh C4 o (^220h - 1) , .

VarKapst(reg)J = - 1) - JJ ^-V)

A conventional ratio estimator &pSt(rat) f°r the finite population variance of the post-stratified sample mean utilizing data from the study's auxiliary variables developed by [13] as: = _ S2

Jpst (rat)

h , Uh h=l

=i^a (4)

The Bias of âpSt(rat) up to the first order of approximation, is given as

ZWh

(^040h — 1) — 0^220h — 1)

w, nh

The MSE of 0pSt(rat) UP to the first order of approximation, is given by

L

'W,

h=l

>2

—^Syh(A400h — 1) + (A04O h — 1) — 2(A22O h — 1) (5)

h= 1 h

The new efficient type estimator of population variance developed [21] as follows: - Y^CZ [9 (s2Aa (8(s2xh-S2xh

■p^-Z, nhSyh 2 [s?J eXP\

h=1

<?2

(6)

)1

'xhj I Sxh — ^xh J

Where a and S are unknown constants whose values are to be determined such that the MSE of the proposed estimator is minimum.

The Bias of GpSt(aS) UP to the first order of approximation, is given as

wY^2 2 (2g + ^)(2g + S-2) r (2a + 5)

Bias[apst(a,S)) - "^rSyh 8 (^o4oh—1) 2 (^22oh-1)

h=l h

The MSE of GpSt(aS) UP to the first order of approximation, is given as

L (^ i)2

MSE(°pst(reg)) - ^ ~~^TSyh(^400h - 1) - (¿^ — 1) (7)

A new ratio type estimator under post-stratified sampling developed [20] as follows:

-2 - Y 2

apst(k) - / j nh Syh

h= 1

k + (1 — k)exp

C2 — 92 c2 — o2

°xh ^xh

(8)

The Bias of fJ^st(k) up to the first order of approximation, is given by

. _ . V^ft2 9 3 1

Bias{$£st(k)) - —-S£h8(A04Oh — 1)(1 — k) — ^(^220h — 1)(1 — k)

h=l h

The MSE of (iipSt(fc)) up to the first order of approximation, is given by

L 4

MSE(a^st(reg)) - £ ^S$h(Xwoh — 1) — 2 (^220h — 1) (9)

h=1

n^" ' (^040, — 1)

III. Proposed Estimator

In the line with the direction of study carried out by [22], we proposed an improved estimator for estimating the finite population variance under the method of post stratified sampling. The bias and MSE of the existing and proposed estimator are derived up to the first order of approximation. The performance of the proposed estimator is the best as compared to existing counterparts in terms of efficiency.

ZL W2 iS2 — s2 \

~r [KihSyh + K2h (Slh — s2h)] exp ( -f—-f -) (10)

h=1 nh \Jxh~ xh/

Where Klh and K2h are unknown constants whose values are to be determined such that the MSE of the proposed estimator is minimum.

For examining the large sample characteristics of the developed estimator, &pst(YS) we define the random variables up to the first order of approximation as:

syh - syh (1 + £2h)- s2xh - S%h (1 + £2h) such that E(£2h) - E(£2h) - 0

Also,

E(e2h) - - 1)' £(£4fi) - ~(^040h - 1) and E(£2h£4h) - — (Ä200h - 1)

1 2 _ 1

nh nh When we use the values of above terms in eq. (10), we have

1

nh

Jpst(YS)

- E-K-

Lu nh LV

h=1

[KlhS^h(1 + £2h) — K2hslh

' ^xh (1 + £4h)j\exp.2 c2

\2 öxh

$xh (1 + £4h)

\.Sxh — Sxh (1 + £4h)

Now expanding the right hand side along with the exponential term of the above equation up to the first degree of approximation, we

Jpst(YS)

L 1 3

^^M{_^lhSyh(1 + £2h) - K2h$xh - Sxh (1 + £4h))] - ^ e4h + 3 £4h) h=1 h

Subtracting Syh from both sidesof(ll),we obtain

(11)

Jpst(YS)

L ?

2, _ ^ Wrf w 2 2 2 1 2

Syh) _ 2_, + + KlhSyhE2h — ^^lh^yhE4h

h=1 h

' ^2h^'xhe4h

;K1hSyhe2he4h + g KlhSyh£4h 1 K2hSxh£4h S2^

By applying expectation on both sides of eq. (12), we get the Bias of ^

pst(YS)

as

(12)

Bias{a,

2

pst(YS)

Syh

) -

h=1 n

+ Klh$yh [1 + g (^040h — 1) - 2(1330h - 1) +1 ^2hS%h Wo40h — 1)|j

(13)

Equation (12) can be squared on both sides we get,

MSE(c

pst(YS)

) -1

nl

C^ 1 TS2 c4 I 1/2 c4 _f 1

^lhöyh "r höyh„ (A400h nh

1)

1 2 4 1

4 ^lh^yfi ~ (^040h ' 4 ^h

1)

2 4 1 1 2 4 1 4

+ ^2 h$yh ~ (^040h — 1) +4 KihSyh ~ (^220h — 1) + h$yh nh 4 nh 16 1 1

— hSyh~~ (^220 h — 1) + q (^040h — 1) + K2hSyhSyh — (A040h — 1)

nh u nh nh

2 4 1 - 2 4 - 2 4 -

— KihSyh~~ (^220h — 1) + 3 KihSyh~~ (^040h — 1) + h$yh~ (^220h — 1) nh Ö

1

1

— 2KlhK2hSyhSyh~ (^220h — 1)

+ Klh^yh „ (^040h 1)

1

(14)

In the above equation, K^ and K2h are unknown constants whose values are to be determined such that the MSE of the proposed estimator is minimum and their optimal values are obtained by differentiating partially equation (14) with respect to K^ and K2h and then equating to zero as:

dMSE(t(YS)/nh)

dK-

1 h

K-

lhopt

= Mo4Q h —

= V 8 )

8 — (^040 h — ^

(^040 h — ^ + (^040 h — 1)(^040 h — 1) — (^220 h — 1)2

dMSE(t(YS)/nh)

dK-

= 0

2 h

K-

2hopt

>yh 8$xh,

4(^220h — 1)2 — (^040h - 1)(^220h — D2 + (^040 h — 1)

0^040h — 1) + (^040 h — 1)(^040h — 1) — (^220h — 1)2

The reduced MSE of G2 pst(YS) the optimum values of K^ and K2h is obtained as

MSE{8-Pst(JS))min

7 64 {(A400h — 1)(A040h — 1J — (X220h — 1)2} — (A040h — 1)3

^4 —16(^040h — 1){(^040h — 1)(^040h — 1) — (^220h — 1)2}

yh 64{(A040h — 1) + Uo40h — 1)(^040h — 1) — U220h — 1)2} IV. Efficiency Comparison

=Kf

(15)

The developed estimator has been theoretically compared to the competing estimators of the population variance. As a result, when (1), (3), (5), (7), (9), (11) and (17) are compared, it is clear that the suggested estimator O2pst(YS)' would be more efficient than [13] usual unbiased estimatorO2Pst, [13] regression estimator O2pst(reg)> [13] regression estimator O2pst(rat)> [21] efficient type estimator &2pst(a,S) > [20] new ratio type estimator &2 pst(k)-1. By taking eq. (1) and (15), we get

[Var (&Xpst) — MSE (9-Xpst(YS))min} > 0

(16)

° Pst

*■ 1(3)

h=l x n '

c4 Jyh

(^400h l)

L

64 {(^400h — 1)(^040h — l) — (^220h — 1)2} — (^040h — 1)3 — 16(Ao4oh — 1){(^040h — 1)(^040h — 1) — (^220h — 1)2}

L

64{(lo40h — 1) + tt

040h

1)(l

040h

1) — (X

XX0h

1)2}

> 0

Jj

2. By taking eq. (3) and (15), we get

[var (a2Pst(reg)) — MSE (d2pst(YS)) } > 0

(17)

ZL (Wh\ 4

( JSyh

( — 1) — (*220h — 1)2 (A400h 1)

(A040h — 1)

L

64 {(^400h — 1)(^040h — 1) — (^220h — 1)2} — (^040h — 1)3

> 0

— 16(^040h — 1){(^040h — 1)(^040h — 1) — (^220h — 1)2}

64{(A040h — 0 + (^040h — 1)(^040h — 1) — (^220h — 1)2} .

L JJ

3. By taking eq. (5) and (15), we get

[Var (a2Pst(rat)) — MSE (tfpstps)) . } >0 (18)

a2 if ■ N I —1 c4

U p^t(ren) 1/ ■ /1 t I J-

Pst(reg) lJ ■ J Jyh

L

(^400h — 1) + (^040h — 1) — 2(^220h — 1)

64 {(^400h — 1)(^040h — 1) — (^220h — 1)2} — (^040h — 1)3 — 16(^040h — 1){(^040h — 1)(^040h — 1) — (^220h — 1)2} 64{(A040h — 0 + (^040h — 1)(^040h — 1) — (^220h — 1)2} .

L JJ

> 0

4. Bytakingeq.(7)and (15), weget

5. {Var (tfpst^g)) — MSE (tfpstps)) . } >0 (19)

-2 -f V/^W

O Pst(a,S) lJ ■ \ ~ I

(^220h — 1)2

(^400h — 0 — '

(^040h 1)

L

64 {(^400h — 1)(^040h — 1) — (^220h — 1)2} — (^040h — 1)3

> 0

— 16(A040h — 11){(^040h — 1)(^040h — 1) — (^220h — 11)2}

64{(A040h — 1) + (^040h — 1)(^040h — 1) — (^220h — 1)2} .

L JJ

5. By taking eq. (9) and (15), we get

{Var (p2Pst(k)) — MSE (d2Pst(YS)) . } >0 (20)

-2 -f X"(Wh\c4 In -n o(A220h- 1)

° Pst(a,8) lJ : ^ \ I ' (Ä400h — 1) — 2

\nh / y C^040h — 1)

L

64 {(A400h - 1)(^040h - 1) - (^220h - 1)2} - (^040h - 1)3

-16(Â040h - 1){(A040h - 1)(^040h - 1) - (^220h - 1)2}

64{(^040h - 1) + (^040h - 1)(^040h- 1) - O*220h - 1)2}

L JJ

> 0

As the conditions (16)-(20) are always satisfied, it is inferred that the proposed estimator is more efficient than the other existing estimators under all cases in theory.

V. Empirical Study

5. Numerical Analysis

We will take into account the data set to assess the efficiency of the suggested estimators. The characteristics of the population are described as below: Population 1:

Let y is the output and x is the fixed capital of 80 factories [23]. The data have classified arbitrarily

into four strata as x < 500,500 < x < 1000, 1000 < x < 2000, and x>2000, respectively.

y: Output x: Fixed capital

Table 1(a): Statistical Description of the Population:

Constants Nh nh Yh xh o2 Syh ^400 h ^040 h ^220h

Stratum I 20 11 3006.55 65.90 572819.20 3.45 1.55 1.49

Stratum II 31 18 4687.62 141.90 433681.58 1.56 3.09 1.73

Stratum III 13 8 6496.23 392.38 162104.69 1.98 1.49 1.56

Stratum IV 16 8 7795.31 749.50 426528.63 2.35 1.91 2.05

Population 2:

we use the data concerning the number of teachers as study variable and the number of students as auxiliary variable in both primary and secondary schools for 923 districts at six regions (as 1: Marmara 2: Agean 3: Mediterranean 4: Central Anatolia 5: Black Sea 6: East and Southeast Anatolia) in Turkey (Source: Ministry of Education, Republic of Turkey). Y: number of teachers X: Number of students

Table 2(a): Statistical Description of the Population:

Constants Nh nh Yh xh syh ^400h ^040h

Stratum I 127 31 703.740 20804.59 781163.9 3.94783 6.251589 3.720488

Stratum II 117 21 413.000 9211.79 415924.8 17.33181 19.35622 18.35209

Stratum III 103 29 573.174 14309.30 1068054 15.87136 16.3073 16.09088

Stratum IV 170 38 424.664 9478.85 657047.8 13.60375 11.67999 11.65605

Stratum V 205 22 267.029 5569.94 162936.9 22.31908 23.14865 22.30021

Stratum VI 201 39 393.840 12997.59 506549 21.49882 24.26014 21.79386

Table 3: Conditional values of different estimators using real data sets:

Conditional values Population I Population II

Conditional values I 2964740.76 628422.27

Conditional values II 1651010.71 29677.75

Conditional values III 2308528.95 41662.86

Conditional values IV 1254126.31 29677.75

Conditional values V 1651010.71 224246.86

Conditional values VI 6579180.77 706578.08

Table 4: Bias values of different estimators using real data sets:

Estimators Population I Population II

a(pst) - -

2 °pst(Reg) - -

rr2 uPst(Rat) 248.85 111.00

2 aPst(a,S) -6154.30 -124519.90

aPst(T) -6392.45 -218847.51

2 aPst(k) -104.74 1024.01

aPst(YS) 247404838.9 4.92057E+19

Table 1(b): MSE and PRE of suggested estimator in relation to a2

No. 1 2 3 4 5 6 7

Estimators a(pst) 2 °pst(Reg) 2 <JPst(Rat) 2 aPst(T) 2 aPst(a,S) 2 aPst(k) 2 aPst(YS)

MSE 3574695.71 2260965.66 2918483.90 1864081.26 2260965.66 7189135.72 609954.95

PRE 100.00 157.40 121.28 191.77 157.40 49.72 586.06

Table 2(b): MSE and PRE of suggested estimator in relation to S2Pst

No. 1 2 3 4 5 6 7

Estimators a(pst) n2 upst(Reg) n2 uPst(Rat) 2 aPst(a,S) aPst(T) aPst(k) _2 "Pst(YS)

MSE 651146.08 52401.56 64386.67 52401.56 246970.67 729301.89 22723.81

PRE 100.00 1242.61 1011.31 1242.61 263.65 89.28 2865.48

Where

Var (&2Pst)

PRE =--- 2-^r- x 100 : i = Re, Rat, (a, S), T, and k.

MSE (a Pst(0)

VI. Conclusion

The population variance of the research variable can be effectively estimated using auxiliary data through an improved estimator under post-stratification. The proposed single and combined classes of estimators, such as bias and mean square error (MSE), are derived approximately to the first order of accuracy. Under specific efficiency conditions, the recommended estimator significantly outperforms existing separate and combined estimators. Additionally, empirical research is conducted using both artificially generated symmetric and asymmetric populations, as well as real-world data, to validate the theoretical findings. The results demonstrate that the proposed estimator is more efficient, with a lower MSE and higher percentage relative efficiency (PRE) than the alternatives. This study provides clear evidence supporting the robustness and practicality of the suggested estimator in experimental surveys. Given its superior performance, we strongly recommend its adoption over traditional estimators for post-stratification variance estimation. The integration of theoretical and empirical analyses makes this research highly credible, insightful, and impactful for statistical applications.

VII. Acknowledgement

The author wishes to extent his sincere gratitude to the Deanship of Scientific Research at the Islamic University of Madinah for support provided to the Post- Publication Program (3).

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M. I. Khan, S. Qurat U1 Ain, M. Younis Shah IMPROVING VARIANCE PRECISION IN POPULATION

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