Construction of Strata for a Model-Based Allocation Under a Superpopulation Model

Bhuwaneshwar Kumar Gupt; Md. Irphan Ahamed

doi:10.2991/jsta.d.210107.001

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Volume 20, Issue 1, March 2021, Pages 46 - 60

Construction of Strata for a Model-Based Allocation Under a Superpopulation Model

Authors

Bhuwaneshwar Kumar Gupt¹^{, *}, Md. Irphan Ahamed²

¹Department of Statistics, North-Eastern Hill University, Shillong, 793022, India

²Department of Mathematics, Umshyrpi College, Shillong, 793004, India

^*Corresponding author. Email: bhuwaneshwargupt@gmail.com

Corresponding Author

Bhuwaneshwar Kumar Gupt

Received 21 April 2019, Accepted 7 October 2020, Available Online 13 January 2021.

DOI: 10.2991/jsta.d.210107.001 How to use a DOI?
Keywords: Approximately optimum strata boundaries; Auxiliary variable; Optimum strata boundaries; Superpopulation models
Abstract: This paper considers the problem of optimum stratification for a model-based allocation under a superpopulation model. The equations giving optimum points of stratification have been derived and a few methods for finding approximately optimum points of stratification have been obtained from the equations. Numerical illustrations using generated data have been worked out and the proposed methods of stratification have been compared with equal interval stratification.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The classical work of Tschuprow [1] and Neyman [2] on allocation of sample size to strata opened a space for further research on allocation and stratification aspects in stratified sampling. However, initial works on problem of allocation of sample size to strata as well as optimum stratification were based on the values of study variable y itself. When the information on an auxiliary variable highly correlated with the study variable y is available, it was demonstrated by Cochran [3] that a superpopulation model could be constructed in which the finite population under study could be treated as a random sample from an infinite population (superpopulation). It could also be used for construction of strata and allocation of sample size to strata.

Hanurav [4] and Rao [5] started using auxiliary information for allocation of sample size to strata in which the following superpopulation model was considered.

iξyi|xi=α+βxiiivyi|xi=σ2xigiiiςyi,yj|xi,xj=0(1)

where α, β, σ2 and g were superpopulation parameters with σ2>0 and g≥0. The script letters ξ, v, ς denoted conditional expectation, variance and covariance given x's respectively.

Hanurav [4] studied the problem of allocation and obtained the allocation as nh∝Nhσh x for simple random sampling with replacement (SRSWR) within each stratum under particular case g=2 of the model (1). This allocation was obtained from Tschuprow–Neyman allocation when the unknown proportionate values of σh y2's were replaced by known proportionate values of σh x2's, which were the estimates of σh y2's. Rao [5] too examined analytically the justification for the assumption that the unknown proportionate values of σh y2's were not quite different from the proportionate values of known σh x2's. He proved that σh y2's could be expected to be in the same proportion as σh x2's, if the squares of the corrected coefficients of variation of x character, defined by σhx2Xh2−δhNh2, where δh=Nh∑j=1NhXh j2−∑j=1NhXh jg, are equal in all strata. He also obtained allocation which minimized the expected variance of strategy consisting πPS sampling scheme, and Narain [6] and Horvitz and Thomson [7] estimator under particular case of the model (1) with intercept α=0.

The problem of optimum allocation of sample size to strata for probability proportional to size with replacement (PPSWR) within each stratum under a particular case α=0 of the model (1) was considered by Gupt and Rao [8].

On the other hand, in stratified sampling, ever since Dalenius [9] had pioneered the work on optimum stratification based on estimation variable for Tschuprow–Neyman allocation, later workers have been extending the work in various perspectives and dimensions till date. Regarding finding of approximate solutions to the equations giving optimum strata boundaries (OSBs), Dalenius and Hodges [10] were the first ones who proposed cum f rule. In case of the problem of optimum stratification using auxiliary variable, the original main works, inter alia, were done by Dalenius and Gurney [11] and Taga [12] who considered it in the case of Tschuprow–Neyman and proportional allocations (PA) respectively. Subsequently, Singh and Sukhatme [13,14], Serfling [15], Singh and Parkash [16] and Singh [17–20] furthered the work on optimum stratification based on auxiliary variable for various allocation methods in which a number of methods of finding approximate solutions to all the equations giving OSB were also obtained.

Whenever an auxiliary variable highly correlated with the study variable is available, Singh and Sukhatme [13] stipulated a superpopulation model as follows —for which the form of regression of estimation variable y on the concomitant variable x and also the form of the variance function were known.

y=cx+e such that ξe|x=0 and ve|x=φx(2)

where cx and φx are real valued functions of x with φx>0, for all values of x in the range (a, b) with b−a<∞.

Singh and Sukhatme [13] derived OSB and approximately optimum strata boundaries (AOSBs) based on the auxiliary variable for Tschuprow–Neyman Optimum Allocation (TNOA) and PA under the superpopulation model (2) and empirically illustrated under model (1) for the particular case g=1.

Yadava and Singh [21] derived equations for OSB for allocation proportional to strata totals of an auxiliary variable and also developed a few methods of obtaining their approximate solutions.

Gupt [22,23] considered sample size allocation problem by modifying the above model (1) in such a way that the element of correlation among units within the same stratum was incorporated; he derived three model-based allocations by assuming some conditions for approximation.

In this paper, we consider problem of optimum stratification for the following allocation which is one of the three model-based allocations, viz.,

nhαNhμhxg(3)

provided θhg=σhxμhxg are equal in all strata.

The equations that give OSB for the allocation (3) for stratified SRSWR have been derived in Section 2. Moreover, these results will hold true for stratified simple random sampling without replacement (SRSWOR) design too when finite population correction is ignored in each stratum. The limiting lower bound of the variance of population mean when the number of strata tends to infinity has been shown in Section 3. The methods of obtaining AOSB to the equations that give OSB have been obtained in Section 4. Numerical illustrations by using generated populations have been worked out in Section 5. Conclusion is given in Section 6.

2. EQUATIONS GIVING OSB

The model-based allocation (3) can be expressed as

nh=nWhμhxg∑Whμhxg(4)

where Wh is the proportion of population units in the hth stratum, μhxg is the mean for xg in the hth stratum.

For the above expression (4), the variance of the estimate of the population mean can be obtained as

vy¯st=1n∑h=1LWhβ2σhx2+σ2μhxgμhxg∑h=1LWhμhxg.(5)

If fx is taken as the marginal density function for the stratification variable x,

We know that

Whμhxg=∫xh−1xhxgfxdx.(6)

2Whσhxδσhxδxh=xh−μhx2fxh−σhx2fxh.(7)

Taking into consideration of the superpopulation model (1) we get,

σh y2=β2σh x2+σ2μh(xg).(8)

The variance expression in (5) is partially differentiated with respect to xhh=1,2,…,L−1 and equated to zero to obtain minimum variance.

On differentiating (5) partially with respect to xh, we get

∑h=1LWhβ2σhx2+σ2μhxgμhxgδδxh∑h=1LWhμhxg+∑h=1LWhμhxgδδxh∑h = 1LWhβ2σh x2+σ2μhxgμhxg=0

⇒∑h=1LWhβ2σhx2+σ2μhxgμhxgWhδh∗δxh+h∗δWhδxh+Wiδi∗δxh+i∗δWiδxh+∑h=1LWhμhxgWhδhδxh+hδWhδxh+Wiδiδxh+iδWiδxh=0

where i∗=h∗+1,h∗=μhxg,i=h+1,h=β2σhx2+σ2μhxgμhxg.(9)

By using the definitions in (6), (7) and relation (8)

Whδh∗δxh+h∗δWhδxh+Wiδi∗δxh+i∗δWiδxh=μhxg+xhg2μhxgfxh+μixg+xhg2μixg−fxh(10)

and

Whδhδxh+hδWhδxh+Wiδiδxh+iδWiδxh=β2σhx2+σ2μhxgμhxg−xhg+2μhxgβ2xh−μhx2+σ2xhg2μixg3/2fxh+ β2σix2+σ2μixgμixg−xhg+2μixgβ2xh−μix2+σ2xhg2μixg3/2−fxh(11)

Then, finally using (10) and (11), the equations giving OSB for model-based allocation (3) can be obtained as

β2σhx2+σ2μhxgμhxg−xhg+2μhxgβ2xh−μhx2+σ2xhg2μhxg3/2∑h=1LWhμhxg+ μhxg+xhg2μhxg∑h=1LWhβ2σhx2+σ2μhxgμhxg=β2σix2+σ2μixgμixg−xhg+2μixgβ2xh−μix2+σ2xhg2μixg3/2∑h=1LWhμhxg+ μixg+xhg2μixg∑h=1LWhβ2σhx2+σ2μhxgμhxg.(12)

The above equations (12) give the OSB of the auxiliary variable x.

3. A FEW METHODS OF FINDING APPROXIMATE SOLUTIONS TO THE EQUATIONS GIVING OSB

In order to find the approximate solutions of the equations (12), we follow the techniques of Singh and Sukhatme [13] and Yadava and Singh [21]. For this purpose, it is required to assume the existence of partial derivatives of fx, cx and ψx, where we have defined ψx=xg, cx=α+βx. Then, we obtain series expansions of this system of equations (Singh and Sukhatme [13]) using the identities (Ekman [24,25]) and Taylor's series expansion, about the point xh which is assumed as the common boundary of hth and h+1th strata. Considering the right hand side of Equation (12) and all the derivatives used hereafter, in this paper, are evaluated at t=xh in the interval t∈xh,xh+1, we have

μiψ=ψ+ψ′2ki+ψ′f′+2fψ″12fki2+ff″ψ′+ff′ψ″+f2ψ‴−ψ′f′224f2ki3+Oki4.

⇒μixg−xhg=ψ′2ki+ψ′f′+2fψ″12fki2+ff″ψ′+ff′ψ″+f2ψ‴−ψ′f′224f2ki3+Oki4.

Now, we can get

β2σix2+σ2μixg=σ2ψ+σ2ψ′2ki+β2f+2σ2fψ″+σ2f′ψ′12fki2+σ2ff″ψ′+σ2ff′ψ″+σ2f2ψ‴−σ2f2ψ′24f2ki3+Oki4.

& β2σix2+σ2μixgμixg−xhg=σ2ψψ′2ki+σ2f′ψψ′+2σ2fψψ″+3σ2fψ′212fki2+ +2σ2f2ψ′ψ″+β2f2ψ′+2σ2f2ψ′ψ″+σ2ff′ψ′2σ2ff″ψψ′+σ2ff′ψψ″+σ2f2ψψ‴−σ2f′2ψψ′+σ2ff′ψ′224f2ki3+Oki4.(13)

Also,

2μixgβ2μix−xh2+σ2xhg=2σ2ψ2+σ2ψψ′ki+3β2fψ+σ2f′ψψ′+2σ2fψψ″6fki2+ +σ2ff′ψψ″+σ2f2ψψ‴−σ2f′2ψψ′3β2f2ψ′+2β2ff′ψ+σ2ff″ψψ′12f2ki3+Oki4(14)

Adding (13) and (14) we get,

β2σix2+σ2μixgμixg−xhg+2μixgβ2μix−xh2+σ2xhg=2σ2ψ2+3σ2ψψ′2ki+ +σ2fψ′2+2β2fψσ2f′ψψ′+2σ2fψψ″4fki2+ +2σ2ff′ψ′2+4σ2f2ψ′ψ″+7β2f2ψ′+4β2ff′ψ3σ2ff″ψψ′+3σ2ff′ψψ″+3σ2f2ψψ‴−3σ2f′2ψψ′24f2ki3+ Oki4(15)

Moreover

12μixg3/2=12ψ3/212 12121−3ψ′4ψki+15fψ′2−4f′ψψ′−8fψψ″32fψ2+ −8ff′ψ″ψ2−8f2ψ‴ψ2+8f′2ψ′ψ2)(20ff′ψψ′2+40f2ψψ′ψ″−35f2ψ′3−8ff″ψ′ψ2128f2ψ3ki3+Oki4.(16)

Multiplying Equations (15) and (16), we get,

β2σix2+σ2μixgμixg−xhg+2μixgβ2μix−xh2+σ2xhg2μixig3/2=σ2ψ+σ2ψ′2+8β2ψ32ψ3/2ki2+2σ2f′ψψ′2+4σ2fψψ′ψ″−8β2fψψ′+16β2f′ψ2−3σ2fψ′3192fψ5/2ki3+ Oki4.(17)

Similarly, we have derived

μixg+xhg2μixg=ψ+ψ′232ψ3/2ki2+2f′ψψ′2+4fψψ′ψ″−3fψ′3192fψ5/2ki3+Oki4(18)

Thus from (9), (17) and (18), we can again express the equations (12) as

g1f kh2−g1f′kh33+Okh4∑h=1LWhh∗+g2f kh2−g2f′kh33+Oki4∑h=1LWhh=g1f ki2+g1f′ki33+Oki4∑h=1LWhh∗+g2f ki2+g2f′ki33+Oki4∑h=1LWhh.(19)

Now, for the purpose of tackling (19), we prove the following lemma:

Lemma 3.1.

If xh,xh+1 are boundaries of the i^th stratum and ki=xh+1−xh, the expressions ∑h=1LWhh∗ and ∑h=1LWhh can, under the conditions of AOSB, be approximated to give fixed values for a given value of L- no of strata, where i=h+1.

Proof:

Using series expansions in powers of interval width ki for Wi, μiψ and σix2 by following certain techniques and strategies (Ekman [24,25], Singh and Sukhatme [13], Yadava and Singh [21]).

We have already got from above

μhxg=ψ1+ψ′4ψki+8fψψ″+4f′ψψ′−3fψ′296fψ2ki2+8ff″ψ2ψ′+8ff′ψ2ψ″+8f2ψ2ψ‴−8f′2ψ2ψ′−8f2ψψ′ψ″−4ff′ψψ′2+3f2ψ′3384f2ψ3ki3+Oki4.(20)

And

Wi=kif+ki22f′+ki36f″+ki424f‴+Oki5.(21)

Multiplying (20) and (21)

Wiμhxg=ψ12 1212kif+fψ′+2f′ψ4ψki2+8fψψ″+16f′ψψ′−3fψ′2+16f″ψ296ψ2ki3+ −10f′ψψ′2+3fψ′3+16f‴ψ324f″ψ2ψ′+24f′ψ2ψ″+8fψ2ψ‴−8fψψ′ψ″384ψ3ki4+Oki5(22)

Considering the expansion of the following term by Taylor's series

∫xhxh+1ψtftdt=ψfki+ψf′ki22!+ψf″ki33!+ψf‴ki44!+Oki5=ψkif+fψ′+2f′ψ4ψki2+2fψψ″+4f′ψ′ψ−fψ′2+4f″ψ224ψ2ki3 + 24f′ψ2ψ″+8fψ‴ψ2+24f″ψ2ψ′−12f′ψψ′2+16f‴ψ3−12fψψ′ψ″+6fψ′3384ψ3ki4+Oki5(23)

Subtracting (23) from (22)

Wiμixg−∫xhxh+1ψtftdt=ki296fψ′2ψψki+4fψψ′ψ″+2f′ψψ′2−3fψ′34ψ2ψki2+Oki3=ki296g1∗f ki+12ddxhg1∗f′ki2+Oki3, where g1∗=ψ′2ψ3/2(24)

The following lemma was proved by Singh and Sukhatme [13], by using Taylor's series expansion at the point t=y.

Lemma 3.2.

∫yxftdtλλ=kλfy+f′y2k+Ok2=kλ−1∫yxftdt1+Ok2

By using Lemma (3.2) in (24) and considering the fact that if we have large number of strata whose strata widths kh are small, the higher powers of kh in the expansion can be neglected. So, by neglecting the terms of order Om5, where m=supa,bkh, we get

Wiμixg−∫xhxh+1ψtftdt=196∫xhxh+1g1∗tft33.

By using cum g1∗tft or g1∗tft3 rule in manner in which Yadava and Singh [21] proceeded for obtaining AOSB - ∫xhxh+1g1∗tft3dt=1L∫abg1∗tft3dt. We get

∑h=1LWhμhxg=∫abψxfxdx+196L2∫abg1∗xfx3dx3.(25)

From (25), we have known that under the conditions of AOSB, ∑h=1LWhh∗ in (19) can be assumed as a fixed value for a given L.

In the same way, the expression ∑h=1LWhh in (19) can also be reduced to a fixed value, for a given L, as follows:

Wiβ2σix2+σ2μixgμixg=1ψσ2fψki+σ2fψ′+2σ2f′ψ4ki2+ +16σ2ff″ψ2−3σ2f2ψ′28β2f2ψ+8σ2f2ψψ″+16σ2ff′ψψ′96fψki3++8σ2f3ψ‴ψ2+3σ2f3ψ′3+16β2f2f′ψ2+16σ2f2f‴ψ3−8β2f3ψψ′−10σ2f2f′ψψ′2−8σ2f3ψψ′ψ″+24σ2f2f″ψ′ψ2+24σ2f2f′ψ″ψ2384f2ψ2ki4+Oki5.(26)

& ∫xhxh+1σ2ψtffdt=1ψσ2fψki+σ2fψ′+2σ2f′ψ4ki2+8σ2f2ψψ″+16σ2f′ fψψ′−4σ2f2ψ′2+16σ2ff″ψ296fψki3+ +16σ2f2f‴ψ3−12σ2f3ψψ′ψ″+6σ2f3f′ψ′324σ2f2f′ψ2ψ″+8σ2f3ψ‴ψ2+24σ2f2f″ψ′ψ2−12σ2f2f′ψψ′2384f2ψ2ki4+Oki5(27)

Subtracting (27) from (26)

Wiβ2σix2+σ2μixgμixg−∫xhxh+1σ2ψtftdt=8β2ψ+σ2ψ′2f96ψ3/2ki3+2σ2f′ψψ′2+4σ2fψψ′ψ″−3σ2fψ′3+16β2f′ψ2−8β2fψψ′384ψ5/2ki4+Oki5

Thus, in this case too, as obtained in (25), we get

∑h=1LWhβ2σhx2+σ2μhxgμhxg=∫abσ2ψxfxdx+196L2∫abg2∗xft3dt3(28)

where g2∗x=8β2ψx+σ2ψ′2xψ3/2x

Under the conditions of AOSB, the expression ∑h=1LWhh in (19) can also be assumed as a fixed value for a given value of L. Thus the proof of lemma is completed.

Now, for the sake of convenience, we put two constant quantities P and Q in place of ∑h=1LWhh∗ and ∑h=1LWhh in (19), and Equation (19) can be rewritten as

g1f kh2−g1f′kh33+Okh4P+g2fkh2−g2f′kh33+Okh4Q=g1f ki2+g1f′ki33+Oki4P+g2fki2+g2f′ki33+Oki4Q,

where g1t=σ2ψ′2t+8β2ψtψ3/2t,g2t=ψ′2tψ3/2t

⇒g3fkh2−g3f′kh33+Okh4=g3fki2+g3f′ki33+Oki4,

where g3t=Pg1t+Qg2t

⇒g3fkh21−g3f′g3fkh3+Okh2=g3fki21+g3f′g3fki3+Oki2

On raising power 3/2 and then applying binomial expansion

⇒g3f3/2kh31−kh2g3f′g3f+Okh2=g3f3/2ki31+ki2g3f′g3f+Oki2(29)

By using the Lemma 3.2 and on further simplification, the system of Equations (29) can be transformed into

kh2∫xh−1xhg3tftdt 1+Okh2=ki2∫xhxh+1g3tftdt 1+Oki2

⇒kh2∫xh−1xhg3tftdt=constant=c1(30)

The Equation (30) can again be easily proved to be equivalent to

∫xh−1xhg3tft3dt=c2(31)

where c2=1L∫abg3tft3 dt(32)

For evaluating the value of constant c2, we can use (32) and approximate solutions may be determined by fixing xh−1 and calculating upper boundaries.

The procedures used in this method of finding AOSB have finally yielded the following theorem.

Theorem 3.1.

If the function g3tft is bounded and first two derivatives for all x in (a, b) exist, for a given number of strata taking equal intervals on cumulative g3tft or g3tft3 yields approximately OSBs on the auxiliary variable.

4. LIMIT EXPRESSION FOR THE VARIANCE

The variance expression in (5) can further be reduced to a form that will give an insight into the pattern of reduction of the variance of the estimate y¯st with the increase in the number of strata. It is also shown that Vy¯st does not tend to zero when the number of strata tends to infinity. It is a sequel to the techniques used by Yadava and Singh [21].

In Lemma 3.1, it is already shown that

∑h=1LWhμhxg=A+1L2B, where

A=∫abψtft dt and B=196∫abg1∗tft3dt3.

And ∑Whβ2σhx2+σ2μhxgμhxg=C+DL2, where

C=σ2∫abψtftdt, D=1L2∫abg2∗tft3dt3.

Therefore,

Vy¯st=1nA+BL2 C+DL2.(33)

In the above expression (33), it is seen that as L→∞⇒Vy¯st→ACn. It shows that Vy¯st does not tend to zero when the number of strata tends to infinity, and hence the following theorem.

Theorem 4.1.

When the approximately OSBs are obtained by using cum g3tft3 rule, it is observed that limL→∞Vy¯st=ACn.

5. NUMERICAL ILLUSTRATION BY USING GENERATED DATA

For numerical investigation, the following three densities of x, which were not only used by Singh and Sukhatme [13] but also by most of the later workers who furthered researches in the area of problem of construction of strata, are used in this paper too. We calculate the solutions of the Equation (12), corresponding solutions of the approximation methods (31) and (32), and the sampling variances of the stratified sampling for equal interval stratification for L=2,3,4,5,6. In the case of exponential density of x, slight deviation from equal interval stratification is considered for L=6 to avoid some inconveniences that came upon in using the generated data. The relative efficiencies of the equations giving OSB and their corresponding a few methods of approximation with respect to equal interval stratification are separately shown in Tables 1–9 by taking g=1,g=1.5 and g=2 successively.

Rectangular fx=1, 1≤x≤2.
Right triangular fx=22−x, 1≤x≤2.
Exponential fx=e−x+1, 1≤x<∞.

No. of Strata (L)	Stratification by Using Equations (12)		Equal Interval Stratification		Relative Efficiency	Stratification by Using Methods of Approximation		Relative Efficiency
No. of Strata (L)	Points	nVY¯st	Points	nVY¯st		Points	nVY¯st
2	1.5170	0.02783	1.5	0.0277	99.68	1.5011	0.0283	97.87
3	1.3259,	0.01641	1.334,	0.0164	100.00	1.3203,	0.0163	100.61
	1.6712		1.667			1.6582
4	1.2564,	0.01278	1.25,	0.0128	99.84	1.2388,	0.0130	98.41
	1.4789,		1.5,			1.49009
	1.7343		1.75			1.7477
5	1.2124,	0.01202	1.20,	0.0132	109.58	1.1904,	0.0119	110.39
	1.3695,		1.40,			1.3861,
	1.5302,		1.60,			1.5865,
	1.7451		1.80			1.7912
6	1.1699,	0.00964	1.1667,	0.0088	90.87	1.1583,	0.0089	98.87
	1.3208,		1.3334,			1.32033,
	1.5200,		1.499,			1.4857,
	1.7233,		1.667,			1.6543,
	1.8767		1.8334			1.8257

Table 1

Uniform distribution, g = 1.

No. of Strata (L)	Stratification by Using Equations (12)		Equal Interval Stratification		Relative Efficiency	Stratification by Using Methods of Approximation		Relative Efficiency
No. of Strata (L)	Points	nVY¯st	Points	nVY¯st		Points	nVY¯st
2	1.5135	0.0279	1.5	0.0278	99.53	1.4800	0.0283	98.20
3	1.3418,	0.0163	1.334,	0.0164	100.68	1.3169,	0.0163	100.68
	1.6792		1.667			1.6557
4	1.2558,	0.0127	1.25,	0.0127	99.92	1.2361,	0.0129	98.53
	1.4785,		1.50,			1.4832,
	1.7337		1.75			1.7403
5	1.2121,	0.0120	1.20,	0.0131	109.26	1.1882,	0.0119	110.46
	1.3695,		1.40,			1.3832,
	1.5302,		1.60,			1.5853,
	1.7447		1.80			1.7940
6	1.1697,	0.0095	1.1667,	0.0086	90.52	1.1571,	0.0089	96.95
	1.3205,		1.334,			1.3183,
	1.5107,		1.499,			1.4854,
	1.7232,		1.6667,			1.6562,
	1.8774		1.8334			1.8305

Table 2

Uniform distribution, g = 1.5.

No. of Strata (L)	Stratification by Using Equations (12)		Equal Interval Stratification		Relative Efficiency	Stratification by Using Methods of Approximation		Relative Efficiency
No. of Strata (L)	Points	nVY¯st	Points	nVY¯st		Points	nVY¯st
2	1.5102	0.0280	1.5	0.0278	99.39	1.4734	0.0283	98.41
3	1.3408,	0.0167	1.334,	0.0164	98.26	1.3118,	0.0165	99.45
	1.6783		1.667			1.6536
4	1.2516,	0.0127	1.25,	0.0127	99.69	1.2300,	0.0128	98.60
	1.4762,		1.50,			1.4813,
	1.7334		1.75			1.7422
5	1.2119,	0.0119	1.20,	0.0130	109.59	1.1840,	0.0118	110.05
	1.3696,		1.40,			1.3774,
	1.5303,		1.60,			1.5853,
	1.7446		1.80			1.7955
6	1.1696,	0.0094	1.1667,	0.0084	90.05	1.1543,	0.0086	97.68
	1.3204,		1.334,			1.3163,
	1.5106,		1.50,			1.4841,
	1.7233,		1.6667,			1.6570,
	1.8784		1.8334			1.8323

Table 3

Uniform distribution, g = 2.

No. of Strata (L)	Stratification by Using Equations (12)		Equal Interval Stratification		Relative Efficiency	Stratification by Using Methods of Approximation		Relative Efficiency
No. of Strata (L)	Points	nVY¯st	Points	nVY¯st		Points	nVY¯st
2	1.3794	0.0226	1.5	0.0278	122.98	1.3926	0.0226	123.04
3	1.2754,	0.0131	1.334,	0.0138	105.11	1.2529,	0.0135	102.15
	1.5898		1.667			1.5504
4	1.1915,	0.0104	1.25,	0.0121	117.29	1.1864,	0.0103	117.64
	1.4006,		1.50,			1.3944,
	1.6349		1.75			1.6373
5	1.1804,	0.0099	1.20,	0.0101	102.33	1.1480,	0.0093	108.73
	1.3812,		1.40,			1.3085,
	1.5800,		1.60,			1.4871,
	1.7367		1.80			1.6960
6	1.1450,	0.0079	1.1667,	0.0093	117.35	1.1507,	0.0081	115.17
	1.2891,		1.334,			1.2842,
	1.4357,		1.50,			1.4300,
	1.5897		1.6667,			1.5920,
	1.7367		1.8334			1.7831

Table 4

Right triangular distribution, g = 1.

No. of Strata (L)	Stratification by Using Equations (12)		Equal Interval Stratification		Relative Efficiency	Stratification by Using Methods of Approximation		Relative Efficiency
No. of Strata (L)	Points	nVY¯st	Points	nVY¯st		Points	nVY¯st
2	1.3759	0.0225	1.5	0.0280	124.73	1.3864	0.0225	124.73
3	1.2737,	0.0131	1.334,	0.0142	108.54	1.2477,	0.0136	104.94
	1.5889		1.667			1.5435
4	1.1842,	0.0103	1.25,	0.0122	118.40	1.1832,	0.0101	120.52
	1.3959,		1.50,			1.3892,
	1.6344		1.75			1.6330
5	1.1799,	0.0099	1.20,	0.0101	102.34	1.1450,	0.0092	109.92
	1.3807,		1.40,			1.3034,
	1.5799,		1.60,			1.4183,
	1.7371		1.80			1.6903
6	1.1447,	0.0078	1.1667,	0.0092	117.92	1.1477,	0.0080	115.70
	1.2888,		1.334,			1.2791,
	1.4356,		1.50,			1.4230,
	1.5897,		1.6667,			1.5850,
	1.7371		1.8334			1.7776

Table 5

Right triangular distribution, g = 1.5.

No. of Strata (L)	Stratification by Using Equations (12)		Equal Interval Stratification		Relative Efficiency	Stratification by Using Methods of Approximation		Relative Efficiency
No. of Strata (L)	Points	nVY¯st	Points	nVY¯st		Points	nVY¯st
2	1.3732	0.0224	1.5	0.0283	126.40	1.3803	0.0221	127.71
3	1.2720,	0.0130	1.334,	0.0142	109.29	1.2431,	0.0129	110.39
	1.5883		1.667			1.5375
4	1.1837,	0.0102	1.25,	0.0121	119.17	1.1790,	0.0098	123.42
	1.3957,		1.50,			1.3830,
	1.6341		1.75			1.6273
5	1.1794,	0.0098	1.20,	0.0101	102.97	1.1430,	0.0087	115.33
	1.3804,		1.40,			1.3002,
	1.5800,		1.60,			1.4775,
	1.7377		1.80			1.6884
6	1.1445,	0.0077	1.1667,	0.0091	118.73	1.1447,	0.0072	126.70
	1.2887,		1.334,			1.2744,
	1.4358,		1.50,			1.4174,
	1.5900,		1.6667,			1.5791,
	1.7377		1.8334			1.7734

Table 6

Right triangular distribution, g = 2.

No. of Strata (L)	Stratification by Using Equations (12)		Equal Interval Stratification		Relative Efficiency	Stratification by Using Methods of Approximation		Relative Efficiency
No. of Strata (L)	Points	nVY¯st	Points	nVY¯st		Points	nVY¯st
2	2.2012	0.1817	2.50	0.1984	109.19	2.0632	0.1747	113.52
3	1. 6974, 2.5992	0.1037	2.0, 3.0	0.1208	116.43	1.6503, 2.5605	0.1055	114.49
4	1.6526,	0.0857	1.75,	0.0888	103.63	1.4686,	0.0802	110.75
	2.3296,		2.50,			2.0625,
	3.1672		3.25			2.8591
5	1.4058,	0.0674	1.6,	0.0807	119.77	1.3663,	0.0657	122.86
	1.9199,		2.2,			1.8066,
	2.5131,		2.8,			2.3489,
	3.2297		3.4			3.0445
6	1.3710,	0.0575	1.5,	0.0644	111.90	1.3007,	0.0504	126.40
	1.8163,		2.0,			1.6503,
	2.3082,		2.3,			2.0626,
	2.7549,		2.8,			2.5602,
	3.2835		3.5			3.1815

Table 7

Exponential distribution, g = 1.

No. of Strata (L)	Stratification by Using Equations (12)		Equal Interval Stratification		Relative Efficiency	Stratification by Using Methods of Approximation		Relative Efficiency
No. of Strata (L)	Points	nVY¯st	Points	nVY¯st		Points	nVY¯st
2	2.1016	0.1715	2.50	0.1997	116.42	2.0247	0.1704	117.16
3	1. 6954,	0.1017	2.0,	0.1219	119.78	1.6264,	0.1041	117.08
	2.5915		3.0			2.5684
4	1.6513,	0.0856	1.75,	0.0891	104.10	1.4466,	0.0762	116.98
	2.3279,		2.50,			2.0249,
	3.1658		3.25			2.8127
5	1.4039,	0.0661	1.6,	0.0806	121.88	1.3484,	0.0645	125.07
	1.9191,		2.2,			1.7741,
	2.5132,		2.8,			2.3082,
	3.2294		3.4			3.0633
6	1.3695,	0.0606	1.5,	0.0682	112.57	1.2856,	0.0540	126.38
	1.1816,		2.0,			1.6221,
	2.3095,		2.30,			2.0250,
	2.7586,		2.8,			2.5191,
	3.2748		3.5			3.1482

Table 8

Exponential distribution, g = 1.5.

No. of Strata (L)	Stratification by Using Equations (12)		Equal Interval Stratification		Relative Efficiency	Stratification by Using Methods of Approximation		Relative Efficiency
No. of Strata (L)	Points	nVY¯st	Points	nVY¯st		Points	nVY¯st
2	2.0613	0.1672	2.50	0.2014	120.90	1.9881	0.1683	119.71
3	1. 6939,	0.1001	2.0,	0.1228	122.73	1.5951,	0.1045	117.55
	2.5846		3.0			2.4780
4	1.6503,	0.0854	1.75,	0.0893	104.56	1.4259,	0.0739	120.89
	2.3271,		2.50,			1.9983,
	3.1667		3.25			2.7732
5	1.4021,	0.0670	1.6,	0.0804	120.11	1.3315,	0.0648	124.10
	1.9186,		2.2,			1.7429,
	2.5144,		2.8,			2.2680,
	3.2311		3.4			2.9714
6	1.3511,	0.0585	1.5,	0.0675	115.31	1.2714,	0.0530	127.23
	1.8094,		2.0,			1.5952,
	2.3114,		2.30,			1.9883,
	2.7637,		2.8,			2.4780,
	3.2886		3.5			3.1141

Table 9

Exponential distribution, g = 2.

The regression function cx is taken to be linear with the slope at 45°. The constant σ 2 is determined in each case for the different values of g in such a way that 90% of the total variation is accounted for by the regression. In the case of the exponential distribution, we truncate the distribution such that the area under the curve to the right of the truncation point is 0.05. The optimum points of stratification are found by successive iterations. In solving the methods of finding AOSB (31) and (32), suitable techniques for solving numerical algebraic and transcendental equations, and numerical integrations are used.

6. CONCLUSION

In this paper, all the proposed methods of stratification—the equations giving OSBs and their methods of approximations—are found to be highly efficient in stratifying heteroscedastic populations. In uniform populations, equal interval stratification is considered to be efficient stratification method and all our proposed methods of stratification perform, in most cases, with nearly same with or, in a few cases, more efficiencies than that of equal interval stratification in the generated populations following uniform density function for all the strengths of heteroscedasticity, i.e., g=1,1.5,2. Moreover the proposed methods of stratifications perform with relatively higher efficiencies than equal interval stratification in stratifying populations of right triangular and exponential probability density functions for all the considered strengths of heteroscedasticity. Hence, it is observed that all the proposed methods of stratification perform efficiently in stratifying less skewed and lower level of heteroscedastic populations as well as highly skewed and higher level of heteroscedastic populations. These methods can be used effectively in stratifying heteroscedastic populations based on auxiliary variable which is highly correlated with estimation variable.

CONFLICTS OF INTEREST

There is no conflict of interest between the authors.

AUTHORS' CONTRIBUTIONS

The authors are equally involved in the work and the joint effort of the authors have led to the making of the paper.

Funding Statement

There is no funding for the research work from any funding agency.

ACKNOWLEDGEMENTS

We, the authors, are grateful to the anonymous reviewers for the inputs and guidance provided to us in improving the quality of the paper.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 20 - 1
Pages: 46 - 60
Publication Date: 2021/01/13
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TY  - JOUR
AU  - Bhuwaneshwar Kumar Gupt
AU  - Md. Irphan Ahamed
PY  - 2021
DA  - 2021/01/13
TI  - Construction of Strata for a Model-Based Allocation Under a Superpopulation Model
JO  - Journal of Statistical Theory and Applications
SP  - 46
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VL  - 20
IS  - 1
SN  - 2214-1766
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