Estimation of more than one parameters in stratified sampling with fixed budget

Math Meth Oper Res (2012) 75:185–197DOI 10.1007/s00186-012-0380-y

ORIGINAL ARTICLE

Estimation of more than one parameters in stratifiedsampling with fixed budget

Rahul Varshney · Najmussehar · M. J. Ahsan

Received: 2 July 2011 / Published online: 15 March 2012© Springer-Verlag 2012

Abstract In a multivariate stratified sampling more than one characteristic aredefined on every unit of the population. An optimum allocation which is optimumfor one characteristic will generally be far from optimum for others. A compromisecriterion is needed to work out a usable allocation which is optimum, in some sense,for all the characteristics. When auxiliary information is also available the precision ofthe estimates of the parameters can be increased by using it. Furthermore, if the travelcost within the strata to approach the units selected in the sample is significant the costfunction remains no more linear. In this paper an attempt has been made to obtain acompromise allocation based on minimization of individual coefficients of variationof the estimates of various characteristics, using auxiliary information and a nonlin-ear cost function with fixed budget. A new compromise criterion is suggested. Theproblem is formulated as a multiobjective all integer nonlinear programming problem.A solution procedure is also developed using goal programming technique.

Keywords Multivariate stratified sampling · Compromise allocation ·Nonlinear programming problem

1 Introduction

Among many available sampling designs stratified sampling is most widely usedfor estimating the population parameters of a heterogeneous population. In stratified

R. Varshney (B) · Najmussehar · M. J. AhsanDepartment of Statistics and Operations Research, Aligarh Muslim University,Aligarh 202 002, U.P., Indiae-mail: [email protected]

Najmussehare-mail: [email protected]

M. J. Ahsane-mail: [email protected]

123

186 R. Varshney et al.

sampling a heterogeneous population is divided into several mutually exclusive andexhaustive groups called strata that are homogeneous as far as possible. Simple randomsamples of suitable sizes are then selected and evaluated to construct the estimates ofthe unknown population parameters. Let a population of size N be divided into Lstrata. Denote for hth stratum (h = 1, 2, . . . , L):

Nh : Stratum sizeWh = Nh

N : Stratum weightnh : Sample sizeyhi : Value of the characteristic under study on the i th

unit of the stratum/sample from stratum

Y h = 1Nh

Nh∑

i=1yhi : Stratum mean

yh = 1nh

nh∑

i=1yhi : Sample mean

S2h = 1

Nh−1

Nh∑

i=1(yhi − Y h)2: Stratum mean square

s2h = 1

nh−1

nh∑

i=1(yhi − yh)2: Sample mean square.

Further let

Y = 1N

L∑

h=1

Nh∑

i=1yhi = 1

N

L∑

h=1NhY h =

L∑

h=1WhY h : Denote the overall population

mean.If the estimation of the overall population mean Y is of interest then the stratified

sample mean

yst =L∑

h=1

Wh yh, (1.1)

provides an unbiased estimate of the population mean Y with a sampling variance

V (yst ) =L∑

h=1

(1

nh− 1

Nh

)

W 2h S2

h . (1.2)

The fundamental problems in a stratified sample survey are

(i) How many strata should be there?(ii) What should be strata boundaries?

(iii) What should be sample sizes from various strata?

In this manuscript we assume that the population under study is already stratified intoL strata, that is the problems (i) and (ii) are already solved.

Problem (iii) is known as the “Problem of allocation” in the sampling literature.Several types of allocations are available. The most popular of them are “proportional”and “optimum” allocations.

Proportional allocation is given as nh = nWh; h = 1, 2, . . . , L . In optimum alloca-tion either the variance of the estimate is minimized for fixed C or the cost is minimized

123

Estimation of more than one parameters 187

for fixed variance. If the cost is linear, that is C = c0 + ∑Lh=1 chnh , where c0 and ch

are the overhead cost and per unit cost of measurement in the hth stratum, then theoptimum allocation is given by

nh = nWh Sh/

√ch

∑Lh=1 Wh Sh/

√ch

; h = 1, 2, . . . , L , (1.3)

where n = ∑Lh=1 nh is the total sample size. Interested readers are referenced

to Cochran (1977) and Sukhatme et al. (1984) for further details. Tschuprow(1923) was first to give the optimum allocation for a fixed sample size. But hisresult remained unknown. Neyman (1934) rediscovered it. Later on Stuart (1954)used Cauchy-Schwarz inequality to work out optimum allocation in more generalconditions.

In multivariate populations where more than one characteristics on each selectedunit of the population are to be measured, the optimum allocation for one character-istic may not be optimum for others. In such situations it is almost impossible to findan allocation that is optimum for all the characteristics simultaneously. Thus one hasto use an allocation that is optimum in some sense for all the characteristics. Suchallocations are called compromise allocations in sampling literature because they arebased on one or the other compromise criterion. Dalenius (1957); Ghosh (1958); Yates(1960); Aoyama (1963); Folks and Antle (1965); Chatterjee (1967, 1968); Kokan andKhan (1967); Ahsan (1975–1976, 1978); Ahsan and Khan (1977, 1982); Bethel (1985,1989); Schittkowski (1985–1986); Chromy (1987); Kreienbrock (1993); Jahan et al.(1994, 2001); Jahan and Ahsan (1995); Rahim (1995); Khan et al. (1997, 2003, 2008);Singh (2003); Semiz (2004); Díaz-García and Cortez (2006); Kozak (2006a,b); Ansariet al. (2009) and many others either suggested new compromise criterion or exploredthe existing criteria under various situations such as the presence of nonresponse,availability of auxiliary information, use of double sampling technique for stratifica-tion etcetera.

Díaz-García and Cortez (2008) proposed optimum allocation in multivariate strat-ified sampling as a matrix optimization problem, for which they studied a numberof possible solutions. They observed that the multiobjective optimization problem isa particular case of the matrix optimization. The optimum allocation in multivariatestratified sampling also can be seen as a multiobjective optimization problem.

If the data on some other characteristic that is highly correlated with the main char-acteristic under study are also available they can be used to enhance the performanceof the estimate in terms of increased precision. Such data are called auxiliary datain sampling literature. Raiffa and Schlaifer (1961); Ericson (1965); Ahsan and Khan(1982) and Dayal (1985) are some of those who made use of the auxiliary data in strat-ified sample surveys. Later Khan et al. (2010) worked out integer optimum allocationusing auxiliary data.

In this manuscript the problem of determining the compromise allocation (denotedby nhc elsewhere) in multivariate stratified random sampling is discussed. The objec-tive is to minimize the coefficient of variations (CV) of the estimates of the p popu-lation means (p > 1) simultaneously under the cost and some other restrictions. Theformulated problem turns out to be a multiobjective all integer nonlinear programming

123


problem (MAINLPP). To solve this MAINLPP a solution procedure is developed withhelp of goal programming technique.

2 Formulation of the problem

The notations used in the formulation are as in Khan et al. (2010). However, for thesake of continuity they are reproduced here. Assume that p(> 1) characteristics aredefined on each population unit and the estimation of unknown population means Y j ;j = 1, 2, . . . , p is of interest. It is assumed that the characteristics are uncorrelated.Let y jhi and x jhi denote the values of i th population unit of the main variable and

auxiliary variable respectively for the j th characteristic and Y jh = 1Nh

∑Nhi=1 y jhi

and X jh = 1Nh

∑Nhi=1 x jhi be the stratum means of y jhi and x jhi respectively. The

corresponding sample means are given as

y jh = 1

nh

nh∑

i=1

y jhi and x jh = 1

nh

nh∑

i=1

x jhi ,

respectively.In stratified sampling, in the presence of auxiliary information, the combined ratio

estimate of the population mean Y j is given by

y j,st =L∑

h=1

Whr jh; j = 1, 2, . . . , p,

where

r jh = R jst X j , R jst = Y jst

X jst

, Y jst =L∑

h=1

Wh y jh and X jst =L∑

h=1

Wh x jh,

with sampling variance

V (y j,st ) =L∑

h=1

W 2h A2

jh

nh−

L∑

h=1

W 2h A2

jh

Nh; j = 1, 2, . . . , p, (2.1)

where

A2jh = S2

y j h + R2j S2

x j h − 2R j Sy j x j h, (2.2)

S2y j h = 1

Nh−1

∑Nhi=1 (y jhi − Y jh)2; j = 1, 2, . . . , p are the stratum variances of the

j th characteristic in hth stratum for main variable,S2

x j h = 1Nh−1

∑Nhi=1 (x jhi − X jh)2; j = 1, 2, . . . , p are the stratum variances of the

j th characteristic in hth stratum for auxiliary variable,Sy j x j h = 1

Nh−1

∑Nhi=1 (y jhi − Y jh)(x jhi − X jh); j = 1, 2, . . . , p are the stratum

co-variances of the j th characteristic in hth stratum,

123


R j = Y j

X j,

Y j = 1N

∑Lh=1

∑Nhi=1 y jhi ; is the overall population mean of the j th characteristic for

main variable,and X j = 1

N

∑Lh=1

∑Nhi=1 x jhi ; is the overall population mean of the j th characteristic

for auxiliary variable.Furthermore, let us assume that we know the population variances and means. It is

obvious that these are unknown in real surveys but can be approximated or knownfrom a recent or preliminary survey (Kozak 2006a).

If the travel cost within the strata to approach the units selected in the sample issignificant then the cost function remains no more linear. Beardwood et al. (1959)showed that the distance between n randomly scattered points is proportional to

√n.

As the travel cost is also proportional to the distance traveled it will also be propor-tional to

√nh . In the above scenario the fixed total cost C of the survey is a nonlinear

function of nh; h = 1, 2, . . . , L

C = c0 +L∑

h=1

chnh +L∑

h=1

th√

nh,

or

C0 = C − c0 =L∑

h=1

chnh +L∑

h=1

th√

nh, (2.3)

where c0 = overhead cost, ch = per unit cost of measurement of all the characteris-tics in hth stratum; h = 1, 2, . . . , L and th

√nh is the travel cost within hth stratum.

The cost function (2.3) is quadratic in√

nh . Note that the RHS of (2.3) includes themeasurement and the travel cost.

Then the required compromise allocation will be the solution to the followingMAINLPP.

Minimize

⎛

⎜⎜⎜⎝

CV1(y1,st )

CV2(y2,st )...

CVp(y p,st )

⎞

⎟⎟⎟⎠

subject toL∑

h=1chnhc +

L∑

h=1th

√nhc ≤ C0

2 ≤ nhc ≤ Nh

and nhc integers; h = 1, 2, . . . , L ,

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(2.4)

where CVj = CVj (y j,st ) =√

V (y j,st )

Y2j

; j = 1, 2, . . . , p and nhc ; h = 1, 2, . . . , L

denote the required compromise allocation.

123


Since different characteristics are usually measured in different units it is not fea-sible to take the objective to minimize

∑pj=1 V (y j,st ), because quantities in different

units of measurements can not be added. Whereas CVj =√

V (y j,st )

Y2j

; j = 1, 2, . . . , p

are unit free and thus can be added.Define Z j as

Z j =√√√√ 1

Y2j

L∑

h=1

W 2h A2

jh

n jh=

√√√√

L∑

h=1

a jh

n jh(2.5)

Where

a jh = W 2h A2

jh

Y2j

; j = 1, 2, . . . , p, h = 1, 2, . . . , L , (2.6)

and n jh; h = 1, 2, . . . , L; j = 1, 2, . . . , p, denote the sample size from hth stratumfor j th characteristic and the finite population correction (fpc) in V (y j,st ), that is thesecond summation in (2.1), which is independent of nh , is ignored. Suffix ‘ j’ standsfor the characteristic number. Thus minimizing CVj is equivalent to minimize Z j .

The authors assume that all the characteristics are equally important. However, ifthe characteristics are significantly different and can be arranged according to theirimportance, there are two ways to handle this situation.

(i) By assigning weight w j to Z j ; j = 1, 2, . . . , p according to their importance,that is taking the objective function as

∑pj=1 w j Z j .

(ii) The problem may be reformulated to minimize Zk where the kth characteristic isthe most important and by fixing tolerance limits to the remaining (p−1) Z j ’s;j = 1, 2, . . . , p; j �= k.

In this manuscript the authors used an approach based on ‘Goal Programming’, asdiscussed in the next section.

3 The solution using goal programming

Let Z∗j be the optimum value of Z j defined by (2.5) under optimum allocation for the

j th characteristic obtained by solving the following All Integer Nonlinear Program-ming Problems (AINLPP) separately for each j = 1, 2, . . . , p.

Minimize Z j =√

L∑

h=1

a jhn jh

subject toL∑

h=1chn jh +

L∑

h=1th

√n jh ≤ C0

2 ≤ n jh ≤ Nh

and n jh; integers; h = 1, 2, . . . , L .

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(3.1)

123


Note that the solution to (3.1) for each j = 1, 2, . . . , p, gives the individual opti-mum allocation n∗

jh for the j th characteristic. No standard nonlinear programmingtechnique is available to solve the MAINLPP (2.4).

However, in the following the authors present an approach to solve the MAINLPP(2.4) by converting it in a single objective problem using goal programming technique.

If Z j denote the value of Z j for a compromise allocation nhc then obviously Z j ≥Z∗

j and Z j − Z∗j ≥ 0; j = 1, 2, . . . , p will give the increase in Z j due to not using

the individual optimum allocation of j th characteristic.Now consider the following problem:

“Find nhc such that for j th characteristic ( j = 1, 2, . . . , p), the increase in the valueof the Z j for each j due to the use of a compromise allocation, instead of its individualoptimum allocation, is less than or equal to d j , where d j ≥ 0 ( j = 1, 2, . . . , p) arethe goal variables”.To achieve these goals we must have

Z j − Z∗j ≤ d j ; j = 1, 2, . . . , p (3.2)

or

Z j − d j ≤ Z∗j ; j = 1, 2, . . . , p

or

√√√√

L∑

h=1

a jh

nhc

− d j ≤ Z∗j ; j = 1, 2, . . . , p (3.3)

The total increase in the value of Z j for not using the individual optimum allocations isgiven by

∑pj=1 d j . To minimize this total increase will be a suitable compromise crite-

rion to obtain a compromise allocation. Consequently we have to solve the followingGoal Programming Problem (GPP) (see Schniederjans 1995)

Minimizep∑

j=1d j

subject to

√L∑

h=1

a jhnhc

− d j ≤ Z∗j ; j = 1, 2, . . . , p

L∑

h=1chnhc +

L∑

h=1th

√nhc ≤ C0

2 ≤ nhc ≤ Nh

and d j ≥ 0; j = 1, 2, . . . , p,

nhc integers; h = 1, 2, . . . , L .

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(3.4)

The GPP (3.4) may be solved by using the optimization software LINGO (LINGO-User’s Guide 2001). For more information about LINGO one may visit the site: http://www.lindo.com.

123

http://www.lindo.com

http://www.lindo.com


4 A numerical example

The following numerical example illustrates the goal programming approach. The dataare from 2002 and 1997 Agricultural Censuses in Iowa State conducted by NationalAgricultural Statistics Service, USDA, Washington DC (Source: http://www.agcensus.usda.gov/) as reported in Khan et al. (2010). The 99 counties in the Iowa State aredivided into 4 strata. Two characteristics: (i) The quantity of corn harvested Y1 and(ii) The quantity of oats harvested Y2, are of interest. The total amount available forconducting the survey is assumed to be C = 300 units with an expected overhead costc0 = 50 units. This gives C0 = C − c0 = 250 units (Table 1).Also

Y1 = The quantity of corn harvested in 2002,Y2 = The quantity of oats harvested in 2002,X1 = The quantity of corn harvested in 1997,X2 = The quantity of oats harvested in 1997.

The values of X1 and X2 are used as the auxiliary information on the main variableY1 and Y2 respectively. It is given that

X1 = 405, 654.19, Y 1 = 474, 973.90, X2 = 2, 116.70, Y 2 = 1, 576.25.

This gives R j ; j = 1, 2 as

R1 = Y 1/X1 = 1.1709 and R2 = Y 2/X2 = 0.7447.

For the given data the values of A2jh and a jh defined in (2.2) and (2.6) respectively

are: (Table 2)For j = 1 AINLPP (3.1) takes the form

Minimize Z1 =√

0.00005314n11

+ 0.00084058n12

+ 0.00129828n13

+ 0.00033426n14

subject to 15n11 + 7n12 + 5n13 + 9n14+5

√n11 + 4

√n12 + 2

√n13 + 3

√n14 ≤ 250

2 ≤ n11 ≤ 82 ≤ n12 ≤ 342 ≤ n13 ≤ 452 ≤ n14 ≤ 12

and n11, n12, n13 and n14 integers.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

Using LINGO optimization software we get n∗1h ; h = 1, 2, 3, 4 as

n∗11 = 2, n∗

12 = 10, n∗13 = 14 and n∗

14 = 5 with Z∗1 = 0.01643820.

123

http://www.agcensus.usda.gov/

http://www.agcensus.usda.gov/


Tabl

e1

Dat

afo

r3

Stra

taan

d2

char

acte

rist

ics

hN

hW

hc h

t hS2 y 1

hS2 x 1

hS2 y 2

hS2 x 2

hS x

1y 1

hS x

2y 2

h

18

0.08

0815

529

,26

7,52

4,19

5.5

21,60

1,50

3,18

9.8

777,

174.

11,

154,

134.

224

,36

0,42

2,80

2.3

904,

170.

6

234

0.34

347

426

,07

9,25

6,58

2.8

19,73

4,61

5,81

6.7

4,98

7,81

2.9

7,05

6,07

4.8

22,00

3,46

6,63

0.3

5,81

3,43

9.5

345

0.45

455

242

,36

2,84

2,46

0.8

27,12

9,65

8 ,75

0.0

1,07

4,51

0.6

2,08

2,87

1.3

33,36

7,59

7,19

2.0

1,28

5,35

5.6

412

0.12

129

330

,72

8,26

5,33

6.9

17,25

8,23

7,35

8.5

388,

378.

573

2,00

4.9

21,03

3,76

9,86

7.3

456,

991.

5

123


Table 2 Values of A2jh and a jh , h = 1, 2, 3, 4; j = 1, 2

h A21h A2

2h a1h a2h

1 1, 836, 063, 750.69064 70, 562.33461 0.00005314 0.00018545

2 1, 607, 795, 241.67013 242, 450.42442 0.00084058 0.01150960

3 1, 417, 602, 304.66257 315, 202.57310 0.00129828 0.02621164

4 5, 132, 570, 858.25721 113, 684.53561 0.00033426 0.00067227

For j = 2 AINLPP (3.1) takes the form

Minimize Z2 =√

0.00018545n21

+ 0.01150960n22

+ 0.02621164n23

+ 0.00067227n24

subject to 15n21 + 7n22 + 5n23 + 9n24++5

√n21 + 4

√n22 + 2

√n23 + 3

√n24 ≤ 250

2 ≤ n21 ≤ 82 ≤ n22 ≤ 342 ≤ n23 ≤ 452 ≤ n24 ≤ 12

and n21, n22, n23 and n24 integers.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

Using LINGO optimization software we get n∗2h; h = 1, 2, 3, 4 as

n∗21 = 2, n∗

22 = 11, n∗23 = 18 and n∗

24 = 2 with Z∗2 = 0.05414231.

Using the values of Z∗j ; j = 1, 2 GPP (3.4) for the given example is

Minimize d1 + d2

subject to√

0.00005314n1c

+ 0.00084058n2c

+ 0.00129828n3c

+ 0.00033426n4c

−d1 ≤0.01643820√

0.00018545n1c

+ 0.01150960n2c

+ 0.02621164n3c

+ 0.00067227n4c

−d2 ≤0.05414231

15n1c + 7n2c + 5n3c + 9n4c + 5√

n1c + 4√

n2c + 2√

n3c + 3√

n4c ≤ 2502 ≤ n1c ≤ 82 ≤ n2c ≤ 342 ≤ n3c ≤ 452 ≤ n4c ≤ 12

and d1 and d2 ≥ 0n1c , n2c , n3c and n4c integers.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

Using LINGO the required optimum allocation is obtained as

n∗1c

= 2, n∗2c

= 10, n∗3c

= 16 and n∗4c

= 4

with d∗1 = 0.00015504 and d∗

2 = 0.00108431 and the optimum value of the objec-tive function d∗

1 + d∗2 = 0.00123935.

The corresponding values of Z j ; j = 1, 2, that is CVj (fpc ignored) are CV1 =0.01659333 and CV2 = 0.05522663. Thus

∑2j=1 CVj = 0.07181996.

123


5 Conclusion

As an alternative to the proposed goal programming approach, when the weightedsum of Z j is minimized, as indicated in Sect. 2, we obtained the following AINLPPto solve:

Minimizep∑

j=1w j Z j =

p∑

j=1w j

(√L∑

h=1

a jhnhc

)

subject toL∑

h=1chnhc +

L∑

h=1th

√nhc ≤ C0

2 ≤ nhc ≤ Nh

and nhc ; integers; h = 1, 2, . . . , L .

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(5.1)

For three different combinations of w j the AINLPP (5.1) is solved using LINGO. Theresults are given in Table 3. Without loss of generality we can assume that the sum ofweights is one, that is

∑pj=1 w j = 1.

It can be seen that the allocation with weights w1 = 0.8 and w2 = 0.2 is equally pre-cise as the proposed allocation. However, the allocations with weights w1 = 0.4, w2 =0.6 and w1 = 0.2, w2 = 0.8 are less precise than the proposed allocation.

Table 4 gives the percentage increase in the coefficient of variations of variouscharacteristics under different individual optimum allocations. Against the row j = 1,columns 2 and 3 of the Table 4 give percentage increase in the coefficient of varia-tion when the optimum allocation with respect to characteristic 1 is used for both thecharacteristics. Column 4 of Table 4 gives the percentage increase in the coefficient of

Table 3 Compromise allocations with different weights

Weights Allocations Coefficient of Variation CV1 + CV2

w1 w2 n1 n2 n3 n4 CV1 CV2

0.8 0.2 2 10 16 4 0.01659333 0.05522663 0.07181996

0.4 0.6 2 9 19 3 0.01731247 0.05454558 0.07185805

0.2 0.8 2 11 18 2 0.01849991 0.05414231 0.07264222

Table 4 Percentage increase in the coefficient of variations of various characteristics under different indi-vidual optimum allocations

Characteristics Percentage increase with respect to

Characteristics Proposed Compromiseallocation

j = 1 j = 2

(1) (2) (3) (4)

j = 1 0 % 12.54164 % 0.94322 %

j = 2 5.30078 % 0 % 2.00272 %

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variation of characteristic 1 when the proposed compromise allocation is used. Similarare the interpretations of figures in the row j = 2.

It can be seen that the percentage increase under the proposed compromise allo-cation is considerably less than the percentage increase when individual optimumallocation for one characteristic is used for both the characteristics.

Thus we conclude that the proposed compromise criterion is a suitable criterion forworking out a usable compromise allocation for multivariate stratified surveys whenauxiliary information are available and the travel cost within strata are significant.

Acknowledgments The authors are grateful to the Editor, Associate Editor and the learned Reviewer fortheir valuable comments and suggestions that helped to revise the manuscript in its present form.

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Estimation of more than one parameters in stratified sampling with fixed budget