[Lecture Notes in Statistics] Topics in Survey Sampling Volume 153 || Miscellaneous Topics

Chapter 8

Miscellaneous Topics

8.1 INTRODUCTION

In this chapter we consider three special topics - (1) calibration estimators (predictors), estimators having some calibration property, (2) poststratification (iii) conditional inference in survey sampling. Calibrationestimators, initiated by Deville and Sarndal (1992) considers estimatorswhich satisfy certain properties with respect to some known auxiliary variables, in certain cases model-unbiasedness property. In post-stratificationanalysis one important consideration is whether one should consider conditional analysis based on the observed sample sizes in different strata orunconditional analysis.

When a superpopulation model is available and is exploited to build upsuitable predictors of population characteristics (mean, total, etc.), the conditionality principle (Cox and Hinkley, 1974) states that the infrence shouldbe based on the model for a given sample, In this chapter, we however,consider conditional inference in the design-based sense, that is, we restrictourselves to a part of the sample space containing samples on which somestatistics often based on x (auxiliary variables)-values have certain fixedvalues (properties) (eg. sample mean Xs = m x , a fixed value) and find conditionally unbiased estimators of population parameters (mean, total, etc.).Under certain circumstances, the conditionally weighted estimators haveless variance than unconditional estimators. This principle of conditionalinference was first invoked by Rao (1985). We review some of these resultsin this chapter.

231P. Mukhopadhyay, Topics in Survey Sampling© Springer-Verlag New York, Inc. 2001

232

8.2

CHAPTER 8. MISCELLANEOUS TOPICS

CALIBRATION ESTIMATORS

Assume that with each unit k there is a vector Xk = (Xkl' ... ,Xkp)' of valuesof auxiliary variables Xl, ... , X p ' It is known that the Horvitz Thompson

estimator HTE, TY1r = T1r = 'EkEs dkYk where dk = l/nk is unbiased forthe population total T. It is desired to find calibration estimator Tyrol ='EkEs WkYk where the weights Wk are as close to dk as possible (in somesense) and which satisfy

L WkXk = LXk = Tz = (Tzll "" Tx,)',kEs

(8.2.1)

Tz; = 'Ef=l Xkj, where it is assumed that the population totals in theright hand side are known. Consider a distance function G with argumentz = wid and with the properties: (a) G is positive and strictly convex (b)G(l) = G'(l) = 0 (c) G"(l) = 1. Here G(Wk/dk) measures the distancebetween Wk and dk and 'EkEs dkG(Wk/dk) is a measure of distance between{wd and {dd for the whole sample. Our objective is to find Wk such thatfor a given s, 'EkEs dkG(Wk/dk) is minimum subject to the condition (8.2.1).The minimising equation is, therefore,

org(wddk ) = X~A = 0 (8.2.2)

where g(z) = dG(z)/dz and A is a vector of constants. From (8.2.2),Wk = dkF(x~A) where F(u) = g-l(u), the inverse function of g(u). Theabove properties of G imply that F(u) exists and F(O) = 1,F'(u) = 1.To calculate the new weights Wk the value of A is determined from thecalibration equation

L dkF(x~A)Xk = Tz

kEs(8.2.3)

where A is only unknown. The systems of equations (8.2.3) can be solvedby numerical methods.

Two functions G and the corresponding F functions are:

(a) The linear method: G(x) = ~(x - 1)2, X E Rj F(u) = u + 1

(b) The multiplicative metho~: G(x) = x log x - x + 1j g(u) = log Uj u =eg(u)jF(u) = eU (> 0). Deville and Sarndal (1992), Deville, Sarndal andSautory (1993) listed seven different distance functions and examined thestatistical properties of the corresponding calibration estimators.

8.2. CALIBRATION ESTIMATORS

In the linear method, the calibrated weights are given by

where). is determined by the solution of the equation

2:= dk(l + XP)Xk = Tx

kEs

or

233

(8.2.4)

(8.2.5)

In this case the estimator Tycal is known as the generalised regressionestimator (GREG) and can be written as

Tygreg = 2:= WkYk = 2:= dk(l + XP)YkkEs kEs

= 2:= dkYk + 2:= dkX~.AYkkEs kEs

= TY1r + (Tx - Tx1r ) ,Bs

where TX1r = L:kEs dkXk denotes the HTEfor the x vector and Bs is obtainedby solving the sample-based equations

(8.2.6)

(8.2.7)

It is well-known that the asymptotic variance (AV) of Tygreg (Section 2.5)is

N

AV(Tygreg) = 2:= 2:= D.kl(dkEk)(dIEI)k,;#=l

N 'B 'B= 2:= 2:= (7r'kl -7r'k7r'I)(Yk - x k )(YI - XI )

k#l=l 7r'k 7r'1

where B is determined as the solution of the normal equations of the hypothetical census fit corresponding to (8.2.6). That is, B satisfies

N N

2:= xkx~B = 2:= XkYkk=l k=l

(8.2.8)

234 CHAPTER 8. MISCELLANEOUS TOPICS

Deville and Sarndal (1992) advocated the use of the variance estimator

(8.2.9)

where ek = Yk - x~Bs.

Any member of the class (8.2.4) is asymptotically equivalent to Tyreg . Thecalibration estimators generated by different F functions share the samelarge sample variance. Thus (8.2.9) can be used to estimate the varianceof any estimator TYcal in this class.

Tille' (1995) considered conditions under which an estimator can be designunbiased as well as calibrated. Dupont (1995) considered calibration estimators of T when the auxiliary information is obtained at different phasesof sampling in a two-phase sampling procedure. Mukhopadhyay (2000 b)derived calibration estimator of a finite population variance. His estimatoris found to be a member of his class of generalised predictor of a finitepopulation variance (Mukhopadhyay, 1990). He (2000 c) also consideredcalibration e:>timator of a finite population distribution function.

The'berge (1999) considered an extension of calibration estimators as follows. Let c be a N x 1 vector of constants, X = ((Xkj», a N x p matrix ofauxiliary values Xkj, the value of Xj on unit k(j = 1, .. . ,pjk = 1, .. . ,N).let

. 1 1D = Dlag (-, ... , -) (8.2.10)

7l"1 7l"N

A matrix or a vector with suffix s will denote the part of the same withelements corresponding to the units in s only. The problem is to estimatey~cs where W s is a n x 1 vector of calibration weights for the sampled units.The weights W s are so chosen as to minimise the distance between W s andDscs under the constraints

X~ws = X'c (8.2.11)

In other words, the constraints state that application of calibration weightsto each auxiliary variable reproduces the known calibration totals.

Let a be a n x 1 vector and U be a N x N positive diagonal matrix. Thesquared norm of a is defined as lIallfJ. = a'Usa. Using the squared norm asa distance, the vector of weights W s should minimise IIws - DscsllfJ. whilesatisfying the calibration equation (8.2.11). The matrix Us serves to weightthe contribution of each unit in the distance measure between W s and Ds~s'Many of the commonly used estimators are obtained by taking U = D-1Q-lwhere Q is a diagonal nonnegative matrix. For Sarndal's (1982) procedure,

8.3. POST-STRATIFICATION 235

U = D-1Q-l where Q is a diagonal variance matrix under a model. InBrewer (1994), U = (D - I)-I Z (assuming 7rk #- 0), where Z is a diagonalmatrix of some size measure.

Often, there does not exist any solution to the calibration problem. Theauthor, therefore, proposes a generalised problem so that if a solution tothe calibration problem exists this comes as a particular case of the generalproblem. Let T be a p x p positive diagonal matrix used to weight thecontribution of each of p constraints in the distance measure between X~W S

and X'c. When T = Ip , equal importance is given to each constraint.Consider the following problem. Among the set of weights W s that minimiseIIX~ws - X'cll}, find the one that also minimises Ilws - Dscslltr .,

The solution is given as

where F+ is the unique Moore-Penrose inverse of F. The estimator of y'cthen becomes

where

, A' ( A)'D AYsws =yc+ y-y ~c

YA = XT1/ 2 (Tl/ 2X'U1X T 1/ 2 )+T1/ 2X'U- 1/ 2 ys s S s s s

(8.2.13)

(8.2.14),

~ being the diagonal matrix of ok> Ok = 1(0) if k E (~)s.

If the calibration equation is solvable, the minimum of IIX~ws - X'cll} is0, regardless of T. In this case T may be set equal to Ip •

If U has the form U = D-1Q-l, where Q is a diagonal variance matrixunder a model and if T = I p and X s is of rank p, then there is a solution tothe calibration equation (8.2.12) and (8.2.13) yields Sarndal's generalisedregression estimator. However, in many situations the calibration equationscan not be solved or X s is of rank less than p. In all these cases The'berge'ssolution holds. His approach is thus very promising, though it needs somestatistical justification.

8.3 POST-STRATIFICATION

Sometimes, after the sample has been selcted from the whole population,the units are classified according to variables like age, sex, occupation, education and similar other facors, information regarding which are not available before sampling and collection of data. The units are thus classifiedinto H post-strata according to these classifying variables. The censusesprovide information on all these variables at aggregrate levels and thus the


post-strata sizes, henceforth called strata sizes Nh are generally known.These provide estimates of population total for each stratum and hence estimate of population total. This procedure is known as post-stratificationor stratification after sampling.

The method is more flexible than stratification before sampling or priorstratification, because after sampling the stratification factors can be chosen in different ways for different sets of variables of study in order tomaximise the gain in precision. The technique is particularly suitable formultipurpose surveys when stratification factors selected prior to samplingmay be poorly correlated with large number of secondary variables.

Let Yhi denote the value of Y on the ith unit belonging to the hth poststratum (i = 1, ... Nh in the population ; i = 1, ... , nh( in the sample ) j h =1, ... , L). Let Z be a stratifying variable (like age group, occupationclass, etc). Clearly, the sample size nh in stratum h is a random variable,L:h nh = n. We assume that n is large enough or stratification is such thatthe probability that some nh = 0 is negligibly small. If nh = 0 for somestrata, two or more strata can be combined to make the sample size nonzero for each stratum. Fuller (1966) gave an alternative procedure which issuperior to that of collapsing strata (Exercise 6). Let

Nh Nh

Yh = LYh;/Nh, Y = L NhYh/N , S~ = L(Yhi - Yh)2/(Nh - 1)i=1 h i=1

nh nh

Ph = Nh/N , Yhs = LYh;/nh, s~ = L(Yhi - Yhs? /(nh - 1)(8.3.1)i=1 i=1

The post stratified estimator of Y is

Ypost = L PhYhsh

Clearly,E[ypostl = EE[Ypost In] = E(y) = Y

where n = (n1, ... ,nH)'. The unconditional variance is

V[Ypost] = L P~S~E(l/nh - l/Nh)h

(8.3.2)

(8.3.3)~ (l/n - l/N) L PhS~ + L(1- Ph)SVn2

h h

using Stephan's (1945) approximation to E(l/nh)' The difference betweenthe variance of the unbiased estimator of Y under prior stratification with


proportional allocation and (8.3.3) is of order n-2 and can be neglected inlarge samples. However, for large value of H the difference between thesetwo quantities may not be negligible. The unconditional variance (8.3.3) isa measure of the long term performance of Ypost and should be considered atthe planning stage of a survey (Deming 1960, p.323) while the conditionalvariance measures the performance of the strategy in the present survey.Durbin (1969) considered n as an anciliary statistic and suggested thatthe inference should be made conditional on n. Royall (1971), Cox andHinkley (1974), Royall and Eberhardt (1975), Kalton [in the discusson ofSmith's (1976) paper], Oh and Scheuren (1983) also advocated the use ofconditional inference under such circumstances.

If we apply the self-weighted estimator

Ys = LnhYhlnh

for the post-stratified sampling procedure,

E(ys In) = Y - LYh(Ph - nhln )h

(8.3.4)

(8.3.6)

MSE(ys In) = L(nhln)2(1 - !h)SVnh + {LYh(Ph - nhln)2} (8.3.5)h h

The conditional bias of Ys is zero when either nhln = Ph or Yh = constant Vh.Also

MSECYs In) - VCYpost In)

= {LYh(Ph - nhln )}2 + L {(nhln? - Ph}(l - fh)SVnhh h

The quantity (8.3.6) may be positive or negative. Thus no definite conclusion can be drawn from the comparison between these strategies. Numericalstudies by Holt and Smith (1979) show that Ypost is superior to Y unless thesample size is small and the ratio of between stratum to within stratumvariance is small.

Similarly, considering the usual variance estimator s2 In for Ys where S2 =

2:~=1 2:7':'1 (Yhi - Ys)2/(n - 1) it is seen that

(n -1)E(s2 In) = L nh(Yh - Ys)2 + L nhS~(l - n-1)

h h

(8.3.7)

Holt and Smith (1979) numerically studied the variations in the values of(8.3.5) and E(s2 In) over values of n for fixed n and concluded that s2Inis unsatisfactory as an estimate of (8.3.5).


Considering the situations of non-response, let Th be the number of respondents in post-stratum h,Th::; nh(h = 1, ... ,H). In this case, n mayor maynot be known, depending on whether Z is known for non-respondents. It isassumed that non-response is ignorable (Little, 1982; Rubin,1976), in thesense that respondents within post-stratum h can be taken as a randomsubsample of sampled cases in post-stratum h. Thus, Th has a binomialdistribution

Th In'" Bin(nh' 'l/Jh)

where 'l/Jh is the response probability in stratum h. A stronger assumptionis that the non-response rate is the same across post-strata, i.e. 'l/Jh = 'ljJ V hand respondents are a random subsample of sampled cases over all h. Thisis called missing completely at random (MACR) assumption by Rubin(1976). The post-stratified sampling estimator of Y is

H

Yps = LPhY~h=l

whereTn

Y~ = LYh;/Th'i=l

the respondent sample mean. Also

T

Yps = LWiYii=l

(8.3.8)

(8.3.9)

where the weight Wi = PhlTh if i belongs to stratum h and T = L:h Th.The conditional variance of Yps, proposed by Holt and Smith (1979) as ameasure of performance by Yps, is

VaT{yps Ir} = LP;(1- fDsVThh

(8.3.10)

where r = (T1' ... , TH), f~ = ThlNh and s~ is the sample variance of Y basedon Th respondents.

We now consider Bayesian approach. Consider the basic normal poststratification model (BNPM)

(Yi IZi = h, Ph, (T~) '" NID(Ph' (T~); f(Ph' log (Th) = Constant (8.3.11)

where Zi is the value of Z on unit i, identifying the post-stratum and f isthe joint density of (.,.) and (Ph, log (Th) follows Jeffrey's non-informative


prior. The iid assumption within post-strata is to be modified for designsinvolving cluster sampling and differential selection rates within post strata.Under BNMP assumption

E(y I Z, Ys) = Yps

V(y I Z, Ys) = L pl(l - f~)8hS~ Irh (8.3.12)h

where Z = (Zl, ... , ZN)', Ys = (Yi, i E s), and 8h = (rh - l)/(rh - 3) is asmall sample correction for estimating the variance.

The Bayesian analysis, therefore, suggests that the randomisation conditional variance (8.3.10) differes from (8.3.12) only in the substitution ofsample estimates for the (unknown) population variance and in the attendant small sample correction.

If we assume the same distribution of Yi across the post-strata

(Yi IZi = h,p,a2) rv NID(p,a2

)

f(p, log a2) = Constant

Bayes estimate of y is

E(y I Z, Ys) = Ys = LLYhi/rh

with posterior variance

(8.3.13.1)

(8.3.13.2)

(8.3.14)

Var(y IZ, Ys) = v = 8(1 - 1)s/2Ir

where s/2 is the sample variance of y based on r units and 8 = (r-1)/(r-3)and t = riN. Alternatively, ignoring the data on Z, we might assume

(Yi I p,a2) rv NID(p,a2

)

together with (8.3.13.2). Here also the Bayesian estimate is y with posteriorvariance v.

Therefore, Bayesian analysis under (8.3.11) leads to the posterior meanyps and variance vps . Bayesian analysis under (8.3.13.1) [ or (8.3.15)] and(8.3.13.2) yields the posterior mean Ys and variance v. The Bayesian approach leads to the two natural measures of precision (vps and v) and alsoprovides small-sample correction (8,8h ) for estimating the variance.

Doss, Hartley and Somayajulu (1979) developed an exact small sample theory for post-stratification (Exercise 7). Some related references are Jagers,et al (1985), Jagers (1986), Casady and Valliant (1993), Valliant (1987,1993).

240

8.3.1


ALGORITHM FOR COLLAPSING POST-STRATA

The small post-strata that contribute excessively to the variance are sometimes collapsed in order to reduce vps (given in (8.3.12)). Frequentist strategies for collapsing post-strata were considered by Tremblay (1986) and byKalton and Maligwag (1991).

If two post-strata i and j are collapsed, the model (8.3.11) is modified byreplacing the means and variances in these post-strata by a single meanand variance for the combined post-strata . The post-stratified estimatorunder the collapsed model is

where

-(ij) _ """"" p - + (n + n )-Yps - L hYhs -Li rj Y(i+j)sh(i-i,j)

(8.3.16)

Y(i+j)s = (TiYi +TjYj)/(Ti +Tj) (8.3.17)

The posterior variance of Y (corresponding to (8.3.12)) is then given by

v(ij) = v - b.v--ps ps tJ

whereA P?8i(1 - f;)sr P]8j (1 - Ji)s;UVij = + -=-~---=-----"--

Ti Tj

If the correction terms 8i and 8j for estimating the variance are ignored,the expected reduction in Bayes risk of Yps is

E(b.Vij) = P?(l- f;)ur + P](l-/j)u] _ (Pi + Pj )2(1 - J;j)utj (8.3.20)

~ ~ ~+~

where E(sD = ut, E(srj) = utj' (srj being defined similarly as sD, expectation being taken wrt the superpopulation distribution of Yi given in (8.3.11).If Y and the stratifying variable Z are independent, then

(8.3.21)

Hence, in this case the variance is always reduced by collapsing, providedthe weights Wi and Wj in the collapsed strata are unequal. However, whenY and Z are associated, this reduction is counteracted by an increase invariance utj from collapsing. The following algorithm has been suggested.


• Order the post-strata so that the neighbours are a priori relativelyhomogeneous.

• Collapse the post-stratum pair (i, j) that maximises (8.3.20), subjectto the restriction that j = i + 1, i.e. only neighbouring pairs areconsidered.

• Proceed sequentially until a reasonable number of pooled post-strataremain or (8.3.20) becomes substantially positive.

Little (1993) gave an algorithm to implement the above rules and madea numerical study of the procedure by collpasing post-strata in the LosAngels Epiodemiological Catchment Area (LAECA) survey.

Some alternative strategies for collapsing are as follows.

If the stratum means can be regarded as exchangeable, one can assume thefollowing prior under normal specification.

p(j.1.h I (Jh) rv N I D(j.1., 72), (J~ = (J2 V h

f(j.1., log 72

, log (J2) = Constant (8.3.22)

This along with (8.3.11) yields Bayes estimate of y that smooths Yps towardsYs.

If variances are not assumed constant, one may assume the model

(8.3.23)

the resulting posterior mean shrinks Yps towards an estimator where contribution from post-stratum h are weighted by Sh2• One should find groupsof post-stratification within which exchangeability is possible and find outBayes estimate of y and its risk.

Another approach to smoothing is to model the response rates. Let () bethe overall probability of selection of a unit and 1rh = ()'lj;h be the probabilityof selection and response in post-stratum h. Assume

(8.3.24)

Assume that 1rh has a beta distribution with mean 1r and variance k1r(l-1r).

Then posterior distribution of 1rh is beta with mean

where1-K

Ah = K KN' (K a suitable constant)1- + h

(8.3.25)


which smooths the observed selection rate 'ITh towards 'IT. Estimating 'IT byT/N, smoothed weights from this model have the form

_ WhWh ex _ _

1 - Ah + AhWh(8.3.26)

(8.3.27)

obtained by replacing Th by Nh'ITh in Wh(= Ph/Th) where :Xh is derived fromAh by replacing K by its estimate. With more than one post-stratifier, thelikelihood of sparse or empty post-strata increases, so the need to modifythe BNMP assumption becomes greater. Let Zl, Z2 be two categorical poststratifying variables and let Phk be the population proportion with Zl = hand Z2 = k. Then

Y = LLPhkYhkh k

where Yhk is the population mean of cell (h, k). Suppose that a random sample is taken, resulting in nhk individuals in cell (h, k), Thk of whom respondto y. Model-based inference about Y involves two distinct components: inference about {Phk} based on a model for the joint distribution of Zl andZ2 and inference about Y based on a model for the distribution of y givenZl and Z2. A direct extention of (8.3.11) for y given Zl, Z2 is the basicnormal two-way post-stratification model

!(Phk, log O"hk) = Constant

The BNPM model (8.3.28) with known Pkh yields

E(y I Z, Ys) = Yps = L L PhkYhkh k

(8.3.28)

(8.3.29)

In some circumstances {Phk} is unknown but the marginal distributionsPh+, PH of Zl, Z2 are known; then a model is needed to predict the Phkvalues. Model inference on Y is then needed using the information on n(Little, 1991 b).

Suppose first that Zl, Z2 are observed for non-respondents so that nhk isobserved. Under random sampling, a natural model for n is

(8.3.30)

the multinomial distribution with index n and probabilities {Phd. Withthe Jeffrey's prior PhkJ the posterior distribution of Phk I nhk is Dirichletwith parameter {nhk + 1/2}.

8.4. CONDITIONAL UNBIASEDNESS 243

The posterior distribution of Phk given Ph+ I P+k I nhk does not have a simpleform. But its posterior mean can be approximated as

(8.3.31)

which denotes the result of raking the sample counts {nhd to the knownmargins. For further details the reader may refer to Little (1993).

8.4 DESIGN-BASED CONDITIONAL UN

BIASEDNESS

One way of taking into account the auxilary information at the estimationstage is to adjust the value of the estimator conditional on the observedvalue of an auxiliary statistic. Post-stratification, where the estimator isbased on the observed sample size in the post-strata, is an example ofusing such a conditional approach. Mukhopadhyay (1985 b) obtained apost-sample estimator of Royall (1970)-type which remains almost (model-)unbiased with respect to a class of alternative polynomial regression models.Robinson (1987) compared the conditional bias of the ratio estimator i =~X for given values of Xs under assumption of normality and correctedx,the estimate by using an estimate of its conditional bias. The conditionallyadjusted estimator proposed by him is

ia = i + CR - b)(xs - x)X/xs

where b = LiES(Yi - Ys)(Xi - xs)/ LiES(Xi - xs?, R = Ys/xs and a conditionally adjusted variance estimator for i is

-2/-2Va = VeX X s

whereV e = (1 - f) L(Yi - RXi)2/n(n - 1)

iEs

and f = niNo Rao (1985, 1994) obtained a general method of estimation using auxiliary information and applied a conditional bias adjustmentfor mean estimator, the ratio estimator in simple random sampling andstratified random sampling. Conditional design-based approach is a hybridbetween classical design-based approach and model-dependent approach.This approach restricts one to a relevant set of samples, instead of the wholesample space and leads to the conditionally valid inferences in the sensethat the ratio of the conditional bias to conditional standard error (conditional bias ratio) goes to zero as n ~ 00. Also, approximately 100(1- a)%


of the realised confidence intervals in repeated sampling from the relevantsubset of samples will contain the unknown total T (Rao, 1997, 1999).

We shall, here, specifically consider the conditionally design-unbiased estimation when sampling is restriced to a relevant set of samples. Tille'(1995, 1998) proposed a general method based on conditional inclusionprobabilities that allows one to construct directly an estimator with a smallconditional bias with respect to a statistic.

Consider TJ(s) = TJ(Xk, k E s) a statistic (where x is an auxiliary variable).Since the population is finite, TJ(s) takes only a finite number of valuesTJl, ... ,TJm (say). Let 0be an unbiased estimator of (). The conditional biasof 0 given TJ(s) is

B(O ITJ(s)) = E(O ITJ(s)) - ()

Hence, a conditionally unbiased estimator of () is

0* = 0- B(O ITJ(s))

Also,V(()*) = V(O) - V(B(O ITJ(s))]

because,Cov(O, B(O ITJ(s))) = V[B(O ITJ(s))]

(8.4.1)

(8.4.2)

(8.4.3)

Therefore, the unconditional variance of conditionally unbiased estimator0* is not greater than that of the original (unbiased) estimator of O. In thisapproach, however, the conditional bias B(O ITJ(s)) has to be estimated.

NOTE 8.4.1

If TJ(s) is a sufficient statistic then E(O I TJ(s)) is the improved estimatorover 0obtained using Rao-Blackwellisation technique. Also,

V[B(O ITJ(s))] = V[E(O ITJ(s))] ::; V(O)

NOTE 8.4.2

The conditional unbiasedness is a special type of calibration imposed onthe estimation procedure.

Consider a linear estimator of ii,

N

Y= L Wk(S)Yk = L wk(s)h(s)YkkEs k=l

(8.4.4)

8.4. CONDITIONAL UNBIASEDNESS 245

where the weights Wk(S) may depend on k, S and a function of x = (Xl, ... , XN)(but is independent of y) and !A;(s) is an indicator function (!A;(s) = 1(0) if s :3

k (otherwise) ). To make y conditionally unbiased we must have

A N 1 N

E(y 11](s)) = LYkE{Wk(S)!A;(S) 11](s)} = N LYkk=l k=l

or1

E{Wk(S)!A;(S) 11](s)} = N V k = 1, ... ,N (8.4.5)

Note that a necessary and sufficient condition for a conditionally unbiasedestimator (8.4.4) for y to exist is

E{!A;(s) 11](s)} > 0 V k = 1, ... , N

A solution of (8.4.5), provided (8.4.6) is satisfied, is

1Wk(S) = NE{fA;(s) 11](s)} Vk E S

Generally, a conditionally unbiased estimator does not exist.

Now,

E(L Wk(S)Yk 11] = 1]i) = ykEs

" P(1] = 1]i)::::} ~ Wk(S)p(S) = N V k such thats3 k;'1( s)='1;

P[k E S 11](s) = 1];] > 0

(8.4.6)

(8.4.7)

(8.4.8)

Therefore, conditions of conditionally unbiasedness can be written in termsof linear constraints (8.4.8).

EXAMPLE 8.4.1

Consider a srswor of size n(s) from a population of size N. Let 1](s) = n(s).Here, E{fA;(s) 11](s)} = n(s)jN V k = 1, ... , N. Therefore, a conditionallyunbiased estimator of y exists. The estimator Ys = nfu I:kEs Yk is conditionally unbiased.

DEFINITION 8.4.1 An estimator y in (8.4.4) is almost or virtually conditionally unbiased (ACU or VCl!) if

E(y 11](s)) = y + L CXk(1])Ykk:E(h(s)I'1(s»=o

(8.4.9)


where the weights CtIe(1]) can depend on 1](s) and k.

We shall now try to obtain conditionally unbiased estimator of y by modifying the method of deriving Horvitz-Thompson estimator y.". = k 'EiES~'

Let us define the conditional inclusion probabilities

E(h(s) 11]) = 1T1e!'/l E(h(s)h'(s) 11]) = 1TIeIe'!7J (k =I- k').

An estimator constructed with conditional inclusion probabilities will becalled a conditionally weighted (eW) estimator. Consider the simpleew (SeW) estimator

(8.4.10)

where we assume that 1T1e17J > 0 V k = 1, ... ,N. The conditional bias ofsew estimator is

where

N

B{y""!7J} = - L ~I[1TIe!7J = OJIe=l

(8.4.11)

[ J {I 1T1e17J = 0

I 1T1e17J = 0 = 0' hot erwise

It readily follows from (8.4.11) that Y""I7J is almost conditionally unbiased.The estimator (8.4.10) is also obtained ifHTE is corrected for its conditionalbias (Exercise 13).

Consider some other CW estimators:

(8.4.12)

(8.4.13)

wherehie = EI[1TIeI7J > OJ = P[1TIeI7J > OJ

The estimator (8.4.12) may be called corrected ew (eeW) estimator. Itsconditional bias is

B(y 11]) = .!- L[I(1TIeI 7J > 0) - IJN Ie hie

The eew estimator is not veu but is unconditionally unbiased. Bothsew and eeWestimators are not translation invariant i.e. these estimators do not increase by the amount e if all the units Yle are increased bye.The following two ratio type estimators are translation invariant.

8.4. CONDITIONAL UNBIASEDNESS

(i)The sew ratio:

"-, '"' Yk / '"' 1Y"'IIJ = L...J 'lrkl L...J 'lrkl

kEs IJ kEs IJ

(ii) The eew ratio:

=-, L Yk /L 1Y = kEs hk'lrkllJ kEs hk'lrkllJ

It may be noted that

E(h(s) I TJ = TJi) = 'lrkP(TJ = TJi IkE s)/P(TJ = TJi)

247

(8.4.14)

(8.4.15)

(8.4.16)

Therefore, to know the conditional inclusion probabilities we must knowthe conditional as well as unconditional probability distribution of TJ.

EXAMPLE 8.4.2

Consider srswor of m units drawn from a population of size N. Let n sbe the number of distinct units. It is known (Des Raj and Khamis (1958),Pathak (1961)) that

E(ns) = N(1 _ (N _1)m ) (8.4.17)Nm

V(ns) = (N - 1)m + (N _ 1) (N - 2)m (N - 1)2m (8,4.18)Nm-l Nm-l N2m-2

Suppose that n s is an anciliary statistic. Given n s , the sampling procedure is srswor and hence 'lrkln. = ns V k. The unconditional inclusion

probabilities are 'Irk = E('lrklnJ = E(ns)/N = 1 - (NN~)m. Now, Prob.('Irk Ins) > 0) = 1 and hence all the four CW estimators are identical andgiven by fls = L-kEs Yk/ns.

The HTE is

"- = nsys = {N(1 _ (N - 1)m )}-l '"'~ E(n) Nm L...J~

s kEs

Note that fls is conditionally unbiased. Again, fls is translation invariantwhile ys is not. Now

(8.4.19)

(8.4.20)


Hence, under certain conditions the CW-estimator Ys has a smaller variancethan y.One criterion of choice of TJ is to choose one for which the unconditionalmean square of the conditionally unbiased estimator is as small as possible.Considering the HTE and its associated CW estimator, it is seen from(8.4.3) that

(8.4.21)

The auxiliary statistic TJ must, therefore, be so chosen that Var[E(y" I TJ)]is as large as possible. The statistic TJ and y" must, therefore, be verydependent. Also, 1rkl'7 should be positive for all k, so that the conditionalbias remains small.

Assuming the joint distribution of y, a design-based estimator of bary and&, a design-based estimator of a q-variate random vector x = (Xl, ... ,xq)'

of mean of auxiliary variables Xj(j = 1, ... , q) is mutivariate normal, Montanari (1999) considered the conditional distribution of ygiven & and hencefound a bias-corrected conditionally unbiased estimator of y and studied itsproperties for several estimators. Mukhopadhyay (2000 b) considered theconditionally weighted estimator of a finite population variance (exercise15).

8.5 EXERCISES

1. Calibration estimator Suppose that a measure of average distancebetween dk and Wk is Dp(w,d) = Ep("£kES(Wk-dk)2/qkdk) where dk = l/1rkand qk are weights uncorrelated with Wk' The problem is to find calibrationestimator teal = "£kES WkYk of T such that Dp is minimised subject to thecalibration conditions

Show that the minimisation leads to the equation

where). is a vector of Lagrangian multipliers and is given by the equation

Ts = L dkqkxkx~kEs

8.5. EXERCISES 249

and is assumed to be non-singular. The resulting calibration estimator of T, , ,A A 1

is, therefore, ty,cal = t y1r+ (tx-tX1r )'B s where B s = Ts- LkEs dkqkXkYk whichis greg- predictor. In particular, when Xk is a scalar (Xk > 0), qk = l/xklshow that>. = t x/iX1r' Wk = dkt x/ix1r and hence iy,cal = t xi y1r /ix1r' ratioestimator of T.

(Deville and Sarndal, 1992)

2. Calibration estimator Consider the estimator T = LiEs bsiYi of population total T where bsi = 0 if i 1. s. Define the distance function

<Ps = L {bsi - b:i}2/qsibsiiEs

where qsi are known positive weights unrelated to bsi ' Show that minimising<Ps subject to calibration conditions

Lb:iXi =XiEs

where Xi = (XiI, . .. , Xip), X = (Xl, ... , X p), Xij is the value of the auxiliary

variable Xj on unit i, X j = L{:l Xij, one gets the calibration estimator

Tgr = T+ (X - X)'B

with

In particular, when bsi = l/7fi and qsi = qi,Tgr reduces to i ygreg given in(8.3.6) .

(Rao, 1994)

3. Calibration estimator of variance Consider the variance of eHT, HTEof population total T(y of Y, VTHT )YG = VYG defined in (1.4.16) and itsunbiased estimator VYG defined in (1.4.17). Consider a calibration estimatorof VyG ,

Vl(eHT) = L L Wij(y;j7fi - Yj/7fyi<jEs

(i)

where Wij are modified weights which are as close as possible in an averagesense to dij = (7f(lrj - 7fij)/7fij for a given distance measure and subject tothe calibration constraint

L L Wij(x;j7fi - Xj/7fj? = VyG(XHT )i<jEs

(ii)


where XHT = I:iEs Xd1ri, the HTE of population total X of an auxiliaryvariable x. Considerthe distance measure D between Wij and chj,

(iii)

(iv)

Show that minimisation of D subject to the condition (iii) leads to theoptimal weights given by

w .. _ A .. + chj%(xd1r; - Xjl1rj)2" - u" '" ",n (I' I) 4LJ LJi<j=l X; pZ; - Xj 1rj

n

VyG(XHT - L L (xd1r; - Xjl1rj?);<j=1

Substitution of (iv) in (i), therefore, leads to the following regression typeestimator

wheren n

B = [L L chj%(xd1r;-xjl1rj)2(Yd1r;-Yi/1rj)2]j[L L chj%(x;1r;-xjl1rj)4];<j=l ;<j=l

(vi)Show that for srswor and % = 1 V i =I- j

v1(T) = N2

(1- j) [s; + b(S; - s;)]n

wheren n

b= L L (y; - Yj)2(x; - Xj)2ILL (x; - Xj)4;<j=l ;<j=l

and f = niN. For srswor and % = (xd1r; - Xjl1rj)-2,

(TA) (TA ) [VyG(XHT)]

VI = VYG HT A

VyG(XHT)

(Singh, et aI, 1999)

4. Consider the problem of estimating a finite population variance

N N

S; = a1 L Y; - a2 L L Y;Y;';=1 #;'=1

8.5.

where

EXERCISES 251

(i)

1 1 1al = N(l- N)' a2 = N2'

It is known that the following Horvitz-Thompson type estimator due to Liu(1974) is unbiased for S;,

where dk = 1/7fk, dkk, = 1/7fkk' and L:s' L:: denoteL:kEs and "L:k;6k'ES'respectively. It is desired to find calibration estimator of S;,

where the weights {Wk} and {Wkk'} are as close as possible to {dk} and{dkk'} (in some sense) and which satisfy the calibration constraints

N

L wk(xD = L(x;)j=l

where

N

L Wkk' (XkXk') = L (XjXj')j;6j'=l

(ii)

(xD = (Xil"" ,xip)', (XkXk') = (XklXk'l>'" ,XkpXk'p)'

and it is assumed that the totals in the right hand side of (i) and (ii) areknown. Here Xkj is the value of an auxiliary variable Xj on unit k in thepopulation (j = 1, ... ,Pi k = 1, ... , N).

Consider a distance function G with argument z = wid measuring thedistance between Wk and dk, Wkk' and dkk, and consider

s

as a measure of distance between the weights Wk, dk and Wkk', dkk' for thewhole sample. Considering the linear method of Deville and Sarndal (1992)where' G(z) = (z - 1)2/2, z E R 1 show that the calibration estimator of S;is

252

where


p N ,

a2 L)L L XijXi'j - L dkklXkjXkljyT.jj=1 i#i'=1

A ( A A )' A ( A A )'B s = B s1 ,···, Bsp , Ts = Ts1 ,··· ,Tsp

and Bs , Ts are obtained by solving the sample-based equations

(Mukhopadhyay, 2000 c)

5. Find a calibration estimator of a finite population variance

1 NFN(t) = N L .6.(t - Yi)

i=1

where .6.(z) = 1(0) if z 2: 0 (elsewhere)

(Mukhopadhyay, 2000 b)

6. Post-stratification Show that under the superpopulation model

Yhi = J1.h + Ehi, i = 1, ... , Nh; h = 1, ... ,L

E(Ehi) = 0, E(E~i) = a~, E(EhiEhi/) = 0, i i= i'

the best linear unbiased prediction estimator of Yh is

y~ = L Yhi + (Nh - nh)fihs = NhYhsiEs.

and hence the BLUP-estimator of Y is

y' = L Nhfih/N = Yposth

The model bias of y' is E[(y' - y) I n) = "'£hJ1.h(nh/n - Nh/N). Theprediction variance of y' is

E[(y' _y)21 n) = L(nh/n)2(1-nh/Nh)ah/nh+{LJ1.h(nh/n-Nh/N)2}2h h

8.5. EXERCISES 253

(i)

+L a~(Nh/n2)(nh/Nh - n/N)2h

If, further, we assume that Ph rv N ID(v, T2 ), Bayes estimate of Y is

fiB = L {nhYh+(Nh-nh)AhYh+(Nh-nh)(I-Ah)Yh+(Nh-nh)(I-Ah)y}/Nh

where7 2

Ys = L AhYhs/ L Ah, Ah = 2+ 2/h h 7 ah nh

Hence, show that when Ah ~ 1, Le. when aVnh is very small compared to7

2 V h, (i) reduces to Ypost.

(Holt and Smith, 1979)

7. Post-stratification Assume that a population is divided into two strata.A srs of size n is drawn and the sampled units are identified as falling intoeither straum 1 or 2. The following types of outcomes are also identified: (i)both the strata contain some sampled units (ii) one of the strata containsno sampled unit.

Consider the following estimator for the population mean y.

For case (i), y= Ypost = W 1Y1 +W2Y2 where W 1 = N1/N, W2 = 1-W 1,Yi =sample mean for the ith stratum.

For case (ii)," {D1Y1S if stratum 2 is emptyy=

D2Y2s if stratum 1 is empty

Show that y is conditionally unbiased if D 1 = W1 / P{ ,D2 = W2 / P;, whereP{ is the probability that the stratum 2 will contain no sample elementgiven that the case (ii) has happened and similarly for P; .Show that the conditional variance of y for case (i) is

2

(A) ['"' 2N i - ni 2]VarI Y = EI L....J Wi nN Si

i=l I I

where EI denotes expectation over all outcomes associated with case (i).Show also that the conditional variance of yfor case (ii) is

2 2

V ( 0.) _ 1 '"'P*D2(Ni - ni)S2 ('"' lV; 0.2 0.2)arII Y -; L....J i i N i + L....J P* Yi - Y

i=l I i=l I

254

Hence, show that


where PI, P II, respectively, are the probabilities that cases (i), (ii) occur.

The common procedure in case (ii) is to combine or collapse the two strata.Denoting this estimator as Yc' show that the conditional bias of this estimator is

Therefore,

( ~) ~) I~*( 2) Ni-ni ) 2MSEII Yc - VarII(Y =:;:;: L.J Pi 1- Di ( N Si +i •

Hence, comment on the relative performance of Yc and y in case (ii).

(Fuller, 1965)

8. Post-stratification A simple random sample of size n is drawn from apopulation of size N divided into L post-strata of size Nh(h = 1, ... , L).Let nh be the sample size in the hth stratum. Show that the estimator

Ys = L ahPhYhs!E(ah)h

where ah is an indicator variable,

at least one sampled unit E stratum helsewhere,

Ph = Nh!N, Yhs = sample mean for the hth stratum if nh f= 0 (f}h if nh = 0),is unbiased for y.

Show that the estimator Ys is conditionally biased,

I:' being over all post-strata with nh f= 0, with n = (nl,.· ., nL)'

8.5. EXERCISES 255

Show also that if each y-value is changed to y + c(c, a constant), V(y~)

(where y~ denotes the estimator of y for the changed values) can be madeas large as possible by the proper chioce of c. Show that the variance ofthe ratio estimator of y,

R = 2::h ahPhYh/E(ah)2::h ahPh/E(ah)

does not depend on the shift in origin of the y-values. Find an exactexpression for the variance of R and its variance estimator.

Also, consider the estimator

Find its variance and variance estimator. Show that k is not unconditionally (over all n) consistent for y.

(Doss, et aI, 1979; Rao, 1985)

9. Post-stratification Suppose the population has been divided into Lpost-strata U1 , ... ,UL and the relationship between y and x in the poststratum (henceforth called stratum) is

(i)

E(fhj) = 0, E(f~j) = oL E(fhjfh1j') = 0, (h,j) =f. (hl,i')

It is assumed that for every sampled unit its x-value and stratum-affiliationis known and for units not in the sample their x-values are known but notthe stratum affiliation. Such a situation may arise if x is not the stratifyingvariable. Hence, the population size N and population mean X are known,but not the stratum means Xh and strata sizes Nh required for use ofseperate regression estimates.

Now an optimal predictor of y is

where ih is the optimal model-dependent predictor of Yj' Again,

L

y(Nh, Xh) = ~ (LYj + L L Ph(j IXj)(&h + ~hXj)} (ii)jE. jEs h=l


where iij (t}hs - /3h Xhs) , /3h is the ordinary least square estimator of(3h, Ph(j IXj) is the conditional probability that a unit j , selected at random out of s belongs to Uh , given that its x-value is Xj, Le.

Ph(j IXj) = Prob. {j E Uh I j E s,Xj = Xj}

Nh(Xj)

N(xj)

where N(xj) = L.hNh(Xj),Nh(Xj) is the number of units j E Uh,j E swith x-values equal to Xj> Hence, show that when Xh and N h are unknown,

an estimate of Y is obtained by estimating Ph(j I Xj) by :~~:~;, where

nt(xj) is the number of units in the sample from stratum h falling ina pre-assigned interval of the x- values which include the value Xj and

n+(xj) = L.~:dl nt(xj)

(Pfeffermann and Krieger, 1991)

10. Post-stratification Consider the post-stratified estimator

L

Y~s = L WhaYshh=l

where Who = o:~ + (1 - o:)~ and 0: is a suitably chosen constant (othersymbols have usual meanings). Show that Y~s is unbiased for Y with variance, to terms of 0(n-2), given by

L

(-I) (1 1 ) L 2 2 1 1V y = - - - WhSh + 0: (- - -)pS n N n N

h=l

L L

(S2- L whSD+(I-o:)2 2~-n )L(1-Wh)S~n N-l

h=l h=l

Also, the minimum variance is given by

where O:opt denotes the optimal value of 0: and is given by

8.5. EXERCISES 257

L L

Al = N;. 1(S2 - LWhS~) and A2 = ~L(l- Wh)S~h=l h=l

Hence, show that Y~s;opt is more efficient than both Yps = z=Ll WhYsh and

Ys(= Z=~=l nhYshln ) to the above order of approximation.

(Agarwal and Panda, 1993)

11. Consider srswr of n draws from a population of size N, resulting in vnumber of distinct units. Let Y" be the mean of the values on distinct unitsand YHT the HTE of y. Suppose we condition on the observed value of v,i.e. the relevant reference set is 51.' of (~) samples of effective size v. Showthat

E2 (y,,) = Yv

E2(YHT) = E(vl =F Y

where E2 , 112 denote, respectively, the conditional expectation and variancegiven v. Show also that an estimate of V2 (Yv) is

where S~y = "~l Z=iES(Yi - Yv)2,S being the set corresponding to the wrsample s. Another alternative estimator of V2(Yv) is

which is, however, conditionally biased.

(Rao, 1985)

12. Outliers Suppose the population contains a small (unknown) fractionW2 of outliers (large observations), Wl >> W2 (Wl +W2 = 1); Y2 >> Yl, Yidenoting the population mean of group i, (i = 1,2). If the observed samplecontains no outliers (i.e. W2 = 0), then E2(WlYls+W2Y2s) = Yl« Y whereE2 is the conditional expectation given (nl> n2), Wi = n;j(nl + n2), nidenoting the number of sampled units in group i and the other symbolshave usual meanings. On the otherhand, if the observed sample containsa outlier (W2 > 0), E2(ys) - Y = (Wl - Wl)Yl + (W2 - W2)Y2 » 0, sinceW2 > W2 and Y2 large. For example, if N2 = 1,W2 = lin » W2 = liN.In this situation consider the modified estimator

N- n2_ n2_Y; = N Yls + N Y2s


Show that the conditional relative bias of Y; is given by

B2~Y;) = 8(w2n _ W2)Y2 N

where 8 = (Yl - Y2)/Y2. Again,

B2~YS) = (W2 - W 2)8Y2

Hence, show that the estimator Y; has less conditional bias than Ys.

(Rao, 1985)

13.Non-response Suppose m responses are obtained in a simple randomsample of size n and let Yl denote the population proportion in the responsestratum (group of units who would have responded if selected in the sample), Y2 the same in the non-response stratum, Y = W1Yl + W 2Y2, W2 =1 - Wi' Let p* be the probability that a person when contacted responds.Show that under this situation, conditionally given m, the sample Sm ofrespondents is a simple random sample of size m from the whole population and hence the sample mean Ym is conditionally unbiased. On theotherhand, the Horvitz-Thompson estimator (P* known)

_ m _ '" YiYHT = E(m) Ym = L..J ;;-;

.Esm p

is conditionally biased, although unbiased when averaged over the distribution of m.

(Oh and Scheuren, 1983; Rao, 1985)

14. Domain Estimation Under srs, the usual estimator of a sub-populationmean Yi is

Yis = LYj/ni' ni > 0jEs,

where Si is the sample falling in domain i and ni is the sample size. Theestimator Yis is conditionally unbiased (given ni) if ni > O. The estimatoris however, unstable for small domains with small ni. Consider a modifiedestimator

_I ai_Yis = E(ai)Yis, ni 2: 0

where ai = 1(0) if ni > 0 (otherwise) and Yis is taken as Yi for ni = O. Theestimator yis is, however, conditionally biased,

E (-I ) ai_2 Yis = E(ai) Yi

8.5. EXERCISES 259

(i)

Sarndal (1984) proposed the following estimator in the context of small-areaestimation:

where Ys = Li niYis/n = Li WiYis is the overall sample mean and lV; =N;fN. The conditional bias of YiS is

where yt = Li WiYi' If ni = 0, the estimator YiS reduces to Ys' However,YiS would have a larger absolute conditional bias (and a larger conditionalMSE) than Ys if Wi > 2Wi •

Hidiroglou and Sarndal (1985) proposed:

if Wi ~ Wi

if Wi < Wi

Y;S is conditionally unbiased if Wi ~ Wi, while its conditional absolute biasis smaller than that of Ys if Wi < Wi'

Drew et al (1982) proposed

YiD = { YisYiS

if Wi ~ Wi

if Wi < Wi

If a concommitant variable x with known domain means Xi is available,then show that the ratio estimator

and a regression estimator

- - + Ys (X- -)Yilr = Yis -=- i-XisXs

are both approximately conditionally unbiased.

(Rao, 1985)

15. Conditionally Weighted Estimator Find an conditionally unbiasedestimator of bias of the HTE, y", and show that a conditionally weightedestimator of Y is Y"'ITJ'

16. Conditionally Weighted Estimators Suppose we want to use theHTE, conditioned on the value of ~'" = *LkES ;; where x is an auxiliary

260

variable. Show that


~ 1 '""" Xl 1 XkE{x". IkE s} = - £..J -1rllk + -- = Xlk (say)N l(#)k 1r1 N 1rk

'""" '""" XIXm 1rlk1rmk+ £..J £..J (1rlmk - --) = V"'lk (say)l#m(#k) 1r11rm1rk 1rk

Also, show that

E{I I" } P {x". = z IkE s}

ks X". = Z = 1rk P{" _ }X". -z

(i)

Assume that X". rv N(x, V(y".) = V", (say) ,x". IkE s rv N(xlk> VzIk) and1rk ~ njN V k. Hence, using (i) show that

N f(x".)7l"klx" ~ - f (" )n k X".

where f(.) is the pdf of £". and ik(.) is the conditional pdf of (x".) givenk E s, as stated above. Hence, write an approximate expression for theconditional weighted estimator of y.

(Tille, 1998)

17. Conditionally Weighted Estimators Find conditionally weighted estimators of a finite population variance S; = L:~=1 (Yk - y)2 j(N - 1) andstudy their properties with special emphasis on simple random sampling.

(11ukhopadhyay, 1999b)

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