ONE WAY TO BUILD AN ESTIMATOR WITH APPLICATIONS TO SAMPLING Ti~ORY 

Steve Woodruff, Bureau of Labor Statistics 

Room 4985, 2 Massachusetts Ave. N.W., Washington D.C. 20212 

variables attached to the i th member of the population. 

Key Words: Nonresponse, Restxmse Error, 

Multivariate Normal 

1) INTRODUCTION 

In survey sampling problems data is usually 

being collected on many study variables, some of which 

(variables of primary interesO are positively correlated 

with the sample design variable(s) and other variables 

which are not (,peripheral variates). A sample design 

which is nearly optimal for estimating (via HorvitzThompson, HT) the means of those variates which are 

positively correlated with the design variable may be 

extremely inefficient for estimating (via HT) means of 

peripheral variates. This problem can be addressed by 

post stratifying on one or more other variates which are 

cormla~d with the peripheral variates being gathered 

from the sample units. This paper describes a way of 

building an estimator, which makes use of these other 

sources of data together with the relationships between 

these data items to reduce the error of estimates of 

peripheral variate means. 

The class of estimators to be built are 

Let Yi = (ai • ti), then a i is known for all ieU (the 

population), t i is unknown for all i not in the sample 

(s), and for each ies some subset of the components of t i 

is known and this subset varies from sample member to 

sample member (item nomesponse). 

4) Suppose it is appropriate to describe every 

Yi (whether its components are observed or no0 in the 

population as the outcome of a vector valued random 

variable ,Y=(A,T), with mean tt =(J~tA,tt T) and 

varianceJcovariance matrix Z. Thus the {Yi} are the 

outcomes of independent and identically distributed 

random variables {Yi}, each distributed as Y--( l.t, Z). 

Using only the data outlined in I) through 3) 

and data relationships described in 4), an estimator of 

the target means (ttT) that minimizes expected squared 

distance between itself and tt T is constructed. 

Expectation is over the distribution generated by the 

distribution of Y (which can include the sampling 

distribution and, if known, the response mechanism). 

The solution descn~ here is a direct 

generally multivariate and utilize data dependencies, application of the Gauss Markov theorem. The 

that can be captured in a covariance matrix. These estimator is linear in the available data and unbiased in 

estimators minimize /vISE in the presence of the appropriate space. With these constraints, 

nonresponse, response bias, and weak relationships knowledge of second moments of Y (Z) will be enough 

with the sample design variables. Since they directly to construct a best (minimum variance) solution to the 

use the type of data dependencies that are exploited by . problem of estimating l.t T. This solution is also 

composite, Bayes, ratio, and regression estimation, they 

can achieve, by default, the same reductions in mean 

square error that these techniques also provide. The 

same is true about nonresponse adjustment, the 

information that is used to pc~orrn this operation is 

also included in the estimation process. 

A vector of finite population means (target 

means) of the study variables (target variables) is to be 

estimated under the following setup. 

1) A sampling frame which identifies 

members of the finite population and contains 

quantitative data on a fixed set of characteristics 

(auxiliary variables) for each population unit. 

2) From this sampling flame, a sample is 

selected. This sample may be selected by a known 

randomization ~ure, possibly a function of the 

auxiliary data. A stratified, clustered, multistage 

sample is the general rule. 

3) For each member of the sample, some 

subset of the target variables is observed and this subset 

may vary from sample member to sample member 

(item nomesponse). Let a i be the row vector of 

auxiliary variables attached to the i th member of the 

population and let t i be the row vector of target 

Maximum Likelihood given the missing sample data 

(see Little & Rubin 1987) and a timed point of the EMalgorithm. 

In repeated surveys, Z may be estimated from 

historical data and relatively stable over time as the 

expected values of the targets and auxiliaries change. In 

such cases, it may be appropriate to use this estimate of 

X as the actual quantity. X may be derived from 

supertx~ulafion models, sampling distributions, or 

combinations of both. The sample indicator function 

and possibly the response indicator functions may be 

tre.zl~ as auxiliary (poststratification) variables. Using 

sample and response indicator variables in this way 

provides the sampler with a way to adjust for 

nonignorable non response (& response bias) in cases 

where the nomesponse mechanism can be given an 

accurate stochastic description. 

Result I in section 3 quantifies the effect on 

MSE of deleting some of the target variables from the 

data structure, described above for Y. Result 2 in 

section 3 quantifies the effect on MSE of deleting 

auxiliary variables from Y. The theory 

presented here is applied in the Johnson, Woodruff 

(1990) paper. 

1013 
2) BUILDING AN ESTIMATOR 

a) An Example- S~, the problem is to 

estimate the number of production workers and women 

workers in a small indusuT. A simple random sample 

of three of the 26 firms in this indusu'y was se.lccmd but 

the firms in this sample wee less than ~ve. 

Only one firm provided both the numbers of its 

Im3ducdon workers (P) and its women woxkm's (W), 

one gave dam on only women workers and one gave 

only production worlmrs. In addition, the total 

employment (E) in each firm in this industry is known 

as is the matrix of variances and covariances for and 

between the random variables ~ing total 

employment (E), women workers (W) and production 

workers (P). This sample dam (ouw, cxnes of (E,W£')) 

ar~: 

E W P 

1 92 35 71 

2 90 44 - 

3 85 - 75 

The average employment in this industry is 76 

and the variance/covariance matrix of (E,W,P) is: 

131 101 120 / 

~,170 120 190J 

Note that all three employment variables are 

positively correlated. Employment (E) in this example 

is the auxiliary variable and the pa~, (Women 

Workers, Production Workers) = (W,P) are the target 

variables. 

Suppose (E,W,P) are approximately normally 

distributed. Since the average employment in the 

industry is 76, the data on the variables of interest from 

all three sample members are positively biased and 

everything needed to compute the bias (conditional on 

the known Employment dam) in the observed values of 

W and P for each sample member is at hand. These 

biases (under normality) for a sample member are the 

sample member's employment minus 76 times .64 for 

W and times .83 for P (for example, the bias in W for 

firm one is 10.24 and the bias in P is 1328). The 

coefficients, .64 and .83, come from the expression for 

conditional expectation of the targets given the 

auxiliary variable under multivariate normality. The 

vector (.64, .83) results from multiplying the reciprocal 

of the (1,1) component of the covariance matrix, 1/7235 

, by the row, (131,170). 

Subtract these biases from the obsezved 

measurements on the targets to get bias adjusted 

obsercations. 

Establishment W p 

1 24.76 57.72 

Q 

Q 

2 35.04 -- 

3 --- 67.53 

These bias adjusted variables are now more 

appropriately thought of as outcomes of lid random 

vectors with the target mean as their common expected 

value and with a common vatiance/covatiance matrix 

that measures a smaller dispersion ammut this target 

mean. This new (conditional) variance/covariance 

matrix is ~11.37 49.02) as opposed to k120 190 

prior to conditioning on E. 

The responding item means of the adjuste~ 

data would be unbiased estimates of the target means. 

These target means may also be estimated by imputing 

for the missing data items and taking the column 

means of both adjusted and imputed data. This second 

estimate of the target means would have a smaller 

variance than the first. Finally, by al~lying the EMalgorithm under normality to reimpum and reestimate 

until convergence a third estimate is obtained with a 

smaller variance than either of the two above. In this 

example, this estimate under the EM-algorithm 

converges to (30.288, 64.188). When the 

vadanceJcovadance matrix is known (as in this 

example) and must not be reestimate~ with each 

iteration of the EM-algorithm, there is a dixe~ closed 

form solution that yields exactly the same resuk as the 

EM-algorithm. This is done via the Crauss theorem on 

minimum variance estimation by using the following 

linear relation between the bias adjusted data and the 

target mean, ~. 

24+51 'i 57. 

35. 

67.53) 

I 17.29 0 0 

11i37 0 0 

17.29 0 

0 49.02 

where Var(~)= 

0 

i P.T+8 

11.37 

49.02 

0 

0 

and E(e)--o. 

Rewriting this as 

where 

,Y = XI.tT + ~ 

Vat'(E)= Z, we get: ~T = (X' Z-1X) -1X' Z-1Y = 

(30.288, 64.188)', exactly as with the EM-algofithm. 

The observed data maximum likelihood 

estimator (MLE) as described in Chapter 7 of Little k 

Rubin is, under normality, the GLS ~[or, 

~t T = (X' E-1X) -1X' E-IY. This estimate is also a 

fixed point of the EM-algorithm by the theorems in the 

1014 
theory section of this chapter. Thus the Normal EMalgorithm with E known will convc=gc m the 

s'iv~ above. 

b) Generalization of the Example in a)- 

There m'e two p~ to ~on once d~ 

se~ of appmpriam mrg~ variables and auxilim'y 

variables has been dem'mmed. Phase one is 

posmncificafion on the awdliary ~les (using the 

r~~on ad~m~ms under conditionality). 

two is ~uival~t m imputation and escim~on wid~om 

explicitly going fire, ugh the coml~l~ional rigors of file 

EM-algcdthm. 

The available dam on d~e population c~ 

inxnv.s~ is relm~md as the ~ oamome of an 

NxCc~+kt) ~ W of random ~ks. N= 

number of uni~ in the popula~n, ]c a = ~h¢ number af 

auxiliary ~les, k t = the number of mrge~ ~1¢~, 

and n= the rumple size. The i thmw of W c.onmin$ the 

random vm-iable.s (auxiliaries and targets) associal~ 

wkh ~h¢ i th popul~on uah. 

By d~fini~n, the aur, fliz~ ~es are 

known for all population units and the target vadable 

ouw.omes are observed only for rumple ~ and, due w 

nonmspon~, we may observe only a subset of the k~. 

mrge.~ far each sample unk and this ~a~t ~~ from 

uait m unit.. 

I.~t Yi = (Ai , TO, the i th mw vectar of W 

and X i ffi , Yi ° = (Ai, Tizi), = 

Zi 

(I o 0)=(Ai, Ti)Xi. (A i , Ti ) Zi 

Let e i model the differmce ~ r~-_-~ 

and expected values of the compcmmm at" Y~", fl~n: 

Y7 ffi(gtA, ~tT)Xi +ei" 

All available infanna~on (polm~m dam 

sample dam, and the smc~m~ic ~,~~'ms 

them) is ~,~,~-~ in a ~ m~ 

(Y~, Y~, Y~ ...... 'in°)= 

(].tA, ].tT)(Xl, X2, X3 ...... Xn) 

where (el' e2'e3 ....... en )-N(0" ]: 0 )' 

Y 

EyO is the block d~ml ~ of ~{X i' ~X i }, 

andZ is the covm'knce man~ c~ ~he ~, (Ai,Ti). 

Writing this in COml~Ct form : yo= I~X + e whcm~ e 

has mean zezo and covafinnce man-ix Z y0. 

' Now anxilim'y ~les are reed to Mjust for 

~hc peculim'ifies of ~h¢ realized rumple (and possibly 

rmpome bias) This is done by post szr~_~~n on 

containing the two random veclors A i , the Ixk a vecu~ 

of atmiliaries (oumomes known), and T i, me ix~ Thea by a Im:medum rhal is similar m impm~cm or the 

random veaor of target vm-iables, the o~mom¢~ of ...EM-algorithm, the observed dam imms are used m 

co~ for mi.m~, g dam. 

which may b¢ unknown (if i is no~ a sample member) ~zZTAA ) 

or ~y known ff i is a sample member. From ~ I.¢t Z = ZAT whe~ ~ A is 

point on, re~finc i slightly m dehorn only smmplc ZT ' 

nmmbem, i runs fzom 1 to n (tim sample size) sinc~ 

only sample membc~ will be considm~l in what 

fdlow~ 

The obscrv~ ~ and all tl~ nnxilimy 

~les for ~ i th sample unic am the compomm~ of 

the ~ of the mndom 

vcxm~. Y~ = (Ai, TiZi ), wh¢~ Zi is the respcmse 

indicator roan'ix far am row ~ of ~ variables 

T i amr.he, d m ~ i th sample uniL Zi is 

from ~ ~ ~ of ord= k~ (the number ¢f 

~l~s) by deleing ~ colnmn j of 

~ty ~ far which the ~h ~ ~le is a 

n~nse. Thus ~" contains the i th units' 

auxiliary variables and ¢mly ~hose target vmriable$ 

observed for the i ~ unit. ~g I be the ideafiW 

ma~-ix of ord~ k a (the number of atmiliary v-~i~le$) 

• .,z~mc~cov'mSanc~ manix of a T i . and ~AT is the 

manix of covm'knces between A i and Ti.. The 

value of Yi=(Ai, T i) is ~(].t A ;],t v ), and by 

defininon ~A is known. When Y is mulfivm-iam 

normal ~ condi~nal pn~n~s of the m~ 

normal dism-m~on am exploimd m make inferences 

abom ~ Imsed on the conditional ~ af tie 

mrs~ ~m given the ama'R~y vambt~s. 

E(T~ 1,4, = a,) = ~'r + (a, - tt~ )Z~Z#. Ttz 

walue of T i is i,,fl,,~~ by ffs 

,,m~ry om~om~ A,~. This ~ ~ ,~ea ai. 

~ta, and E ~ all known, the ~ dam ~ l~ 

n-anslamd as follows: 

tJ~ conditional e~pec.~ value of z i giv= ~-=a i is ~t r 

,r~quanfiv/m be~ The {Zi} ~~on 

[£r and the dispersion of the {Zi} about [£r is fighzer 

1015 
by ~rAE~ ! EAT; that is, the conditional 

varianceJcovariance matrix of (ZilAi=a i) is 

z8 =ET- ZTAE:, E . 

The observed comtments of Z i (responses) are 

separated f~em the unobserved components (non 

responses) by post multiplying Z i by Xi and in order to 

avoid introducing another variable name, now let 

Zi= [T~ -(A, -~t A )Z~t~Zar ]Xi , (2.0) 

then E tAi)= trZi and VCZitAi)= 

Zi'(~r " ~r~A~Lr)Zi --'--'~i~i" Letting ~i model 

the difference between the reai/zed and ~ted value 

of (ZilAi): 

Zi= BtZi+~i and summarizing over all sample 

units: 

(Zl, Z2, .... Zn) --" lJ't (Zl, Z2 ..... Xn) 

+(~1, ~2,''" 8n) 

or Z=IJ.rX + 8, (2.1) 

where 8 has mean zero and covariance matrix 7.. z . 

and E z is the block diagonal matrix of the {X~EaXI }, 

0 . . 

0 Z2 :8 '2 0 . 

• 0 • • 

0 . . 0 

0 '~ 

o 

0 

X,,EsZ',, ., 

From (2.1), the generalized least squares 

(GLS) estimator for IJ. r is: 

^ = (~: ,~: . The 

of (XZ: X ' 

This estimator seems to ignore the sampling 

distribution. As explained in the next section, the 

sampling distribution is often implicitly included in the 

estimation process. Section 3 contains two results (or 

theorems) which quantify the effect on bias and 

variance of omitting target and auxiliary variables from 

the construction of the estimator outlined above. 

3) STRU~E THEOREMS 

Let Af(A 1,A2), and T = (T1,T 2) be partitions 

of A and T into subv~ In this section, two results 

are proved which, quantify the effect on MSE of 

deleting the data on T ~ from W and the construction of 

the GLS for I.tr~= Ecr 1) and the effect on MSE of 

deleting the data on A 2 from the construction of the 

GLS for ~t T = EO'). Superscripts denote subvectors. 

Bias and variance of the GLS estimators under 

these two forms of reduced data structure are 

evaluated with respect to the probability space that 

these Gl.~s inherit from the stochastic structure on 

(A,T) and all the data on these variates. 

Definition 1. A 2 is redundant for estimating I~T if 

and only ff the conditional random vector, T given A1, 

is independent of , A 2 given A 1 , 

[(TIA ~ ).L(A21A ~ )]. 

Definition 2. T 2 is ~dant for estimating IJ.T~ if 

and only ff T 2 given A, is independent of T 1 given A 

[(T~IA).L(T21A)]. 

This last definition is reciprocal, it also 

implies that T1 is redundant for estimating I.l.~ • 

A target or auxifiary variable is redundant for 

estimating a particular target mean if neither its 

inclusion in nor omission from the consm~ction of the 

GLS has any effect on the estimator of that target 

mean. 

Definition 3. A variate (or vector of variates) that is 

not redundant is called pertinent. 

The next two results are the structure 

theorems. Their proofs in the case of multivariate 

normality are by direct computation and the Gauss (- 

. Markov) theorem on minimum variance estimation. 

Bias and variance of these "deleted" GLS estimators are 

with respect to the full stochastic structure on (A,T), all 

the data on these variates, and conditioned on the 

known sample outcomes for A. The proofs below 

depend on (and possibly clarify) this last statement. 

. Result I - Deleting pertinent target variables, T 2 from 

W. will increase the variance of the components of 

A 

~tr~t, the GLS for J.LT1 compared to the variance of 

the corresponding components of ~ and will have 

no effect on their bias. 

Proof of 1 

a) By ~ging the components of Z, (2.1) 

can be rewritten with block diagonal X-matrix as: 

Z=(ZI'z2)=(BTI'ttT2)(Z~I Z220 )+(81,82)(3.1) 

where Z 1 contains o~y members of the first set of 

target variables, T 1, Z 2 contains only members of the 

second set of targets, T 2 , and the covariance matrix of 

(~1,82) is ~y21 ~z2 ' a rea3mangement of Y-'z. 

b) Note that by ignoring T 2 in the construction 

of (3.1) we would have: 

zl =/2r, Zll + ~ 1 (3.2) 

where the variance/coy•fiance matrix of ~1 

islE11. 

1016 
e) The GLS for ]2 r, under (3.2) is /2r, ~. It is 

linear and unbiased under (3.1). By the Cmnss 

theorem, the first T 1 components of ~ are minimum 

variance linear unbiased for ]274 under (3.1). 

Therefore, V(~.~) ~ V(~r, ta ) where Q is the 

matrix of zen~ and ones that picks out the T 1 

components of ~TIA , V denotes variance of the 

enclosed vector, and < is component by component 

comparison. 

Finally, since E(~r,~t) = E(~Q) under 

(3.1), deleting targets has no effect on bias. END PF1. 

Corollary 

F~n~Q = f%~ ¢~ Cov(r',T21A) = O. This 

corollary shows that the definition of redundant target 

variable makes sense; when T 2 is redundant, the 

estimate of I~rt can borrow no strength from data on 

T 2 . 

]Result 2. - Let ]271, 0 be the GLS for the mean of T 

when the auxiliary variables in A 2 are omitted fixnn W 

and the estimator construction described in the previous 

section. Conditional on A-(A1,A2), Var(~,~ . 

Var(~Lrt a) and ~ZnA, is a biased estimate for the mean 

of T, ~r. ">" is component by component 

comp~n. 

Proof of 2 

Let the varianceJcovariance 

(A,T)=(A I,A2,T) be: 

(E~ Z~2 Z~T 

E =/Era Z22 Z2T . Then ignoring A 2 and 

~ET1 ET2 ETr 

$ 

conditioning on A1 alone, the analog to Z i is Z i = 

W i - 1 it h (a~-~1., )~:1~:1T , where a i is the unit's 

realized value for A 1 and ~l.t is the exlx~ted value of 

A 1. The varianceJcovariance matrix of Z~ is 

]ETT -- Y'T~ ]Ex~ ~T" The varianceJcovariance mau'ix, 

Z z , given in (2.2) for the vector 

z=(z~.z2,z3 ............ ,Zn) has ZO • • lid • =(Z, ,Z2,Z 3, ........ ,Z.), 

,Z z" and is given by 

Zs = Zs. = ZTT -- ZT~ ZI-~ Z~T" 

an analog for 

that is denoted 

(2.2) with 

Conditioning on 

[(aI - !~ ~),(a 2 [E,t 

- ~,~ )lie2, 

'rZ T 

Z22 Lz2T] 

= (a] - Ix., ) Z~'~ Z,T + (a I - It. d )GFG' Z,T 

-(aI - I.t~ )GF~2T - (a 2 - Ix.2 )FG' E,T 

+(a 2 - B,2 )F~2T 

)-I whereG = Zi~Zi2,F = (Z22 - Z21Z~'~Zt2 . 0 

2 denotes transpose, and at and I.t,z are defined 

1 analogously to a t and ~l.d. Thus the difference in bias 

adjustment between conditioning on A and 

conditioning on A1 alone is given by:. 

I Diff i = (a i -- ~t,, )GFG' EIr -- (a~ - It.,, )Gl~zr - 

2 (ai 2 -g,~ )FG' E1T + (a t --I.t, )F~zr = 

s 

Z i - Z i, 

Conditioning on both A1 and A 2, Diff i is the 

bias in Z t when A 2 is ignored. Let D = 

(Difflz I ,Diff2z 2 , ........... ,Diff, z . ), then, after 

deleting the nomes~nses from Z i and Z i , (see 2.0) 

and forming Z and Z*, 

Z*=Z + D. The GLS given both A1 and A 2 is: 

~v~ = Z~zXZ ' (Z~zlZ ' )-x , and the GLS ignoring 

matrix of ' A2 is: 

~r~' = CZ + D)E~ X' (ZZz x. Z' )-'= 

-~Z' )-1+ (ZZ~ 

7~ z. (XEz ~- Z' DEz'- Z' X' )-' = 

-~ z' ( zz~ z' )-~ 7~z. + BIAS, where 

conditional on both A I and A 2 , BIAS is the constant 

DZ~'. Z' (ZZ z t Z' )-~. The ~.c~co~.nc~ 

matrix of ~, given A is the varianceJcovariance of 

-IZ, Z-IZ,)-I 7~z. (Z z" conditional on A=(AI,A2), and 

this matrix is 

-1Z, Z-1Z')-I (ZZzl. X, )-I Zy~z,.BZZz. (Z z" . Note that 

term-by-term, the diagonal 0£ 

Z' Z' )-1 (Z~: ~ ~, )-1ZZz 1. ZZy~z ~ (Z$:z I. is greater 

than the diagonal of (Z~zlZ') -l , the 

variance/covariance matrix of ~ . This inequality 

follows because ~v~ is the GLS conditional on A and 

both subvectors of A, T i uansforms to Z i by subtracting 

the following expression from T i. 

1017 
Z~ z 1. ~' (XI: z 1. ~:' )-1 is linear and unbiased. 

Thus MSE(~TIAt) = 

(~Z z 1. ~' )-1 Z-I -1 ~, ~, )-1 z.ZZZz. (~Zz 1. + 

03IAS)'(BIAS) 

and the diagonal elements of the first term in this MSE 

are greater than the corresponding diagonal elements of 

^ 

Var(tt~A). 

~ID Pf2 

-1 ~, Z-1 Z' )-1 Z~z. (Z z" in the above proof is an unbiased 

estimate of the mean of T but, conditional on 

A=(A 1,A2), this estimator has less than optimal weight 

-1 Z' Z' )-1 vectors, I:z. (ZZz 1. , and thus a larger 

variance than ~tTiAt. GLS estimators are generally 

robust against misspecification of the optimal weight 

Vectors, ZZ 1~' (~ZZ 1 ~' )-1 • In these cases, the 

increase in MSE due to omitting pertinent auxiliary 

variables is dominated by bias and the additional 

variance due to using the wrong weight vectors is 

relatively negligible. 

Corollary - Diff i =0 ¢:# Cov[(A2,T)IA1]--0. Thus 

the definition of redundant ( not pertinen0 auxiliary 

variable at the beginning of this section makes sense. 

In summary, target variables control variance 

and auxiliary variables control mostly bias. There are 

three ways to reduce variance: increase the sample 

size, increase the number of pertinent target variables 

in T, and increase the number of pertinent auxiliary 

variables in A. Increasing the number of pertinent 

auxiliary variables can reduce bias. 

If one includes target or auxiliary variables in 

the data structure (data matrix, W) that are redundant 

for a particular target mean, then the GLS estimator of 

that target mean is algebraically identical to the GLS 

estimator derived from the data structure, W, that 

excludes them but in every other way is the same. This 

means that there is no penalty (except possibly 

computer time) for including unnecessary (redundant) 

variables in the data matrix, W. In particular, the 

sampling distribution may be included in the estimation 

process by using the sample indicator function as an 

auxiliary variable, but in many common situations it 

will prove to be redundant. Many estimators which 

seem to ignore the sampling distribution implicitly use 

it through other auxiliary variables in the data matrix 

that already contain all the pertinent information that 

the sample indicator provides. 

The observed data maximum likelihood 

estimator (MLE) as described in Chapter 7 of Little and 

Rubin is, under normality, the GLS estimator, 

^ -1 , . This estimator is also a #,,d = ZZ, Z (ZZ~lZ') -1 

fixed point of the EM-algorithm (see theorems in the 

theory section of that chapter). Thus the Normal EMalgorithm with Z given, will converge to this GLS. 

The GLS estimator contains it's own nonresponse 

adjustment by default. 

5) CONCLUSIONS 

Although the sampling distribution is a 

necessary part of inference from sample survey data, it 

is rarely sufficient because of many features of applied 

sampling. These features include nonresponse, 

response bias, and data relationships that make applied 

sampling a multivariate discipline where univariate 

methods generally fail to produce optimal inferences. 

In spite of this, sample survey inference has remained 

largely univariate with an encyclopedia of corrective 

techniques to handle these negative features of sample 

data. 

This paper discussed theory and applications 

of multivariate methods for estimating finite population 

mean vectors assuming data deficiencies like 

nonresponse (both item and total, ignorable and 

otherwise) and response bias, but exploiting data 

dependencies modeled by the covariance matrix of 

survey variables (both design and target variables). 

These data dependencies are used to minimize mean 

square error in the presence of the data deficiencies. 

The estimator so derived automatically handles many 

missing data problems that practitioners face by fully 

exploiting known data dependencies. Its use is 

indicated in repeated surveys where nonresponse is a 

problem and strong data dependencies are present. 

The theory presented here is at least two 

centuries old, Gauss (1809). Sampling theory is much 

newer; its standard methodologies work and can be 

applied with primitive computational aids (say 1950s 

technology). The material presented here would have 

been totally impractical in 1950 but in the 1990s the 

computer revolution has made Gauss' methods quite 

practical. Present day computers allow instant 

availability of huge supplementary data bases and the 

computational power to make short work of complex 

estimators. The estimation process described in this 

paper is applied in Johnson and Woodruff (1990). 

REFERENCES 

Gauss K.F. (1809), Werke 4, 1-93, Gottingen. 

Johnson C. and Woodruff S. (1990). An Application of 

Regression Superpopulation Models in the Current 

Employment Statistics Survey, Proceedings of the 

American Statistical Association, Survey Research 

Methods. 

Little J.A. and Rubin D. B. (1987). Statistical Analysis 

with Missing Data, Wiley. 

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