_____________________________________________________________________________________

This paper was prepared for presentation at the 1992 convention of the American Statistical Association

held in Boston MA. Any opinions expressed are those of the authors and do not reflect policy of the

Bureau of Labor Statistics.

Conditional Properties of Post-Stratified

Estimators Under Normal Theory

Robert J. Casady

Richard Valliant

U.S. Bureau of Labor Statistics

441 G St. NW

Washington DC 20212

Revised 10/26/92
CONDITIONAL PROPERTIES OF POST-STRATIFIED

ESTIMATORS UNDER NORMAL THEORY

ROBERT J. CASADY and RICHARD VALLIANT1

ABSTRACT

Post-stratification is a common technique for improving precision of estimators by using

data items not available at the design stage of a survey. In large, complex samples, the

vector of Horvitz-Thompson estimators of survey target variables and of post-stratum

population sizes will, under appropriate conditions, be approximately multivariate

normal. This large sample normality leads to a new post-stratified regression estimator,

which is analogous to the linear regression estimator in simple random sampling. We

derive the large sample design bias and mean squared errors of this new estimator, the

standard post-stratified estimator, the Horvitz-Thompson estimator, and a ratio estimator.

We use both real and artificial populations to study empirically the conditional and

unconditional properties of the estimators in multistage sampling.

KEYWORDS: Asymptotic normality; Regression estimator; Defective frames; Ratio

estimator; Horvitz-Thompson estimator.

1. INTRODUCTION

1.1 Background

A major thrust in sampling theory in the last twenty years has been to devise ways

of restricting the set of samples used for inference. In a purely design-based approach, as

described in Hansen, Madow, and Tepping (1983), no such restrictions are imposed.

Statistical properties are calculated by averaging over the set of all samples that might

have been selected using a particular design. Although it is generally conceded that some

type of design-based, conditional inference is desirable (Fuller 1981, Rao 1985,

Hidiroglou and Särndal 1989), satisfactory theory has yet to be developed except in

relatively simple cases. Alternative approaches are prediction theory, developed by

Royall (1971) and many others, and the Bayesian approach, found in Ericson (1969),

which avoid averaging over repeated samples through the use of superpopulation models.

 1Robert J. Casady and Richard Valliant, U.S. Bureau of Labor Statistics, 2 Massachuetts Ave. N.E.,

Washington D.C., 20212-0001.
A design-based approach to conditioning was introduced by Robinson (1987) for the

particular case of ratio estimates in sample surveys. Robinson applied large sample

theory and approximate normality of certain statistics to produce a conditional, designbased theory for the ratio estimator.

In this paper, we extend that line of reasoning to the problem of poststratification. Convincing arguments have been made in the past by Durbin (1969), Holt

and Smith (1979), and Yates (1960) that post-stratified samples should be analyzed

conditional on the sample distribution of units among the post-strata. However, as Rao

(1985) has noted, the difficulties in developing an exact, design-based, finite sample

theory for post-stratification in general sample designs may be intractable. Model-based,

conditional analyses of post-stratified samples are presented in Little (1991) and Valliant

(1993). The alternative pursued here is design-based and uses large sample, approximate

normality in a way similar to that of Robinson (1987) as a means studying conditional

properties of estimators.

1.2 Basic Definitions and Notation

The target population is a well defined collection of elementary (or analytic)

units. For many applications the elementary units are either persons or establishments.

We assume the target population has been partitioned into first stage sampling units

(FSUs). For person based surveys the FSUs are commonly households, groups of

households or even counties, while for establishment based surveys it is not uncommon

that the individual establishment is an FSU. In any event, the collection of FSUs will be

referred to as the first stage sampling frame (or just sampling frame). It is assumed

that there are M FSUs in the sampling frame and they are labeled 1, 2, ..., M. We also

assume that the population units can be partitioned into K "post-strata" which can be used

for the purposes of estimation.
We let y represent the value of the characteristic of interest (e.g. weekly income,

number of hours worked last week, restricted activity days in last two weeks, etc.) for an

elementary unit. Associated with the i

th FSU are 2K real numbers:

yik = aggregate of the y values for the elementary units in the i

th FSU which are in

the kth post-stratum,

Nik = number of elementary units in the i

th FSU which are in the kth post-stratum.

For each post-stratum we then define

Y y k ik

i

M

◊

=

= = Â

1

 aggregate of the y values for all elementary units in the kth post-stratum,

N N k ik

i

M

◊

=

= = Â

1

 total number of elementary units in the kth post-stratum.

In what follows we assume that the N◊k are known. The population aggregate of the y

values is given by Y Yk

k

K

◊◊ ◊

=

= Â

1

 and the total population size by N Nk

k

K

◊◊ ◊

=

= Â

1

 . In sections 1-

3, we assume that the sampling frame provides "coverage" of the entire target population.

In section 4, we consider the problem of a defective frame, i.e. one in which the coverage

of the frame differs from that of the target population.

1.3 Sample Design and Basic Estimation

Suppose that the first stage sampling frame is partitioned into L strata and that a

multi-stage, stratified design is used with a total sample of m FSUs. In the following, the

subscript representing design strata is suppressed in order to simplify the notation. For

the subsequent theory, it is unnecessary to explicitly define sampling and estimation

procedures for second and higher levels of the design. However, for every sample FSU,

we require estimators y$ik and N$

ik so that E

2+

y y $ik ik = and E

2+

= N N $

ik ik where the notation

E

2+

 indicates the design-expectation over stages 2 and higher. Letting p i be the

probability that the i

th FSU is included in the sample and wi i = 1 p , it follows that the
estimator Y wy $ $ k i ik

i

m

=

= 1 is unbiased for Yk and the estimator N wN $ $

k i ik

i

m

=

= 1 is unbiased

for Nk .

1.4 An Analogue to Robinson's Asymptotic Result

Following Krewski and Rao (1981), we can establish our asymptotic results as

L fi ¥ within in the framework of a sequence of finite populations l q P L with L strata in

P L . It should be understood that we implicitly assume (without formal statement) the

sample design and regularity conditions as specified in Krewski and Rao and more fully

developed in Rao and Wu (1985). Details of proofs add little to those in the literature

and are omitted.

Converting to matrix notation, we let Y = Y Y ¢ 1L .K , N = N N ¢ 1L .K ,

$ $ $ Y = . Y Y ¢

1L K , N$ = N N $ $ ¢1L K and V YN = ¢ R

S

|

T

|

U

V

|

W

| var $ $ where Y Y $ $ =1b g N and

N N $ $ = b1 N g . Note that $

Y, which uses N in the denominator, is a notational

convenience and does estimate means in the post-strata. Analogous to conditions C4 and

C5 of Krewski and Rao (1981), we assume that

lim

L

k

k

k

Y

fi ¥ N

= m , for k K =1 2,, , L , (1)

lim

L

k

k

N

fi ¥ N

= > f 0 for k K =1 2,, , L , and (2)

lim

L

m fi ¥= = L

N

M O

Q

V P S

S S

S S

11 12

21 22

 (positive definite) (3)

where S is partitioned in the obvious manner. Note that we have again suppressed the

subscript representing design strata. Assumptions (1)-(3) simply require that certain key

quantities stabilize in large populations. Condition (2), in particular, assures that no poststratum is empty as the population size increases. We now state the following,

Result: Assume the sample design and regularity conditions specified in Krewski and

Rao and that S22

-1

 exists; then, given $N, the conditional distribution of $Y is
asymptotically N m M NM V 1 12 22 c

1

2

1 + - F

H IK - - S S $ , e j , where Vc = - - S SSS 11 12 22

1

21,

M Y 1 11 2 2 = = ¢

fi ¥

lim $

L

fm fm f m L K K and M N 2 12 = = ¢

fi ¥

lim $

L

ff f L K .

Proof. This result is analogous to the result for K=1 given by Robinson (1987) and

follows directly from the fact that the random vector m1 2 1 12 22

1

2

2

$ ( $ )

$

YM NM

N M

-- -

-

L

N

M

M

O

Q

P

P

- S S

tends in distribution to N

0 c

0

V 0

0

L

N

M

O

Q

P

L

N

M O

Q

P F

H

G I

K

J , S22

. Strictly, as in Robinson, we consider the

conditional distribution of $Y for $N in a cell of size em-1 2 for small e. . Note that in

some sample designs 1 N¢ = $ N (such as those in which a fixed number of elementary

units are selected with equal probabilities) in which case S22

-1

 does not exist; in such cases

only the first K-1 post-strata are considered for the purpose of conditioning.

In the next section, the asymptotic mean of $Y is used to motivate a linear

regression estimator of the population mean of the y's.
2. CONDITIONAL PROPERTIES OF ESTIMATORS FOR THE

 POPULATION MEAN

2.1 Estimators for the Population Mean

The population mean is, by definition, m = fi ¥

lim

L

b g Y N =

lim

L k k k

K

fi ¥ = a f 1Y1N ¢ ¢ = f m

1

 where 1¢ is a row vector of K ones. Four estimators of the

population mean will be considered. The first three are standard estimators found in the

literature while the fourth is a new estimator motivated by the asymptotic, joint normality

of $ $ Y N and :

(1) Horvitz-Thompson estimator

$ $ $ YHT = 1Y1N 1Y ¢ ¢ = ¢

(2) ratio estimator

$ $ $ $ $ YR = 1Y1N 1Y 1N ¢ ¢ = ¢ ¢

(3) post-stratified estimator

$

$

$ $

.

Y N N

N PS Y k

k

k

k

K

= F

H

G I

K

J = ¢ -

=

1

1

r Y where r¢= NN N N 1 1 K K

$ , , L $

(4) linear regression estimator

$ $ $ YLR = ¢ - - F

H IK L

N

M O

Q

P - 1Y NM S S12 22 2

1

e j

The linear regression estimator is motivated by the form of the large sample mean of the

conditional random variable $ $ Y N listed at the end of section 1.4 and is very similar to the

generalized regression estimator discussed by Särndal, Swensson and Wretman (1992).

The linear regression estimator (4) was also discussed in the context of calibration

estimation by Rao (1992). It should be noted that the ratio estimator does not require

that Nk or their sum N be known. The Horvitz-Thompson estimator only requires that

N be known, whereas the post-stratified and linear regression estimators require that

l q Nk K k =1, , L be known. In practice, the linear regression estimator has the additional

complication that the covariance matrices S S 12 22 and are unknown and must be
estimated from the sample. In implementing $

YLR in section 3, the known finite

population quantities b g 1 N N will be used in place of the limiting vector M2.

2.2 Conditional expectations and variances of the estimators

Using the asymptotic setup given earlier, the conditional expectations and

variances of the four estimators can be computed. First, define the following three

matrices:

H = - S S12 22

1

,

R HD =-ma f, and

P HD =-mk b g

where Daf a f m mm = diag , , L and D m mm k K bg b g = diag 1, , L are K K· diagonal matrices .

Below, we state the mean and variance of the four estimators without providing any

details of the calculations. When the sample of first-stage units is large, each of the

estimators has essentially the same conditional variance. The Horvitz-Thompson, ratio,

and post-stratified estimators are, however, conditionally biased, whereas the linear

regression estimator is not. Thus, the linear regression estimator has the smallest

asymptotic mean square error among the four estimators considered here. Rao (1992)

also noted the optimality of the regression estimator within a certain class of difference

estimators and its negligible large sample bias.

(1) Horvitz-Thompson estimator:

E $ $ $ LYHT N 1HN M NM O

QP = + ¢ - LNM OQP m e j 2

var $ $ Ym m V HT N 1 1 1V1c c HT( )

L

NM OQP = ¢ - = ¢ = - -- 1

11 12 22

1

21

1 c h S SSS

(2) ratio estimator:

E $ $

$

$

$

Y N

N

o m

R N 1RN M

1RN M

2

2

L

NM OQP = +F

H

G I

K

J ¢ - L

NM OQP

= + ¢ - L

NM OQP + -

m

m

e j

e j c h1
var $ $ $

( )

( )

Y NN V

V om

R HT

HT

N c

c

L

NM OQP =

= + -

d i

d i a f

2

3 2

(3) post-stratified estimator:

E $ $ $

$

Y

o m

PS N rPN M

1PN M

L

NM OQP = + ¢ - LNM OQP

= + ¢ - L

NM OQP + -

m

m

2

2

1

e j

e j c h

var $ $

( )

Y m

V om

PS

HT

N rVrc

c

L

NM OQP = ¢

= +

-

-

1

a f 3 2 d i

(4) linear regression estimator:

E $ $ YLR N L

NM OQP = m

var $ $ Y V LR HT N ( ) c

L

NM OQP =

As noted in section 1, some minor modifications of the above formulas are necessary for

designs, such as simple random sampling, in which 1 N¢ = $ N . The derivation of the

requisite modifications is straightforward and is not detailed here.

The large-sample biases of the first three estimators depend on $N M- 2 . In other

words, their biases are determined by how well the sample estimates the population

distribution among the post-strata. In some special cases each of the first three can be

conditionally unbiased. The post-stratified estimator, for example, will be approximately

unbiased if 1H D 0 ¢c h - = b g mk ¢. This occurs in simple random sampling and is possible,

though certainly not generally true, in more complex designs. The matrix H can be

interpreted as the slope in a multivariate regression of $ $ Y N on , or of Y N on when the

sample estimates are close to the population values. Thinking heuristically in

superpopulation terms, if Ey N xb g ik k ik = m , as in Valliant (1993), with Ex denoting an

expectation with respect to the model, then EY N xb g k kk = m . The slope of the regression
of Y N k k on is then mk and, in the unusual case in which the $

Yk ' s are independent, H is

diagonal. In fact H D= mk b g, so the conditional design-bias of the post-stratified

estimator would be zero. If, on the other hand, the model has an intercept,

i.e. if EY N xb g k k kk = + a m , then the post-stratified estimator may have a substantial

conditional design-bias. We will use this line of reasoning in the empirical study in

section 3 to devise a population for which $Yps is conditionally biased.

Similar model-based thinking can be applied to the Horvitz-Thompson and ratio

estimators to identify populations where the conditional design-biases will be predictably

small for large samples. Suppose, as above, that the $

Yk ' s are independent. If each poststratum total is unrelated to the number of units in the post-stratum, i.e. a peculiar

situation in which E Y x k b g does not depend on Nk , then $

YHT is conditionally designunbiased. If EY N xb g k k = m , implying that all elementary population units have the same

mean regardless of post-stratum, then $

YR is conditionally design-unbiased.

2.3 Unconditional expectations and variances of the estimators

Unconditionally, all estimators are approximately design-unbiased as noted below.

The relative sizes of the variances depend on the values of S S 12 22 , ,, m m and D k b g. This

is similar to the case of simple random sampling of a target y and an auxiliary x. In that

case, whether the ratio estimator, yxx s s , or the regression estimator, y bx x s s + - b g, has

smaller design-variance also depends on the values of certain population parameters.

(1) Horvitz-Thompson estimator:

E $YHT = m

var $Y m HT = ¢ -1

11 1 1 S

(2) ratio estimator:

E $Y om R = + - m 1 c h

var $Y m om R = ¢ -+ + -1 -

11 21

2

22

3 2 1 1 SSS 2m m a f d i

(3) post-stratified estimator:
E $

Y om PS = + - m 1 c h

var $Y m om PS k k k = ¢ -+ + -1 -

11 21 22

3 2 1 D D D1 S SS 2 bg bg bg m mm d i a f

(4) linear regression estimator:

The unconditional expectation and variance are the same as the conditional

expectation and variance.

3. SIMULATION RESULTS

The theory developed in the preceding sections was tested in a set of simulation

studies using three separate populations. The population size and basic sample design

parameters for the three studies are listed in Table 1. The first population consists of a

subset of the persons included in the first quarter sample of the 1985 National Health

Interview Survey (NHIS) and the second population consists of a subset of the persons

included in the September 1988 sample from the Current Population Survey (CPS). Both

the NHIS and CPS are sample surveys conducted by the U.S. government. The variable

of interest for the NHIS population is the number of restricted activity days in the two

weeks prior to the interview and the variable of interest for the CPS population is weekly

wages per person.

Post-strata in the NHIS and CPS populations were formed on the basis of

demographic characteristics (as is typically done in household surveys) in order to create

population sub-groups that were homogenous with respect to the variable of interest. For

the NHIS population the variables age and sex were used to define 4 post-strata and for

the CPS population the variables age, race, and sex were used to define 8 post-strata.

The third population is artificial; it was created with the intention of producing a

substantial conditional bias in the post-stratified estimator of the mean. As noted in

section 2.2, $

YPS will be conditionally biased if the FSU post-stratum totals for the

variable of interest, conditional on the number of units in each FSU/post-stratum, follow
a model with a non zero intercept. With this in mind, we generated the population in

such a way that

Ey N N N ik ik k ik ik c h =+ + ab g 2 (4)

where Nik is the number of units in the kth post-stratum for the i

th FSU and ab g k , , and

are constants. Specifically, five post-strata were used with ak =100k (k=1,...,5), b =10,

and g=-.05. In total two thousand FSUs were generated with the total number of units

in the i

th FSU, say Ni , being a Poisson random variable with mean 10. Then, conditional

on Ni , the numbers of units in the five post-strata (i.e., NN N ii i 12 5 , ,, L ) for the i

th FSU

were determined using a multinomial distribution with parameters Ni and

p k k = = . ,, , 20 1 2 5 for L .

Finally, the value of the variable of interest for the j

th unit in the kth post-stratum

for the i

th FSU was a realization of the random variable

yN N N ijk k ik ik = ++ + + + a bg e e e 1i 2 3 ik ijk i

where ee e 1i 2 3 ik ijk , , and are three independent standardized chi-square (6 d.f.) random

variables. This structure implies that Ey Nik ik c h is given by (4). Furthermore, the values

of the variable of interest for units within an FSU are correlated and the correlation

depends upon whether the units are in the same post-stratum or not.

A single-stage stratified design was used for the NHIS population with

"households" being the FSUs. Ten design strata were used and an approximate 10%

simple random sample of households was selected without replacement from each

stratum. Each sample consisted of 115 households and each sample household was

enumerated completely. A total of 5,000 such samples was selected for the simulation

study.

Two-stage stratified sample designs were used for both the CPS and artificial

populations. For the CPS population, geographic segments, employed in the original

survey and composed of about four neighboring households, were used as FSUs and

persons were the second-stage units. In both populations, 100 design strata were created
with each stratum having approximately the same number of FSUs and a sample of

m = 2 FSUs was selected with probability proportional to size from each stratum using

the systematic sampling method described by Hansen, Hurwitz, and Madow (1953, p.

343). Thus, 200 FSUs were selected for both populations. Second stage selection was

also similar for both populations. For the CPS population a simple random sample of 4

persons was selected without replacement in each sample FSU having Ni > 4 and all

persons were selected in each sample FSU where Ni £ 4. For the artificial population

the within FSU sample size was set at 15 rather than 4 which resulted in the complete

enumeration of most sample FSUs. A total of 5,000 samples were selected from each of

the populations for the simulation study.

In each sample, we computed $YHT , $YR , $YPS, and two versions of $YLR . For the first

version of the regression estimator, denoted $YLR (emp) in the tables, H was estimated

separately from each sample as would be required in practice. Each component of

S S 12 22 and was estimated using the ultimate cluster estimator of covariance, appropriate

to the design, as defined in Hansen, et. al. (1953, p.419). The second version, denoted

$

YLR (theo) , used the same value of H in each sample, which was an estimate more nearly

equal to the theoretical value of the H matrix. For the CPS and artificial populations, the

theoretical H matrix was estimated from empirical covariances derived from separate

simulation runs of 5,000 samples. For the NHIS population the design was simple

enough that a direct theoretical calculation of H was done. As the sample of FSUs

becomes large, the performance of $

YLR (emp) should approach that of $YLR (theo). The

performance of $YLR (theo) is, consequently, a gauge of the best that can be expected from

the empirical version of the regression estimator for a given sample size.

Table 2 lists unconditional results summarized over all 5,000 samples from each

population. Empirical root mean square errors (rmse's) were calculated as

rmse Y Y Y S s s

$ $ S

ej e j = - L

N

M O

Q = P

2

1

1

2

 with S = 5,000 and $

Ys being one of the estimates of the
population mean from sample s. In the CPS and artificial populations, results for the

Horvitz-Thompson and the ratio estimators were nearly identical so that only the former

is shown. Across all samples, the bias of each of the estimators was negligible. As

anticipated by the theory, $YLR (theo) was the most precise of the choices, although the

largest gain compared to $YPS was only 4.7% in the artificial population. The need to

estimate H destabilizes the regression estimator as shown in the results for $YLR (emp).

For the NHIS and CPS populations, $YLR (emp) has a larger root mse than both $YLR (theo)

and $YPS. The most noticeable loss is for the NHIS population where the root mse of

$

YLR a f emp is about 15% larger than that of either $YLR (theo) or $YPS. This result is

consistent with the smaller FSU sample size and hence less stable estimate of H for the

NHIS population.

Figures 1-3 present conditional simulation results. The 5,000 samples were

sorted by the theoretical bias factors presented in section 2.2. The sorting was done

separately for each of the estimators of the population mean. In the cases of the two

regression estimators, which are theoretically unbiased in large samples, the bias factor

for $YPS was used for sorting. The sorted samples were then put into 25 groups of 200

samples each and empirical biases and root mse's were computed within each group. The

group results were then plotted versus theoretical bias factors in the figures. The upper

sets of points in each figure are the empirical root mse's of the groups, while the lower

sets are empirical biases. The two regression estimators are conditionally unbiased as

expected. The other estimators, however, have substantial conditional biases that, in the

most extreme sets of samples, are important parts of the mse's. For the CPS population,

the range of the bias factors for $

YHT is so much larger (-10 to 10) than that of the other

estimators that we have omitted $YHT from the plot for clarity. In the neighborhood of the

balance point, $N N= , all estimators perform about the same, but, because of a lack of

data at the design stage, we have no control on how close to balance a particular sample

may be. The safest choice for controlling conditional bias is, thus, $YLR (emp). This
finding is similar to that of Valliant (1990), who noted that, in one-stage, stratified

random or systematic sampling, the separate linear regression estimator is a good choice

for controlling bias, conditional on the sample mean of an auxiliary variable.

4. DEFECTIVE FRAMES

4.1 The Basic Problem of Defective Frames

In most real world applications not all of the elementary units in the population

are included in the sampling frame. In household surveys, it is not unusual for some

demographic subgroups, especially minorities, to be poorly covered by the sampling

frame. Bailar (1989), for example, notes that in 1985 the sample estimate from the CPS

of the total number of Black males, ages 22-24, was only 73% of an independent

estimate of the total population of that group. Corresponding percentages for Black

males, ages 25-29 and 60-61, were 80% and 76%.

To formalize the discussion of this type of coverage problem, suppose that Nk

now refers to the number of elementary units in the frame and that N&

k is the actual

number of population elements in the kth post-stratum. In the discussion below terms

with a dot on the top are population values while terms with no dot are frame values.

Letting Y

&

k be the aggregate of the y values over all population elements in the kth poststratum, then it follows that the true population mean is given by

& lim

&

&

lim

&

&

&

&

m fm == = & & fi ¥

=

=

fi ¥ = = L

k

k

K

k

k

K L

k k

k k

K

k k

k

K Y

N

N

N

Y

N

1

1

1 1

.

Obviously, all four of the estimators of the mean given in section 2 are biased (both

conditionally and unconditionally) for m&; the additional bias term being given by m m - &

for all of the estimators. It should be noted that this bias term is oa f1 so it will dominate

the other bias terms listed in section 2.2 as the number of FSUs increases. There is

another even more basic problem; namely, in most cases the individual frame values Nk
are not known so only the ratio estimator is well defined. For example, the HorvitzThompson estimator of the mean as defined in section 2 requires N , the total number of

units in the frame, but N may be unknown. On the other hand, the N&

k (or least the

proportions f&

k ) may be known from independent sources and hence be available for the

purposes of estimator construction. In household surveys, for instance, the N&

k may

come from intercensal projections of population counts.

Before attempting to construct unbiased estimators for m& it should be noted that

m m f f m m f fm fm m -= - - + - + - = ==

& & & & & & & k k

k

K

k k kk

k

K

k k

k

K

d i k k b g d i b g

1 11

.

So, if we assume that for each post-strata the mean of the units in the frame is equal to

the true population mean, (i.e. m m k k = & for every ) then the bias term reduces to k

m m f fm f fm -= - = - = =

& & & & k k

k

K

k kk

k

K

di di k

1 1

.

This is very strong (and also very expedient) assumption; however, addressing the

problem of defective frame bias without such a condition is virtually impossible.

4.2 Alternative Estimators

The basic strategy is to construct an estimator for the defective frame bias, m m - &,

and then subtract this estimator from the estimators studied earlier. Two cases need to be

considered:

Case 1. The frame parameters fk l q , 1£ £k K are unknown, and

Case 2. The frame parameters fk l q , 1£ £k K are known.

Case 1. For this case only the ratio estimator is well defined and the only obvious

candidate for an estimator of the bias is

$ $

$

& $

$

$ & $

$ . .

B N

N

Y

N

Y Y

N

k

k

k

K

k

k

R k

k

k k

K

1

1 1

= - F

H

G I

K

J = - = =

f f .

Using the strategy given above, the resulting estimator for m& is

$ $ $ & $

$ YYB Y

N R k

k

K

k

k

1 1

1

=-= =f .
This is the "post-stratified" estimator usually found in practice. It is straightforward to

verify the following properties of $Y1 :

E $ $ & $ Y om 1 1

1 L N pPN M NM O

QP = + ¢ - LNM OQP + - m e j c h where ¢= L

N

M O

Q

P p

& ,& , , f &

f

f

f

f

f

1

1

2

2

L K

K

var $ $ Y m om 1

1 3 2 N pVpc

L

NM OQP = ¢ + - -a f d i

E $Y om & 1

1 = + - m c h

var $Y m om 1 k kk

1

11 21 22

3 2 = ¢ -+ + 2 - - p D D Dp S SS bg bg bg m mm d i a f

The attempt to correct for the defective frame bias is successful in the sense that $Y1 is

unconditionally unbiased for m&. However, the conditional bias is still present.

Case 2. For this case it can be verified that the estimator

$ $ $

$ B

N 2 12 22

1 = - 2 ¢ - - F

H

G I

K

J

L

N

M

M

O

Q

P

P - 1p Y N a f S S M

is approximately, conditionally unbiased for m m - & and, as $YLR is conditionally unbiased

for m , it follows directly that the estimator

$ $ $ $ $

$ YY B

N 2 2 12 22 LR

1 = -= ¢ - - 2

F

H

G I

K

J

L

N

M

M

O

Q

P

P - p Y N

S S M

 is both conditionally and unconditionally, approximately unbiased for m&. It can also be

verified that

var $ $ var $ Y Ym 2 2

1 N pVpc

L

NM OQP = = ¢ - .

In addition to the problems of the linear regression estimator cited earlier, this estimator

is usually not even well defined as the frame parameters fk l q , 1£ £k K are rarely, if

ever, known when the frame is defective.

5. CONCLUSION

This study has generalized the asymptotic techniques suggested by Robinson

(1987) to study the problem of post-stratification from a design-based, conditional point-
of-view. An important paper in the conditional study of post-stratification was that of

Holt and Smith (1979), one of whose basic premises was that $YPS is conditionally

unbiased. This will be true (at least asymptotically) only if 1H D 0 ¢c h - = b g mk ¢; so, in

general, this premise is false. In fact, simple random sampling of elementary units may

be one of the few realistic cases where this basic premise is true.

From a conditional point of view the linear regression estimator is preferable

among the four studied here. Only the regression estimator is conditionally unbiased.

The post-stratified estimator is no better (or worse) than either the Horvitz-Thompson or

the ratio estimator; all have conditional bias terms of order m-a f 1 2 . All of the estimators

have the same conditional variance to terms of order m-1; furthermore, the conditional

variance does not depend on $N, the vector of estimated proportions in the post-strata.

Consequently, because of its conditional unbiasedness, the regression estimator has the

smallest conditional mean square error.

The Horvitz-Thompson, ratio, and post-stratified estimators are unconditionally

unbiased. Although somewhat illogical, one might attempt to make a case for the

estimators by comparing their unconditional properties with the conditional properties of

the linear regression estimator. But even from this mixed perspective, the $

YLR a f theo

estimator is clearly superior to the others. Not only is it conditionally unbiased, but the

conditional variance of the linear regression estimator can be no larger than the

unconditional variance of any of the other estimators. In large FSU samples, the

empirical version of the regression estimator will inherit these good properties of

$

YLR a f theo and also perform well.

The problem of a defective frame introduces complications not found otherwise.

Each of the estimators of the mean studied here is biased both conditionally and

unconditionally. Bias adjustments are possible only under the restrictive assumption that

the mean of units within each post-stratum is the same for all population units whether

they are included or excluded from the frame.
An area we have not addressed is variance estimation. A design-based variance

estimator for the regression estimator can be obtained using the methods of Särndal,

Swensson, and Wretman (1989).

ACKNOWLEDGMENT

Any opinions expressed are those of the authors and do not reflect policy of the U.S.

Bureau of Labor Statistics.

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Table 1. Population size and basic sample design

parameters for three simulation studies.

Population

Pop.

Size

N

No. of

FSUs

M

No. of sample

FSUs

m

HIS 2,934 1,100 115

CPS 10,841 2,826 200

Artificial 22,001 2,000 200
Table 2. Simulation results for three

populations. 5,000 samples were selected

from each population.

Estimator

Relbias $

Y

(%)

rmse Y$e j 00 1 * -

L

N

M

M

M

O

Q

P

P

P

rmseY

rmseYPS

$

$

e j

e j

HIS population

$

YHT .12 .141 .05

$

YR .10 .141 .02

$

YPS .11 .141 0

$

YLR (emp) .19 .162 14.71

$

YLR (theo) .08 .140 -.96

CPS population

$

YHT -.01 10.25 15.8

$

YPS 0 8.85 0

$

YLR (emp) -.03 9.11 3.0

$

YLR (theo) -.01 8.79 -.6

Artificial population

$

YHT .02 2.30 -2.93

$

YPS .12 2.37 0

$

YLR (emp) .04 2.31 -2.41

$

YLR (theo) .02 2.26 -4.70