Poisson Sampling, Regression Estimation, 

and the Delete-a-Group Jackknife

Phillip S. Kott

))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

When coupled with the simple expansion estimator, Poisson sampling leads

to estimators with higher-than-necessary variances. That problem vanishes

when the expansion estimator is replaced by a randomization-consistent

regression estimator. A simultaneous estimator for the model variance and

randomization mean squared error of this estimation strategy is developed.

It is nearly identical to the weighted residual variance estimator, but can be

slightly better at estimated the model variance when finite population

correction matters. When finite population correction can be ignored, an

appropriately-defined delete-a-group jackknife variance estimator is shown

to have desirable asymptotic properties making it a practical alternative in

many applications. 

KEY WORDS Asymptotic, finite population correction, model variance,

randomization consistent, randomization mean squared error.

))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Phillip S. Kott is Chief Research Statistician, Research and Development Division,

National Agricultural Statistics Service, 3251 Old Lee Highway, Room 305, Fairfax,

Virginia, 22030. This paper was originally prepared for the Joint Statistical Meetings,

August 2000, Indianapolis, Indiana.
1

1. Introduction 

Poisson sampling is perhaps the simplest form of unequal probability selection. Its use

often leads to inefficient estimation, which is why it is not more widely used. When

combined with a regression-type estimator, however, the advantages of Poisson

sampling can be realized. That is why the National Agricultural Statistics Service

(NASS) has recently overhauled its major crop survey program and adopted Poisson

sampling (see Kott and Bailey, 2000). 

This paper reviews and extends the theory supporting the use of Poisson

sampling coupled with a randomization-consistent regression estimator. Section 2

introduces the basic setup. Section 3 explores the randomization and model-based

properties of the estimation strategy. Since the large-sample and large-population

properties of variance estimators will be a main focus here, some care is taken in the

development of the asymptotics. Section 4 proposes a simultaneous estimator of model

variance and randomization mean squared that is slightly better than the one proposed 

in Särndal et al. (1989). Section 5 addresses the issue of small-sample bias in variance

estimation. Section 6 discusses the applicability of the delete-a-group jackknife

variance estimator when the sample size is large and the population even larger. 

Finally, Section 7 offers a broader discussion of the topics covered in the text.

2. Background 

Suppose we want to estimate a population (U) total, T = 3U yk based on a sample (S) of

y-values. If the probability that population unit k is in the sample is Bk, then the simple

expansion of T is t = 3S yk /Bk. Another useful way to render t is as t = 3U ykI

k

/Bk, where

I

k

 is a random variable equal to 1 when k0S and 0 otherwise. This means E(Ik) = Bk. 

Under randomization-based inference the yk

 are fixed constants, while the Ik

 are

random variables. It is easy to see that t is a randomization-unbiased estimator of T;

that is Ep

(t) = T, where the subscript p denotes the expectation with respect to the Ik

(this is a convention; the p derives from “probability”).
2

The randomization variance of t is 

Vrp(t) = Ep[(t ) T)2] = 3U (yk /Bk)(yg /Bg)(Bkg ) BkBg), 

where 3U

 denotes 3k0U 3g0U

 in this context, and Bkg = E(Ik

Ig

) is the joint selection

probability of units k and g. When k = g, Bkg = Bk. The randomization variance of t very

much depends on how exactly the sample is drawn, and in particular of the joint

selection probabilities.

Under Poisson sample, each unit k is sampled independently of every other unit

in the population. Consequently, Bkg = BkBg when kg. This simplifies the

randomization variance of t immensely:

Varp

(t) = 3U

 (yk /Bk

)

2

(Bk ) Bk

2

) = 3U

 (yk

2

/Bk

)(1 ) Bk

),

and leads to the simple unbiased randomization variance estimator:

varp

(t) = 3S

 (yk /Bk

)

2

(1 ) Bk

).

The problem with Poisson sampling in this context is that it can lead to a largerthan-necessary randomization variance. This is because the sample size of a Poisson

sample is random. It has an expected value of n* = E(3U I

k

) = 3U Bk, and a variance of 

Var(3U I

k

) = 3U Bk (1 ) Bk). 

Under a sample design where the sample size is fixed at n = n*, the unit

selection probability of each unit k set equal to Bk, and each yk is proportional to Bk, the

randomization variance of t would be zero (because if, say, y1 /B1 = b, then t / bn = T). 

Under Poisson sampling, by contrast, this variance would be b2 3U Bk (1 ) Bk

). 

 The problem caused by random sample size can be eliminated when Poisson

sampling is coupled with this regression estimator: 
3

tR = t + (3U xk ) 3S Bk

-1xk

)(3S ckBk

-1xk

'xk)

 -1 3S ckBk

-1xk

'yk, (1)

where xk = (xk1, ..., xkQ) is a row vector of values known for all S, ck is a constant, 3U xk

is known, and 3S ckBk

-1xk

'xk is invertible. For the simple example given above where yk

is proportional to Bk , we can now let xk be the scalar Bk. It is easy to see that tR will

always be n*(y1 /B1), which is also what T is. 

The regression estimator in equation (1) is a very slight variation of the so-called

general regression estimator (GREG). See, for example, Särndal, Swensson, and

Wretman. (1992). A good review of regression estimators in the survey sampling

context is Brewer (1994). The GREG is poorly named because it does not include

purely model-based regression estimators. 

The regression estimator in equation (1) can be rewritten as tR = 3S akyk, where

ak

 is the regression weight of k:

ak

 = Bk

-1 + (3U xi ) 3S Bi

-1xi

)(3S ciBi

-1xi

’xi

)

 -1 ckBk

-1xk

'. (2)

It is well known (and easy to see) that the ak satisfy the calibration equation: 3S akxk =

3U xk (Deville and Särndal 1992; tR is identical to the regression estimator in their

equation (1.6) when their qk

 is set to equal our ck

).

3. Properties of the Estimation Strategy 

The regression estimator, tR, under Poisson sampling has both desirable

randomization-based and model-based properties under mild conditions. 

3.1. Randomization-based Properties

The randomization-based properties of tR

 are asymptotic (we use the more accurate

modifier “randomization” in place of the often-used “design”). That is to say, they

depend on the expected sample size, n*, being large. A sufficient condition for an
4

estimation strategy (an estimator coupled with a sampling design) to be randomization

consistent is that its relative mean squared error should approach 0 as n* grows

arbitrarily large. 

Let N be the population size of U. We want to entertain the possibility that O(n*)

is less than O(N). Consequently, we assume the following as N and n grow arbitrarily

large and Q remains fixed: 

0 < Ly # 3U yk

*

/N < By < 4, *= 1, ..., 8; (3.1)

0 < Lxq # 3U xkq

*

/N < Bxq < 4, q = 1, ..., Q; * = 1, ..., 8; (3.2)

0 < Lc # 3U ck

*

/N < Bc < 4, * = 1, ..., 8 (3.3)

0 < LB # 3U [(N/n*)Bk]

-*

/N < BB < 4, * = 1, ..., 8. 

 

 (3.4)

The relative randomization mean squared error of the expansion estimator, t, under

Poisson sampling is 3U (yk

2

/Bk

)(1)Bk

)/(3U yk

)

2 < 3U (yk

2

/Bk

)/(3U yk

)

2

. Equations (3.1), (3.4),

and Scharwz’s inequality tell us that the numerator of this last expression is O(N2

/n*),

while its denominator is O(N2). Thus, the relative randomization mean squared of t

under Poisson sampling is O(1/n*), and the estimation strategy is randomization

consistent. Furthermore, since EP

[(t ) T)2

]/T2

 = O(1/n*), (t ) T)/T = Op

(1/3⁄4n*), and 

t ) T = Op(N/3⁄4n*)

The regression estimator, tR, from equation (1) under Poisson sampling and the

assumptions in equation (3) is equal to t + Op(N/3⁄4n*). This is because, using similar

argument as above (and Scharwz’s inequality repeatedly), the components of 

3U xk ) 3S Bk

-1xk

 are Op(N/3⁄4n*), while the absolute values of the components of 

3S ckBk

-1xk

'xk and 3S ckBk

-1xk

'yk are Op(N). Thus, like t, tR is randomization consistent. 

Furthermore, (tR ) T)/T = Op(1/3⁄4n*), and the relative mean squared error of tR is

O(1/n*). 

Assuming, as we will from now on, that N-1(3U ckxk'xk) is invertible, let 

B = (3U ckxk'xk)

-1 3U ckxk

'yk, and ek = yk ) xkB, so that 3U cixi'ei = 0. We can now

express the error of tR as
5

tR ) T = 3S aiyi ) 3U yi

 = 3S ai(xiB + ei) ) 3U (xiB + ei) 

 = 3S aiei ) 3U ei 

 = 3S ei /Bi + (3U xk ) 3S Bk

-1xk

)(3S ckBk

-1xk

'xk)

-13S ciBi

-1xi

'ei ) 3U ei

 = 3S ei /Bi + (3U xk ) 3S Bk

-1xk

)(3S ckBk

-1xk

'xk)

-1(3U cixi

'ei + OP(N/3⁄4n*)) ) 3U ei 

 = 3S ei /Bi

 + (3U xk ) 3S Bk

-1xk

)(3S ckBk

-1xk

'xk

)

-1 OP

(N/3⁄4n*) ) 3U ei

 

 = 3S ei /Bi ) 3U ei + Op(N/n*) .

This tells us that the randomization mean squared error of tR under Poisson sampling is

dominated by VarP( 3S ek /Bk) = 3U (ek

2

/Bk)(1)Bk). This is identical to the variance of the

expansion estimator under Poisson sampling except that ek has replaced yk. 

3.2. Model-based Properties

Suppose the yk were random variables that satisfied the following model:

yk = xk$ + ,k, (4)

where $ is an unknown column vector, E(,k|xk, Ik) = E(,k,g|xk,xg, Ik, Ig)) = 0 for k  g, and 

E(,k

2

|Ik) = Fk

2

 = f(xk, zk) < 4, where zk is a vector of values associated with k. The Fk

2

need not be known. Moreover, there is no reason that Ik cannot be a function of the

components of xk and zk.

It is easy to see that as long as the regression weights satisfy the calibration

equation, 3S akxk

 = 3U xk

, tR

 will be model unbiased in the sense that E,

(tR ) T) = 0. 

Moreover, its model variance is 

E,

[(tR ) T)2

] = E,

[(3S aiyi ) 3P yi

)

2

]

 = E,

[(3S ai,i ) 3P ,i

)

2

]

 = 3S ai

2Fi2

) 23S aiFi

2

 + 3U Fi

2

. 

 = 3S ai

2Fi

2

) 3S aiFi

2

) (3S aiFi

2

) 3U Fi

2

). 
6

When Fi

2

 has the form xih, for some not-necessarily-specified vector h, then 

3S aiFi

2

 = 3U Fi

2

, and the model variance of tR

 collapses to 3S (ai

2 

) ai

)Fi

2

. Alternatively, if

we add to ours asymptotic assumptions the following: 

0 < LF # 3U Fk

2r /N < BF

 < 4, r = 1, ..., 4, (3.5)

then one can see that the model variance of tR is O(N2/n*), while 3S aiFi

2

) 3U Fi

2

 is

OP(N/3⁄4n*). Observe that although we are interested in model-based expectations here,

we plan to invoke a large-sample, randomization-based equality. Model-based theory

does not deny the applicability of the law of large numbers to random samples. It

simply resists taking averages (expectations) across all possible samples.

Our last equality suggests the following asymptotic approximation for the model

variance of tR

:

E,[(tR ) T)2] . 3S ai

2Fi

2

) 3S aiFi

2

, (5)

 

which drops OP(N/3⁄4n*) terms. In so doing, it assumes that O(N) is greater than O(n).

What about likewise replacing ai

2

 by Bi

-2 (and ai

 by Bi

-1) in equation (5)? Such a

substitution would effectively drop OP(N2/[n*]3/2) term. To see why, observe that 

3S ai

2Fi

2

 = 3S Bi

-2Fi

2

 + 2 (3U xi ) 3S Bi

-1xi

)(3S ciBi

-1xi

'xi)

 -1 3S

 ciBi

-2xi

'Fi

2

 + 

 (3U xi ) 3S Bi

-1xi

)(3S ciBi

-1xi

’xi)

 -1 3S Fi

2

ci

2Bi

-2xi

’xi (3S ciBi

-1xi

’xi)

 -1 (3U xi ) 3S Bi

-1xi

)’ 

 = 3S Bi

-2Fi

2

 + OP(N2

/[n*]3/2). 

In subsequent analyses, such asymptotic arithmetic will often be left to the reader.

Suppose finite population correction really matters. At the extreme, 

O(N) = O(n), and OP

(N2/[n*]3/2) is of the same asymptotic order as the OP

(N/3⁄4n*) term

dropped by equation (5). Finite population correction still matters somewhat when 

O(N) = O([n*]3/2); that is when the population is relatively large (Kott 1990). Under this
7

setup equation (5) appropriately drops a OP(n*). term. Observe, however, that

replacing ai

2

 by Bi

-2 would effectively drops a larger, OP

([n*]3/2) term. 

3.3. Anticipated Variance 

The model variance of tR is a function of the realized sample and does not depend at all

on the sampling design. As noted in the previous section, it is OP

(N2/n*) under the

(extended) asymptotic assumptions of equation (3). In fact, if we are willing to drop

OP(N2

/[n*]3/2) terms, the model variance can be approximated by 

E,[(tR ) T)2] . 3S (Fi

2

/Bi

2

)(1 ) Bi). 

The randomization expectation of the model variance of tR

 is then 

Ep{E,[(tR ) T)2

]} . 3U (Fi

2

/Bi)(1 ) Bi). (6)

This quantity can be called the “anticipated variance” of tR; that is, the model variance

anticipated before random sampling . The term is due to Isaki and Fuller (1982),

although equation (6) goes back considerably further in the literature. They use use it

to mean E,{Ep[(tR ) T)2

]}, what that model anticipates the randomization mean squared

error to be. The expectation operators can be switched, and the two concepts of

anticipated variance coincide, when ,k and ,k

2

 are uncorrelated with Ik given xk and zk,

where Fk

2

 = f(xk, zk), as we have assumed. This is weaker than the requirement that the

,i and Ii be independent, as stated in Isaki and Fuller. Maintaining the latter condition

would rule out designs where Bk % Fk for some hypothesized Fk

2

. This selection

probability rule minimizes the asymptotic anticipated variance on the right hand side of

equation (6) for a fixed expected sample size, n* = 3U Bi

. Brewer (1963) makes a

similar point.

 From equation (6), we can also see that the anticipated variance of the

randomization-consistent regression estimator is (asymptotically) a function of the unit
8

selection probabilities but not the joint selection probabilities. Every design with the

same unit selection probabilities produces an regression estimator with the same

anticipated variance. If minimizing anticipated variance is the goal, then there is no

penalty from using Poisson sampling. 

4. Simultaneous Variance Estimation

It is a simple matter to estimate the (approximate) model variance of tR expressed in

equation (5):

v = 3S (ai

2

) ai)ri

2

, (7)

where ri

 = yi ) xib, and b = (3S ckBk

-1xk

'xk

)

-1 3S ckBk

-1xk

'yk

. Now

ri = ,i ) xi(b ) $) = ,i ) xi(3S ckBk

-1xk

'xk)

-1 3S ckBk

-1xk

',k, 

so 

E(ri

2

) = Fi

2

 + 2xi(3S ckBk

-1xk

'xk)

-1 ciBi

-1xi

'Fi

2

 + 

 xi(3S ckBk

-1xk

'xk)

-1 (3S ck

2Fk

2 Bk

-2xk

'xk)

-1 (3S ckBk

-1xk

'xk)

-1xi

', 

After a little work, we can conclude that v is asymptotically model unbiased: 

E,(v) = 3S (ai

2

) ai)Fi

2

 + Op(N2

/[n*]2

). (8)

Observe that the terms we are ignoring in equation (8) are smaller than the Op(N2/[n*]3/2

terms we would have ignored had we replaced ai with Bi

-1

. 

We can likewise show that v is an asymptotically unbiased estimator for the

randomization mean squared error of tR under Poisson sampling. In this context,

however, we are willing to drop Op(N2/[n*]3/2) terms. The equalities 
9

ri = ei ) xi(b ) B) = ei ) xi(3S ckBk

-1xk

'xk)

-1 3S ckBk

-1xk

'ek

 = ei ) xi O(q x q)p(1/N)O(q x 1)p(N/3⁄4n*),

 = ei ) O p(1/3⁄4n*) (9) 

 

ultimately imply that v = 3S (ai

2

) ai

)ri

2

 = 3S (Bi

-2

) Bi

 -1 )ei

2

 + Op

(N2/[n*]3/2). From

which, we conclude

Ep

(v) = 3U (Bi

-1

) 1)ei

2

 + O(N2/[n*]3/2). (10)

 The relative model bias of v (as an estimator of E,

[(tR ) T)2

] . 3S ai

2Fi

2

) 3S aiFi

2

)

is Op(1/n*); see equation (8). Its relative randomization bias (as an estimator of 

Ep

[(tR ) T)2

] . 3U (Bi

-1

) 1)ei

2

) is O(1/[n*]1/2); see equation (10). Empirical analyses like

that in Wu and Deng (1983) have showed that this emphasis on the model bias can

lead to superior coverage estimates.

4.1. Two Alternatives 

The variance estimator v in equation (7) is very close to the weighted residual variance

estimator of Särndal, Swensson, and Wretman (1989), which is vSSW = 3S ai

2

(1 ) Bi

)ri

2

for tR under Poisson sampling. Like our simultaneous variance estimator, the weighted

residual variance estimator was designed to estimate both model variance and

randomization mean squared error with an emphasis on getting the model variance

more-nearly unbiased.

Kott (1990) offers another estimator of both model variance and randomization

mean squared error. The idea there is to multiply the traditional “randomization” mean

squared error estimator, vRB = 3S (rk /Bk)

2

(1 ) Bk) it this context, by E,[(tR ) T)2

]/E,(vRB)]. 

Kott’s estimator is cumbersome to compute and requires the Fi

2

 be specified up to a

constant. To my knowledge, it has never been used in practice. 
10

4.2. An Example

The following simple example will clarify the differences among v, vSSW, and vRB. 

Suppose all the selection probabilities where equal, Bk = B, and all xk = 1. Let n be the

realized sample size and N the population size. The expected sample size, n*, equals

NB. When all the ck

 are 1, ak

 becomes N/n the randomization consistent regression

estimator is tR = 3S (N/n)yk. This is exactly the same as the expansion estimator under

simple random sampling given a fixed sample size n. 

In our asymptotic environment, n* is large, and the probability that n is zero is

itself zero. The variance estimator proposed here is v = 3S (N/n)2

(1 ! n/N)(yk ! 3S yi /n)2

. 

 It is easy to show that vE = [n/(n )1)]v is model unbiased when all the Fk

2

 are equal. 

Moreover, the model unbiased variance estimator corresponds exactly the

randomization estimator conditioned on the realized sample size n. This is the best

variance estimator for confidence interval construction. 

The traditional (unconditional) randomization mean squared error estimator for tR

under Poisson sampling is vRB = 3S (N/n*)2(1 ! n*/N)(yk ! 3S yi /n)2. This estimator has

an unfortunate property. As the sample size increases making tR more accurate, vRB

gets larger rather than smaller. The near randomization unbiasedness of vRB refers to

an average squared difference between t and T taken all possible samples. For

particular sample sizes, vRB need not be very good.

In order-of-probability notation, v = vE ( 1 + OP (1/n*)). In contrast, 

vRB = vE (1 + OP (1/3⁄4n*)). Treating vE as the gold standard, v is easily seen to be

superior to vRB. 

The weighted residual mean squared error estimator for tR under Poisson

sampling is vSSW = 3S (N/n)2

(1 ! n*/N)(yk ! 3S yi /n)2

. This is approximately equal to v

when N is relatively large (i.e., N = O([n*]3/2)) , but vSSW can be noticeably different from

v when n is not that close to n* and n/N is not negligible. 

The example is extremely simple but it makes an important point. All three

estimators estimate the unconditional randomization mean squared error adequately,

but v does the best job at estimating the conditional randomization mean squared error. 
11

Like all model-based variance estimators, it is conditioned on the realized sample not

averaged over all possible samples. In the context of this example, it is a simple matter

to construct a conditional randomization mean squared error estimator (it would be v).

That is not the case in general. Moreover, although the literature on conditional

randomization-based theory for regression estimators is growing, without a model one

does not really know on what ancillary statistics to condition. 

5. A Note on Adjusting for Small-sample Bias

It is a tempting to scale v in equation (7) by n/(n - Q) to account for the fact that ri 

2

, a

squared residual from a Q-variate regression, is a slightly biased estimator for Fi 

2

. 

This ad hoc adjustment would only be reasonable in our context when n is relatively

small (technically, O(n) = 1), and either N is large (so that the model variance of tR is

approximately 3S ai

2Fi

2

) or Fi 

2= xih for some h (so that 3S aiFi

2

 - 3U Fi

2

 can be still ignored

even when n is small). 

A better approach than the ad hoc adjustment of v would be to replace the ri

2

with unbiased estimators for the components of F

2

 = ( F1 

2

, ..., Fn 

2

)', namely, 

r(2) = M

-1( r1 

2

, ..., rn 

2

)', where the i,jth element of then n x n matrix M is 

mij = [1 - xi(3S ckBk

-1xk

'xk)

-1cjBj

-1xj

]

2

. See Chew (1970). Calculating r(2) involves

inverting an n x n matrix, but we are presuming n is relatively small. The ad hoc

adjustment to v can also produce a model-unbiased variance estimator, but only if

additional assumptions are made (e.g., when ckBk

-1 is constant across k, and Fk

2

 is

likewise constant across k as in the example in Section 3.2). 

6. Delete-a-group Variance Estimation

Many surveys have multiple variables of interest. The problem with v in equation (7) is

that is requires rk to be calculated separately for each such variable, even when a

common regressor vector, xk, is employed. That is one reason why a delete-a-group

jackknife variance estimator can prove helpful in practice. The term can be found in
12

Kott (2001), while the variance estimator itself in some form has long been used, not

always with theoretical justification. A NASS research report, Kott (1998), discusses a

wide variety of uses for the delete-a-group jackknife. 

In this section, we assume that finite-population correction can be ignored. 

Formally, 1/N #O(1/[n*]2). This means when OP

(N) terms are dropped, the model

variance in equation (5) is approximately V0 = 3S ai

2Fi2

, which is Op(N2/n*). Moreover, 

v0

 = 3S ai

2

ri

2

 becomes asymptotically indistinguishable from the simultaneous variance

estimator, v.

 Let the Poisson sample be randomly divided into G replicate groups, denoted

S1, S2, ..., SG (some groups can have one more member than others). The complement

of each Sg is called the jackknife replicate group S(g) = S ) Sg. A sets of replicate

weights is computed for each replicate group. For the gth set: ai(g) = 0 when i 0Sg

; and 

ai(g) = ai

 + (3U xk ) 3S(g) akxk

)(3S(g) ckakxk

'xk

)

-1ciaixi

' (11)

otherwise. The ai(g) have been computed to be reasonably close to the corresponding ai

for i 0S(g) and to satisfy the calibration equation 3S ak(g)xk = 3U xk for all g. We return to

equation (11) in the concluding section. 

The delete-a-group variance estimator for tR

 is :

vJ = (G ) 1/G) 3

G (3S ai(g)yi ) tR

)

2

, (12)

which WESVAR (Westat 1997) calls JK1. This can be re-expressed as 

vJ

 = (G ) 1/G) 3

G (3S ai(g)ui ) 3S

 aiui

)

2

 

where ui may be either ,i or ei. 

For ease of exposition, let us assume that n/G equals an integer, d. To do

otherwise, complicates the subsequent formulae without adding insight. 

Result are again asymptotic. We assume G grows arbitrarily large along with n*,
13

but allow the possibility that Op(G) < O(n*

). As a result, O(d) $ O(1). Nevertheless,

O(d) must be less than O(n*) because n = Op(n*) and Op(G = nd) must be greater than

O(1). 

The sets that Sg

 and S(g) can be viewed as simple random subsamples of S. 

With that in mind: 

3S ai(g)ui

 ) 3S aiui

 = ) 3Sg aiui

 + (3U xk ) 3S(g) akxk

)(3S(g) ckakxk

'xk

)

-13S(g) ciaixi

'ui

 

 = ) 3Sg aiui + (3Sg ak xk)(3S(g) ckakxk'xk)

-13S(g) ciaixi

'ui

 = ) 3Sg aiui

 + {[d/n] 3S ak xk

 + O(1 x q)p(3⁄4d)}O(q x q)p(1/N) 3S(g) ciaixi

'ui

 = ) 3Sg aiui

 + O(1 x q)p(d/n*) 3S(g) ciaixi

'ui

. (13) 

 

Consequently, 

E,

[( 3S ai(g),i

 ) 3S ai,i

)

2

] = 3Sg ai

2Fi

2

 + O(1 x q)p(d/n*) 3S(g) ci

2ai

2 Fi

2 xi

'xi O(q x 1)p(d/n*)

 = 3Sg ai

2Fi

2

 + Op

(N2d

2

/[n*]3

)

 = 3Sg ai

2Fi

2

 + Op

(G -2 N

2

/n*), 

where the dominant term is Op(N2d/[n*]2

) = Op(G-1 N

2

/n*). From this, we can see that the

delete-a-group is asymptotically model unbiased estimator for V0 = 3S ai

2Fi

2

: 

E,

(vJ

) = ([G ) 1]/G) [ 3

G 3Sg ai

2Fi

2

 + Op

(G -2 N

2

/n*)] 

 = 3S ai

2Fi

2 [1 + Op

(1/G)]

Establishing the asymptotic randomization-based properties of vJ is a bit more

difficult. From equation (13): 

3S ai(g)ei ) 3S aiei = ) 3Sg aiei + O(1 x q)p(d/n*) 3S(g) ciaixi'ei. 

 = ) 3Sg aiei + O(1 x q)p(d/n*){[n/(n ) d)] 3S ciaixi'ei + O(q x 1)p(N/3⁄4[n ) d])} 

 = ) 3Sg aiei + O(1 x q)p(1/G){[1 + O(1/G)] 3S ciBi

-1 xi

'ei + O(q x 1)p(N/3⁄4n*)} 

 = ) 3Sg aiei + O(1 x q)p(1/G) {[1 + Op(1/G)] 3U ci xi'ei + O(q x 1)p(N/3⁄4n*)} 
14

 = ) 3Sg aiei + Op(G -1 N/3⁄4n*), (14) 

 

where the dominant term is Op(dN/n*) = Op(G -1 N). 

Combining equations (12) and (14):

vJ = ([G ) 1]/G) 3

G (3S ai(g)ei ) 3S

 aiei)

2

 

 = ([G ) 1]/G) 3

G [ ) 3Sg aiei

 + Op

(G -1 N/3⁄4n*)]2 

 = ([G ) 1]/G) [ 3

G ( 3Sg aiei

)

2

 + Op(G-1 N2

/3⁄4n*)]

 = 3

G ( 3Sg aiei

)

2

 + Op(N2/G2) + Op(G-1 N2/3⁄4n*). (15)

 

We now turn our attention to the randomization expectation of vJ

 under the

random subsamplings of sample S. We need an addition assumption; namely, 

ci

 = 1/(xi() for some vector (. Under this assumption, 3U ei

 = 3U ('xi

'ciei

 = ('3U cixi

'ei

 = 0 

The replicate group Sg

 can be viewed as a random subsample of S. In fact,

dropping Op

(N2/G2) and Op

(G-1 N2/3⁄4n*) terms, equation (15) implies that for any g, 

E2(vJ) . E2(wg

2

/G), where wg = G3Sg aiei= (n/d)3Sg aiei,, and the subscript 2 refers to the

subsampling. 

 Now 

E2(wg

2

) = Var2(wg) + [E2(wg)]2 

 = {n2/[d(n-1)]}[3S (aiei)

2

 ) (3S aiei)

2

/n] + (3S aiei)

2

 = G(n/[n-1])3S (aiei)

2

 + (1 ) n/[d(n-1)]) (3S aiei)

2

 = G(n/[n-1])3S (aiei)

2

 + (1 ) n/[d(n-1)]) (3U ei + Op(N/3⁄4n*))2

 = G 3S (aiei)

2

 + Op(N2

/n*) . (16)

From which we can conclude E2(vJ) = 3S (aiei)

2

 + Op(G-1N

2

/n*), which is asymptotically

indistinguishable from v0. 

From the derivation of equation (16), we see that when ci  1/(xi(), so that 

3S aiei  Op

(N/3⁄4n*), vJ

 can have an upward bias as an estimator of the randomization

mean squared error of tR. 
15

7. Discussion

Let us first review the main results discussed so far. 

Under the linear model in equation (4), the regression estimator, tR, in equation

(1) has the same (asymptotic) anticipated variance under Poisson sampling as it does

under any sampling design having the same set of first-order selection probabilities

(equation (6)). “Anticipated variance” here means the randomization expectation of the

model variance, but under mild conditions that value is identical to the model

expectation of the randomization variance.

When the expected sample size, n*, is large, both the model variance and

randomization mean squared error of tR under Poisson sampling can be estimated by 

v = 3S (ai

2

) ai

)ri

2

. Under mild asymptotic assumptions (in equation (3)), the relative

bias of v as an estimator of model variance is Op(1/n*) if N is relatively large (i.e.,

O([n*]3/2) or if Fi

2

 = xih for some not-necessarily-specified h. The relative bias of v as an

estimator of randomization mean squared error is O(1/[n*]1⁄2). 

When n is not large, randomization-based properties lose much of their

relevance. Section 4 describes a method of modifying v to produce an unbiased

estimator for the model variance of tR

 when Fi

2

 = xih. This method is also effective

when N is large, and the model variance asymptotically approximated by VO

 = 3S ai

2Fi

2

. 

When both n and N are large, but N is so large that finite-population correction is

ignorable (N $ O([n*]2), the delete-a-group jackknife variance estimator (equation (12))

can be used to estimate both the model variance and randomization mean squared

error of tR provided that G, the number of jackknife replicate groups, is also large. The

asymptotic unbiasedness of the latter requires on additional assumption: ci = 1/(xi() for

some vector (. The ( need bear no relationship to the h discussed above. The

simplest way for this assumption to be satisfied is for ci to be set equal to 1 and xi to

contain unity as one of its components. 

Several observations are in order.

The choice of ck in equation (1) has no impact on the anticipated variance

expressed in equation (6). Consequently, there is no reason to prefer on set of ck over
16

another on efficiency grounds. A secondary concern is that the ak all be bounded from

below by unity. Following Brewer (1994), this suggests the choice ck = 1 ) Bk (although

this limits the possibility that some ai < 1, it does not remove it). When N is relatively

large, the resultant estimator is asymptotically indistinguishable from that when ck

 = 1. 

The variance of any model-unbiased weighted estimator (tR

 = 3S akyk

, where 

3S akxk

 = 3U xk

) can be estimated with v. This variance estimator is asymptotically

unbiased when Fi

2

 = xih. A strictly model-unbiased estimator can be computed

following the suggestion in Section 4. In addition, when finite-population correction is

ignorable, the delete-a-group jackknife, vJ , with replicate weights calculated using

equation (11) is an asymptotically model-unbiased variance estimator provided that G is

large. 

It is also tempting to use v or vJ to estimate the randomization mean squared

error for a randomization consistent tR based on a non-Poisson sample even though

there may be a nonignorable randomization bias. This is especially true when there

are O[(n*)2] cross terms in a plug-in mean-squared-error estimator such as v* = v +

3i,i0S (ij) [(Bij ) BiBj

)/Bij](ri /Bi

)(rj /Bj

). In that case, the asymptotic randomization

unbiasedness of v* has not been clearly established except under strong conditions. 

See Breidt and Opsomer (2000). Moreover, with so many terms, the variance of v*

itself becomes an issue. The potential bias in v may not be as much a practical

concern as the variance of v*. 

Finally, observe that just as equation (11) can be used to compute replicate

weights for any model-unbiased estimator, it can also be used to compute replicate

weights for a randomization-consistent, model-unbiased estimator when 

ak = Bk

-1 + (3U xi ) 3S Bi

-1xi

)(3S ciBi

-1xi

'xi)

 -1 ckBk

-1xk

' + Op(1/n). (17)

The resultant delete-a-group jackknife will have the usual model and randomizationbased properties. Deville and Särndal (1992) and Singh and Mohl (1998) discuss a

number of calibration estimators with weights satisfying equation (17). 
17

8. References

Breidt, F.J. and Opsomer, J. D. (2000). Local polynomial regression in survey

sampling. Annals of Statististics, 28, 1026-1053.

Brewer, K.R.W. (1963). Ratio estimation and finite populations: some results

deductible from the assumption of an underlying stochastic process. Australian

Journal of Statistics, 5 93-105.

Brewer, K.R.W. (1994). Survey sampling inference: some past perspectives and

present prospects. Pakistan Journal of Statistics, 10(1)A 213-233.

Chew, V. (1970). Covariance matrix estimation in linear models. Journal of the

American Statististical Association, 65 173-181.

Deville, J-C. and Särndal, C-E. (1992). Calibration estimators in survey sampling. 

Journal of the American Statistical. Association, 87 376-382. 

Isaki, and Fuller, W.A. (1982). Survey design under the regression superpopulation

model. Journal. American. Statistical. Association, 77, 89-96.

Kott, P.S. (1990). Estimating the conditional variance of a design consistent regression

estimator. Journal of Statistical Planning and Inference, 24 287-296. 

Kott, P.S. (1998). Using the Delete-a-Group Jackknife Variance Estimator in NASS

Surveys, RD Research Report No. RD-98-01. Washington, DC: National

Agricultural Statistics Service.

Kott, P.S. (2001). The delete-a-group jackknife. Journal of Official Statistics,

 forthcoming.

Kott, P.S. and Bailey, J. T. The theory and practice of maximal Brewer selection.

Proceedings of the Second International Conference on Establishment Surveys,

Invited papers, 269-278. 

Särndal, C-E, Swensson, B., and Wretman, J. (1989). The weighted residual technique

for estimating the variance of the general regression estimator of a finite population

total. Biometrika, 76, 527-537, 

Särndal, C-E, Swensson, B., and Wretman, J. (1992). Model Assisted Survey

Sampling. New York: Springer.
18

Singh, A.C. and Mohl, C.A. (1996). Understanding calibration estimators in survey

sampling. Survey. Methodology, 22 2, 107-115.

Westat, Inc. (1997). A User's Guide to WesVarPC®, Version 2.1. Rockville, MD: Westat.

Wu, C.F.J. and Deng, L.Y. (1983). Estimation of variance of the ratio estimator: an

empricial study. In Scientific Inference, Data Analysis and Robustness, ( Eds.

G.E.P. Box, et al.) New York: Academic Press, 245-277.