USDA Forest Service Res. Note RMRS–RN–4. 1998 1

Poisson Sampling – The Adjusted 

and Unadjusted Estimator Revisited

Michael S. Williams, Hans T. Schreuder, and 

Gerardo H. Terrazas 

United States

Department of

Agriculture

Forest Service

Rocky Mountain

Research Station

Research Note

RMRS–RN–4

September 1998

Introduction

Hajek (1958, 1964) introduced Poisson sampling into the statistical literature. It is defined as a 

sampling design in which the sample units have unequal probabilities of selection,πi . In addition 

the units in the population are independent and the sample size, n , is a random variable. Hajek 

proposed the unbiased estimator for the population total given by



Y y u

i

i i

n

= =

∑

1 π [1]

where yi = the value of interest for uniti,πi = ne xi / X is the probability of selecting uniti based 

on a covariate xi ,n is the achieved sample size with expected sample size E n ne [ ] = , and X is the 

sum of all xi in the population.

Grosenbaugh (1964) introduced 3-P sampling into the forestry literature as an alternative to 

Poisson sampling. He used a slightly biased estimator defined by

  Y , n

n

y n

n a Y e i

i

e

u

i

n

= = =

∑

1 π [2]

which will be referred to as the adjusted estimator.

with 

Ya and Yu and shown to be not as sensitive to such 

data points. Simulations on a small population illustrate 

these ideas.

Keywords: poisson sampling, adjusted estimator, 

unadjusted estimator, generalized regression estimator, 

approximate Srivastava estimator 

Williams, M.S.; Schreuder, H.T.; Terrazas, G.H. 1998. 

Poisson sampling – The adjusted and unadjusted 

estimator revisited. Res. Pap. RMRS–RN–4. Fort 

Collins, CO: U.S. Department of Agriculture, Forest 

Service, Rocky Mountain Research Station. 10 p.

Abstract—The prevailing assumption, that for Poisson 

sampling the adjusted estimator 

Ya is always substantially 

more efficient than the unadjusted estimator 

Yu , is shown 

to be incorrect. Some well known theoretical results are 

applicable since 

Ya is a ratio-of-means estimator and Yu a simple unbiased estimator. We formalize an additional 

realistic situation for high-value timber estimation for 

which 

Yu is more efficient. Here y x i ≈ β i for all but a 

few units in a population for which yi is very large and 

xi very small. This is a common situation in estimating 

the net volume of high-value standing timber such as 

that found in the Pacific Northwest region of the United 

States. Basically this means that 

Ya is sensitive to some 

types of valid data. The generalized regression estimator 

and an approximate Srivastava estimator are compared 
2 USDA Forest Service Res. Note RMRS–RN–4. 1998

The literature contains many examples showing that 

Ya is more efficient than Yu (Grosenbaugh 

1964, Schreuder et al. 1968, Furnival et al. 1987). Recently, occasions were observed where 

Yu is at 

least as efficient as 

Y . a These situations arose for high-value timber stands in the Pacific Northwest where an accurate estimate of net volume was desired. In order to obtain a good estimate of 

net volume in a stand, cutting down and destructively measuring trees for their actual volume 

may be desirable. In this case, Poisson sampling may be appropriate where an ocular estimate of 

net volume while the tree is still standing is the covariate xi . Thus, sampling is proportional to 

estimated standing tree net volume. It can be very difficult to ascertain net volume on a standing 

tree and even experienced timber cruisers severely overestimate or underestimate net volume of 

some trees, particularly for larger trees. However, these estimates are valid data. Thus, there is no 

reason to remove these points from an estimate net volume. It is in situations where the ocular 

estimate of net volume (x )i is very small and the actual net volume ( y )i is large that problems 

with the adjusted estimator arise.

Literature Review

Poisson sampling is a strategy based on using unequal probabilities of selection for each of the 

elements in the sample. If the sample membership indicator Ii is given by 

I i

I

i

th

i

=

=

1

0

 if the element is in the sample

 otherwise

for each i = 1,2,N, then

P Ii i [ = 1] = π

and

P Ii i [ = 0] = 1− π .

The probability that any sample s is chosen under Poisson sampling is given by

P s i i

i s i s

[ ] = ( − ). ∈ ∉

∏π ∏ 1 π

In Poisson sampling the sample size n is random with mean E n i

N

i [ ] = ∑ =1 π and variance 

V n i

N

i i [ ] = ∑ ( − ). −1 π 1 π

Under Poisson sampling the variance for the unadjusted estimator given in [1] is given by

V Y y y u

i

i

i

i

N

i

N

(  ) = − = =

∑ ∑

2

2

1 π 1 . [3]

The adjusted estimator, given in [2], was proposed by Grosenbaugh (1964). This estimator is 

slightly biased, but generally has a smaller variance than the unadjusted estimator. The approximate variance for the adjusted estimator is given by

V Y y Y

n

V n

n a

i

i i e e

N

(  )  [ ]( ( ) = − + ), =

∑

2 2

2

1

1

π [4]
USDA Forest Service Res. Note RMRS–RN–4. 1998 3

where

V n i

i

N

i

i

N

( ) = − . = =

∑π ∑π

1

2

1 [5]

Since its introduction, 

Ya has been considered more efficient than Yu (Grosenbaugh 1964, 

Schreuder et al. 1968, Furnival et al. 1987). Schreuder et al. (1968) compared the efficiency of the 

adjusted and unadjusted estimators using two highly correlated populations. When the correlation 

coefficient(ρ ) xy between xi and yi exceeded .95, they found the ratio of Monte Carlo estimated 

variances,Vu Va / , for the unadjusted and adjusted estimators to range from 4.20 to 9.38 for one 

population and 6.74 to 12.17 for the other population. The differing ratios were generated using 

different sampling fractions. Van Deusen (1987) states that the 3-P adjusted estimator has nearly 

optimal properties under a regression superpopulation model and can be expected to perform 

well except under extreme deviations for this model. Schreuder (1987) describes a situation where 

the model between xi and yi violate the superpopulation model. The results in Van Deusen (1987) 

and Schreuder (1987) indicate that 

Ya can perform poorly, but the performance of Yu in these situations is undocumented.

Furnival et al. (1987) and Van Deusen (1987) note that 

Ya can be rewritten as

 (  /  Y Y X )X , a = u u [6]

where X

u is defined analogously to 

Yu in [1] and X is the sum of the covariate values. Hence, the 

comparison in efficiency of the ratio-of-means estimator 

Ya and the unadjusted estimator of the 

population total 

Yu is applicable. Thus, the relative efficiency of 

Ya to Yu is

RE V Y

V Y

R V X

V Y

R

V X

V Y

u

a

u

u

u

u

= ≈ + − − (  )

(  ) ( (  )(  )

(  )

(  )

1 2 ) 2 1 ρ

[7]

where ρ = Cov X Y

V X V Y

u u

u u

(  ,  )

(  ) (  )

, R Y

X = , andY and X are the population totals for the variable of interest 

and covariate respectively. Then , 

Yu is more efficient than Ya providedρ <

1

2

( (  )

(  )

) CV X

CV Y

u

u

 (Cochran 1977).

These results seem straightforward but do not explain why 

Yu is more efficient than Ya for populations like the one described later. It can be shown that if the following minimum list of assumptions 

is met then the adjusted estimator will be at best as efficient as the unadjusted estimator.

1. ne N 2 ≥ , where ne is the expected sample size and N is the total number of trees in the 

population.

2. There exists at least one point y X e k ≥ + k with βxk small relative to yk for at least one k.

3. The coefficient β is such that β2

2

2 1

≤ −

X

N (n )

.

4. X −1 ≈ X.

5. (β(X −1) + X ) ≈ β X . 2 2 2

The proof is given in the appendix.
4 USDA Forest Service Res. Note RMRS–RN–4. 1998

Alternative estimators exist for Poisson sampling. Ouyang and Schreuder (1993) describe an unbiased Srivastava estimator for Poisson sampling in a forestry setting. The Srivastava estimator



| |

( )

| | '

' Y

s

y s

s

Sra

i

i

i

i s i s

1

1 = = ∈ ∈

∑ ∑ π

π ρ with 

[8]

has zero variance ifπi α i y . If pps sampling is used, the Srivastava estimator can be approximated 

by the following ratio estimator

 *

* Y

y

y

Sra Y

i

i

i

n

i

n i

i

2

1

1

= =

=

∑

∑

π

π [9]

(Ouyang et al. 1992) where y x i i

* = α + β and Y y iN

i

* * = ∑ . =1 This ratio estimator is not an unbiased 

estimator, but Ouyang and Schreuder (1993) found the bias to be small provided a linear relationship exists between y and x.

Särndal (1980,1982) gives the asymptotically unbiased generalized regression estimator



Y ( ) ( ). y N X x

GR

i

i i

i

i i

n

i

n

i

n

= + − + − = = =

∑ ∑ ∑ π

α

π

β

π

1

1 1 1 [10]

Van Deusen (1987) shows that 

Ya and Yu are special cases of the Y , GR where Yu uses no model 

and 

Ya uses a ratio model with variance proportional to xi

2

.

This paper includes these alternative estimators to demonstrate which of these estimators might 

be a good choice in similar survey situations.

Data Description and Results

For a data set described by Johnson and Hartman 

(1972), ocular estimates of net volume (= x )i and the 

net volume obtained by destructive sampling (= y )i

are given. The size of the data set was N = 131. For 

the simple linear regression model R2 = 0.9505 and

y = 1.0469x. Figure 1 shows the net volume y versus 

the ocularly estimated net volume x. This data set 

contains one outlying point where xi is very small in 

relation to yi . Given the high correlation of the model 

and the results of Schreuder et al. (1968), the adjusted 

estimator was expected to produce standard deviations 

which were substantially smaller than the unadjusted 

estimator. The first line of table 1 contains the true 

standard deviation as a percentage of the total net 

volume for the population of N = 131 units. 

Figure 1. Net scale volume – Net cruse volume 

relationship
USDA Forest Service Res. Note RMRS–RN–4. 1998 5

Table 2. Magnitudes of terms in equations [5] and [6].

Population T1 T2 T3 T4 T5

N = 131 1.381x1011 1.304x109 1.381x1011 3.308x109 1.017

N = 130 3.396x109 1.289x109 3.396x109 3.235x109 1.017

Table 1. Population standard deviations as a percentage of Y.

Population 

Ya Yu

N = 131 108.832 108.699

N = 130 3.840 13.641

The true variance for the population, N = 131, is slightly larger for the adjusted estimator than 

for the unadjusted estimator. The cause of this is the one outlying point where xi is very small 

in relation to yi . This produced a very small probability of selection,πi , in conjunction with an 

average value of yi . By deleting this point from the data set, the standard deviation values in the 

second line of the table are produced(N = 130). These values fall in line with the traditional belief 

that the adjusted estimator is more efficient than the unadjusted estimator. To better understand 

the cause of this situation, the variance equations for the adjusted and unadjusted estimator need 

to be studied. For the unadjusted estimator the variance is given as

V Y y u y T T i

i

i

i

N

i

N

(  ) = − =( − ), = =

∑ ∑

2

2

1

1 2

1 π [11]

whereT y i

N

1 1 i i

2 = ∑ = / π andT y iN

2 1 i

2 = ∑ = . For the adjusted estimator the approximate variance is 

given by

V Y y Y

n

V n

n a T T T i

i i e e

N

(  )  [ ]( ( ) = − + ) = ( − ) , =

∑

2 2

2 3 4 5

1

1

π [12]

whereT3 T1 T4 Y ne T V n ne

2

5

2 = , = / , and = 1+ ( ) / . The magnitudes of each term of equations [11] and 

[12] are listed in table 2 for the data set with and without the outlier data point included. The first 

line of the table indicates that termsT1 andT3 dominate the variance equations when the outlier 

data point is included. When T3 is multiplied by the variable sample size correction term, T5 , the 

adjusted estimator can be less efficient than the unadjusted estimator.

Table 3 contains iterated standard deviations as a percentage of the total net volume after 200,000 

simulations using the two data sets previously mentioned. As the results indicate, 

YSra2 is substantially more efficient than either 

Yu or Ya for N = 131. The reason YSra2 is more efficient is that when 

the outlier data point is selected yk and yk

*

 divided byπ k produces very large overestimates of 
6 USDA Forest Service Res. Note RMRS–RN–4. 1998

Table 3. Iterated standard deviations as a percentage of ∑i=

N

i y 1 .

Population 

Ya Yu YSra1 YSra2 YGR

N = 131 114.094 111.328 115.353 3.568 3.780

N = 130 3.151 13.835 3.164 3.479 3.708

theY. Since these terms appear in the numerator and denominator, the effect of the overestimate 

is reduced. The generalized regression estimator, 

Y , GR also shows a considerable increase in efficiency over 

Ya and 

Y . u The reason YGR is more efficient is because when the outlier point is selected

yk k / π , α / π , k and βxk π k / all produce very large overestimates. When the terms ∑i=

n

1α πi / and

∑i=

n

i i x 1 β / π are subtracted from ∑i=

n

i i y 1 / π , the effect of the outlier point is reduced. Estimator



YSra1 is as inefficient as Ya and Yu because there is no compensation for outlier points as occurred 

for 

YSra2 and Y . GR It should be noted that 200,000 simulations were not sufficient to fully stabilize 

the estimates of 

Ya and Y , u but due to limited computing time no attempt was made to perform 

additional simulations.

For N = 130 Ya YSra1 YSra2 YGR ,  ,  ,   and are about equally efficient.

Conclusions and Recommendations

Since its introduction, 

Ya has been considered more efficient than Y . u In situations where an outlier point xk is small and βxk much less than the corresponding yk value, it is reasonable to expect 

Ya to be less efficient than Y . u This occurs, for example, when ocular estimates of net volume are 

used to predict actual net volume. 

YSra2 and YGR perform quite well in the presence of these types 

of outlier observations. 

YGR would generally be a better choice than 

YSra2 since it has a reliable variance estimate (Schreuder and Ouyang 1992) and is asymptotically unbiased and efficient.

Literature Cited

Cochran, W.G. 1977. Sampling Techniques. Third ed. John Wiley and Sons, New York. 428 p.

Furnival, G.M., T.M. Gregoire, and L.R. Grosenbaugh. 1987. Adjusted inclusion probabilities with 3P sampling. 

For. Sci. 33:617-631.

Grosenbaugh, L.R. 1964. Some suggestions for better sample-tree measurement. Proc. Soc. Am. For., Oct. 1963, 

Boston, Massachusetts. Soc. Am. For. Bethesda, Maryland.

Hajek, J. 1958. Some contributions to the theory of probability sampling. Bull. Int. Stat. Inst. 36(3):127-134.

Hajek, J. 1964. Asymptotic theory of rejective sampling with varying probabilities from a finite population. Ann. 

Math. Stat. 35:1491-1523.

Johnson, F.A., and G.B. Hartman, Jr. 1972. Fall, buck, and scale cruising. J. For. Sci. Sept. 1972. 566-568.

Ouyang, Z., and H.T. Schreuder. 1993. Approximate Srivastava estimation in forestry. For. Sci. 39:309-320

Ouyang, Z., J.N. Srivastava, and H.T. Schreuder. 1992. A general ratio estimator and its application to regression 

model sampling. Ann. Inst. Stat. Math. (in press).

Särndal, C.E. 1980. A two-way classification of regression estimation strategies in probability sampling. 

Can. J. Stat. 8:165-177.
USDA Forest Service Res. Note RMRS–RN–4. 1998 7

Särndal, C.E. 1982. Implications of survey design for generalized regression estimation of linear functions. J. of 

Stat. Plan. Inf. 7:155-170.

Schreuder, H.T., J. Sedransk, and K.D. Ware. 1968. 3-P sampling and some alternatives, I. For. Sci. 14:429-454.

Schreuder, H.T. 1987. Design-dependent and model-dependent procedures in timber sale and updating surveys. 

Proc. Soc. of Am. For., National Communication, Birmingham, Alabama. Oct. 5-8, 1986. 74-76.

Schreuder, H.T., and Z. Ouyang. 1992. Optimal sampling strategies for weighted linear regression estimation. Can. 

J. For. Res. 22:239-247.

Van Deusen, P.C. 1987. 3-P sampling and design versus model-based estimates. Can. J. For. Res. 17:115-117.
8 USDA Forest Service Res. Note RMRS–RN–4. 1998

Appendix

Schreuder et al. (1968) give the true varianceV Yu Yu (  )  of and an approximate variance of  , (  Y V Y ) a a as

V Y y y u

i

i

i

i

N

i

N

(  ) = − = =

∑ ∑

2

2

1 π 1 [1]

and

V Y y Y

n

V n

n a

i

i i e e

N

(  )  [ ]( ( ) = − + ) =

∑

2 2

2

1

1

π [2]

where V n i i

i

N

i

N

( ) = − . = =

∑π ∑π 2

1 1 [3]

We wish to show that under certain conditions V Ya V Yu (  ) (  ≥ ). For the adjusted estimator the variable sample size factor can be approximated by

( ( )) ( ) ( ( )1 1 1 ) ( ) 1 1 1

2

1 1

2

2

1

2

1

2 + = + ∑ − ∑ = + − ∑

∑ ≈ + = = =

=

V n

n n n

x

x n e

i

N

i i

N

i

e e

i

N

i

i

N

i e

π π

[4]

since for even moderately sized populationsV n ne ( ) ≈ because

x x i i

i

N

i

N

2 2

1 1

<< = =

∑ (∑ ) .

[5]

Let V Y V Y y Y

n n a a i

N i

i e e

(  ) (  ) [ ]( ), * ≈ = ∑ =1 − +

2 2

1 1

π

 then in order to have V Ya V Yu (  ) (  ), * ≥ we want

V Y V Y

n

y

n

Y Y

n

y a u

e

i

i e e

i

i

N

i

N

(  ) (  ) ( ) * − = − + + ≥ = =

∑ ∑ 1 1 0

2

2

2

2

1 π 1 [6]

which implies

1 2

2

1

2 2

2

n 1

y yY

n

Y

e n

i

i

i

i

N

i e e

N

π

+ ≥ + = =

∑ ∑ .

[7]

Now

y

Y

n

i n N

e

e

i

N

2

2

2

2

1

≥ ≥ =

∑ , provided .

[8]
USDA Forest Service Res. Note RMRS–RN–4. 1998 9

If this condition holds, these two terms may be dropped from [7]. All that remains to be shown is 

that under the simplifying assumptions and a given population model

y Y i

i i

N 2

1

2

π

≥ =

∑ .

[9]

Some simplifying assumptions need to be made. A widely used model in forestry is

y x e i = β i + i [10]

with E ei [ ] = 0, E e x i i [ ]

2 2 2 = σ with E e ei j [ ] = 0 fori ≠ j. In order to formalize conditions under 

whichV Ya V Yu (  ) (  ), * ≥ consider the following alternative model

y x e i = β i + i [11]

fori = 1,2,k −1,k +1,N with

y X x k = and k = 1 [12]

with E e E e x i i i [ ] = 0, [ ] = , 2 2 2 σ and E e ei j [ ] = 0 fori ≠ j for alli = 1,2,N. In [12] X is the mean of 

the x values in the population and xk = 1 is used for simplicity. Any case where xk > 0 and βxk is 

sufficiently small compared to yk will produce the same results. The model given in [11] and [12] 

is common in forestry when ocular estimation has to be used (D. Bruce - personal communication). 

The results which follow can be generalized to a population where more than one outlier exists, 

but for the sake of simplicity only one outlier will be considered. Using models [11] and [12] in 

conjunction with equation [9], we have

1 1 2

n

X x e x e

n x

X X e X e

n x

X X

n e

i i i i

e i

k k

i k e k e

N

( ( )( ) ( )( )) ( ( ) ) , β + β + β +

+ + ≥ − +

≠

∑

[13]

which can be written as

1 2 1 2 1 2 2 2 2

2 2

2

2

2

n

X x

n x n

X x e

n x n

Xe

n x

XX

n

XXe

n

Xe

n

X X

e n

i

e i e

i i

e i e

i

i k e i e

N

k

e

k

i k e e

N

i k

N β β β + + + + + ≥ − +

≠ ≠ ≠

∑ ∑ ∑ ( ( ) ) .

[14]

Due to the assumed error structure, this reduces to

β σ σ β 2

2

2

2

2

2

2

2

2 X X 1 1 1

n

X X

n

XX

n

X

n

X X

n e e e e e

( ) ( ) ( ( ) ) . − + − + + ≥ − +

[15]

Ignoring the error terms, it is sufficient to show

β β 2

2

2

2

2 X X 1 1

n

XX

n

X X

n e e e

( ) ( ( ) ) . − + ≥ − +

[16]
10 USDA Forest Service Res. Note RMRS–RN–4. 1998

Equation [16] can be rewritten as

β β 2

2

2

2

2 2 X X 1

n

XX

n

X

n e e e

( − )

+ >

[17]

assuming (β(X −1) + X ) ≈ β X , 2 2 2 which reduces to

X

N

n 2 e

2 > ( −1)β ,

[18]

assuming X −1 ≈ X. For the data set discussed in this paper, xi and yi are both estimates of net 

volume so β ≈ 1 >> −1 2 , and X N (n ). e If the following minimum list of assumptions is met, then 

the adjusted estimator will be at best as efficient as the unadjusted estimator.

1. ne N 2 ≥ , where ne is the expected sample size and N is the total number of trees in the population.

2. There exists at least one point y X e k ≥ + k with βxk small relative to yk for at least one k.

3. The coefficient β is such that β2

2 1

≤ −

X

N ne ( )

.

4. X −1 ≈ X.

5. (β(X −1) + X ) ≈ β X . 2 2 2
USDA Forest Service Res. Note RMRS–RN–4. 1998 11

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media. Please send the publication title and number.

Telephone (970) 498-1719

E-mail rschneider/rmrs@fs.fed.us

FAX (970) 498-1660

Mailing Address Publications Distribution

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3825 E. Mulberry Street

Fort Collins, CO 80524-8597

The Authors

Michael S. Williams is a mathematical statistician with the USDA 

Forest Service Rocky Mountain Research Station Fort Collins, 

Colorado.

Hans T. Schreuder is a mathematical statistician with the USDA 

Forest Service Rocky Mountain Research Station, Fort Collins, 

Colorado.

Gerardo H. Terrazas is a Ph.D. with the Department of Statistics, 

Colorado State University, Fort Collins, Colorado.
12 USDA Forest Service Res. Note RMRS–RN–4. 1998

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