Abstract 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Coordinating the PRN: Combining Sequential and Bernoulli-Type 

Sampling Schemes in Business Surveys 

Ronit Nirel, Aryeh Reiter*, Tzahi Makovky* and Moshe Kelner* 

*Central Bureau of Statistics, Israel 

66 Kanfey Nesharim St., Jerusalem 95464, Israel 

nirelr@cc.huji.ac.il; areiter@cbs.gov.il; tzahim@cbs.gov.il; moshek@cbs.gov.il. 

In this paper we use permanent random numbers (PRNs) for the maintenance of a single sample over a long period of time. 

The sample is stratified by industry and size levels and units are selected with equal probabilities within strata. Two types of 

updates are carried out: (a) annual updates reflecting all changes in the frame during the preceding year and (b) intra-annual 

updates accounting for newborns during the year. A sequential PRN scheme is used for the annual sample and a Bernoulli 

scheme for the intra-annual samples. This design causes two problems. First, because of the differences between the 

sequential and Bernoulli schemes, the upper bound of PRNs of newborns may differ from that of the old units, thus resulting 

in different selection probabilities for the two types of units in the next annual sampling. Second, since generally the number 

of newborns in each update is small, the cumulative sample size over updates may have large deviations from its expected 

value. Here we propose adjusted PRNs for units included in the intra-annual frames that have a uniform distribution in the 

unit interval, and a maximal value for the sampled units that is approximately equal to that of the annual sample. We also 

suggest a sampling scheme that controls the overall sample size of the intra-annual updates, while keeping the expected 

weights unchanged over time. Finally, we illustrate the proposed method for data from the Israeli Manufacturing Indices 

Survey. 

Keywords: Beta distribution; Collocated sampling; Coordination of samples; Order statistics. 

1. Introduction 

National Statistical Agencies (NSAs) publish time series of economic indicators based on periodic surveys of business 

establishments. While efficient estimation of periodic change requires maximal overlap between successive samples, samples 

should also be continuously updated to account for rapid changes in the business population (e.g. births, changes in size). 

Other common problems include the need to coordinate between several samples and between units within a single sample, to 

reduce response burden. Sampling schemes based on Permanent Random Numbers (PRNs) provide simple and flexible 

solutions to such problems. Ohlsson (1995) reviews the main schemes based on PRNs and describes their application in a 

number of NSAs. The basic idea is that each unit in the frame is assigned a random number in the interval (0,1) that is

permanently associated with that unit. Within strata, units are ordered by their PRNs and the sampling is based on the ordered 

list. For example, in a “sequential” scheme for selecting a simple random sample without replacement (srswor) of n units 

from a population of size N the first n units in the ordered list are selected. In a Bernoulli scheme, with the same sampling 

rate, the sample comprises all units with a PRN not larger than n/N. 

Here we use PRNs for the maintenance of a single sample over a long period of time (e.g. a decade). The sample is stratified 

by industry and size levels and units are selected with equal probabilities within strata. Two types of updates are carried out: 

(a) an annual update reflecting all changes in the frame during the preceding year (e.g. deaths, splits, and changes in industry) 

and (b) intra-annual updates accounting for newborns during the year. The design of the annual update sample is guided by a 

requirement for a maximal overlap of successive annual samples. The design of the intra-annual samples is guided by the 

need to keep the selection probabilities within strata constant throughout the year. To attain the first requirement a sequential 

PRN scheme is applied each year to an updated frame. Uniformity of the probabilities over the year is achieved by using a 

Bernoulli scheme for the intra-annual updates. This design causes two problems. First, because of the differences between 

the sequential and Bernoulli schemes, the upper bound of PRNs of newborns may differ from that of the old units 

(persistants) resulting in different selection probabilities for the two types of units in the next annual sampling. Second, since 

generally the number of newborns in each update is small, the cumulative sample size over updates may have a large 

deviation from its expected value. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

In this paper we address these problems analytically. In Section 2 we propose a method for combining two frames from 

which a sequential and a Bernoulli samples were selected, respectively, with the same probabilities. The combined frame is 

then used to select a new sequential sample that has a sizeable overlap with the two former samples. This problem is related 

to the issue of “birth bias” discussed by Ernst, Valliant and Casady (2000) and by Ohlsson (1995, p. 166). Both papers are 

concerned with bias in the selection probability of newborns under sample rotation when the newborns and persistants are 

sampled together. For specific situations these papers suggest solutions such as a correction factor for the selection 

probability of newborns, and an adjusted sampling scheme. In our problem, persistants and newborns are sampled separately 

and the two frames are then combined for a later sampling occasion. Our method is based on an adjustment of the newborns’ 

random numbers. The adjusted numbers have a uniform distribution in the unit interval, and the maximal value for the 

sampled units is approximately equal to that of the sequential sample. 

Section 3 deals with the problem of sample size variability over several intra-annual updates. The PRN literature considers 

the question of choosing between fixed sample sizes and fixed inclusion probabilities, particularly in the context of Poisson 

sampling. Brewer, Early and Hanif (1984) suggest collocation of the random numbers so that they are more evenly 

distributed in the unit interval and thus reduce, but not eliminate, the variation in sample size. Ohlsson (1995) suggests a 

sequential Poisson scheme that ensures a fixed sample size but results in approximate selection probabilities. The 

approximation is improved by Saavedra (1995) but no solution is suggested for the calculation of the true selection 

probabilities. The problem of controlling the cumulative sample size of a sequence of Bernoulli samples imposes an 

additional challenge. We propose a constructive method that calculates the desired sample size for each update, and ensures 

that the cumulative sample size does not deviate from its expected value by more than one unit. Furthermore, the selection 

probabilities are kept at their expected values. The method is based on collocation of the random numbers and on 

randomization of the conditional inclusion probabilities. In Section 4 we illustrate the suggested method for the Israeli 

Manufacturing Indices Survey. Finally, Section 5 contains some concluding remarks. 

2. A Single Intra-Annual Update 

We deal here with annual business samples that are stratified by industry and size levels. Generally there is a take-all stratum 

and several (e.g. one to five) take-some strata for each industry. To achieve statistical efficiency and reduce fieldwork burden 

the same base sample is used for several years. This sample is updated periodically (e.g. once in several months) for new 

establishments (newborns) and once a year for other changes in the frame, including merges and splits as well as changes in 

industry and size level. Formally, for each stratum h we have a base frame list Lh0

 comprising Nh0 units. An annual 

sample Sh0 of nh0 units is selected with equal probabilities ( nh0 = Nh0 for the take-all strata). Units are selected using a 

sequential PRN scheme; that is, each unit i in stratum h is assigned independently a permanent random number xhi from a 

uniform distribution on (0,1). A random starting point rh0 is selected, and we assume here without loss of generality that 

rh0 = 0 . Hence, units with the nh0 smallest random numbers are included in the sample. Let x( nh 0 ) be the largest random 

number in the sample. To abbreviate we drop hereinafter the index h. 

We begin with the simple case where the sample is updated for newborns once during a given year. A frame update 

L1 comprising N1 newborn units is obtained, and a PRN is assigned to each unit in L1 . To simplify the estimation procedure it 

is required that the update selection probabilities are retained at the annual sample value π 0 = n / 0 N0 . A Bernoulli scheme 

may be used to select the intra-annual sample update S1 ; that is, all units with xi ≤ π 0 are included in the sample. The update 

sample size n1

 is thus random, with expected value N1 0 π and variance N 10 π (1−π 0 ). Collocation of the random numbers 

may reduce the variability of n1 . For example, we can sort the units in L1 in a random order and assign to unit k a random 

number from a uniform distribution on ( (k −1) / N 1

, k / N 1 ). As before, all units with PRN not greater than π 0 are selected, but 

the sample size variability is reduced greatly because the random numbers are more equally spaced. In fact, n1 is equal to 

[N1 0 π ] or [N 10 π ] +1 , with respective probabilities 1− N 10 π and N1 0 π , where [a] and a are the integer and fractional 

part of a, respectively. We refer to this modified scheme as a Bernoulli-Type (B-T) scheme. 

In preparation for the selection of the next annual sample, a new frame L2 is constructed combining living units in L0 and L1, 

and units that were born in-between L1 and L2 . Generally, a coordination of some degree is desired between subsequent 

annual samples, ranging from a complete overlap to no overlap. A sequential srswor PRN scheme is applied to the new 
We construct now a transformation of the random numbers of units in L1

 so that the transformed numbers have two 

properties: (a) they have a uniform distribution on the unit interval and (b) the maximal value of the transformed numbers of 

units in S1 is approximately equal to x( ) n 0

. Let Xi be the PRN assigned to unit i L ∈ 1 and define a rescaled random 

variable X

* 

i

by 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

frame, and the starting point r2controls the degree of overlap. In our case, a maximal overlap is desired and we thus assume 

r2 = 0 . However, units in S1 have generally different likelihood of being included in the new sample as compared to units 

in S0 , since generally x() n ≠ π

0

0 . The reason is that the order statistic X() k

 of a series of N uniform (0,1) random variables has 

a Beta( kN , − + k 1) distribution (e.g. Stone, 1996, p. 178). In particular, X( ) n 0

 has a Beta( nN 0 , 0 − + n01) distribution with 

expectation p n = 0

/(N0 +1) and variance p(1− p) /( N0 + 2) . Thus, the scales of random numbers in L0

 and L1 have to be 

coordinated before sampling from L2

. 

 x( ) 

n0 Xi if i ∈ S 1

* *  π

= X x n ) 

0 Xi i (1)

(

() =

X −π

0 

(1− x ) i 0 + x if i ∉ S .

() () n n 0 0 1 



1−π 0

For i S X

* maps X from the interval (0,π ) to (0, x n0

) , and for ∉ from ( ,1) to (x ∈ 1 ( ) , i i 0 () i S1 π 0 n0 

,1) . The conditional 

distribution of X |X i

* 

(n ) 0 

= x is a mixture of U(0, x) and U x( ,1) distributions with respective probabilities π and 1−π 0 . The 0 

unconditional distribution is given by 

n

f * ( ) =

0 Pr{Beta( n −1,N − n +1) > t}+Pr{Beta( n ,N − n ) ≤ t}, 0 t 1 (2)

t ≤ ≤ Xi 

0 0 0 0 0 0

n0 −1 

(For more details see the Appendix). Analysis of the first and second derivatives of f

Xi 

* ( )t indicates a minimum at 

t = π and two turning points 0 < t < π < t < 1 , between which the function is convex, and otherwise it is concave. Thus, 0 1 0 2 

Xi

* is not uniformly distributed on (0,1). Define the U(0,1) random variable Yi = FXi 

* ( Xi 

*

) , where F( ) ⋅ is the cumulative 

distribution function (CDF). The values of Yi

can be computed from 

n n0 −2 N0 −1 

yt ( ) = F 1 

{ − ≤ , 0 ≤ ≤ 1 (3)

i X

* (t) = 

0 ∑ Pr {Beta( i +1, N0 − i) ≤ t} + ∑ Pr Beta( i +1, N0 i) t}

t 

i N n 1 i=0 i=n 0  0 − 0 

(See the Appendix for details). The transformed variable Yi

has a uniform (0,1) distribution by definition and therefore holds 

property (a). 

To verify property (b) we study empirically the relative deviation of ( ) X( ) y p from p, where p is the expected value n . Table 

0

yp p p n

1 shows the relative deviates RD = ( ( ) − ) / and the relative standard deviation of X( ) n , RSTD = Var( X( ) ) / p , for 

0 0

selected values of n0

and N0 . It is seen that the values of RD are considerably smaller than the respective values of RSTD. 

For the least favorable row in the Table, n0 = 2, the ratio RSTD/RD is equal to 4.4, 4.9, and 7 for N0 = 20 50 and 100 , , 

respectively. We conclude that the differences between x( ) and π 0 may be substantial and that as, on average, the maximal n0

value of X

*

 for units is S1is p, and as the deviation of the CDF of X

*

 from p is small, the adjusted PRNs satisfy property (b). 

3. Several Intra-Annual Updates 

In this section we deal with the more realistic case of several intra-annual updates of the frame (e.g. quarterly). We assume 

that for each update the number of new business units in a given stratum, if any, is small. As before, it is required that the 

selection probability in each stratum is kept at its annual sample value, and that the sample size variation for each update and 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 1. Expected values of x( ) n0( p), values of y(p), relative deviates of y(p) from p (RD), and relative standard deviation of 

X( ) n0

 (RSTD) for selected values of N0 and n0. 

N0 =20 N0 =50 N0 =100 

n0 p y(p) RD RSTD p y(p) RD RSTD p y(p) RD RSTD 

2 0.095 0.109 0.15 0.66 0.039 0.045 0.14 0.69 0.020 0.023 0.14 0.98 

5 0.238 0.256 0.08 0.38 0.098 0.105 0.07 0.42 0.050 0.053 0.07 0.61 

10 0.476 0.497 0.04 0.22 0.196 0.204 0.04 0.28 0.099 0.103 0.04 0.42 

15 0.714 0.738 0.03 0.13 0.294 0.302 0.03 0.21 0.149 0.153 0.03 0.33 

20 0.392 0.401 0.02 0.17 0.198 0.202 0.02 0.28 

25 0.490 0.499 0.02 0.14 0.248 0.252 0.02 0.24 

50 0.495 0.500 0.01 0.14 

over updates is minimal. We begin by explaining how to determine the size of the sample for each update and then show how 

to sample from the updates using PRNs. Finally we describe the adjusted PRNs for the annual update. 

3.1 Sample Size Determination 

For any take-some stratum let Nj be number of sampling units in the j-th update, = , ,, nj

j 1K J the actual sample size in 

update j and π 0 the desired inclusion probability. As before, denote by [a] the integral part of a, that is [a a ] ≤ < [a] +1, and 

by a the fractional part of a, so that a = [a] + a . The expected sample size for the first update (j=1) is E1 = N1π0. Thus, 

the actual sample size n1 is equal to [E1 ] or [E1 ] +1 with respective probabilities 1− E1 and E1

. For j>1, let 

m = n + + K n E , = En () = N π , and F = E + + K E be the actual and expected cumulative sample sizes after j 1 j j j j 0 j 1 j 

sampling from the j-th update, respectively. Assume that the cumulative sample size mj−1 for update j−1 has the desired 

expected value with minimal variation, i.e. 

Pm ( = F  = 1− F and Pm = F  +1 = F . j−1  j−1 ) j−1 ( j−1  j−1  ) j−1 (4) 

Clearly E mj−1 = F . We show now how to determine nj

so that m has the same optimal distribution as j 1 ( ) , that is, j−1 j m −

with the index j replacing the index j−1 in (4). Consider first the case F   

j > Fj−1 . Given mj−1

, nj is selected at random    

from the following conditional distribution: 

( = j − F +1|

j 1 = Fj− ) Pn           j F j−1 m − 1 = F ,Pn j ( j = Fj − Fj−1 |mj−1 = Fj−1 ) = 1− F ,           j

P n( =   F −  m   , and Pn      1 j j Fj−1 |

j−1 = Fj−1 + = 1) Fj ( j = Fj − Fj−1 − 1| mj−1 = Fj−1 + =) 1− F .           j

This ensures that the unconditional distribution of m m + n is the desired one. If F   = F  then n =1 with j = j−1 j  j j   −1  j 

conditional probabilities 

( = 1|m =   Pn F ) = E (1− F ) and Pn ( =1|m = F  +1) = 0,

j j−1   j−1 j j−1 j j−1   j−1

and nj = 0 otherwise. The expectation of nj is thus equal to Ej

. 

3.2 Sampling for the Intra-Annual Updates 

Once the sample size is determined for each stratum within update we sample using the Bernoulli-Type method. For the first 

update, the units 1,...,N are randomly permuted and a random number between (i−1)/N1 and i/N1 is assigned to unit i in the 1 

permuted list. Units with numbers not greater than π are sampled. The random numbers are U(0, 1) distributed and the 0 

sample size n1

 is as desired. 

For the j-th update we assign the random numbers as for the first update, with Nj replacing N1

. The sampling process 

depends, again, on the relationship between   F    

  Fj and 

 j−1 



. If Fj > Fj−1 then for every value of mj−1 we have    
 

 

 

 

 

 

 

 

 

 

 

 

 

 

4. An Illustrative Example: The Manufacturing Indices Survey 

 

 

 

 

 

 

 

 

 

 

E n | m = − F m 

− ( . Denote by π = F − m / N so that E n |m = N π . Units with random numbers not j j−1 ) j j 1 j ( j j−1 ) j ( j j−1 ) j j 

greater than π j are sampled. Note that here we have a sampling scheme with a random sampling probability and a fixed 

expected probability. The sample size variability is minimized, with a maximal deviation of one unit between the actual and 

expected cumulative sample sizes. The values of π are ( F − F ) N and ( F − F  −1) N with probabilities j j  j−1  j j  j−1  j 

1− Fj-1 and Fj-1 , respectively. These values are non-negative, and in rare cases may be equal to zero, or be equal or 

greater than one. Such a case happens when m 1 = Fj−1  1 Fj , and then π = 0 and nj = 0. If π > 1 j− + = j j , we truncate it to  

one and sample n  

j = Nj units. If Fj = Fj−1  we basically proceed as before, except for the case mj−1 = Fj−1 

+1 where   

we take π j = 0 and = 0, and the case when mj−1 = F  where we take π n j = E {Nj (1− Fj −1 )} . We refer to this j  j−1  j 

sampling scheme over updates as a conditional Bernoulli-Type (CB-T) scheme. 

3.3 Sampling for the Annual Update 

For the annual update a new frame based on L 0 , , L1 K, LJ and additional newborns is constructed. As in Section 2, in 

preparation for the next annual sampling we adjust the random numbers assigned to units in the update frames. First, let X * 

ji 

be the random variable defined for the i-th unit in the j-th update frame as in (1), with π j and S j

 replacing π 0 and S1

, 

respectively. The variable X * 

ji is well defined when 0 < π j <1 . When π j = 0 , no units are sampled for the j-th update and 

hence X * 

ji is computed only for units i S ∉ j . Similarly when π j = 1 , X * 

ji is computed only for units i S ∈ j

. 

As we have seen in the single update scenario of Section 2, X * 

ji does not have a U(0,1) distribution and thus we further adjust 

it using Yji = F * ( X * 

ji ) . To extract F * ( )t , we first note that the conditional distribution of X * 

ji | X(n )

,π j is, as before, a X ji X ji 0 

mixture of U(0, x(n0 ) ) and U( x(n0 ) , 1) distributions, but with respective probabilities π j and 1 −π j . Using the independence 

of X ji , π j and X(n0 ) , we write their joint distribution as a product of f

X *| X ,π

 and the marginal distributions of X(n0 ) and 

ji (n 0 ) j 

π j . Integrating over the values of π j it is seen in the Appendix that the unconditional distribution of X * 

ji is as in (2), because 

E(π j ) = π 0 . Therefore, the CDF of X * 

ji is as in Equation (3). 

The proposed method is implemented for the monthly Manufacturing Indices Survey. The survey provides measurements of 

the development in manufacturing and in the economy of Israel in general by estimating the total number of employees, labor 

cost, turnover and other economic indicators, by industry and sector. 

The sampling frame is extracted from the Israeli Business Register (BR). Data for each sampling unit include the industry 

classification, the annual turnover (T) and the annual average number of employees (E). To determine the unit size, a 

regression model of T on E is fitted (in square root scale) for each industry. The unit size is equal to the observed annual 

turnover Ti

, except for cases with a negative Studentized residual with absolute value larger than a predefined value. In these 

cases the size is set to the model prediction plus a random noise. For the 2004 base sample, units were stratified by 77 

manufacturing industries and by 2 to 4 size strata. The number of size strata depends on the number of units in each industry, 

where the top size group is a take-all stratum. The Lavallée-Hidiroglou (1988) stratification algorithm with a uniform 

coefficient of variation (CV) across industries is used to set the strata boundaries and to allocate the sample to the take-some 

strata. The allocation may then be adjusted to meet additional methodological constraints, such as a minimal number of 

sampled cases in a stratum, a maximal sampling weight and a maximal CV of the estimated number of employees. In each 

take-some stratum a srswor is drawn using a sequential scheme. The total number of sampled units for the 2004 base sample 

was 2,200. The sample is updated for newborns once in two months, using the conditional B-T (CB-T) scheme described in 

Section 3.2. 
Table 2 compares the relative standard deviations RSTD = Var(m5

) / F5 of the cumulative sample size obtained up t o and 

including the fifth update, for selected industries and three sampling schemes: Bernoulli (B), Bernoulli-Type (B-T) and the 

proposed conditional B-T (CB-T) scheme. The variances of the cumulative sample size m5

 are computed as follows: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

VarB (m5 ) = ∑

5

Njπ 0 (1−π 0 ), Var B-T (m5 ) = ∑

5 

Ej (1- Ej ), Var CB-T (m5 ) = F5 (1− F5 ). 

j=1 j=1 

We present industries that had a substantial number of newborns throughout 2004. It is seen that on average using the 

Bernoulli scheme yields sample sizes with RSTDs ranging between about 50 to 120 percent for the selected industries, as 

compared to 10-85% for the CB-T method. Over all take-some strata, the B-T scheme reduces the RSTD by about 17 percent, 

and the CB-T by further 30 percent. Equality between values for the B-T and CB-T, as seen for industry 265 stratum 2, 

occurs when all newborn units are in the same update. Equality between the Bernoulli and B-T RSTD values (industry 360, 

stratum 3) occurs when there is at most one newborn unit per update. 

5. Concluding Remarks 

Revisions of monthly business surveys that account for new units are common practice. Frequently, however, it is desired 

that these sample revisions interfere as little as possible with the ongoing main sample. The technique suggested by Ohlsson 

(1995, Section 9.1.1) for coordination over time is based on recurring standard sequential sampling from an updated main 

frame. This scheme may affect persistants and change the selection probabilities. We have seen that it is possible to sample 

newborns separately from the main sample, while preserving fixed selection probabilities throughout the year as well as 

minimal sample size variability. We have also shown how to combine correctly sequential and Bernoulli samples to form a 

subsequent sample with a desired overlap. Once all units have random numbers that are U(0,1) and have a comparable scale 

for all previously sampled (and non sampled) units, we can continue with the standard sequential scheme. 

Ernst, Valliant and Casady (2000) discuss various PRN-based techniques in the context of births and deaths and argue that 

“Though the methods are in common use, there appears to be a limited literature on their properties, particularly 

regarding the treatment of population changes due to births and deaths” 

In this paper we have suggested a workable scheme that is based on exact distributional considerations. The deviate y(p)-p is 

not defined analytically and we leave this question for future work. It would also be interesting to study the bias induced by 

using X or X * for sampling, rather than using Y . 

Table 2. Relative standard deviations (RSTD) of the cumulative sample size for take-some strata in selected industries using 

the Bernoulli, Bernoulli-Type (B-T) and Conditional B-T (CB-T) schemes. Data from five updates, the Manufacturing 

Indices Survey 2004. 

Industry Stratum Cumulative Expected RSTD 

number of cumulative Bernoulli B-T CB-T 

newborns sample size 

180 1 59 2.91 0.57 0.34 0.10 

2 17 3.31 0.49 0.27 0.14 

202 1 24 0.85 1.06 0.96 0.41 

2 4 0.59 1.21 1.15 0.84 

265 1 34 0.93 1.02 0.88 0.28 

2 6 0.91 0.97 0.32 0.32 

360 1 135 1.94 0.71 0.51 0.13 

2 36 1.86 0.71 0.50 0.19 

3 4 0.84 0.97 0.97 0.43 

All take-some strata 1144 67.68 0.12 0.10 0.07 
 

 

 

 

 

 

Appendix: Density and Cumulative Distribution Function of X

* 

The density of X( ) is n0

N !

0 n −1 N −n 0 0 0 

X g x () = x (1− x) ,

( n0) (n −1)!(N − n )! 0 0 0 

and the conditional density of Xi 

*

| X() = x is n0 

π 0 if 0 ≤ t x ≤  x

f

X * ( | t X() n = x) = (5) 0  i 

1−π 0

 if x<t ≤ 1 .  1− x 

( α β + −1)! α −1 β −1 Noting that the density function of a Beta(α, β ) variable on (0,1) with integer parameters is x (1− x) , the 

(α −1)!(β −1)! 

unconditional density is equal to 

1 

f * () = ∫f t( | x t

() = x g x n ) ( )dx

Xi 0

0 

N !

1 t 

0 n −2 N −n n −1 N n − −1 0 0 0 0 0 0 =

(n −1)!(N − n )! {π 0 ∫ x (1− x) dx + − (1 π )∫ x (1− x) dx} t 

0

0 0 0 0 

n 

=

0 Pr{Beta(n −1, N n − + > 1) t}+Pr{Beta(n , N − n ) ≤ t} 0 0 0 0 0 0 (n0 −1) 

* * The random variable X has the same density. To see this, recall that the conditional density of X |X = x,π = π is as ji ji ( n0 ) j 

*

in (5) with π replacing π 0 . Let h (π ) denote the probability function of π j . The joint distribution of X ji , X(n ) and π j is 

0 

π 1−π 

f

X ,X ,π

( )  I

[ ] 0,x ( ) + I

( x,1]

* t,x,π = t t ( ) g x h ( ) ( ) π ,

ji ( ) n j 0  x 1− x 

where IA(t) is the indicator function of a set A at t. Integrating the joint distribution over π yields f * ( | tX x () = x) g n ( ) Xi 0

since E(π j ) = π 0 . To extract the CDF it is useful to recall that Pr{Beta( , km − k +1) ≤ t} = Pr{Bin(m t, ) ≥ k} . Hence, 

t 

* () = ∫ Ft u f * ( )du

Xi Xi

0 

n t t

=

0 ∫ Pr{Bin(N0 −1,u) < n0 −1}du + ∫ Pr{Bin(N0 −1,u) ≥ n0

}du 

(n0 −1) 0 0 

t n n 0 −2 t N0 −1

0 N −1 i N0 − −1 i 1 i N0 − −1 i

= ∑ 0

i (1- ) u ∑

N0

i 

−

∫ ( ) u u d +∫ ( ) u u (1- ) du 

(n0 −1) 0 i=0 0 i=n0 

n −2 N −1 1  n0

0 



0 

Pr Beta( { i 1, N0 i) t} { i +1, N0 i) t  = ∑ + −≤+ ∑ Pr Beta( − ≤ } .

N n −1 i=0 i=n

0  0 0 

References 

Ernst, L. R., Valliant, R. and Casady, R. J. (2000). Permanent and collocated random number sampling and the coverage of 

births and deaths. Journal of Official Statistics, 16, 211-228 . 

Lavalée, P. and Hidiroglou, M. A. (1988). On the stratification of skewed populations. Survey Methodology, 14, 33-43. 

Ohlsson, E. (1995) Coordination of samples using permanent random numbers. In Business Survey Methods Ed B. G. Cox, D. 

A. Binder, D. N. Chinnappa, A. Christianson, M. J. Colledge, and P. S. Kott. New York: John Wiley, 153-169. 
Saavedra, P. J. (1995). Fixed sample size PPS approximations with a permanent random number. In Proceedings of the 

Section on Survey Research Methods, American Statistical Association, Orlando . 

http://www.amstat.org/sections/srms/Proceedings/papers/1995_120.pdf 

Stone, C. J. (1996). A Course in Probability and Statistics. Belmont: Duxbury Press.