661Journal of Applied Statistics, Vol. 22, Nos 5&6, 1995

A capture± recapture survival analysis modelfor radio-tagged animals

KENNETH H. POLLOCK1, CHRISTINE M. BUNCK

2, SCOTT R.

WINTERSTEIN3

& CHIU-LAN CHEN1,

1Department of Statistics, North

Carolina State University, USA,2Patuxent Wildlife Research Center, US Fish and

Wildlife Service, MD, USA and3Department of Fisheries and Wildlife, Michigan State

University, USA

SUMMARY In recent years, survival analysis of radio-tagged animals has developed

using methods based on the Kaplan± Meier method used in medical and engineering

applications (Pollock et al., 1989a,b). An important assumption of this approach is that

all tagged animals with a functioning radio can be relocated at each sampling time with

probability 1. This assumption may not always be reasonable in practice. In this paper,

we show how a general capture± recapture model can be derived which allows for some

probability (less than one) for animals to be relocated. This model is not simply a

Jolly± Seber model because it is possible to relocate both dead and live animals, unlike

when traditional tagging is used. The model can also be viewed as a generalization of the

Kaplan± Meier procedure, thus linking the Jolly± Seber and Kaplan± Meier approaches to

survival estimation. We present maximum likelihood estimators and discuss testing

between submodels. We also discuss model assumptions and their validity in practice. An

example is presen ted based on canvasback data collected by G. M. Haramis of Patuxent

Wildlife Research Center, Laurel, Maryland, USA.

1 Introduction

In wildlife ® eld studies, there are now basically three common methods of

estimating survival rates. These are ring-return methods (Brownie et al., 1985),

capture± recapture methods (Seber, 1982; Burnham et al., 1987; Pollock et al.,

1990; Lebreton et al., 1992) and radio-tagging survival analysis methods

(May® eld, 1961, 1975; Trent & Rongstad, 1974; Heisey & Fuller, 1985; Pollock

et al., 1989a,b; White & Garrott, 1990).

It is well known that the ring-return models of Brownie et al. (1985) and the

0266-476 3/95/050661-12 1995 Journals Oxford Ltd

662 K. H. Pollock et al.

capture± recapture models which began with the Jolly± Seber model (Jolly, 1965;

Seber, 1965) are very closely related statistically, although the ® eld sampling

methods are quite different. Ring returns are from dead animals either killed in the

process of hunting (or ® shing) or found dead, whereas capture± recapture methods

are used where live recapture or resighting is possible. With live recapture thepossibility of multiple recaptures exists whereas with ring returns clearly no release

and subsequent recapture is possible.

Until recently, we had assumed that the radio-tagging survival analysis models

were very distinct from the capture± recapture models. As we subsequently show in

this paper, that is not the case. There is, in fact, a very strong connection betweenthe two types of models. We show in this paper that it is possible to formulate a

general model which includes both Jolly± Seber type capture± recapture models

(Seber, 1982; Pollock et al., 1990; Lebreton et al., 1992) and Kaplan± Meier type

survival analysis models (Kaplan & Meier, 1958; Pollock et al., 1989a,b) as special

cases.The data used in capture± recapture models include only capture or observation

of live animals whereas the Kaplan± Meier type estimators for telemetry studies

include observations of live and dead animals. Capture± recapture models allow

capture probabilities less than 1 while the traditional Kaplan± Meier type estima-

tors do not. We formulate and present a general model which allows for captureprobabilities of less than 1 and follows both live and dead animals.

In a recent paper, Burnham (1993) has presented a general method for

combining ring-return and recapture data in one comprehensive analysis. His

model includes site ® delity rate parameters, as well as survival and capture rate

parameters, as ours does. Thus, while the type of data used is different the modelsare related.

First, we present an outline of model structure, notation and assumptions. This

is followed by a detailed development of maximum likelihood estimators (MLEs).

Submodels obtained by imposing parameter constraints (for example, constant

survival over time) are discussed together with model selection using likelihoodratio tests (LRTs) and the Akaike Information Criterion (AIC). An example is

presented which is based on canvasback radio-tagging data collected by G. M.

Haramis of the Patuxent Wildlife Research Center on Chesapeake Bay, Maryland,

USA. We close with a general discussion of important issues raised and sugges-

tions for future research.

2 Model notation and assumptions

The notation used here is a generalization of the capture± recapture notation of

Seber (1982). We consider a K-sample capture± recapture setting where new

radio-marked animals can enter at any of the K times. We allow relocation of live

and dead animals, which is not feasible for traditional tagging studies.

M i 5 the number of marked animals in the population that are alive at

the time the i th sample is taken (i 5 1, . . . , K ; M 1 ; 0).

D i 5 the number of marked animals in the population that have newly

died at the time the i th sample is taken (i 5 1, . . . , K ; D 1 ; 0).

That is, these animals have died between i 2 1 and i .

m i 5 the number of (live) marked animals located in the ith sample

(i 5 2, . . . , K ) (0 # m i # M i ).

Capture± recapture model for radio-tagged anaimals 663

d i 5 the number of (dead) marked animals located in the i th sample

(i 5 2, . . . , K ) (0 # d i # D i).

m*i 5 m i 1 d i 5 the total number of marked animals (alive or dead) located in the

ith sample (i 5 2, . . . , K ).

R i 5 the number of animals released after the ith sample and includesnewly and previously marked animals (i 5 1, . . . , K 2 1).

r i 5 the total number of animals relocated after the ith sample of the

R i released. The animals may be relocated alive or dead

(i 5 1, . . . , K 2 1).

z i 5 the total number of animals captured before i, not captured at i,and captured again later. The animal may be relocated after i

alive or dead (i 5 2, . . . , K 2 1).

m w (d w ) 5 the number of live (newly dead) animals relocated in a particular

sample with the capture history w. For example, w 5 11 implies

m 11, which is the number of radio-tagged animals from time 1which are still alive and relocated at time 2 or d 11, which is the

number of radio-tagged animals from time 1 which die between

time 1 and time 2 and are relocated at time 2.d i 5 the probability any animal alive at time i remains in the study area

between time i and i 1 1 and does not have a radio loss or failurein that period (i 5 1, 2, . . . , K 2 1).

S i 5 the probability any animal alive at time i survives from time i to

i 1 1 (i 5 1, 2, . . . , K 2 1).

p i 5 the probability any animal alive or newly dead at time i is

relocated at time i, conditional on it having a functioning radio.c i 5 the probability any animal released at time i is never relocated

again (i 5 1, 2, . . . , K 2 1).Á i 5 S i d i 5 the probability that any animal alive at time i survives to time

i 1 1 and remains in the study area with a functional radio.

The assumptions of this model are obvious extensions of the capture± recapture

assumptions of the Jolly± Seber model to the case where there are live and dead

animals. The assumptions are now presented.

(1) Every marked animal present in the population at a particular sampling time

(i ) has the same probability (p i) of being relocated if alive or if newly dead(i.e. dying in the interval i 2 1 to i ).

(2) Every marked animal alive and present in the population at a particular

sampling time (i ) has the same probability (S i ) of surviving until the next

sampling time (i 1 1). (Note that this is true for newly marked and previ-

ously marked animals.)(3) Every marked animal alive and present in the population at a particular

sampling time (i) has the same parameter d i . This parameter is the probabil-

ity of the animal remaining in the study area with a functioning radio

attached. If it can be further assumed that loss of radio tag because of

malfunction is 0, then this parameter can be interpreted as the site ® delityrate, which is a very important biological parameter.

(4) Emigration is permanent for any animal which leaves the area.

(5) Marked and unmarked animals have equal survival rates. That is, there is no

effect of the radio tag on survival and also animals chosen to be tagged are

representative of the whole population. This assumption is not necessary to

664 K. H. Pollock et al.

obtain a likelihood but it is necessary for the survival parameters to be useful

to the biologist.

(6) All relocation periods are assumed to be short (ideally instantaneous).

(7) All animals behave independently with respect to capture, survival and

emigration processes so that we can model the likelihood as a productmultinomial distribution.

These assumptions are very similar to the assumptions of capture± recapture

models (see, for example, Seber, 1982; Pollock et al., 1990; Pollock, 1991). We

now brie¯ y discuss each assumption.In the ® rst assumption, we assume that animals alive at a sampling time (i ) have

the same probability of relocation as animals that have just died right before that

time, conditional on them having a functional radio. This may be reasonable but

sometimes movement makes it easier to detect animals with functioning radios and

dead animals cannot move. If it were possible to establish the earlier times of deathof relocated animals, we could possibly generalize our model to allow different

probabilities of relocation for live and dead animals, but we do not think this will

be realistic because only very small numbers of animals will die in each interval in

most practical applications.

The second assumption basically involves homogeneity of survival rates. Theremay be situations where newly marked animals have different survival rates from

previously marked animals due to differences in age (or perhaps other factors).

There are two parts to the third assumption. We are assuming that site ® delity

rates are equal for animals which are alive and animals about to die. This may not

be reasonable. Animals may leave an area to search for food due to weakness andthen die at a higher rate than animals remaining. The other part of this assumption

is that the probability of a radio tag remaining on the animal and functioning is the

same for live and newly dead animals. We suspect in some applications that

animals may die and have their radio destroyed in the process, causing a poten-

tially serious violation of this assumption. We address this further in Section 5.Violation of the fourth assumption involves so-called temporary emigration.

While this may often occur in practice, our model does not allow for it. We discuss

this issue further in the discussion section. It turns out that it is possible to weaken

this assumption and still obtain valid survival rate estimates.

One way to mitigate failure for the ® fth assumption is to include only animalsin the marked population after they have survived some period (often 2 weeks).

The premise here is that animals that have carried a radio successfully for 2 weeks

are not likely to have their future survival impeded.

Typically, relocation occurs on a regular basis with a short interval so that the

assumption of an instantaneous relocation period should therefore be reasonable.The last assumption of independence of animals is necessary for the multinomial

likelihood which follows. Violation of this assumption is likely. Positive depen-

dence does not cause bias but does mean calculated standard errors for estimators

are smaller than they should be (Pollock et al., 1989a).

3 Maximum likelihood estimation

3.1 The likelihood function

The likelihood is a product of multinomial distributions, similar to that of the

Capture± recapture model for radio-tagged anaimals 665

Jolly± Seber model for capture± recapture data. We present a form similar to

Brownie et al. (1986).

L 5 S m 11, d 11,R 1

m 101, d 101, . . . D5 {d 1 S 1 p 2}m 11 {d 1(1 2 S 1)p 2}d 11 {d 1 S 1(1 2 p 2) d 2 S 2 p 3}m 101

5 {d 1 S 1(1 2 p 2) d 2(1 2 S 2)p 3}d 101 . . . c R 1 2 r 1

1

5 S m ´11, d ´11,R 2

m ´101, d ´101, . . . D5 {d 2 S 2 p 3}m ´11 {d 2(1 2 S 2)p 3}d ´11 {d 2 S 2(1 2 p 3) d 3 S 3 p 4}m ´101

5 {d 2 S 2(1 2 p 3) d 3(1 2 S 3)p 4}d ´101. . . c R 2 2 r 2

2

5 S m ´´11,R 3

d ´´11, . . . . . . D {d 3 S 3 p 4}m ´´11 {d 3(1 2 S 3)p 4}d ´´11 . . . c R 3 2 r33

:

5 S R K 2 1

m ´11, d ´11 D {d K 2 1 S K 2 1 p K }m ´11 {d K 2 1(1 2 S K 2 1)p K }d ´11 c R K 1 1 2 rK 2 1K 2 1

In this likelihood, a ´ indicates a summing of 0 and 1 capture history. For example,

m 011 1 m 111 5 m ´11. For the last term, all capture occasions are summed in that

manner except K 2 1 and K so that m ´11 5 m 00´´´011 1 m 11´´´111 and similarly for d ´11.

The nature of the model can be seen clearly by examining the cell probability

structure. Take for example d 101. The animal was ® rst captured and radio-taggedat time 1; its radio continued to function and it stayed in the area from time 1 to

time 2 ( d 1); survived to time 2 (S 1); was not relocated at time 2(1 2 p 2); stayed in

the area and had its radio continue to function from time 2 to time 3 ( d 2); died

between time 2 and time 3 (1 2 S 2); and was relocated at time 3 (p 3).

The likelihood can also be re-expressed in terms of a set of conditional binomialdistributions based on the minimal suf® cient statistic, but in the interest of brevity

we do not develop this here. Computation of MLEs is facilitated by using a

package like SURVIV (White, 1983), but explicit estimators can be found as well

and are now presented.

3.2 MLEs

Explicit MLEs can be derived for this full-rank model and they can also be derivedintuitively as for the Jolly± Seber model. M i and D i are not unknown parameters of

the likelihood but rather unknown random variables. However, it is useful to use

them as the basis for the other estimators.

3.2.1 Marked population size (M i) alive. Equating the two ratios

Z 1

M i 2 m i

. r i

R

gives

MÃ i 5 m i 1R i z i

r i

, i 5 2, . . . , K 2 1 (1)

666 K. H. Pollock et al.

These are the usual Jolly± Seber estimators except that m i only involves live

relocations while z i and r i involve live and newly dead relocations.

3.2.2 Marked population size (D i) dead. Equating the two ratios

d i

D i

5m i

M i

gives

DÃ i 5 MÃ i

d i

m i

, i 5 2, . . . , K 2 1 (2)

These estimators have no counterpart in the Jolly± Seber model.

3.2.3 Survival rate estimators. The survival rate estimators can be written equiva-

lently as

SÃ i 5MÃ i 1 1

MÃ i 1 1 1 DÃ i 1 1

or

SÃ i 5m i 1 1

m i 1 1 1 d i 1 1

, i 5 1, . . . , K 2 1 (3)

which are the conditional survival rate estimators used as the basis for the

Kaplan± Meier estimator except that M i and D i are now estimated. These estima-

tors have no counterpart in the Jolly± Seber model.

3.2.4 Capture rate estimators.

PÃ i 5m i

MÃ i

, i 5 2, . . . , K 2 1 (4)

These are analogous to the Jolly± Seber estimators.

3.2.5 Site ® delity rate estimators.

d à i 5Mà i 1 1 1 Dà i 1 1

MÃ i 1 R i 2 m i

, i 5 1, . . . , K 2 2 (5)

These are new estimators which can be viewed as estimators of the probability of

site ® delity provided there is no radio loss or radio failure, which may be

reasonable in some short-term studies. It is not possible to estimate these quanti-

ties if you have traditional capture± recapture data (i.e. are using the Jolly± Sebermodel).

In addition to these estimators is the equation

d K 2 1Ã p K 5rK 2 1

R K 2 1

because it is not possible to separately estimate d K 2 1 and p K .

We recommend that variance estimates of the estimators be calculated by using

simulation (i.e. a parametric bootstrap procedure). Approximate Taylor series

methods usually give variance estimators which are too small.

Capture± recapture model for radio-tagged anaimals 667

TABLE 1. Capture history information on the modi® ed canvasback data

i 5 1 i 5 2 i 5 3 i 5 4 i 5 5 i 5 6 i 5 7

Animal Release WK 1 WK 2 WK 3 WK 4 WK 5 WK 6

1 1 1 0 1 1 1 1

2 1 1 1 1 1 1 1

3 1 1 1 1 0 4 0

4 1 1 0 4 0 5

5 1 1 1 0 1 1 1

6 1 0 1 1 1 9

7 1 1 1 1 1 1 1

8 1 1 0 5

9 1 1 0 1 1 1 1

10 1 1 1 0 0 0 0

11 1 1 0 0 0 4 0

12 1 1 1 1 1 1 1

13 1 9

14 1 1 0 4 0 4 0

15 1 1 0 0 1 0 1

16 1 0 0 1 0 0 0

17 1 1 0 4 0 4 0

18 1 1 1 1 1 1 1

19 1 1 1 9

20 1 1 1 1 1 1 1

21 1 0 9

22 1 1 1 1 0 4 0

23 1 1 1 1 1 1 1

24 1 1 0 0 0 0 0

25 1 1 1 1 0 1 1

26 1 9

27 1 1 1 9

28 1 1 1 1 1 1 1

29 1 1 1 0 1 9

30 1 1 0 4 0 4 0

31 1 1 1 1 1 1 1

32 1 1 1 1 9

33 1 1 0 1 1 1 1

34 1 1 0 1 9

35 1 1 0 0 0 1 9

36 1 1 1 1 1 1 1

37 1 1 1 1 1 0 0

38 1 1 1 0 1 1 1

39 1 0 1 0 0 0 0

40 1 1 1 1 1 1 1

41 1 0 1 1 1 1 1

42 1 1 0 1 9

43 1 1 1 1 1 1 1

44 1 1 1 1 1 1 1

45 1 9

46 1 1 0 4 0 0 0

47 1 1 1 9

48 1 1 1 1 1 1 1

49 1 1 9

50 1 1 1 1 1 1 1

1 5 relocated on study site; 0 5 failure to locate; 9 5 known death on study site;

4 5 relocated off study site; 5 5 known death off study site.

668 K. H. Pollock et al.

3.3 Relationship to traditional Jolly± Seber estimators

In the previous section, we found that

SÃ i 5MÃ i 1 1

MÃ i 1 1 1 DÃ i 1 1

and d à i 5Mà i 1 1 1 Dà i 1 1

MÃ i 1 R i 2 m i

so that

Á à i 5 Sà i d à i

5MÃ i 1 1

MÃ i 1 R i 2 m i

This establishes the relationship between the general capture± recapture model and

the traditional Jolly± Seber model with live animals only. Here, we can estimate

true survival rate (Sà i ) and the probability of remaining in the area ( d à i) whichtogether give Á à i , the probability of surviving and remaining in the study area. In

the traditional Jolly± Seber model, we can only estimate Á Ã i and it has the form

given above.

4 Example

In this section, we present a detailed example based on a modi® cation of

canvasback radio telemetry data collected by G. M. Haramis on Chesapeake Bay,Maryland, USA. We consider a 7-week study with a release in the ® rst week and

then 6 weeks for relocation. In Table 1 the capture history data are presented. In

Table 2, summary statistics are presented. The analysis assumes an a priori de® ned

study site such that birds located off the study site were considered missing. In

Table 3 the important parameter estimates are presented for the full modeldescribed in Section 3.2. These estimates use the explicit equations presented in

Section 3.2. Note that one estimate of emigration rate (1 2 d à 1) is negative. It is

possible to have estimators outside the range (0, 1) although it does not make any

biological sense.

TABLE 2. Summary statistics for the modi® ed

canvasback radio-telemetry study

i m i d i m*i R i r i z i

1 Ð Ð Ð 50 50 Ð

2 42 3 45 42 34 5

3 29 2 31 29 27 8

4 27 3 30 27 24 5

5 24 3 27 24 23 2

6 22 2 24 22 22 1

7 22 1 23 22 Ð Ð

We then input the full likelihood to program SURVIV (White, 1983), a general

maximization program for multinomial likelihoods. In Table 4, we present theimportant parameter estimates and their standard errors. Note that these estimates

may be a little different from those in Table 3 because they are constrained to

be in the range of 0 to 1. Usually the estimates are very close or identical.

The precision of the estimates is high because the capture probabilities (p i) are

close to 1.

Capture± recapture model for radio-tagged anaimals 669

TABLE 3. Explicit estimators of important model parameters for

the canvasback example using the full model

Week (i) Mà i Dà i Sà i Pà i 1 2 d à i

1 Ð Ð 0.93 Ð 2 0.03 a

2 48.18 3.44 0.94 0.87 0.17

3 37.59 2.59 0.90 0.77 0.04

4 32.63 3.63 0.89 0.83 0.10

5 26.09 3.26 0.92 0.92 0.04

6 23.00 2.09 0.96 0.96 Ð

aNote that the explicit estimators of S i , p i or (1 2 d i) do not

necessarily have to be between (0, 1).

TABLE 4. Estimators (standard errors) of important parameters for the

canvasback example using program SURVIV (White, 1983) on the

full model

Week (i ) Sà i Pà i 1 2 d à i

1 0.94 (0.035) Ð 0.00a (0.025)

2 0.94 (0.044) 0.90 (0.048) 0.15 (0.062)

3 0.90 (0.054) 0.77 (0.071) 0.04 (0.053)

4 0.89 (0.060) 0.83 (0.070) 0.10 (0.062)

5 0.92 (0.056) 0.92 (0.054) 0.04 (0.042)

6 0.96 (0.043) 0.96 (0.043) Ð

aThe estimators with program SURVIV are constrained to be between

0 and 1. Therefore, these estimates are different from the explicit

estimators which are not so constrained. With the explicit estimator in

Table 3, this estimate was 2 0.03.

We next considered all the submodels obtained by constraining survival, capture

and site ® delity rates to be constant over time in all possible combinations. Use of

the AIC (Lebreton et al., 1992) and LRTs suggested the preferred parsimoniousmodel was one with only survival rates constrained to be constant over time (Table

5). In Table 6, all the parameter estimates under this model are presented. There

is almost no effect on the estimates of capture and emigration rates or their

precisions. The estimate of survival has much improved precision as expected

because there is only one survival rate parameter to estimate instead of six.

5 General discussion

This paper merges the current capture± recapture and radio-telemetry research into

a global model for survival estimation. We believe this is useful because manybiologists inform us that relocation probability is not one (for example G. M.

Haramis, canvasbacks, R. Labisky, deer, K. Potak, ® sh; all personal communica-

tions).

Although the explicit estimators have a nice intuitive form, as we showed in

Section 3, it is better to use a program like SURVIV because:

(1) estimates are constrained to (0, 1);

670 K. H. Pollock et al.

TABLE 5. Summary of log-likelihood and AIC values for a variety of models with

increasing restrictions

Parameters

Model ® tted Log-likelihood AIC

Full 17 2 29.7763 93.5526

Constant site ® delity 13 2 35.6380 97.2761

Constant capture 13 2 35.4409 96.8818

Constant survivala 12 2 30.4383 84.8766

Both constant ® delity and capture 8 2 42.5579 101.1157

Both constant ® delity and survival 8 2 36.4539 88.9078

Both constant capture and survival 8 2 36.1818 88.3635

All constant 3 2 43.2679 92.5358

aBoth LRTs and the AIC suggest this is the preferred parsimonious model.

TABLE 6. Estimators (standard errors) of important parameters for the

canvasback example using program SURVIV (White, 1983) on a

reduced model where survival is assumed constant from week to week

Week (i ) Sà i Pà i 1 2 d à i

1 0.92 (0.012) Ð 0.00 (0.026)

2 0.92 (0.012) 0.90 (0.048) 0.14 (0.062)

3 0.92 (0.012) 0.77 (0.071) 0.04 (0.051)

4 0.92 (0.012) 0.83 (0.069) 0.10 (0.061)

5 0.92 (0.012) 0.92 (0.052) 0.04 (0.042)

6 0.92 (0.012) 0.96 (0.043) Ð

(2) submodels without explicit solutions can be ® tted as we did in the example(Section 4);

(3) it is possible to use the AIC and LRTs to decide on the best parsimoniousmodel from a large set of possible submodels;

(4) c 2 goodness-of-® t tests are calculated for each model ® tted.

Despite the relevance of this paper, we wish to emphasize that the model usedmakes two very strong model assumptions which may not be valid in somepractical settings. These are homogeneity of relocation probabilities (p i) and site® delity and functioning radio probabilities ( d i ) over all animals, irrespective ofwhether they have survived or have newly died. We recommend that more researchwork will need to be done evaluating violation of these assumptions. We havebegun to consider a more general model with p i and p

*i , the probabilities of

relocation at time i of live and newly dead animals, respectively. These probabili-ties are conditional on the animals having remained in the area and continued tohave functioning radio tags. Also, we allow the probabilities of remaining in thearea with a functioning radio tag for period i to i 1 1 to be d i and d *

i for live andnewly dead animals at time i 1 1. The expectation of equation (3) is

E (SÃ i) 5 E {m i /(m i 1 d i )}

. E (m i)/E (m i 1 d i)

. R i S i p i d i /(R i S i p i d i 1 R i (1 2 S i )p*i d *

i )

. S i /{S i 1 (1 2 S i )(p*i d *

i /p i d i )}

Capture± recapture model for radio-tagged anaimals 671

which is only approximately equal to S i if p i d i 5 p*i d *

i which effectively means

p i 5 p*i and d i 5 d *

i . Simple numerical work on the size of the resulting approximate

bias of SÃ i is being carried out together with simulation studies to obtain the exact

bias for ® nite samples. Further work could also attempt to obtain corrections to

eliminate the bias. This would effectively involve trying to set up special studies toestimate d *

i and p*i separately from d i and p i .

Although our model assumes permanent emigration (as for traditional capture±

recapture models), it turns out that the estimate of survival given in equation (3)

is valid under a certain type of temporary emigration. It is necessary to assume that

animals which temporarily emigrate do not have different survival rates from thosethat do not temporarily emigrate.

Acknowledgements

We would like to thank J.-D. Lebreton and J. D. Nichols for their assistance in

reviewing earlier versions of this paper.

Correspondence: K. H. Pollock, North Carolina State University, Department of

Statistics, Box 8203, Raleigh, NC 27695, USA.

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