Dick London

SURVIVAL MODELS

AND THEIR ESTIMATION

Third Edition

Dick London, FSA

xii Contents

PREFACE TO THE SECOND EDITION

Survival Models and Their Estimation is a general textbook describing theproperties and characteristics of survival models, and statistical proceduresfor estimating such models from sample data. Although it is written primarilyfor actuaries, it is also intended to be of interest to a broader mathematicaland statistical audience. Academically, the text is aimed at the fourth yearundergraduate or the first year graduate level.

Actuaries and other applied mathematicians work with models whichpredict the survival pattern of humans or other entities (animate or inani-mate), and frequently use these models as the basis for calculations of con-siderable financial importance. Specifically, actuaries use such models to cal-culate the financial values associated with individual life insurance policies,pension plans, and income loss coverages. Demographers and other socialscientists use survival models for making predictive statements about thefuture make-up of a population to which the model is deemed to apply.

This text is not primarily concerned with the of survival models,usesbut rather with the question of how such models are established. This exer-cise is sometimes referred to as survival model or survivaldevelopmentmodel ; in this text, however, we prefer the more descriptiveconstructionphrase survival model estimation.

It cannot be noted too strongly that the “real” survival distribution (orsurvival probabilities) which apply to a group of persons is unknown, andprobably will forever be so. What we, therefore, attempt to do is thatestimatedistribution, based on the data of a sample and a chosen estimation procedure.It is vitally important that this be clearly understood. Since the name of thegame is estimation, there are no “right” answers. There are only sound (orunsound) procedures.

Because the result of our exercise is an estimate of the theoretical,underlying, operative survival distribution, based on the particular experienceof a sample, we recognize that the estimate is a realization of a randomvariable, called an . In turn, this random variable has propertiesestimatorsuch as expected value and variance, and these properties tell us somethingabout the quality of the estimator. Note that we do not judge the “accuracy”of the resulting estimate, but rather the quality (or validity) of the procedurewhich produced the estimate. Properties of estimator random variables aredefined in Appendix A. Readers who are not entirely familiar with these

properties may wish to review Appendix A before studying the specific esti-mators developed in the text.

Frequently the estimated survival model produced directly from astudy is not entirely suitable for practical use, and is, therefore, systematicallyrevised before such use. The process of revising the initial estimates intorevised estimates is called . This step in the development of agraduationuseable estimated survival model is the topic of a companion text to this oneentitled .Graduation: The Revision of Estimates

Survival Models and Their Estimation is said to be a general text inthat it treats survival model estimation from the viewpoint of several differentpractitioners, including the actuary, the demographer, and the biostatistician,without attempting to be an exhaustive treatment of any one of thesetraditions.

A more thorough treatment of the actuarial tradition, from a differentperspective, can be found in texts by Gershenson [32], Benjamin and Pollard[11], and Batten [8]; demographic approaches are the main theme of theworks by Keyfitz and Beekman [46], Spiegelman [71], and Chiang [19]; themedical, or biostatistical, tradition is more deeply pursued by Elandt-Johnsonand Johnson [25]. Additional texts, which deal with the statistical analysis ofsurvival data at the graduate level, include those by Lawless [50], Lee [51],Miller [56], and Kalbfleisch and Prentice [42].

How is an initial estimated survival model determined from sampledata? There are many approaches to this. A survival model estimation prob-lem will generally have three basic components: (1) the form and nature ofthe sample data (which might also be called the ); (2) the chosenstudy designestimation procedure simplifying assumptions; and (3) any made along theway. All of these concepts will be further developed in this text. The tradi-tional actuarial approach, for example, is characterized by a cross-sectionalstudy design using the transactional data of an insurance company or pensionfund operation, a method-of-moments estimation procedure, and the Balduccidistribution assumption. In this text we will consider, as well, other studydesigns, primarily those encountered by the clinical statistician or the reliabil-ity engineer. In addition, we will consider other estimation procedures, espe-cially the maximum likelihood and product-limit methods. Finally, we willconsider other simplifying assumptions, such as the uniform and exponentialdistributions.

The text presumes a basic familiarity with probability and statistics,including the topics of estimation and hypothesis testing. The application ofthese ideas specifically to the estimation of survival models is then developedthroughout the text. An effort has been made to keep the mathematics andthe pedagogy at a level which does not require a prior familiarity with thetopic. Whenever a choice between mathematical rigor and pedagogic effec-

xiv Preface to the Second Edition

tiveness appeared to be necessary, we opted for the latter. As a result, thelevel of mathematical rigor in the text may be somewhat less than that desiredby the precise mathematician, but the increased clarity which results fromsacrificing some rigor will hopefully be welcomed by the student reader.

The first edition of this text, published in 1986, included the subjectmatter contained in the first eight chapters of the new edition. Chapters 5and 6 have been completely rewritten from the first edition, Chapter 7 hasbeen substantially revised, and three entirely new chapters have been added.The first edition was adopted by the Society of Actuaries as a reference forits examination program in 1987, and many valuable suggestions for im-provement were contributed by students and educators.

Drafts of the material in both editions of the text were submitted to areview team, whose many valuable comments are reflected in the finalversion. The indispensable assistance of this group is hereby gratefully ack-nowledged.

Warren R. Luckner, FSA, of the Society of Actuaries, coordinatedthe efforts of the review team and made many valuable comments himself.

Stuart A. Klugman, FSA, Ph.D., of the University of Iowa, wasparticularly adept at detecting mathematical errors in the drafts, and much ofthe precision that the text has attained is due to his careful efforts.

Stanley Slater, ASA, of Metropolitan Life Insurance Company, did aremarkable job of editing the drafts for improvements in writing style andclarity, especially for the benefit of the student reader.

Other members of the review team who made valuable contributionsto the final text include Robert L. Brown, FSA, and Frank G. Reynolds, FSA,both of the University of Waterloo, Cecil J. Nesbitt, FSA, Ph.D., of theUniversity of Michigan, Geoffrey Crofts, FSA, of the University of Hartford,and Robert Hupf, FSA, of United of Omaha Life Insurance Company.

Much of the research and writing time invested in this project wassupported by a grant from the Actuarial Education and Research Fund. Theauthor would like to express his appreciation to the directors of AERF forthis support.

Special thanks and appreciation are expressed to Marilyn J. Baleshi-ski of ACTEX Publications who did the electronic typesetting for the entiretext, through what must have appeared to be an endless series of revisions.

Despite the efforts of the review team and the author to attainpedagogic clarity and mathematical accuracy, errors and imperfections areundoubtedly still present in the text. For this the author and the publishertake full responsibility and sincerely apologize to the reader. We respectfully

xvi Preface to the Second Edition

PREFACE TO THE THIRD EDITION

It has been nearly ten years since the publication of the second edition ofSurvival Models and Their Estimation, and its adoption by the Society ofActuaries as the principal references for its Course 160 examination. It ap-pears that the text has proved satisfactory in that role from the point of viewof exam candidate and examiner alike. The mathematics of survival models themselves, and how theymight be estimated from sample data, has been a fairly stable topic, so that arevision of the theory presented in the first eight chapters of the textbookdoes not seem to be required at this time. Consequently, the reader familiarwith the second edition will note various clarifications and improvements inpresentation in these chapters, but no substantive change in the overall con-tent. Why, then, is a new edition appearing at this time? Beginning in 1994, a Society of Actuaries Board Task Force onEducation has been working toward a new model of actuarial education forthe twenty-first century. Among many other important principles, the TaskForce has established that actuarial education in the future should includeguidance for the application of standard actuarial techniques in disciplinesbeyond the traditional actuarial areas of insurance and pensions. SinceSurvival Models and Their Estimation plays its small part in the actuarialeducation arena, as the reference text for Course 160, it naturally follows thata revision of the text, guided by the Task Force principle of broadenedapplication, is now appropriate. The result is the appearance of new Chapters10, 11, and 12, which present applications of the general theory of survivalmodels in such fields as epidemiology, facilities planning, economics, invest-ments, reliability engineering, and others. Two other issues have affected certain changes from the prior to thecurrent edition of this text as well. The first is that Chapter 9 in the prior edition, which described thedemographer’s process of estimating survival models from general popula-tion data, has been deleted from the text. The material in the prior editionwas based on out-of-date studies, namely the Canadian census of 1981 andthe U.S. census of 1980, and is not included in the course of reading for theCourse 160 exam. Furthermore, the content of that chapter is also included,in up-dated form, in the new (third) edition of Robert L. Brown’s Introduc-tion to the Mathematics of Demography [16], and interested readers aredirected to that reference. The second important change reflected in the new edition of this textis a recognition that the traditional theory and data processing mechanics for

large-scale actuarial studies, developed in the pre-computer age, should nolonger receive the emphasis that it has in the past. Accordingly, thedescription of the theory of the traditional actuarial approach and a critiqueof that theory, as presented in Section 6.4, has been appropriately reduced.In addition, Chapter 11 of the prior edition, which dealt with the now out-of-date practice of calculating actuarial exposue as a by-product of lifeinsurance liability valuation, has been deleted. On the other hand, afterconsiderable deliberation it was decided that the prior Chapter 10 should beretained as Chapter 9 in the new edition. Although dated in some respects, adescription of the actual data processing mechanics involved in the actuarialtechnique of estimating survival models from insurance company and pen-sion fund data, was deemed to still be an important component of actuarialeducation. The author would like to acknowledge the contributions of severalcolleagues to the new edition of the text. A review of Chapter 10 was provided by Bruce Leonard Jones, FSA,FCIA, Ph.D., of the University of Western Ontario. Frank G. Bensics, FSA,Ph.D., of the College of Insurance suggested much of the content of Chapter11, and reviewed the drafts of that chapter as well. An importantcontribution to the development of Chapter 11 was also made by Matthew J.Hassett, ASA, Ph.D., of Arizona State University. A similar role for Chapter12 was played by Rohan J. Dalpatadu, ASA, Ph.D., of the University ofNevada at Las Vegas. Overall guidance for the content of the new editionwas provided by Robert A. Conover, FSA, the Education Actuary for Course160 at the Society of Actuaries. The text layout design and typesetting was again handled by MarilynJ. Baleshiski at ACTEX Publications. The author is very appreciative of herconsiderable skills, professionalism, and patience. We hope that readers familiar with the prior edition of this text willagree that the changes reflected in the new edition make a valuable contri-bution to actuarial education as we enter the twenty-first century. As always,corrections and suggestions for improvements are welcome.

Winsted, Connecticut Dick London, FSAApril, 1997

CHAPTER 6TABULAR SURVIVAL MODELS ESTIMATED

FROM INCOMPLETE DATA SAMPLESMOMENT PROCEDURES

6.1 INTRODUCTION

The foundation which underlies estimation by a moment procedure is thestatistical principle that the expected number of deaths to be observed in( , 1] from a certain sample should be equated to the number actuallyx xobserved. Assuming that the number of actual deaths in the age interval, tobe denoted by is readily available, then the estimation problem consists ofd , xtwo steps:

(1) Finding an expression for the expected number of deaths in ( , 1];x x(2) Solving the moment equation for the desired estimate.

Generally a simplifying assumption will be required to solve the momentequation.

6.2 MOMENT ESTIMATION IN A SINGLE-DECREMENT ENVIRONMENT

Assume that the basic data regarding an observation period, dates of birthand membership into the group under observation, and dates of death havebeen analyzed to produce the ordered pair ( , ) for person ’s sched-x r x s i i iuled contribution to ( 1], as illustrated in Figure 5.6. Recall that 1 isx, x s i œpossible but 0 is not, whereas, 0 is possible but 1 is not.s r ri i iœ œ œ For person , the probability of death while under observation in theisingle-decrement environment is the conditional probability of death beforeage , given alive at age , which is , and this is also the ex-x s x r q i i s r x rii i

pected number of deaths from this sample of size one. This is easilyseen, since the number of deaths is one (with probability ), ors r x ri i i q

zero (with probability ) Thus the expected number iss r x ri i i p .

1 0 (6.1)† †s r x r s r x r s r x ri i i i i i i i i q p q . œ

114 Incomplete Data Samples Moment Procedures:

6.2.1 The Basic Moment Relationship

If is the total number of persons who contribute to ( , 1], then the totaln x xx

number of expected deaths is (For convenience we will use !ni

s r x rx

i i iœ

1

q . n

for in our summations.) When equated to the actual number of observedn xdeaths, we obtain the moment equation

E D q d[ ] , (6.2)x s r x r x

n

iœ œ"

œ

1i i i

where is the random variable for deaths in ( , 1], and is the observedD x x d x xnumber. To solve (6.2) for our estimate of , we will use the approximationqx

( ) (6.3)s r x r i i xi i i q s r q . ¸ †

Then (6.2) becomes

E D q s r d[ ] ( ) , (6.4)x x i i x

n

iœ † œ"

œ1

from which we easily obtain

q ^ ds r

xx

n

ii i

œ!

œ1( )

, (6.5)

the general form of the moment estimator in a single-decrement environment.

6.2.2 Special Cases

If 0 and 1 for all persons who contribute to ( , 1], then wer s n x xi i xœ œ have , and (6.2) becomes q q s r x r xi i i œ

E D n q d[ ] , (6.6)x x x xœ † œ

so that (6.5) becomes

q .^ dnx

x

xœ (6.7)

Recall that this is Special Case A, as defined in Section 5.2. We recognize (6.7) as the binomial proportion estimator alreadyencountered in Chapter 4. We also recognize it as the maximum likelihood

Moment Estimation in a Single-Decrement Environment 115

estimator of the conditional mortality probability , where the model for theqxlikelihood is a simple binomial model. Thus the number of persons in thesample, , can be thought of as a number of binomial trials. The standardnxcharacteristics of a binomial model are assumed to apply. Thus each trial isconsidered to be independent, and the probability of death on a single trial( ) is assumed constant for all trials. In such a situation, the sample pro-q xportion of deaths, which is given by (6.7), is a natural estimator for thisparameter q .x If 1 for all, but 0 for some of the persons who contributes r n i i xœ to ( , 1], then we have , and (6.2) becomesx x q q œs r x r r x ri i i i i 1

E D q d .[ ] (6.8)x r x r x

n

iœ œ"

œ

11 i i

The general approximation given by (6.3) then becomes

1 r x r i xi iq r q¸ †(1 ) , (6.9)

which is the same as (3.77). Substituting (6.9) into (6.8), we obtain the result

q ^ dr

xx

n

ii

œ !œ1

(1 ), (6.10)

the moment estimator for Special Case B. If 0 for all, but 1 for some of the persons who contributer s n i i xœ to the interval ( , 1], then we have , and (6.2) becomesx x q q œs r x r s xi i i i

E D q d .[ ] (6.11)x s x x

n

iœ œ"

œ1i

The general approximation given by (6.3) then becomes

s x i xiq s qœ † , (6.12)

which is the same as (3.54). When (6.12) is substituted into (6.11) we obtainthe result

q ^ ds

xx

n

ii

œ !œ1

, (6.13)

the moment estimator for Special Case C.


EXAMPLE 6.1 From the sample of five persons whose basic data aregiven below, estimate , using an observation period of calendar year 1997.q30

Person Date of Birth Date of Death1 July 1, 1967 —2 April 1, 1967 —3 January 1, 1967 October 1, 19974 July 1, 1966 —5 April 1, 1966 April 22, 1997

SOLUTION The data are shown in Figure 6.1:

- - - - - - - - | | qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqñ ñ ñ ñ ñ30.25 30.50 30.75

30 31 1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ñ 2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ñ 3 ~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~Œ ñ 4~~~~~~~~~~~~~~~~ñ 5 - - -~~~~~~~ñ Œ

FIGURE 6.1

The ordered pairs ( , ) are (0, .50), (0, .75), (0, 1), (.50, 1), and (.75, 1), forr s i ii œ 1,2,3,4,5, respectively. Thus the denominator of Estimator (6.5) is.50 .75 .1.00 .50 .25 3.00. The numerator of (6.5) is 1, be- œ dx œcause person 5 died reaching age 31. Thus .after q̂30

13œ

6.2.3 Exposure

The concept of exposure was defined in Section 3.3.5 in the context of a lifetable model. At this time, and at several other places throughout the text, wewill make use of this concept in the context of a study sample. The ordered pair ( , ) for person represents the interval of age,r s i i iwithin ( , 1], over which person is potentially under observation. We sayx x i that person is from age to age i x r x s .exposed to the risk of death i iNumerically, is the length of time (in years) that person is exposed, sos r i i iit also represents the amount of exposure, measured in units of life-years,


contributed by that person. The total exposure contributed by the sample is!n

ii i

œ1( ), the denominator of Estimator (6.5).s r

The exposure interpretation of the denominators of Estimators (6.7),(6.10), and (6.13) is the same as that for Estimator (6.5). It is important to recognize that the denominator of the momentestimator represents the of the sample. When a samplescheduled exposuremember dies while under observation, not all of the scheduled exposure isrealized. Henceforth in this text we will refer to the realized exposure asexact exposure, to distinguish it from scheduled exposure. Note that exactexposure cannot exceed scheduled exposure.

EXAMPLE 6.2 Calculate the exact exposure, for the interval (30,31], ofthe sample described in Example 6.1.

SOLUTION Only person 3 has exact exposure less than scheduled expo-sure, and it is less by .25 years. Thus the total exact exposure of the sampleis 2.75.

Exact exposure plays an extremely important role in estimation, aswe shall see in Chapter 7. Intuitively, the observed deaths divided by exactexposure provides an estimate of the central death rate, , for the interval ( ,m xxx m q ^1]. From the derived value of , we could then estimate byx x

q ^ m̂m̂x

xx

œ1

, (6.14) †1

2

which uses the linear distribution assumption for . Alternatively, if we as-jx s

sume an exponential (the constant force of mortality assumption), thenjx s

this constant force, , is the same as , so that , and thus^ ^. .m mx xœ

q e e .^ 1 1 (6.15)xmœ œ .̂ ^x

In the next chapter we will see that this (and thus ) are maximum likeli-^ ^. q xhood estimators. To summarize, scheduled exposure is divided into observed deaths toestimate in a single-decrement environment by the moment estimationq xapproach, coupled with the approximation ( ) Exact expo-s r x r x q s r q . ¸ †sure is divided into observed deaths to estimate , or to estimate a constantmxforce of mortality over ( , 1]. This latter case will arise regularly inx xChapter 7.


6.2.4 Grouping

Consider Special Case B in which the persons who contribute to ( , 1]n x xx enter this interval at various ages, but all are scheduled to exit the interval atage 1. That is, all are scheduled to survive ( , 1]; none are scheduled tox x x be study enders at an age such that 1. It may reasonably bez x z xi i assumed that many of these persons enter the interval at exact age ( ,n x i.e.xhave 0); let us say that is the subset of which has 0. ( isr b n r b i x x i xœ œchosen to stand for , since these persons are exposed from thebeginnerbeginning of ( , 1].)x x The remainder of the sample, say , all have such thatk n b r x x x iœ 0 1. Let us assume that the value of these ’s is , 0 1. r average r r ri i (Alternatively, it might be possible that the study was designed such that kxpersons all had , and no averaging is needed.) This situation isr r i œillustrated in Figure 6.2.

bx Æ ~~~~~~~~~~~~~~~~~~~~ñ kx Æ ~~~~~~~~~ñ

- - - - - - | | |qqqqqqqqqqqqqqq

x x r x 1

FIGURE 6.2

As a result of this grouping, the expected number of deaths in theinterval ( , 1] is , so the moment equation becomesx x b q k q † †x x x r x r1

E D b q k q d . [ ] (6.16)x x x x r x r xœ † † œ1

Under the hyperbolic assumption, (1 ) , so (6.16) is easily1r x r xq r q œ †solved for the estimator

q .db r k

^(1 ) (6.17)x

xx x

œ †

The exposure interpretation of the denominator of (6.17) is readily apparent. Next we consider Special Case C, in which all persons are ex-n xposed from age ( , have 0), but are scheduled to exit ( , 1] at vari-x i.e. r x xi œ ous ages such that 1. Let represent the subset of z x z x c ni i x x Ÿ scheduled to exit at 1 ( , 1); the remainder, , arez x i.e. s n ci i x x scheduled to remain under observation to age 1.x Let us assume that the value of the ’s for the subset is ,average s c si x0 1. This situation is illustrated in Figure 6.3. s


n cx x ~~~~~~~~~~~~~~~~~~~~~~~ Æ ñ cx ~~~~~~~~~~~~~Æ ñ - - - - - - | | |qqqqqqqqqqqqqqqqqqqq x x s x 1

FIGURE 6.3

The expected number of deaths in ( , 1] is now ( ) , sox x n c q c q † †x x x x s xthe moment equation becomes

E D n c q c q d .[ ] ( ) (6.18)x x x x x s x xœ † † œ

Under the linear assumption, , (6.18) is easily solved for the esti-s x xq s qœ †mator

q .dn s c

^(1 ) (6.19)x

xx x

œ †

Again the exposure interpretation of the denominator is apparent. Our Special Case C is one that frequently arises in clinical studies.As seen in Chapter 4, these studies often are designed so that all members ofthe study sample come under observation at time 0. It follows that allt œmembers who reach the estimation interval ( , 1] must enter it at exact timet tt r i. , so that 0 for all But if the study is terminated before all have died,i œthen there will be enders at various times, so that 1 for some Becauses i. i of the frequent occurrence of Special Case C in practice, we will refer to itseveral more times as we develop the theory of estimation in this text.

6.2.5 Properties of Moment Estimators

In this section we will explore the bias and the variance of the momentestimators derived thus far in the chapter. It is assumed that the reader is fam-iliar with these two properties of estimators. Formal definitions of them aregiven in Appendix A. A characteristic of the moment estimators for which we willq xderive is that they are always unbiased under the assumption, or other ap-proximation, used to derive them. We will demonstrate this for the generalform of the moment estimator given by (6.5). To investigate bias, we need to compare the expected value of the es-timator with the true value of the parameter, , which it seeks to estimate.qxWe are using to represent the realized value of the estimator randomq̂ xvariable when the random variable for number of deaths, , has realizedDx

value Thus we use for the estimator random variable, so that^d . Q x x


Q Ds r

^( )

(6.5a)xx

n

ii i

œ!

œ1

is the relationship between the random variables and for the generalQ̂ D x x

single-decrement moment estimator. Then the expected value of is givenQ̂ x

by [ ] , where [ ] When is ap-^E Q E D q . q x x s r x r s r x rn

iœ œ

E D

s r

[ ]

( )x

n

ii iD

œ1

!œ

1

i i i i i i

proximated by (6.3), we obtain

E Q q . q q s r

s r s r[ ] (6.20)^

( ) ( )

( )x

n n

i is r x r x i i

n n

i ii i i i

xœ œ œ

† ! !!œ œ

œ

1 1

1 1

i i i

œ

!This shows that Estimator (6.5) is unbiased under the approximation used toderive it. It is biased to the extent that (6.3) deviates from the mortality distri-bution to which the sample is actually subject. This last point is illustrated inthe following example.

EXAMPLE 6.3 Analyze the degree of bias contained in Estimator (6.10)for Special Case B.

SOLUTION For Special Case B, we have [ ] and the esti-E D qx r x r n

iœ !

œ

11 i i

mator Then [ ] If we assume^ ^Q . E Q . x xœ œ œD

r r rE D q

x

i i i

x r x ri i

D D Dn n n

i i i

n

i

œ œ œ

œ

1 1 1

1

(1 ) (1 ) (1 )[ ]

D 1

that (1 ) , then [ ] , so Estimator (6.10) is unbiased^1r x r i x xxi iq r q E Q q œ † œ

under this assumption. However, as shown in Chapter 3, this assumptionimplies that the force of mortality decreases over ( , 1). Thus if the forcex xis constant or increasing, as is true over most ages other than juvenile ages,

then (1 ) In this case we have [ ] (1 ) , and1r x r i x x i xi iq r q . E D r q † † !niœ1

thus [ ] , which shows that, for most ages, Estimator^E Q qx x œ r q

r

D

D

n

in

i

œ

œ

1

1

(1 )

(1 )

i x

i

†

(6.10) is positively biased.


To determine the variance of the general moment estimator given by(6.5), note first that the random variable for number of deaths in ( , 1], ,x x D xcan be written as

D Dx i

n

iœ "

œ1, (6.21)

where is the random variable (either 0 or 1) for number of deaths fromD iperson We hypothesize that all , for 1,2, , , are mutually inde-i. D i n i œ ápendent. Since is binomially distributed, with sample size 1 and proba-D ibility of occurrence , we haves r x ri i i q

Var D Var D q q( ) ( ) (1 ), (6.22)x i s r x r s r x r

n n

i iœ " "

œ œ

1 1œ i i i i i i

so that

Var Q n r s . q q

s r( , { , }) (6.23)^

(1 )

( )x x i i

n

is r x r s r x r

n

ii i

l

œ

!œ

œ

1

1

2

i i i i i i

” •!The notation ( , { , }) is used to remind us that the variance is con-^Var Q n r sx x i ilditional on the size of the sample, , and the set of times at which personsnxenter and are scheduled to exit ( , 1]. Expression (6.23) could then be sim-x xplified by using approximation (6.3). Alternatively, the variance of (6.5) can be approximated by notingthat is approximately a binomial proportion with sample size ( )Q̂ s r .x

n

ii iD

œ1

Thus the approximate variance is

Var Q n r s .p q

s r( , { , }) (6.24)^

( )x x i i

x xn

ii i

l ¸!

œ1 Expressions for the variances of Estimators (6.7), (6.10) and (6.13),for Special Cases A, B, and C, respectively, are analogous to Expressions(6.23) and (6.24), and their derivations are pursued in the exercises.

EXAMPLE 6.4 Find an exact expression for the variance of the SpecialCase C estimator given by (6.19).

SOLUTION Let be the random variable for deaths out of the D cx xw

group, and be the random variable for deaths out of the ( ) group.D n c x x xww


Note that , that and are both binomial randomD D D D D x x x x xœ w ww w ww

variables, and that they are independent. From these properties we find thatVar D Var D Var D c q q n c q q( ) ( ) ( ) (1 ) ( ) (1 ), sox x s x s x x x x xx x

œ œ † †w ww

that ( , )^Var Q n c .x x xl œc q q n c q q

n s cx s x s x x x x x

x x

† † †

(1 ) ( ) (1 )[ (1 ) ]

2

6.2.6 A Different Moment Approach for Special Case C

We saw in Section 6.2.4 that the expected deaths out of the , all scheduledcxto exit at age was given by The fractional-interval probabilityx s, c q . †x s xrequired an assumption (such as the uniform) in order to solve the resultingmoment equation. As an alternative to making such a distribution assump-tion, we now consider a different approach. Assume that the scheduled enders are subject to the same proba-c xbility of death in ( , 1] as are the remaining , namely Now con-x x n c q . x x xsider of the deaths out of in ( , 1]; let us call this Some of theseall c x x d . x x

cdeaths occur before exit from the study, and are thus observed, whereasothers occur after exit, and are not observed. Let denote the number ofdx

w

observed deaths out of , and let represent the of that arec proportion d x xc!

observed. Thus d d .x xcw œ †!

Note well that is not known, so is not calculated. Rather, isd xc ! !

an input parameter, presumably derived from other data, or merely assumed.The assumption of takes the place of a distribution assumption.! Of the group, are expected to die in ( , 1], and c c q x x c qx x x x x† †! †is the expected number of observable deaths. ( ) is the expectedn c q x x x †observable deaths out of , so total expected observable deaths, equatedn cx xto the actual observed deaths, gives the moment equation

( ) (1 ) (6.25)n c q c q n q c q d .x x x x x x x x x x † † † œ † † œ! †!

A person scheduled to exit observation at age as a study ender,x s and who survives to that age, is then observed to be an actual ender. Let exdenote the number of actual enders in the sample. Now let us consider theexpected number of observable enders in the sample. We have (1 ) notc q x xexpected to die in ( , 1] at all, and (1 ) expected to die, but x x c q after † †! x xdeparture so that they are observed enders. Then total expected observableenders, equated to actually observed enders, gives the moment equation

c q c q c c q e .x x x x x x x x† † œ † † œ(1 ) (1 ) (6.26) †! !

Note that there are no observed enders in ( , 1] from the group.x x n c x x


The estimator of is derived by eliminating from Equationsq c x x(6.25) and (6.26). From Equation (6.26), we have ; substitu-cx œ

eq

xx1 †!

ting this into (6.25) gives us

n q de qqx x x

x xx

† œ†(1 )

1 , † †!!

which leads to the quadratic equation

! !† † † œn q n e d q d x x x x x xx2 [ (1 ) ] 0, (6.27) †!

the solution of which is

q b b n d

n^ 4

2, (6.28)x

x x

xœ

†È 2 !

!

†

†

where (1 ) The negative radical is used in (6.28)b n e d . œ †x x x †! !because the positive radical would result in 1.q̂x It turns out that Estimator (6.28) is also a maximum likelihoodestimator, which we will show in Chapter 7 (see Exercise 7-9). We also deferto Chapter 7 the derivation of the variance of this estimator (see Exercise 7-22). One advantage of Estimator (6.28) over Estimator (6.19) is that(6.28) does not require knowing the value of , but rather depends on thecxnumber of actually observed enders, Also, it has some of the desirablee . xproperties of an MLE. Although it does not require a distribution assump-tion, as does (6.19), it does require an assumed value for .! This estimator was derived by Schwartz and Lazar [66].

EXAMPLE 6.5 In a Special Case C study, 1000, 100 (all ofn cx xœ œ

whom are scheduled to be enders at age ), and 20, of which x d d œ14 x x

w

œ 2. What value of will produce the same estimate of by Estimator! q x(6.28) as that produced by Estimator (6.19)?SOLUTION Estimator (6.19) gives . Sinceq̂x œ 20 4

1000 (.75)(100) 185 œ

e c d bx x xœ œw œ œ 98, then 1000 98 98 20 902 118 . It is easier! ! !to solve for from Equation (6.27) than from Equation (6.28). Substituting,!

we have (1000 ) (902 118 ) 20 0, which produces! !Š ‹ Š ‹4 4185 185

2 œ

! !œ .23864. Since is the proportion of deaths out of which occur in thecxfirst quarter of ( , 1], this result, which is close to .25, is very reasonable.x x


6.3 MOMENT ESTIMATION IN A DOUBLE-DECREMENT ENVIRONMENT

As in the case of estimation in a single-decrement environment, we assumethat the sample data have been analyzed to produce the ordered pair( , ) for person scheduled contribution to ( , 1], as illustrated inx r x s i’s x x i iFigure 5.6. The probability of death for person while under observation in thei double-decrement environment is , and this is also the expected num-s r x r

di i i q( )

ber of deaths from this sample of size one, as demonstrated by (6.1) for thesingle-decrement case.

6.3.1 The Basic Moment RelationshipsAs explained in Section 5.3, we now consider both death and withdrawal tobe random events. Thus, analogous to the moment Equation (6.2) in thesingle-decrement case, we now write the of moment equationspair

E D q d[ ] (6.29a)x s r x

n

i

dx rœ œ"

œ

1

( )i i i

and

E W q w[ ] (6.29b)x s r x

n

ix rwœ œ"

œ

1

( )i i i

where is the random variable for withdrawals in ( , 1], and denotesW x x w x xthe observed number of withdrawals in the sample. A simple way to solve Equations (6.29a) and (6.29b) for estimates ofthe double-decrement probabilities and would be to use approxima-q q x x

d w( ) ( )

tions analogous to (6.3), namely q s r qs r i ix r

dxd

i i i ( ) ( )¸ †( ) (6.30a)

and

( ) , (6.30b)s r i ix rw

xw

i i i q s r q( ) ( )¸ †

producing

Moment Estimation in a Double-Decrement Environment 125

q d s r

^( )

(6.31a)xd x

n

ii i

( )

1

œ!

œ

and

q .ws r

^( )

(6.31b)xw x

n

ii i

( )

1

œ!

œ

However, our objective is usually to estimate the single-decrement proba-bilities and , in this case from data that have been collected in aq qx x

d ww w( ) ( )

double-decrement environment. To accomplish this, we could calculate esti-mates of and from the estimates and , produced by (6.31a)^ ^q q q qx x x x

d w d ww w( ) ( ) ( ) ( )

and (6.31b), using one of the approximate relationships between single-decrement and double-decrement probabilities presented in Section 5.3. Alternatively, we could calculate estimates of and directlyq q x x

d ww w( ) ( )

from our sample data by expressing and directly in terms ofs r s rd w

x r x ri i i ii i q q ( ) ( )

q qx xd ww w( ) ( )and , under a chosen assumption, without using the preliminary

approximations (6.30a) and (6.30b). This approach normally results in equa-tions which must be solved numerically for and ^ ^q q .x x

d ww w( ) ( )

6.3.2 Uniform Distribution Assumptions

If the random events, death and withdrawal, are each assumed to have uni-form distributions over ( , 1) in the single-decrement context, we knowx x

from Section 3.5 that , 0 1, and also thatu r x rd

q r uw ( ) œ Ÿ

( )1u r q

r q †

†x

d

xd

w

w

( )

( )Ÿ

.x u x r x ud d d

u r( ) ( ) ( )œ

q qu q r qx x

d d

x xd d

w w

w w

( ) ( )

( ) ( )1 1 † †, so that (1 ) Then Equation

wq .. œ

(5.14a) becomes


q q q du

duq u qr q r q

qr q

s r u r u rx r x r x r x ud w d d

s

rs d w

r

x x

x xd w

xd

xd

w w

w w

w w

w

w

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

( )

( )

œ

œ †† †

†

œ†

( ’ “’ “(

’ “

1 1

1 11

1

.

r q u q du

q s r s r q

r q r q.

’ “ ( ’ “’ “

’ “ ’ “

11

(6.32a)( ) ( )

1 1

œ

††

†

† †

xw

s

rx

w

x xd w

x xd w

w

w

w w

w w

( )( )

( ) 12

2 2 ( )

( ) ( )

Similarly, Equation (5.14b) would lead to

q .q s r s r q

r q r qs r x r

w x xw d

x xd w

w w

w w

( )( ) 1

22 2 ( )

( ) ( )œ †

† †

’ “’ “’ “

( ) ( )

1 1 (6.32b)

Substitution of (6.32a) and (6.32b) into the basic moment equations (6.29a)and (6.29b), respectively, results in

E D d q s r s r q

r q r q[ ] (6.33a)

( ) ( )

1 1x x

n

i

x i i xd

i iw

i ix xd wœ œ

†

† †"œ

w w

w w1

( ) 12

2 2 ( )

( ) ( )

’ “’ “’ “

and

E W w q s r s r q

r q r q[ ] . (6.33b)

( ) ( )

1 1x x

n

i

x i i xw

i id

i ix xd wœ œ

†

† †"œ

w w

w w1

( ) 12

2 2 ( )

( ) ( )

’ “’ “’ “

The complexity of Equations (6.33a) and (6.33b), necessitating a numericalsolution, is obvious. If Special Case A, in which 0 and 1 for all prevails, thenr s i i iœ œestimation of and is considerably simplified. We first note thatq qx x

d ww w( ) ( )

Equations (6.31a) and (6.31b) simplify to

q̂ dnx

d xx

( ) (6.31c)œ

and

q̂ wnx

w xx

( ) , (6.31d)œ


respectively, under Special Case A. Then we can substitute these estimatesof and for and themselves in Equations (5.20a) andq q q qx x x x

d w d w( ) ( ) ( ) ( )

(5.20b), producing the estimates

q̂n w d n w d n d

nxd x x x x x x x x

xw ( )

1 1 1 12 2 2 2 (6.34a)

( ) ( ) 2œ

É 2

and

q̂n d w n d w n w

nxw x x x x x x x x

xw ( )

1 1 1 12 2 2 2 . (6.34b)

( ) ( ) 2œ

É 2

In Chapter 7 we will see that Estimators (6.34a) and (6.34b) are maximumlikelihood estimators. Alternatively, we could simplify Equations (6.33a) and (6.33b) underthe Special Case A values of 0 and 1 and solve the resultingr si iœ œequations directly for our estimates of and given by Equationsq qx x

d ww w( ) ( )

(6.34a) and (6.34b). This alternative approach is pursued in the exercises.

EXAMPLE 6.6 100 persons enter the estimation interval (20,21] at exactage 20. None are scheduled to exit before exact age 21, but one death andtwo random withdrawals are observed before age 21. Assuming a uniformdistribution of both random events, estimate q .20

( )w d

SOLUTION Using Equation (6.34a), we have 100 (2) (1) 99.5, so œ1 12 2

that .0101015.q̂20( )w d

œ 99.5 (99.5) 200

100 È 2

œ

The simplification of (6.33a) and (6.33b), and the resulting solutionof them for and , under Special Cases B and C, are left to the exer-^ ^q qx x

d ww w( ) ( )

cises.

6.3.3 Exponential Distribution Assumptions

If both death and withdrawal are assumed to have exponential distributions(constant forces) over ( , 1), then, from Section 3.5, we have the valuesx x

. .x u x r x rd d wd u r u r

u r u r( ) ( ) ( )( ) ( ) ( )œ , 1 , and 1 Equa-

w w q e q e . œ œ. .( ) ( )d w

tion (5.14a) then becomes


s r u r u rx r x r x r x ud w d d

s

r

d u rs

rd

d ws r

w w

q q q du

e du

e

( ) ( ) ( ) ( )

( ) ( )( )

( )

( ) ( )( )( )

œ

œ

œ

( ’ “’ “(

’ “

1 1

1

.

.

.. .

.( ) ( )

( ) ( )

d w

d w

.

. . . (6.35a)

Similarly, Equation (5.14b) becomes

1 (6.35b)s r x rw w

d ws r

q e . ( ) ( )

( ) ( )( )( )œ

.

. .’ “ . .( ) ( )d w

Substitution of (6.35a) and (6.35b) into the basic moment Equations (6.29a)and (6.29b), respectively, results in

E D e d[ ] 1 (6.36a)x x

n

i

d

d ws rœ " ’ “

œ

1

( )

( ) ( )( )( ).

. . œi i

d w. .( ) ( )

and

E W e w . [ ] 1 (6.36b)x x

n

i

w

d ws rœ " ’ “

œ

1

( )

( ) ( )( )( ).

. . œi i

d w . .( ) ( )

Again, the necessity of a numerical solution of these equations for and.̂( )d

.̂ is obvious.( )w

If Special Case A prevails, with 0 and 1 for all then wer s i, i iœ œcan easily obtain our estimates of and by substituting the estimatesq qx x

d ww w( ) ( )

of and , given by Equations (6.31c) and (6.31d) for and q q q qx x x xd w d w( ) ( ) ( ) ( )

themselves in Equations (5.28a) and (5.28b), producing,

q ^ n d wnx

d x x xx

d d ww Î ( ) ( )œ 1 (6.37a)’ “ x x x

and

q ^ n d wnx

w x x xx

w d ww Î ( ) ( )œ 1 . (6.37b)’ “ x x x

In Chapter 7 we will see that Estimator (6.37a) and (6.37b) are maximumlikelihood estimators.


EXAMPLE 6.7 Use the data of Example 6.6 to estimate , assumingq20( )w d

an exponential distribution for both death and withdrawal.

SOLUTION Using Equation (6.37a), we have the values 97n d wn

x x xx

œ .

and . Then 1 (.97) .0101017.^dd w

xx x œ œ1

3 q 20( ) 1 3w d

œÎ

The simplification of (6.36a) and (6.36b), and the resulting solutionof them for and , under Special Cases B and C, are left to the exer-^ ^q qx x

d ww w( ) ( )

cises.

6.3.4 Hoem’s Approach to Moment Estimation in a Double-Decrement Environment

The mathematical difficulty inherent in the solution of (6.29a) and (6.29b)for and , under either the uniform or exponential distribution, moti-^ ^q qx x

d ww w( ) ( )

vates us to find a different, simpler, approach to moment estimation in thedouble-decrement environment. One such approach is the actuarial approach,to be described in Section 6.4; another is Hoem’s approach (see Hoem [36]),which we present at this time. For person who enters ( , 1] at age , Hoem contemplates ai, x x x r itheoretical age scheduled for , say , in the same manner thatwithdrawal x t ithere is an age scheduled for exit due to the ending of the observation period,which we have called The difference is that is known whenx s . x s i iperson enters ( , 1], whereas is not known (except, according toi x x x t iHoem, to a Laplacian demon). For those persons with , the values of t s ti i ibecome known, , as the sample experience unfolds and thea posterioriwithdrawals are observed. Using the exposure interpretation of the moment estimator developedin Section 6.2.3, we find exposure of ( ) if person survives to the sched-s r i i iuled ending age, and exposure of ( ) if person withdraws at age t r i x ti i i x s i.e. i ( , survives to the “scheduled” withdrawal age). But what is theexposure for those who die in ( , 1]?x x In Section 6.2.3 it was emphasized that the appropriate exposure fora death was the exposure ( ), in a single-decrement environ-scheduled s r i iment. Now in our double-decrement environment, we will still use the sched-uled exposure for a death, but the scheduled exposure is now ( ) ors r i i( ), whichever is less. In other words, if a person had an unknownt r i ischeduled withdrawal time of , and died before time , this scheduledt s ti i iwithdrawal time would never become known. This person contributeshould( ), but we can never know what is (or would have been).t r t i i i


Hoem deals with this dilemma by simply assuming that no one whodied in ( , 1] had a value that was less than In other words, deathsx x t s . all i i

contribute ( ) to the exposure. Then the estimate of is produced ass r q i i xd w ( )

the ratio of number of deaths to the exposure. An example will make this procedure clear. The important point isthat all persons have a known value of at entry into ( , 1], regardless ofs x xi the outcome for person (survival, death or withdrawal). Each person’s valuei of becomes known that person actually withdraws. Since cant only if t i inever become known for a person who dies, values are used for all deaths.s i

EXAMPLE 6.8 The basic data on a sample of four persons has beenanalyzed to produce the following ordered pairs ( , ) with respect to ther s i iestimation interval ( , 1]. In addition, it is known that one person died andx xone person withdrew while under observation, at the times shown. Estimateq .x

dw ( )

Time of Time ofPerson ( , ) Death Withdrawalr si i

1 ) — —(0, .402 (0, 1) .70 —3 (.30, 1) — —4 (.50, .80) — .75

SOLUTION Persons 1 and 3 contribute exposure of .40 and .70, respec-tively. Person 4 contributes .25, the theory being that he was scheduled towithdraw at age 75, since, in fact, he did! Person 2 contributes 1, underx .the assumption that she was scheduled to withdraw before age 1. Thenot xtotal exposure is .40 70 25 1 2.35, so .42553.^ œ. . qœ œx

dw ( ) 12.35

6.4 THE ACTUARIAL APPROACH TO MOMENT ESTIMATION

Of the various approaches to the estimation of over the interval ( , 1]q x xx described in this text, the actuarial approach was the first to be developed,dating from the middle of the nineteenth century. There are two significantimplications of this early origin:

(1) The method predates the formal, scientific development of sta-tistical estimation theory.

(2) The method predates any kind of mechanical or electronic cal-culating equipment.

The Actuarial Approach to Moment Estimation 131

It will be easy to see the reflection of these two observations in the featuresof the actuarial method. The method was developed primarily in an intuitivemanner, in the absence of a guiding statistical theory, and a key theoreticalconcession was made in the interest of simpler record-keeping and calcula-tion. The traditional actuarial approach remained in use well into the secondhalf of the twentieth century, even after the availability of both high-speeddata processing capabilities and modern statistical theory had made the pro-cess nearly obsolete. Over the past ten years, however, we have seen mount-ing evidence to suggest that this pioneering method of survival modelestimation should become, and is becoming, of historical significance only.Therefore the presentation of the method here is in the nature of a historicalsummary. The reader interested in a fuller history can pursue this throughthe many references given here. We will present the actuarial method assuming a double-decrementenvironment. The simplification that results if the environment is single-dec-rement will be easily seen.

6.4.1 The Concept of Actuarial Exposure

The key difference between the traditional actuarial approach to estimationand the moment approaches presented thus far in this chapter is that theactuarial approach has not generally made use of the concept of scheduledexposure, as defined in Section 6.2.3. Rather it made use of a type ofobserved exposure, which we will call , to distinguish itactuarial exposurefrom the two other types of exposure (scheduled and exact) described earlier. Recall that person is scheduled to be an ender to the study at age ,i ziand this is known at entry to the study. Recall also that if 1, thenx z x iwe say that person is scheduled to exit ( , 1] at age , wherei x x z x s œ i i0 1. On the other hand, if 1, we say that person is scheduled s z x i i i to survive ( , 1], which is the same as to say scheduled to exit at ,x x x s iwhere 1.si œ Suppose person enters ( , 1] at age , 0 1. Under thei x x x r r i iŸactuarial method, if person is an at , he contributesi observed ender x s iexposure of ( ) If he is an at , where,s r . observed withdrawal x ti i i necessarily, , he contributes exposure of ( ) But what if person ist s t r . i i i i iŸ an in ( , 1]? We have seen that a proper momentobserved death x xestimation procedure would have person contribute exposure to age ,i x s ithe scheduled exit age, and early in the historical development of the actu-arial method this was intuitively recognized. However, the identification ofscheduled exit age, for a person who died before reaching it, required moreextensive data analysis than the early manual procedures allowed. To resolve


the problem of an unknown for an observed death, it was simplyx s iassumed that 1 for all deaths.si œ The important point here is that the traditional actuarial approach tohandling the data, as thoroughly described by Batten [6] and Gershenson[26], simply did not identify . Rather became known x s a priori x s a i iposteriori i only if person was, in fact, an ender, just as Hoem’s “scheduled”withdrawal age becomes known, , only if person is anx t a posteriori i iobserved withdrawal. Thus, in the actuarial approach, did not becomex s iknown for a death, so it was assumed to be 1, just as, in Hoem’s approach,xa death that might otherwise have been a withdrawal has an unknown ,x t iand is therefore assumed to be x s . i Note that if, in fact, 1, then the actuarial method of givingsi œexposure to age 1 is correct; it is only if a death has 1 that the methodx s i“over-credits” exposure to age 1 rather than only to age x x s . i

EXAMPLE 6.9 Three persons in a study sample all have the ordered pair( , ) (0, .75). Person 1 remains in the sample to age 75, Person 2r s = x .i i withdraws at age 50, and Person 3 dies at age 50 How much expo-x . x . . sure does each person contribute (a) under the actuarial approach; (b) underHoem’s moment approach?

SOLUTION (a) Person 1 contributes .75, from age to age 75; Personx x .2 contributes .50, from age to age 50, where exposure ceases due tox x .withdrawal; Person 3 contributes 1.00, from age to age 1. (b) Person 1x xand Person 2 again contribute .75 and .50, respectively; Person 3 contributesonly to scheduled exit age, which is .75, under the moment approach.

It cannot be overemphasized that the historical reason for creditingPerson 3 in Example 6.9(a) with exposure to age 1, rather than only to agexx . 75, was that the scheduled exit age was not conveniently available infor-mation in the actuarial approach to processing the sample data. Assumingsuch scheduled exit age to be 1 was simply a practical expedient.x Since age is person s age when the observation periodz x s i’i iœ closes, then it follows that crediting exposure to age 1, whenever 1,x s imeans that person is viewed as contributing exposure after the observationi period has closed. Ackland [1] referred to this as an “odd procedure” whichis “in strictness” incorrect, but nevertheless acceptable for reasons of expedi-ency. Seal [67], [68] and [69] repeatedly criticized this facet of actuarialmoment estimation, especially later attempts to justify it by a statistical prin-ciple. These attempts at statistical justification appear to have originated with

The Actuarial Approach to Moment Estimation 133

Cantelli [18] in 1914, and Balducci [5] a few years later. (Cantelli’s faultyargument is not presented here; the interested reader will find it in [18].) Cantelli’s argument, although faulty, appeared to have been widelyaccepted, at least in North America. Subsequent papers by Wolfenden [75],Beers [9] and Marshall [54], and textbooks by Gershenson [32] and Batten[8], in effect repeated his argument. But the preponderance of modernstatistical thought indicates that Cantelli’s approach, although correct forSpecial Cases A and B, is seriously flawed in the cases involving enderswithin the estimation interval. (See, for example, Hoem [36] and Seal [68].) In a single-decrement environment the possibility of withdrawalsdoes not exist. Then observed enders contribute exposure to their observedending ages, and observed deaths contribute exposure to age 1, regardlessxof a possibly earlier scheduled ending age.

6.4.2 Properties of the Actuarial Estimator

In the single-decrement environment, the actuarial estimators for SpecialCases A and B are the same as the moment estimators, so the discussion inSection 6.2.5 regarding bias and variance is applicable to the actuarial esti-mators as well. In situations involving enders within ( , 1], that is, enders at x x x s iwhere 1, various investigations have shown under simulation, or other-si wise, that the actuarial estimator is . It has also been shownnegatively biasedto be . (See, for example, Breslow and Crowley [13], and Broffittinconsistent[14].) In large-scale actuarial studies of insurance data, there will always bea considerable number of enders for the observation period, and it followsthat there could therefore be a significant number in any estimation interval(unless the study design places them at the interval boundary, which, as wewill see in Chapter 9, is frequently the case). The moment estimator willalways produce a larger estimate than will the actuarial estimator, so aq̂ xnatural way to counteract the negative bias in the latter is thereby suggested. Finally it should be repeated that all estimators of the form ,Deaths

Exposurewhich includes most of the ones in this chapter, are approximately binomialproportions, so all are approximately unbiased with variance given by

p qExposure

x x† . This fact plays a crucial role in the next section.

6.5 ESTIMATION OF S x( )

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