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A NOTE ON LIFE TABLE AND MULTIPLE-DECREMENT TABLE FUNCTIONSAuthor(s): W. F. ScottSource: Journal of the Institute of Actuaries (1886-1994), Vol. 117, No. 3 (DECEMBER 1990), pp.671-675Published by: on behalf of theCambridge University Press Institute and Faculty of Actuaries

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7/23/2019 A Note on Life Table and Multiple-Decrement Table Functions

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A

NOTE ON

LIFE

TABLE AND

MULTIPLE-DECREMENT

TABLE

FUNCTIONS

By

W. F.

Scott,

M.A.,

Ph.D.,

F.F.A.

ABSTRACT

This note onsiders

he

mortality

able

functions,

nd

shows hat heusual formulae old under ess

restrictive

ssumptions

han those

usually

made. The foundations

f the

theory

of

multiple-

decrementables are also considered,n thecontext fprobabilityheory.

KEYWORDS

Life

Tables;

Decrement ables.

1. INTRODUCTION

The

properties

f the

mortality

able functions re

usually

derivedunder the

assumption

hat

i

exists nd is

continuous

see

Neill,

Section

1

6).

We

shall how

that

heusual

formulae oldunder

essrestrictive

ssumptions,

iz. that

x

nd

fix

exist nd are continuous. hemathematical asis for his pproach s similar o

that sed n

connection

ith he

force f

nterest

n

McCutcheon&

Scott,

ection

2.4.

We

also consider

he

foundations f

the

heory

f

multiple-decrement

ables,

whichwe

attempt

o

place

n

the

ontext f

probability

heory.

n

particular,

e

prove

the

identity

f

the forces'

see

Hooker

&

Longley-Cook,

ection

20.7).

2. SOME

MATHEMATICAL

RESULTS

Theorem .1

Let /=

[A,B)

or

-

oo,

2?),

where

B

may

be oo

Suppose that/(x)

s continuous

on

/and that ts

right-hand

erivative,/+x),

is zero on /.

Then/(;c)

s constant.

Proof (Adapted

from

Hobson,

p.

365)

Let us consider he

ase when

/=

(

-

oo,

/?);

he

ase when

/=

[A,B)

s

similarly

dealt with. f

the

onclusion f the

theorem e

false,

heremustbe a

and b

in

/,

with

(x,k)=f(x)-f(a)-k(x-a)

which s

continuous n /.

Note that

t>{a,k)

0. Let k

be

positive

ut o small

hat

(/>(b,k)2q>0.Let M =

{x:

a

7/23/2019 A Note on Life Table and Multiple-Decrement Table Functions

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672

A Note on

Life

Table

and

Multiple-

ecrement

a(,&)

q.

Now let

xn-+

- Since

q

for

ll

n,

/>(,

)>q,

and so

(j)(,k)

q.

Also:

/i

x

X,?

-

But

this ontradicts

he fact hat

p'+(x,k)=

-k

7/23/2019 A Note on Life Table and Multiple-Decrement Table Functions

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Table

Functions

673

We also define:

x

=

1-tPx

=

Pr{a

life

ged

x

does

not survive o

age

x

+

t}.

(3.4)

(If

t=

1,

we

may

write

qx

qx

and

tpx=px.)

We assume

that,

for

x>a:

Pr{a

life

ged

x dies

within ime

A}

ix

=

hm

//-0+

A

= lim ^ (3.5)

/i-o+ A

exists,

and is

a continuous function n

[a,oo).

We also assume

that,

for

OL

7/23/2019 A Note on Life Table and Multiple-Decrement Table Functions

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674

A Note on

Life

Table and

Multiple-

ecrement

Formula 3.8

now follows

rom

his

nd formula .7.

Q.E.D.

It now follows

asily

on substituting

=

'nx+sds)

that:

o

hqxJ'tPxHx

dt.

(3.9)

0

Now let

x and t be

fixed,

with

t25,

say,

s(x)

is continuous nd

that:

Pr{a

life

ged

x

dies

within ime

h}

=

Bcx

h

+

o(h).

This

implies

hat:

i.e.

Gompertz'

aw

holds

for

x>25,

and

xx

s

clearly

ontinuous.

f

we also

suppose

thatformula .6

holds,

Theorem3.1 showsthat,forx>25 and />0:

,p,

exPr-fr*+'&l

=

exp[-5rv(c'-l)/logc].

4.

THE MULTIPLE-DECREMENT

TABLE

Let usconsider nly womodesofdecrement,andy, ndsupposethatwe are

mainly

nterested

n

mode

.

(If

there

re

more han

wo

modes,

ll

except

mode

may

be

combined.)

t s

supposed

hat

ach mode

ofdecrement

as the

life able'

functions

x?, px,

tc.,

as

in

Section

3.

Lives are

considered

to continue

n

existence

ith

espect

o a

given

mode after

xit

by

theother

mode ofdecrement.

To avoid

philosophical

roblems

when

one of the

modes of

decrement

s

death,

one

may,

for

xample,

ake

mode

as exit

by marriage

nd

mode

y

as

exit

by

withdrawal

rom

ervice

mong

thebachelor

mployees

f

a

large

organisation,

mortality

eing gnored.

Let

Tu T2

denote

he imes

o exit

by

modes

and

y respectively

or

life

ged

x.

It follows

by

formula

3.9

that

Tu T2

have

probability ensity

functions

txP%Px

u

(t'

>

0)

and

t2pl'i'

t2

t2

>

0) respectively.

e assumethatTx nd T2

are

independent,

nd

define:

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Table

Functions

675

t(ap)x

the

probability

hat life

ged

jc,

subject

o both

modes of

decrement,

survives o

age

x+t

=

tp.'tpi (4.1)

by

the

ndependence

f

T'

and

T2.

If

/

1

we

may

omit

t.)

We also

define:

t(aq)x

the

probability

hat life

ged

x,

subject

o both modes of

decrement,

will

xit

by

mode

before

ge

x+

1,

xit

by

mode

y

not

having reviously

occurred

=

Pr{r