Refinements and distributional generalizations of Lundberg's inequality

MATHEMATICS & ECONOMICS ELSEVIER

Insurance: Mathematics and Economics 15 (1994) 49-63

Refinements and distributional generalizations of Lundberg’s inequality * Gordon Department

of Statistics and Actuarial

E. Willmot

Science, University of Waterloo, Waterloo, Ont., N2L 3G1, Canada

Received January 1994; revised June 1994

Abstract An upper bound is obtained for the tail of the total claims distribution in terms of a ‘new worse than used’ distribution. A simple bound exists when the claim size distribution is also new worse than used. The classical exponential bound is also refined when the claim size distribution is new worse than used in convex ordering, as well

as for other classes of claim size distributions characterized by properties of the failure rate and the mean residual lifetime. Pareto bounds may be used when claim size moments are known. The compound geometric case and ruin probabilities are then considered. Keywords: Failure rate; New worse than used; New worse than used in convex ordering; Mean residual lifetime; New worse than used in expectation; Martingale; Ruin probability; Pareto distribution; Markov’s inequality

1. Introduction Consider

and notation

the classical

model for the total claims distribution where N is the number of claims random

variable. Let n=0,1,2...

p,=Pr(N=n);

(1)

and cc a, =

c

pk;

n=0,1,2....

(2)

k=n+l

Furthermore, let the claim sizes be denoted by {Y,, Y,, . . . 1, an independent and identically distributed sequence of positive random variables independent of N with common distribution function (df)

F(y)

=Pr(y,
yro,

(3)

* This work was supported by the Natural Sciences and Engineering Research Council of Canada. Thanks also to Xiaodong Lin for helpful comments and discussions. 0167-6687/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0167-6687(94)00019-B

50

G.E. Willmot /Insurance:

Mathematics and Economics 15 (1994) 49-63

and let F(y) = 1 -F(y). For convenience, it is assumed that Y, > 0, although this assumption can be relaxed to yi 2 0. Also define F*n(y) = Pr
‘PY(X>X)

= f

pn .F*n(x),

x20.

(4)

n=l

We assume the existence of a positive number r$ < 1 such that a n+l 2 +a,,

n =o,

1,2 ) . . . .

(5) See Willmot and Lin (1994) for more details about this assumption. Then Willmot and Lin (1994) showed that

1 --PO

iT(x)s~e

where

_-K*

,

(6)

x 2 0,

0 satisfies

K >

c#-‘=

I0

ffie’ydF(y).

(7)

Gerber (1994) also proved this generalization of the classical Lundberg inequality of ruin theory using martingales. In this paper a further generalization of (6) is proved. Central to the discussion is the concept of a new worse than used (NWU) distribution (Barlow and Proschan, 1975, p. 159). That is, if B(x) is the df of a non-negative random variable and B(x) = 1 -B(x), then B(x) is Nwu if B(xc)&y)
B(x)B( y)

1. Suppose 4 < 1 satisfies (5), and B(x) S B(x + y) for x 2 0, y 2 0. Suppose B(x)

j,l{B(~)}-’

dF(y)

is a non-negative function for x 2 0 which satisfies also satisfies

54-l.

(8)

If, in addition, F(x) where c(x) G(x)


x20,

(9)

is a non-decreasing function for x 2 0, then <+-‘(l-p,)c(x)B(x),

x20.

(10)

G.E. Willmot /Insurance:

Proof.

Define Ck(

x)

the sequence

=

51

of functions

-F”“(x)},

am{F*(m+‘y x)

;

Mathematics and Economics 15 (1994) 49-63

x20

m=O

for k = _’ 0 1,2,. . . , where F*‘(x) = 0 for x 2 0. We will show by induction p,)c(x)B(x), x r 0. For k = 0 we have Go(x)

on k that

Gk( x) 4 4 - ‘(1 -

= (1 -POP(x)

-P~)c(x)~(~)~~{B(Y))-~ dF(y)

I

(1

I

(1 -~,)c(x)B(x)~~{~(~)}-'

54-l(1

dF(y)

-p,)c(x)B(x).

AlSO,

k%k(

dF( y) =

x -y)

; am{F*@+*)( m=O

x) - F*(“+‘)(

x)},

and so using (5) it follows that k+l Gk+l(x)

sa,F(x)

= (1 By the inductive

++

-p,)F(x)

hypothesis,

m=lu,_l{F*(m+l)(x) c

-F*“(x)}

+$G,(x-Y) ??,(x - y) 5 $-‘(1

WY). -po)c(x

- Y)B(x - y). Thus

-

y) + (1

m=dF( qy)

=

qx)

G,(x)

=

5

p,,)c(x)$$$-dk(

y>

WY)

(1 -~o)c(x)~(x)jo~(~(y))-I

54-‘(1 Therefore, yields

-

-pa)c(x)B(x).

I $-‘(1

-p,>c(x)&x)

am{F*(m+l) (x)

for x 2 0 and

-F*“(x)}

k = 0,1,2,. . . . But summation

by parts

on (3)

= hmCk(X)

m=O

and (10) follows.

0

In many applications it is convenient to use Theorem 1 with c(x) constant that F(x) < 1 for x
k =

LmP(Y)l-l dF(Y) inf O~X
’

in (9). To fix ideas, suppose

52

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Mathematics

and Economics

15 (1994) 49-63

it is clear that (9) is satisfied with c(x) = k-‘, yielding

i?(x)

l-P,-

I

-p(x),

x20.

The following theorem shows that k 2 1 quite generally. Theorem 2. Suppose 4 < 1 satisfies (5), and B(x) is a non-negative, non-increasing function for x 2 0 which satisfies B(x)B(y) I @x + y) for x 2 0, y 2 0. If B(x) also satisfies (8), then G(x) Proof.

I+-‘(1

-p,)E(x),

x20.

(II)

One has

F(x) =jmdF(y)

~B(x)j~{@y)}-~ dF(y) x x since B(x) is non-increasing. Thus (9) holds for c(x) = 1 and the theorem follows from Theorem 0

1.

Although it is not necessary, it is sufficient in Theorems 1 and 2 that 1 - B(x) be a NIVU df satisfying (8). Although we can always choose c(x) = 1 in Theorem 1 to satisfy (9) if B(x) is non-increasing, for some choices of F(x) and B(x) we may refine the inequality by choosing c(x) < 1. Note, however, that if B(x) I B(O) then from (9) one has

~B(of=~m{~(o))-l WY) ${~(y)}-'dF(y)

I+-‘,

i.e. +{&O)}-’ I 1. For (8) and (9) to hold simultaneously, therefore, we have from (9) with x = 0 that 1 = F(O) I c(O)B(O)f$ - 1, i.e. c(O) 2 +{&O)}-‘. Thus, to refine the inequality, we should have 4(3(O)}-’ < 1 and choose a non-decreasing function c(x) such that 4{@0))-’ I c(x) I 1, x 2 0. In some cases we may choose c(x) = +{@O)}-‘, and we note that in this situation (10) yields G(O) I 1 -pO, obviously an equality. This is the case in the following theorem, which also provides an obvious choice of B(x). Theorem 3. G(x)

Zft#~ < 1 satisfies (51, and F(x) is absolutely continuous and hWU, then 5 (1 -p,,){F(x)}l-Q,

x20.

(12)

Proof.

If F(x) is NWU, then so is 1 -B(x) where B(x) = (F(x)}‘-@. We will use Theorem choice of B(x). Since F(x) is absolutely continuous,

$lrn{F( Y)}'-'

dF(y) =

{F(X)}‘-

x

Then

F(x) = 4{F(x)}‘-‘L-{F( Y)]-

dF(y),

i.e. (9) is an ‘equality with c(x) = 4. Let x = 0 to obtain

c{“( Y))‘-~dF(

y) =4-l,

i.e. (8) is also an equality. Then (12) follows from (10).

q

1 with this

G.E. Willmot / Insurance: Mathematics and Economics 1.5 (1994) 49-63

53

It is interesting to note that if p, = (1 -p&l - +)b”-’ for y1= 1,2,3,. . . , and F(x) = epP”, x 2 0, then one can show that c(x) = (1 -p,)e-Pcl-@‘)X, x 2 0, i.e. (12) is exact. Theorem 4.

Suppose C#J < 1 satisfies (.5), and 1 -B(x)

B( y)D(x)

x 2 0,

5 {F(Y)/F(x)}T,

If, in addition, either F(x)

or 1 - B(x)

y 2x.

(13)

is absolutely continuous, then

G(x)G$-‘(l-7)(1-po)B(x), Proof.

is a hW7J df which satisfies (8) as well as

x20.

(14)

First note that for (8) to hold one must have r < 1. This follows from the fact that F(x) -= B(x)

mdF(y) / X B(x)

mdF(Y) / S X B(Y)

’

= 0. But (13) requires that B(x) I @O){F(x)P which may be rewritten implying that lim, am F(x)/B(x) as F(x)/@x) 2 {F(x))‘-‘/B(O). Clearly, the right-hand side can only remain bounded at infinity if r < 1. Now, if F(x) is absolutely continuous, one has ~(x)~a{~(y)]-1dF(y)~{~(~)}TjmIF(y)}-TdF(y)=(l-~)-1~(x), x X i.e. F(x) I (1 - ~)B(x)/~~B( y>}-’ dF( y). Thus (9) holds with c(x) = 1 - r. Conversely, suppose that = 0, and integration by parts 1 - B(x) is absolutely continuous. As above, one has lim, _ F(x)/B(x) yields B(x${B(y)}-‘dF(y)

-F(x)

=B(x)lmF(y)

d{B(y)}-‘.

>F(x)

+~(x){~(~)}‘-~‘~/:{~(y)}~‘~d{~(y)}-~

=F(x)

+F(x){7/(1-r)}

(15)

Thus, from (13) and (1.5), @x)~~{@Y)}-‘dF(y)

=F(x)/(l-7). Again, therefore, (9) holds with c(x) = 1 - T and the result follows from Theorem 1. q It is worth noting that (13) holds if both F(x) and 1 -B(x) are absolutely continuous with failure rates p&x) = -d/dx In F(x) and pcLB(x)= -d/dx In B(x) respectively which satisfy /..L~(x)2 7puF(x). Also, (13) holds if F(x) is AWU and satisfies F(x) 2 (@t +x)/&t)}“’ for x 2 0, t 2 0. As in the proof of Theorem 4, it is of interest to examine when the right-hand side of (15) divided by F(x) is non-increasing in x, or has a non-increasing lower bound. This technique is used in the next section. Many subclasses of the NWU class are closed under mixing (e.g. Barlow and Proschan, 1975, pp. 103-104, 186-187). Also claim size distributions which are themselves mixtures arise quite naturally in connection with models incorporating inflation and other phenomena (e.g. Willmot, 1989). The present results are often applicable in these situations. Note that if q(x) is non-negative and non-decreasing, Markov’s inequality (e.g. Galambos, 1988, p. 294) yields G(x)

~E{s(X)}/q(x),

x20.

(16)

54

G.E. Willmot /Insurance:

Mathematics

and Economics

15 (1994) 49-63

But if (8) is an equality and B(x) is non-increasing, c(x)K’(I

-PO) =c(x)(l

-~n)jga{%y)}-i

=c(x)

dF(y)

5 ~,,jol{o(y)}-‘dF(y) n-1

SC(X)

5

P,~~{~(Y)}-’

dF*“(y)

n=l 5 E{l,‘@ X)}

since F*“(x) 2 F(x) for II 2 1 and {B(x)}-’ is non-decreasing in x (e.g. Ross, 1983, p. 252). Thus, if (8) is an equality, (10) is a refinement of Markov’s inequality applied to q(x) = (B(x))-‘, and in some cases E{l/&X)} = ~0. 3. Refinements

of the exponential

bound

In this section we consider the special case B(x) = ePKx where K > 0 is the solution to (7), and 4 < 1 is chosen to satisfy (5). It is of interest to improve on the upper bound (6). Since B(O) = 1, we wish to find c(x) satisfying 4 < c(x) I 1 so that c(x) 5 ~#-l(l -pO)c(x)ePKX. We shall see that it is possible to choose c(x) = 4 for a large class of claim size distributions. To begin with, note that (15) becomes, in this case, meKydF( y) = eK*F( x) + Kjme”‘F( y) dy, x 2 0. (17) x Also, ie note that the existence of K > 0 satisfying (7) implies that F(x) has a moment generating function and hence moments of all orders. In particular, the mean E(Y) = jrF(t) dt exists, and we define the absolutely continuous df F,(x) = 1 -F,(x) by /

F,(x)

= (p(t)

d+(Y),

x20.

(IS)

Moreover, we have eKXFe( x)

= eKxlm{F( y)/E(Y)} x I

{E(Y)}-l~me”YF(y)

dy dy

= {KE(Y)}-‘{/ae”

dF(y) -eKxF(x)). x Since (7) holds, /reKy dF( y) goes to zero as x goes to infinity, and as noted in the proof of Theorem 4, lim x ,,eKxF(x) = 0. Thus, lim, _,e”“F,
x

integration by parts yields meKyF( y) dy = E(Y) eKXFe(x) + KlmeKyFe( y) dy , x 2 0. /x x 1 i Eqs. (17) and (19) will prove to be of use later in this section.

(19)

G.E. Willrnot /Insurance:

We also define rF(x)

=

the mean

residual

{p(t) d@(x),

Mathematics

lifetime

and Economics

associated

with F(x),

5.5

15 (1994) 49-63

namely

x20.

The df F(x) is said to be increasing mean residual lifetime (ZMRL) is rF(x) is a non-decreasing function for x 2 0. A subclass of the ZMRL class is the DFR class (Gertsbakh, 1989, p. 108). Conversely, F(x) is decreasing mean residual lifetime (DMRL) if rF(x) is non-increasing, and the class of increasing failure rate (ZFR) distributions are contained in the DMRL class (e.g., Lawless, 1982, pp. 44-45). The df F(x) is said to be new worse than used in expectation (NB’UE) if r&x) 2 r,(O) = E(Y). The NWU and DFR classes are contained in the NB’UF class (e.g. Barlow and Proschan, 1975, p. 159), as is the ZMRL class. Cao and Wang (1991) introduced the new worse than used in convex ordering (NB’UC) class of distributions. This subclass of the NWUE class contains both the NB’U and ZMRL classes. The df F(x) is NWUC if jrn

F(t)

dtzF(x$F(t)

dt Y

XfY

for all x 2 0, y r 0. We have the following Theorem 5.

Suppose

G(x) 5 Proof.

C$ <

1 satisfies

(1 -pO)eeKX,

result. and K > 0 satisfies (7). Then ifF(x)

(51,

is NWUC,

x 2 0.

(21)

From (17) and (19), m e --K.K eKYdF( y) = F(x) /x

+ KeeKX/.-eKyF( X

y) dy

=F(x)

+KeeKXE(Y)

eKXFe(x) + KjmeKyFe( y) dy x I

=F(x)

+KE(Y)F~(x)

+fc2E(Y)e~x”~meKy~~(y)

=F(x)

+KE(Y)F(x)

+K2E(Y)~me”yF~(y+x)

Since F is NWUC, one has FJy +x) 2 F(x)F,(y) and NWUC F,(x) 2 F(x). Thus, again using (19) with x = 0, we have e

--K.r /

meKy dF(y) X

>F(x)

+KE(Y)F,(x)

1

l+~/g~e”‘F(y)

1 + Kj”e”‘F( 0

dy. NWUE which may be restated

dy

+K’E(Y)F(x)~~e”‘F,(y)

=F(~)+KE(Y)F,(x)+KF(x)

=F(x)

implies

dy

imeKyF(y)dy-E(Y)}

dy)

y) dy

+KE(Y){F,(x)

-F(x)}

as

G.E. Willmot /Insurance:

56

Mathematics

and Economics

15 (1994) 49-63

But, using (17) with x = 0, one has K

LmeKYF( y) dy = /‘u,.,

dF( y) - 1 = f - 1

0

from (7). Thus F(x)

s 4e-KX/meKy dF( y), x

and (9) is satisfied with c(x) = 4. Then Theorem

1 applies.

0

Again, we remark that (21) is exact when x = 0 and when P, =(l -p&l and F(x) = e-OX, x L 0. We now obtain a refinement of (6) in the case when F is DMRL,. C#J< 1 satisfies (5), and K > 0 satisfies Theorem 6. Suppose for x
G(x)

I$-l(l-po){l

i-Ka(X)}-le-“X,

-c$>c&“-’ for n = 1,2,3,...

(7). If F(x) is DMRL and satisfies F(x) < 1

X20,

(22)

with y)

dy/(e”“F(

x)],

X
(23)

x2x,. Proof. From (18) it is clear that if x 0 but F,(x) = 0 for x 2x,. from (23) and (19) that a(x)

=

W-)@I

m

For x
1 +K

i

But from (18) and (201, rJx)

=

E(Y)F,(x)/F(x)Thus

a(x)=rF(X){l+KY(X)}, X<-%,

(24)

where ]xoezyFe( y) dy

Y(X) =

X

eKxFe(x)

(25)

.

Note also from (18) and (20) that the failure rate associated with F,(x) is {rF(x)]-’ Then it is not hard to show that

= -d/dx

In F,(X).

(26) We wish to show next that y’(x) 5 0. This is clearly true if {rF(x)jpl (rF(x))-’ - K > 0. From (2.51,

-

K

IO,

so now assume that

G.E. Willmot /Insurance: Mathematics and Economics 15 (1994) 49-63

57

=(-&K}-‘.

Thus, 7(x)(1/r&x> -K) s 1 and from (26) y’(x) 5 0. By assumption, rF(x) is x
= { 1 + Ka( x)} -‘ePKxjXOeK’ x

dF( y),

non-increasing for x
x
(27)

Thus, the inequality F(x) I {l + Ku$x))-‘ePKX/~eKY dF(y) holds for all x 2 0. Since 11 + K~(x)}-’ non-decreasing, (9) is satisfied with c(x) = (1 + K(Y(x)}-l, and (22) follows from Theorem 1. 0

is

The bound (21) is applicable for the NWUC class of distributions. More generally, we consider the larger class of distributions with r&x) 2 r > 0. The NWCIE class is the special case r = E(Y). We have the following result. Theorem 7. Suppose C$< 1 satisfies (51, and K > 0 satisfies (7). Then if F(x) rF( x) which satisfies rF(x) 2 r > 0, it follows that G(x)

The condition

Proof.

that

dykr

jme”‘F(Y)

x

Thus,

-Kr)ePKX,

I4-‘(l-p,)(1

x20.

(28)

r,Ax) 2 r may be restated

e”“F(x)

has mean residual lifetime

+K/“eKyF(y) X

dy

as E(Y)F$x) =r/-,KY X

r @(xl.

Thus

from (19) and (171,

dF(y).

again with (17), m eKy dF( y) 2 eK*F( x) + Kr/-eKY dF( y), x

/x

a restatement

of

F(x)~(l-~r)e-“‘lmdF(y). X Then

(9) is satisfied

We remark F(x)

with c(x) = 1 - Kr and Theorem

that Jensen’s 5

inequality

e-KrF(+ajmeyY

implies

1 applies.

q

that

dF(y). x

Thus, if r*(x)= infoSySx c(x) = e-“r*(x). This yields

r,(y), it follows that ePK’*cX) is non-decreasing and (9) is satisfied with a general refinement to (6) for any claim size distribution. This may be

58

G.E. Willmot /Insurance:

Mathematics

and Economics

15 (1994) 49-63

simpler to use than c(X)={1 +KLY.+(X)}-~ where ‘~,(X)=infa.,., a(y). The justification for this latter choice of c(X) comes from (27). Note, however, that if F(x) is DMRL then T*(X) = TV for x 0, for some values of x (at least) one has T(X) = r. But since e --Kr > 1 - Kr, Theorem 7 gives a tighter bound in this case. It can also be shown that if F(X) is DMRL with TV > 0, then (28) applies with r = r&m), but (21) is a tighter bound. The inequality (6) may be refined if F(x) is absolutely continuous with bounded failure rate. Corollary 1. Suppose C#J < 1 satisfies (5), and K > 0 satisfies (7). If F(x) failure rate ~Jx) = - d/dx In F(x) which satisfies pcLF(x)I m < 03, then G(X)I+-r(l-p”)(l-Km-‘)e-“X, Proof.

is absolutely continuous with

x20.

(29)

One has from (20),

and Theorem

7 applies

with r = rn -l. Alternatively,

Theorem

4 applies.

0

4. Pareto bounds Suppose that is,

it is known

E( Yj) = lpyi 0

only that moments

dF( y) < 03,

of the claim size distribution

exist up to order

PO)

a simple Pareto bound may then be derived for the tail c(x) 0 and B(x) = (1 + KX)-r, X 2 0, then 1 - B(x) is Nwu. Then K equality. That is, K > 0 is chosen to satisfy

K >

Then we employ G(X)

Theorem

{4-‘W2)

distribution. If so that (8) is an

that (32)

X20.

to be a positive

= (f-1 -

K=

of the compound 0 may be chosen

(31)

2 to conclude

if r is taken

which is a polynomial obtains

>

dF(y).

-
It is convenient

r > 01,

jlr.

Then

(f-1 = /6;(1 + K$

r (where

in

K

integer,

in which case (31) may be re-expressed

as

1,

of degree

(33) r. If r = 1, then

K =

(1 - $){4E(Y)}-l

whereas

if r = 2 then

one

- Var( Y)}1’2 - E(Y)

W2)

{c# -

For general r 2 1, Jensen’s inequality yields K I ‘I’ - 1)/E(Y) from (31). Clearly, the Pareto bound (32) with K > 0 satisfying (31) is analogous to the exponential bound (6) with K > 0 satisfying (7). The exponential bound requires that Y have a moment generating function whereas the Pareto bound requires only that yi have finite moments up to order r (at least).

G.E. Willmot /Insurance: Mathematics and Economics 15 (1994) 49-63

59

We note that Markov’s inequality is also applicable in this situation. Since the existence of 6 < 1 satisfying (5) implies that N has a moment generating function (Willmot and Lin, 1994) and hence finite moments of all orders, it follows that E(Xj) < CCfor j I r as well. Then Markov’s inequality (16) with q(x) = xr yields G(x)

I E(X’)/x’,

The Pareto

bound

4+(1

x 2 0.

(32) is tighter

-p,)(l

+KX)-‘IE(X~)X-‘,

which may be restated 1 -X 2

r

(34) than (34) for all x for which

1 -p. 4E(X’)

as 1’r

(35)

--*

i

Thus, (32) is tighter than (34) for x sufficiently small, or for all x 2 0 if the right-hand strictly positive. If the mean claim size is finite, then with r = 1 (32) becomes

(1 -POP(Y)

-

G(x)s

side of (35) is not

x20.

@(y)+(l_+)x

(36)

Also, with r = 1 (3.5) becomes 1 I-P, ->x - 4E(Y) since E(X)

1

1 -Po

1 E(N)

= E(N)E(Y).

E(N)=

l-4

ta,s n=O

But (Feller,

1968, p. 265) from (5)

?ao4”=(1-~o)/(l-4), n=O

with equality if p, = (1 -po)(l - 4)@-’ for n = 1,2,3,. only in this case. We also have the following result.

. . . Thus (36) is a refinement

of (34) with r = 1

Theorem 8. Suppose X has a compound geometric distribution with number of claims probabilities {p, = (1 - 4)4”, n = 0,1,2,. . . ), finite mean E(X), and non-negative claim sizes. Then Pr(X>x)

= 5 (l-+)+“F*“(x) il=l


+x},

x20.

(37)

Proof. One can always express a compound geometric df with non-negative claim amounts as a compound geometric df with positive claim amounts (e.g. Panjer and Willmot, 1992, p. 219). Thus assume without loss of generality that the claim sizes are positive. Substitute p. = 1 - $, E(N) = +(l 4)-l, and E(X) = E(N)E(Y) into (36) and rearrange. q It is interesting to note that (37) implies that B-IX> E(X)} I l/2, which demonstrates compound geometric mean is at least as large as the (smallest) median. The following example illustrates how the above ideas may be used to refine the Lundberg some situations. Example. F’(y)

Consider

the generalized

=A4yA-‘e-*y-p/y,

inverse

y > 0,

Gaussian

claim size density

that bound

the in

60

G.E. Willmot /Insurance:

where I_L> 0, p > 0, and - 03< A < 03.The The hazard rate is bounded, say by m < A > 0 one can always find K > 0 satisfying as discussed by Embrechts (1983). Thus, A < 0 and E(e@‘) < 4-‘, one has

Mathematics

(+M)-’

=

a restatement

of

4-l =

/f(K

15 (1994) 49-63

normalizing constant is M where M-l = 2(P/~)h’2K,,(2~). 1982, p. 102). If Y has density F’(y), then for (7) but if A < 0 no such K exists if E(epy) = Mj3”r( -A) < 4-l, if (7) is satisfied, the refined inequality (29) may be used. If 03 (Jorgensen,

G(x)I~-l(l-p,)(l-~m-‘)(l+~X)-re-~LX, where 0 < r < -A and

and Economics

x20

0 are chosen to satisfy

K >

dx,

+x)rx-h-r-le-Px

m(l +Ky)refiyF’(y) /0

dy.

In this case we have used c = 1 - pm - ‘, since we have chosen B(x) = (1 + KX)-re-LLX, and pB(x) = /1 + KX)-‘. Thus rpF(x) I am is satisfied with r = pm-’ and Theorem 4 applies. This technique of multiplying NWU tails is well suited for the class of distributions considered by Embrechts and Goldie (1982). q

r~(l +

5. Application

to ruin theory

Consider the classical model of ruin theory. That is, assume that IN,; t 2 O} is an ordinary Poisson process, independent of the independent and identically distributed claim amounts (W,, W,, . . .}. Suppose that Wi has mean E(W) and df H(x) = 1 -R(X) with H(O) = 0. Premiums are payable continuously at the rate c = (1 + B)E(N,)E(W)/t per unit time, where 8 > 0. Beginning with an initial reserve X, the ruin probability q(x) is defined as t+k(x)=Pr(x + ct < C~:l14$ for some t > 0). Then (e.g. Bowers et al., 1986, chapter 12) @(x) = f

f3(1+0)_“-‘F*“(x),

x 2 0,

(38)

n=l

is the df of the n-fold convolution of the distribution with df 1 - F(x) where

where 1 - F*“(x) -

F(x) = /m+dy,

(39)

x 2 0.

X

Evidently, 1 - 4(x) is a compound geometric df, and it is easy to verify that the previous results apply with c(x) = e(x), 4 = (1 + 19~’ = 1 -po; F(x) given by (39), and E(Yj) = E(Wj”>/{(j + l)E(W)}. For example, if the mean claim size E(W) and variance E(W ‘I- {E(W)}2 are finite, (37) becomes

$(x)

5

ww E( W’) + 28E( W)x ’

x

2

0.

(40)

The special form (39) allows for further insight. Define the mean residual lifetime associated with H(x) to be rH(x)

={fi(~))-~/~H(y)

X

dy,

x20.

(41)

G.E. Willmot /Insurance:

Mathematics

61

and Economics 15 (1994) 49-63

Note that the failure rate associated with F(x) is ZJ~(X) = -d/dx In F(x) = {rH(x))-l. Consequently, H(x) is ZMRL is equivalent to F(x) being DFR, and H(x) is DMRL is equivalent to F(x) being ZFR. Moreover, H(x) is NWUE is equivalent to F(x) >J?(x), x 2 0. We have the following result. Corollary 2.

If

H(x) is ZMRL, then

(44 Proof.

Since am

= (TV}-‘,

F(x)

is DFR and Theorem 3 applies.

0

Note that if p(x) = e-“/E(W), then (42) is an equality (e.g. Bowers et al., 1986, p. 354). Also, (42) is exact when x = 0. Recall that the adjustment coefficient R > 0 satisfies 1 + (1 + B)E(W)R

= imeRX dH(x)

(43)

(e.g. Bowers et al., 1986, section 12.3). We have the following result. Corollary 3.

If

H(x) is ZMRL and R > 0 satisfies (431, then

~(x)~(1+13-‘e-~“, Proof.

(44)

x20.

Note that (43) is a restatement

of

1 + /3= J”eRX dF( x)

(45)

0

where F(x) is given by (39). Since H(x) is ZMRL implies that F(x) is DFR and hence NWUC, Theorem 5 may be applied. 0 We remark that in Corollaries 2 and 3, the condition that H(x) is ZMRL may be replaced by the weaker conditions that F(x) is Nwu and NWUC, respectively. Again, (44) is exact when H(x) = e-“/E(W’ and when x = 0. Also, under the slightly stronger assumption that H(x) is DFR, Gerber (1979, p. 135) gives the bound $(x)1{1+(1+0)RE(W)}-~e-~~,

x20.

(46)

See also Ross (1974), Stoyan (1983, Section 5.3), and Taylor (1976). But 1+8rl+(l+B)RE(W)

(47) is equivalent to JteRX dF(x) 2 &‘eRx dH(x), as follows from (43) and (45). But this is clearly true (e.g. Ross, 1983, p. 252) since DFR implies NWUE, i.e. F(x) 2 H(x). Thus (44) is a refinement of (46). We also obtain a refined Lundberg inequality in the Dh4RL. (including ZFR) case. Corollary 4. Suppose R > 0 satisfies (43). Zf H(x) H(x) = 1 for x 2 x0 where x0 5 ~0, then 4(x)

~{l+Rn(~)}-‘e-~~,

x20,

is DMZU

and satisfies

H(x)

-C 1 for x
but

(48)

G. E. Willmot /Insurance:

62

Mathematics and Economics 15 (I 994) 49-63

with

v(x) = ii \

/,

X”eRY“‘(Fqt> dt

1 Y

dy}/jeR~pqr) dy},x


(49)

X2X".

0,

Proof. From (39), F(x) > 0 for x 0, and the uniform distribution Corollary G(x) Proof.

H(x)is DMRL may be replaced by the weaker condition that F(x) H(x) = e-X/Eo’r”) and when x = 0. An example of a DMRL df for apply easily is H(x) = 1 - (1 -x/x,Y for 0
If R > 0 satisfies (431, and H(x)

5.

I (1 -rR)epRX,

One has am

has mean residual lifetime rJx)

L r > 0, then

x 2 0.

= {TV}-’

(50) I Y-‘. The result

follows from (45) and Corollary

As is clear from the proof of Corollary 1, a sufficient condition for TV continuous with failure rate p,(x) which satisfies pLH(x) I r -l. Also, applies with r = E(W). However, (47) may be restated as

so that (50) with r = E(W)

that F(x)

implies

is a weaker

inequality

than (46) and hence

1.

0

2 Y is that H(x) is absolutely if H(x) is NWUE then (50)

(44) when the latter

two bounds

apply. 6. Further remarks It is worth noting Under the conditions

that Gerber’s (1994) martingale approach may be utilized to prove of Theorem 2, {S,; k = 1,2,3,. . . } is a supermartingale where

Theorem

2.

k

‘kc

n (E(~))-llN>j~ j=l

and this may be used with the stopping

time

k

klN
C~>X j=l

.

I

Furthermore, Lin (1994) has obtained related results using a generalization of Wald’s stopping time argument. This approach also yields lower bounds under certain conditions, most notably the exponential case considered in Section 3 of this paper.

References Barlow, R., and F. Proschan Winston, New York.

(1975).

Statistical Theory of Reliability and Life Testing: Probability Models. Holt,

Rinehart and

G.E. W&tot /Insurance:

Mathematics and Economics 15 (1994) 49-63

63

Bowers, N., H. Gerber, J. Hickman, D. Jones and C. Nesbitt (1986). Actuarial Mathematics. Society of Actuaries, Itasca, IL. Brown, M. (1990). Error bounds for exponential approximations of geometric convolutions. Annals of Probability 18, 1388-1402. Cao, J., and Y. Wang (1991). The NBUC and NWUC classes of life distributions. Journal of Applied Probability 28, 473-479. Embrechts, P. (1983). A property of the Generalized Inverse Gaussian distribution with some applications. Journal of Applied Probability 20, 537-544. Embrechts, P. and C. Goldie (1982). On convolution tails. Stochastic Processes and their Applications 13, 263-278. Feller, W. (1968). An Introduction to Probability Theory and its Applications, Vol. 1 (3rd ed., revised). John Wiley, New York. Galambos, J. (1988). Advanced Probability Theory. Marcel Dekker, New York. Gerber, H. (1979). An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation, University of Pennsylvania, Philadelphia, PA. Gerber, H. (1994). Martingales and tail probabilities. Astin Bulletin 24, 145-146. Gertsbakh, I. (1989). Statistical Reliability Theory. Marcel Dekker, New York. Jorgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution, Lecture Notes in Statistics 9. Springer Verlag, New York. Lawless, J. (1982). Statistical Models and Methods for Lifetime Data. John Wiley, New York. Lin, X. (1994). Tail of compound distributions and excess time. University of Toronto, Technical Report, Department of Statistics. Panjer, H., and G. Willmot (1992). Insurance Risk Models. Society of Actuaries, Itasca, IL. Ross, S. (1974) Bounds on the delay distribution in GI/G/l queues, Journal of Applied Probability 11, 417-421. Ross, S. (1983). Stochastic Processes. John Wiley, New York. Steutel, F. (1970). Preservation of Infinite Divisibility under Mixing and Related Topics. Math. Centre Tracts 33, Math. Centre, Amsterdam. Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. John Wiley, New York. Taylor, G. (1976). Use of differential and integral inequalities to bound ruin and queueing probabilities. Scandinavian Actuarial Journal, 197-208. Willmot, G. (1989). The total claims distribution under inflationary conditions. Scandinavian Actuarial Journal, 1-12. Willmot, G. and X. Lin (1994). Lundberg bounds on the tails of compound distributions. Journal of Applied Probability. To appear.