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Bounds for compound distributions based on mean residual lifetimes and equilibrium distributions
ELSEVIER
Insurance: Mathematics and Economics 21 (1997) 25-42
Bounds for compound distributions based on mean residual lifetimes and equilibrium distributions Gordon E. Willmot Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ont., Canada N2L 3GI Received March 1997; received in revised form May 1997
Abstract Equilibrium distributions arise naturally in ruin theory and may be characterized in terms of the mean residual lifetime, an analytic and statistical tool for analyzing insurance claim size distributions. Bounds for the right tail of the total claim distribution which refine and generalize previous results are constructed which depend on properties of the equilibrium distribution of the claim size distribution and the mean residual lifetime in particular. Moment based Pareto bounds are considered in some detail. Extensions are considered to higher-order equilibrium distributions. 0 1997 Elsevier Science B.V. Keywords: Lundberg bound; Failure rate; Mean residual lifetime; Ruin theory; s-Equilibrium distribution
1. Introduction Suppose that N is a counting random variable with probability P,, = Pr(N = n),
n = 0, 1,2,.
..,
mass function (1.1)
and tail probabilities cm
a, =
c
pkr
n=0,1,2,...
(1.2)
k=n+l Y2,. . .) be an independent and identically of N with common distribution function (df)
Let VI,
F(y) = 1 - F(y)
= Pr(Yi
5 y),
distributed
y L 0,
sequence of positive random variables independent
(1.3)
and mean E(Y) = &” y dF(y) < 00. Also, let F*“(y) = Pr(Y1 + Y2 + f . . + Y,, > y) be the tail of the n-fold convolution. Define the random sum X = Y1 + Y2 + . . . + Y~v(with X = 0 if N = 0). Define -d(x) = Pr(X > x) = e~,F*~(x),
x > 0.
n=l 0167-6687/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PIISO167-6687(97)00016-4
(1.4)
26
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Upper and lower bounds on G(x) have been obtained by Lin (1996) and others. See Willmot and Lin (1997) and references therein. Suppose 4 E (0, 1) and B(y) = 1 - B(y) is a df satisfying
(1.3
Then bounds for G(x) may be given in terms of B(y) as follows (Lin, 1996). The df B(y) is NWU (NBU) if x(x + y) 1 (c)B(x)B(y) for all x s 0 and y 2 0 (e.g. Barlow and Proschan, 1975). If 4 E (0, 1) satisfies %+I
5
n=0,1,2
4%3
and B(y) is an NWU df satisfying
G(x)
5
1-
PO
-
4
(1.6)
,...) (1 S), then -1
1
inf a(z,x) 05zix,F(z)>O
I
I
x L 0,
(1.7)
where
s 00
a(z, x) =
L
(z(x + y - z)l-’dF(y).
F(z)
Conversely,
(1.8)
Z
if 4 E (0, 1) satisfies n=0,1,2
an+1 ? #a,,
,...)
(1.9)
and B(y) is a NBU df satisfying (1.5), then -1
CC4
1-
> -yjy
PO
a(z,x) sup i o~zix,F(z)>O
1
9
x > 0,
(1.10)
where cr(z, x) is given by (1.8). We remark that 4 is determined by (1.6) or (1.9) and is a parameter associated with the distribution {pn; n = 0, 1, 2, . . .}, and B(y) is subsequently chosen to satisfy (1.5). A detailed discussion of 4 and its relationship to discrete reliability properties may be found in Willmot and Lin (1994). Furthermore, condition (1.6) (( 1.9)) implies that a, I (?)@a~, i.e. that N is stochastically smaller (larger) than a geometric random variable. Therefore, X is stochastically smaller (larger) than a compound geometric with severity df F(y). See, e.g., Kaas et al. (1994, p. 16). Consequently, numerical (rather than analytic) bounds may be obtained by evaluation of the corresponding compound geometric distribution, e.g. using the Panjer recursion (Panjer, 198 1). In this paper we will use these results to construct bounds on G(x) based on properties of the equilibrium df
F,(y) = 1 -‘e(y)
=
y 2
0.
(1.11)
0
Closely related to Fe(y) is the mean residual lifetime
Y > 0.
(1.12)
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Clearly r-~(0) = E(Y), and TF(Y)
=
WWdY) ‘(Y>
(1.13) .
Moreover, the failure rate associated with the df F,(y) is {TF(~)}-’ = F,‘(y)/Fk(y),
F,(Y)
=
so that (1.14)
exp
See, e.g., Gertsbakh (1989). Various reliability classifications of distributions are based on these properties. See, e.g., Pellerey (1995) and Fagiuoli and Pellerey (1993, 1994) for details. A subclass of the NWU (NBU) class is the decreasing (increasing) failure rate or DFR (IFR) class. The df F(y) is DFR (IFR) if F(x + y)/F(x) is nondecreasing (nonincreasing) in x 2 0 for fixed y > 0. A larger class than the DFR (IFR) class is the increasing (decreasing) mean residual lifetime or IMRL (DMRL) class for which TF(Y) is nondecreasing (nonincreasing). A larger class than each of the DFR (IFR), IMRL (DMRL), and NWU (NBU) classes is the new worse (better) than used in convex ordering or NWUC (NBUC) class for which Fk(x + y) > Fe(y)F(x) for all x >_0 and y 2 0. These classifications are very useful from the standpoint of measuring the thickness of the right tail (of great interest to insurers), and the first named class in each case is the heavier tailed of the two mentioned. In this paper we shall generalize and refine previous results by demonstrating that many of these hold for larger classes of distributions. In Section 2 we present some technical preliminaries which are needed subsequently for both upper and lower bounds. In Section 3 we consider upper bounds, and apply these in the case where B(y) is a (moment based) Pareto df in Section 4. Lower bounds are then considered in Section 5. Finally, in Section 6 we consider extensions to higher-order s-equilibrium distributions (see Nanda et al., (1996) and references therein).
2. Some technical preliminaries In this section we present two lemmas which are needed in Sections 3 and 5 in connection with the general upper and lower bounds. In general, we will require that the df B(y) be absolutely continuous with failure rate (2.1)
y ? 0.
cLis(y) = -dlnB(y), dy
The first lemma is an integral identity. Lemma 2.1. Suppose A(y) = 1 - A(y) is a df on (0,~) with mean mA = bM A(y) dy and equilibrium df A,(y) = 1 - x,(y) = sl A(t) dt/mA. Ifthe dfB(y) is twice diferentiable with (B(y)]-’ convex and satisjies co s 0
Iz(x
+ y))-’ dA(y)
swx+
(2.2)
< 00,
Cc
dA(y)
0
Y>
= 1 + mAwE
m
+mA
sasPgx+ ax++ Y)
0
bue(x + VII*Y)
&(Y)
dy.
(2.3)
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Proof. This identity is effectively a special case of the probabilistic Taylor series expansion of Massey and Whitt (1993) and follows by repeated integration by parts. For notational convenience, let h(y) = (B(x + y))-’ = = PB(X + y) and h”(~)lMy) = L&(X + Y) + IELB@ + u)12. Since NY) exp(.&t+YpLg(f)drl. Then h’(y)lh(y) is nondecreasing, co 05
MY)%Y)
=
m
h(y)dN)
s Y
which implies that lim y,,h(y)A(y)
= 0 since (2.2) holds. Then integration by parts yields
00
cc
h(y)
s
dA(t),
h(t)
5
s Y
d-4(y)
= h(O) + mA
h’(y)
d&(y).
s 0
0
Since (2.2) holds, h(0) = lo” h(O) dA(y) 5 lo” h(y) dA(y) c 00, which implies that lo” h’(y) dAe(y) < co. But h(y) is also convex and so h’(y) is nondecreasing. Thus 00
00 05
h’(y)-;ik(y)
=
h’(y)
d&(t)
h’(f)
I
s Y .
.
.
.
which m turn implies that llmy+,c
s
d&(t),
s Y
= 0. Integration by parts again yields
co d&(y)
= h’(O)
+
~“(Y)&(Y)
dy.
s 0
0
Since h’(y) is nondecreasing, h’(0) = joooh’(O) dAe(y) 5 lam h’(y) dAe(y) < 00, implying that lo” h”(y)&(y) dy < 00. Combining the two integration by parts results yields
The second lemma concerns integrability. Lemma 2.2. Suppose that A(y) = 1 - A(y) is a df on (0, oo), and h(y) 1 0 is difSerentiable with derivative h’(y) L 0 which satisfies Jo” h’(y)A(y) dy < cc. Then so” h(y) dA(y) cl 00 and limy,,h(y)l(y) = 0. Proof. Let t 2 0 and integrate by parts to obtain t
t h’(y)x(y)
s 0
dy + h(O)
= h(t)%t)
+
h(y)
dA(y).
s 0
As t + 00 the left-hand side approaches a finite nonnegative limit and so therefore does the right-hand side. The integrand on the right-hand side is nonnegative, so $ h(y) dA(y) is nondecreasing in t, and therefore has a limit as
29
G.E. Willmot/lnsurance: Mathematics and Economics 21 (1997) 2542 t -+ 00 which must be finite since both terms on the right-hand MY)%Y)
Also, 0 I
=
lim ,,,h(y)A(y)
h(y)
_/y”
dA(w)
I
s?p”
= 0 and the result follows.
0
w 1
h(w)dA(
side are nonnegative,
i.e., &” h(y) dA(y)
since h(y) is nondecreasing,
< co.
which implies that
3. Upper bounds The main upper bound will now be given. In full generality, the conditions of the theorem are quite complicated, but for particular choices of B(y) and F(y) they can and will be simplified. In particular, identification of a df H, (y) whose equilibrium df He,,(y) satisfies (3.1) will be done based on reliability classifications for F(y). Theorem 3.1. Suppose t#~E (0, 1) satisfies (1.6), and the MVUdfB(y) is twice differentiable with {B(y)]-’ convex and satisfies (1 S). If H, (y) = 1 - H,(y) is a dfwith mean m, = so” y d H, (y) satisfying H, (0) = 0 and whose df H,,,(y)
equilibrium He,,(y)
= 1 - H,,,(y)
E(Y)
i
-
mx
= #ox
dt/m,
F,(Y +z1
inf
F(z)
oizix.F(z)>~
y
satisjies (3.1)
1%
-
then -1
cc
5 ~1
C(x)
-
PO
+ y))-’
s 1%
4
,
d&(y)
x
(3.2)
10.
0
Proof. We will use (1.7). Thus assume that 0 5 z F x, F(z) NWU, it follows that
(F(z)&x - z)}-’ (&)I-’ M(y)
a(~, x) i
5
> 0 and consider a(z, x) in (1.8). Since B(Y)
is
W’(z)~(x - z>l-’< 00
using (1.5). Now consider the df A(y) = 1 - A(y) = 1 - -(Y + z)/F(z). Then U(y) = dF(y + z)lF(z), and changing the variable of integration from y to y+z results in cx(z, x) = ~ooo{~(x+y)l-’ dA(y). Sincea(z, x) < co, we may apply Lemma 2.1 with this choice of A(y). In the notation of Lemma 2.1, mA = so” F(y + z) dy/F(z) rF(z) using (1.12). For the associated equilibrium distribution, we have
=
using (1.1 l)-( 1.13). Therefore, from (2.3)
(lI(z,
x>
CISIlgJ
1 + rF(z)KB(x)
=
+
rF(Z)
+
(CLBb
Rx
+
+
Y)12
E;,(Y
+
z> dy.
s 0
Rx)
Y) +
Y)
(3.3)
F,(z)
From (3.1) and (1.13) we have %x(Y)
I
wm4Y m,
+z> F(z)
_
rF
k> mx
Feb’ -e(z)
+
z)
(3.4)
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Therefore,
s
m&(x+Y) +I/JcLB(x + YN2H ax + Y> 0
s
y&(x + y) +
-
mx
Rx
0
e,x
(Y)
dy
I/JB(X + YN2 I;,(Y + d + Y)
dy < cc
-F,(z)
since (3.3) holds. From Lemma 2.2 with h(y) = PB(X + y)/B(x + y) and A(y) = ??e,x(y) we conclude that + Y>)%(Y) dy < CCL From Lemma 2.2 with h(y) = (B(x + y>)-’ and A(y) = H,(y) we
Jo” (PB(X + Y)/%
conclude that J,“(B(x
s
+ y))-’ dH,(y)
< 00. Then from Lemma 2.1 we obtain
ccl
dHx(y)
0
B(x + y)
From (3.3)-(3.5)
a(z, x) ?
1 + mxm(x) + m x
Rx)
=
s
as&(x +Y) +IPB(X + Y>12-H ax+Y> 0
e,x
(Y> dy.
(3.5)
we have
1 + TF(Z)LLB(X) + m mlL~(X+Y)+b4X+YN2~ x
Rx)
J
Rx + Y)
0
(y)dy
e,x
co
s O” s
1 + md-o(x)
dHx(y>
= 1+ rF(Z)PB(X)+
Rx)
0
= PB(~>~-F(z)- mxl + Rx)
B(x + y)
-
B(x)
dHx(y)
0
B(x + y>.
Using (1.13) and (3.1) with y = 0, we obtain
IF(Z) =
-wFe(z) > m F(z)
-
X’
and so
a(z, x) > m(B(n + y))-’ dHx(y). J 0
Then (3.2) follows from (1.7).
•i
We remark that it is not difficult to replace m, by a more general function of n on the right-hand side of (3. l), but the choice m, is sufficient for our purposes. We now identify suitable choices of Hx (y) for various classes of claim size distributions. Corollary 3.1. Suppose cf~E (0, 1) sati@es convex and satis-es (1 S). Zf F(y)
(1.6), and the hWU df B(y) is twice diferentiable
is IVWUC, then
with {B(y))-’
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+ y))-t dF(y)
s 1%~ 0
,
(3.6)
x 2 0.
and (3.1) is satisfied by H,(y)
Proof. Since F(y) is NWUC F,(y + z) 1 F(z)‘e(y) applies. ? ?
= F(y),
and Theorem 3.1
We now turn to the DMRL case. Corollary 3.2. Suppose t$ E (0, 1) satis$es (1.6), and the NWU df B(y) is twice differentiable with (B(y))-’ (1 S). If F(y) is DMRL and F(x) > 0, then
convex and satis$es
-1
co
G(x)
I 1
l-PO5 -F(x) 4
J
x
dF(Y) B(Y)
x > 0.
(3.7)
’
Proof. Using (1.13), one has E(Y);,(y + z)/F(z) = rF(z)Fe(y + z)/F,(z). But (e.g. from (1.14)), F(Y) is DMRL is equivalent to Fe(y) is IFR, i.e. F&y + z>/F~(z) is nonincreasing in z. Thus, for 0 5 z I x one obtains WY)F,(Y
+ z)
F(z) Consider??,(Y)
= rF(z)
E(Y + z) E?(Y +x) 1 rF(x) Te(z) F,(x)
.
= F(x + y)/;(x).
Then m, = rF(x) and as in the proof of Theorem 3.1, one finds that??,,,(y) = Thus, (3.1) is satisfied by ??,(y) = F(x + y)/F(x) and the result follows from Theorem 3.1. 0
F&X + y)/F&).
The next corollary gives a general bound in terms of the mean residual lifetime. Corollary 3.3. Suppose 4 E (0, 1) satisjies (1.6) and the NWZJ df B(y) is twice diferentiable (1.5). IfrF(y) > r > 0, then
with (B(y)}-’
convex and satisfies
C(x) 5 !I/?
(;[,l;:,dy}-‘,
xL0.
(3.8)
Proof. From (1.13) and (1.14), for0 4 z 5 x,
E(Y)l;,(y
+ z)/F(z) = rF(z)F,(y
+ z)ll;,(z)
= rF(z) exp -
s
kAt))-t
z
Thus, (3.1) is satisfied by H,(y)
=
1 - e-J’lr, and Theorem 3.1 applies.
0
dt
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G.E. Willmot/lnsurance: Mathematics and Economics 21 (1997) 25-42
Lin (1996) used Jensen’s inequality to show that if (B(y)]-’ IS . convex (we remark that his results are slightly more general), then
l-PO4J
G(x) I
--(x
+rj,
x 2 0,
(3.9)
where TF(Y) 2 r > 0. But the df H,(y) = 1 - e-ylr is that of a random variable with mean r, and for {B(y)}-’ convex, Jensen’s inequality yields e-Yl’ 1 -loo dy > r s B(x + Y) B(x + r) ’
0
But this implies that the inequality in (3.8) is tighter than (3.9) and hence a refinement. This is particularly useful when the integral can be easily evaluated, as in the case when B(y) is an exponential tail (in Corollary 3.5) or when B(y) is a Pareto tail (in Corollary 4.2). We remark that a sufficient condition for r-F(y) > r to hold is that - ( d/dy) In F(y) 5 1/r . The new worse than used in expectation or NWUE class is obtained with r = E(Y). The special case where B(y) is an exponential tail is of special interest. That is, when K > 0 satisfies (3.10)
Corollary 3.4. Suppose 4 E (0, 1) satisjies (1.6) Andy > 0 satisfies (3.10). ZfHX(y) = 1-H,(y) is a dfwith mean m, = ir y dZf,(y) satisfying H,(O) = 0 and whose equilibrium df H,,,(y) = 1 - H,,,(y) = # H,(t) dt/m, satisfies (3. l), then (1 - p0) eeKx
(3.11)
Proof. Theorem 3.1 applies with E(X) = eeKX. 0 Willmot and Lin ( 1997) proved (3.7) for the smaller IFR class, and Lin ( 1996) proved (3.6) for the smaller NWU class. Willmot (1994) proved the special cases of Corollaries 3.1-3.3 when B(y) = e+Y. Often no K > 0 satisfies (3.10), however, and a moment based Pareto tail for B(y) is often useful. This special case is now considered.
4. Moment based Pareto bounds For very long tailed claim size distributions, there may only be a finite number of moments (e.g. Hogg and Klugman, 1984). Thus, we assume that moments exist up to order m, i.e. 00 E(Yj)
=
s
yj dF(y)
< co,
Oljlm.
(4.1)
0
In this case choose B(y) = (1 + ~y)-~, the tail of a DFR Pareto distribution. For m > 1, (B(y))-’ is convex and the results of the previous section apply. Then (1 S) becomes
=
(1 +
~y)~
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Dci
(1 +~y)~dF(y)
= ;.
(4.2)
s 0
If m is a positive integer,
K
0 may be obtained as the solution to the polynomial
>
(4.3) We now apply results of the previous section. Corollary 4.1. Suppose 4 E (0, 1) satisfies (1.6) and (4.2) holds with
C(x)
Proof.
5
COrOhy
s
0 and m > 1. If F(y) is hWUC,
then
(1 +KX + KYjm dF(y)
,
x > 0.
(4.4)
0
applies with B(y) = (1 +
3.1
>
-1
00
1 - PO 4
K
~y)-~.
Cl
In the special case when m is a positive integer, (4.4) may be expressed as
(4.5)
The incomplete gamma function (e.g. Abramowitz and Stegun, 1965, p. 260) is given for o > 0 by x
s
P(cr, x) =
p-1
,-t
r(a)
0
x 1 0.
dt,
___
(4.6)
We have the following result in terms of the mean residual lifetime. Corollary 4.2. Suppose r$ E (0, 1) satisfies (1.6), and (4.2) holds with
@(Kr)mr(m
where rF(x)
+
,
1)
1: r B 0.
Proof. Corollary 3.3 applies with B(y) = (1 +
~y)-~.
Let
03 t(x) = +
s
(1 +
KX
+
~y)~
eCyi’
dy,
0
and changing variables from y to t = (1 + s(x)
=
(Kr)m
e(‘+KX)IKrr(m
KX
+
+ 1)
Then (4.7) follows by substitution into (3.8).
0
Ky)/(Kr)
results in
K
>
x>O,
0 and m 1 1. Then (4.7)
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34
As mentioned
in Section 3, the bound (4.7) is a refinement
-d(x) I y(l+KX+KT)m,
of the bound
x 1 0,
obtained by Lin (1996) via Jensen’s inequality. Furthermore, when m is a positive integer simplification results. One has (e.g. Tijms, 1986, p. 18), P(cr, x) = 1 - eAX c,“Id xj/j! for (Y = 1,2,3, . . . Thus, form a positive integer, (4.7) may be expressed as
(4.8)
5. Lower bounds The main lower bound is a dual to Theorem 3.1.
Theorem 5.1. Suppose $J E (0, 1) @s-es and satisfies (1.5). If HI(y) = 1 - H,(y) equiEibrium df H,,, (y) = 1 - p,,,(y)
E(Y)
KdY)
> -
sup mx o~zcx,‘(z)>o
1-
PO
(1.9) and the NBU dfB(y) is twice dzsrentiable with (B(y)}-’ convex is a df with mean m, = &” y dH, (y) satisfying H, (0) = 0 and whose
= s$ ??,(t) _,(Y
+ z)
satisfies
(5.1)
y 10,
’
F(z)
dt/m,
then -I
cc
G(x)
> -
I&x
,
+ y)l-‘d&(y)
(5.2)
x ? 0.
s 0
4
Proof. The proof uses (1.10) and is similar to that of Theorem 3.1. If i,“(B(x is trivially true. Thus, we assume that ~,“(B(x Also, from (5.1), rdz) Fk(y
X,x(Y)
L -
mx
+
+ y))-’ dH,(y)
z)
+ y)}-’ dH,(y) = co then (5.2) < 00 and from Lemma 2.1 relation (3.5) holds.
(5.3)
E(z) .
Thus,
w(z)
s 05~~(X + + s O” &
Rx + Y>
0
Y)
-< mx
0
Since
+ y) + IPB(X + Y>12F,(Y + z)
1 - F&y
_&O” (PUB@ + Y)/%
IPB(X + YN2-
Rx + Y)
+ z)/Fk(z)
‘e(z)
He,x(y)dy
is the equilibrium
+ Y))(F(Y + Z)/‘(Z))
dy
dy
<
00.
df of 1 - F(y + z)/F(z), it follows from Lemma 2.2 that < 00. Again from Lemma 2.2, this means that cr(z, X) =
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3.5
z 1 < oo.Thus,fromLemma2.1,(3.3)holds.Thus,(3.3),(5.3),and(3.5)hold.The + y)l-’ dF(Y +z)/F( remainder of the proof follows by reversing the inequalities in the portion of the proof of Theorem 3.1 following (3.5). 0
/,“I%
It is worth noting that in the special case when B(y) is twice differentiable and IFR, then (B(y))-’ is convex. We have the following corollaries. Corollary 5.1. Suppose 4 E (0, 1) satisfies (1.9), andthe MU&B(y) and satisfies (1.5).
IfF (y) is NBUC, then
G(x)
z -
PO
4
convex
-1
cc
1-
is twice differentiable with {z(y))-’
{x(x
+
y))-’ dF(y)
,
(5.4)
x > 0.
s 0
Proof. Reverse the inequality in Corollary 3.1.
??
Corollary 5.2. Suppose 4 E (0, 1) satisfies (1.9) and the NBV df B(y) is twice differentiable with (??(y))-’ convex and satisfies (1.5). If F (y) is IMRL then
G(x)
l--PO1 7F(x)
(5.5)
Proof. Reverse the inequality in Corollary 3.2.
0
Corollary 5.3. Suppose 4 E (0, 1) satis$es (1.9), and the NBU df B(y) is twice d@erentiabZe with (B(y))-* and satisfies (1.5). Zfr-F(y) 5 r c 00 then
convex
(5.6)
Proof. Reverse the inequality in Corollary 3.3.
??
Simplifications result for the exponential case. Corollary 5.4. Suppose $I E (0, 1) satisfies (1.9) and K > 0 satisjes (3.10). ZfHX(y) = l-2, m, = Jo” y dH,(y) satisfying H,(O) = 0 and whose equilibrium df He,x(y) = 1 - -i?,,,(y) satisjes (5. l), then (1 - po) eSKX ‘(‘)
’ 4 lo” eKYdH, (y) ’
x >_0.
Proof. Theorem 5.1 applies with B(y) = ePKY. 0
(y) is a df with mean = ll pX(t) dt/m,
(5.7)
36
G.E. Willmot/lnsurance: Mathematics and Economics 21 (1997) 2542
Lin (1996) proved (5.4) for the smaller NBU class, as well as in the special case when B(x) = eCKX.Willmot and Lin (1997) proved (5.5) for the smaller DFR class. Lin (1996) also gave the special case of (5.6) when B(x) = eMKX and r = E(Y), i.e. F(y) is new better than used in expectation or NBUE.
6. Higher-order equilibrium distributions Recently, there has been much attention given to higher-order equilibrium distributions. See, e.g., Fagiuoli and Pellerey (1993, 1994) and Nanda et al. (1996) and references therein. Suppose that W = Wt is a nonnegative random variable with mean E( WI) and df A(w) = A 1(w) = 1 - A(w) = 1 - At (w). Then suppose that W2 has the (equilibrium) df AZ(W) = 1 - x2(~) = lowAt(f) dt/E(Wt). In general, let Wj+l have df Ai+l(w) = 1 - xj+t (w) = so” ;Ij(t) dt/E(Wj), which is referred to as the jth equilibrium df of A(w). This requires that E( Wj) -C 00, and by repeated integration by parts (e.g. Hesselager et al., 1997) xj+l(x)
I?(Y
=
-
x)’dA(y)
lo”yj dA(y)
’
(6.1)
x ’ ”
Various reliability classifications are based on properties of the higher-order equilibrium distributions. The approach employed in this paper may be extended to the present situation by employing the probabilistic Taylor series expansion of Massey and Whitt (1993). While the approach is applicable for the general case, for simplicity and due to the importance we shall consider the exponential case B(y) = e- KY.The Massey and Whitt (1993) approach results in
s 00
00
n-l
eKYdA(y) = 1 + c
sE(Wj)
j=l
0
’
eKY&, (y) dy.
+ J
(6.2)
0
The above result can easily be seen to hold if lo” eKYdA(y) < 00 by repeatedly applying the argument used in Lemma 2.1, and if lo” eK-Y&, (y) dy < 00 by repeatedly applying the argument used in Lemma 2.2. For notational convenience, define for x 1 0 and n = 0, 1,2, . . . cc m,(x)
=
s
(Y - x)” dF(y).
(6.3)
Also let Fj(y) = 1 - Fj(y) be the (j - 1)th equilibrium df of F(y) = Fl (y), as described above. We are now in position to give the main results which give upper and lower bounds on the tail of G(x) based on properties of Fn (y). We shall exclude results for IZ= 1 and 2 since these simpler situations have been considered previously and yield simpler bounds. Theorem 6.1. Suppose that C$E (0, 1) satisfies (1.6) and K > 0 satisfies (3.10). Suppose also that H,(y) is a df sati@& H,(O) = 0 with moments kj(x) = lo” yj dH,(y), j = 1,2, . . . , n - 1, where n > 3. ZJ in addition, H,(y) has (n - 1)th equilibrium dfH,,,(y) = 1 - 7f,,,(y) which satisjies (6.4) for some c,,(x) =- 0, then
G.E. Willmot/lnsurance:
Mathematics and Economics 21 (1997) 2542
-1
co
C,(X)
h-1 6)
31
s
e-KX
eKY dff, (y)
,
XLO,
(6.5)
0
where
(6.6) and Yj(x) =
(6.7)
inf mj(Z)/'(Z) Oqix.F(z)zO
Proof. Since (3.10) holds, it follows that for 0 I z 5 x, F(z) > 0, 00
cc s
i
e"("+J'-")dF(y)/F(z)= eKcx-z) s
eKY dF(y)/F(z)
5 e”‘“-“‘/{$‘(z)}
< co.
i
&t(y) < 00 where A(y) = 1 - F(y + z)/‘(z). Since A(y) has jth equilibrium df That is, (.y(z,n) = lo” eK(X+Y) Aj+, (y) = 1 - Fj+l (y + z)/Fj+l (z) as shown in the proof of Theorem 3.1, it follows from (6.2) that cc
s
my eKY
0
But, using (6.1), F,(z) = m,_l(z)/m,_1(0), = 1+
‘?I(z)
dy.
and so cc
n-l
eCKXa(z,X)
+z>
Kj
m.(Z)
C ---_I
j=* j! F(z)
Fn(Y
+
eKY 0
cl
+z1 (2)
dy.
(6.8)
From (6.4),
s
mn-l(O)
eKYHn,,(y)dy I ~
0
s Co
C-2
c,(x)
+z)
.,EdY
0
e
.
dy < cc
F??(z)
since (6.8) holds. Thus, from (6.2),
s 00
eKYdH,(y)
0
n-l
‘kj(x) j=, j!
= 1+ C
Therefore, from (6.4), (6.8), and (6.9),
+ &kn-1
Cx) 7 eKYHn,,Cy) dy. 0
(6.9)
G.E. Willmot/lnsurance:
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Mathematics and Economics 21 (1997) 2542
But from (6.4) with y = 0, cn(x> I m,-t(O)F;,(z)/‘(z)
= ma-l (z)/F(z),
and SO
00
e -KXa(Z,x) z 1 -
s
eKYOK
0
and the result follows from (1.7).
Cl
The df F(y) is n-IFR if F,(y) is IFR. We have the following result. Corollary 6.1. Suppose hat 4 E (0, 1) sutis$es (1.6), and K > 0 satisjies (3.10).
and’(x) G(x)
If F(y) is n-ZFR where n > 3
> 0, then 5
1 -PO
(6.10)
-
4 where yj (x) is given by (6.7) and n-2
mj (X)
Yj(X) - %-2(X)-
&z(x) = cc j=1
(6.11)
*
m-2(x)
i
j!
1
Proof. For notational convenience, let pj (x) be the mean residual lifetime of Fj (x). Then, using (1.13) together
with Nanda et al. (1996), one obtains Fj+l(x)
mj(O) -.-=
Pi(X) =
jmj
(0)
Fj
mjb)
6)
(6.12)
jmj-l(x)'
Then ‘n(Y
m-1
(0)
+z) F(z)
Fn(z) I;,-1 (z)
-.~.
n _l(0)FE1(zr
=m
F(z)
+d Fn(z>
R
Fn(Y
= (n - l)m,_2(0)‘n-1(Z)
-F,(z)
FoPn-l(Z) = (n -
‘n(Y
,)n21_2(Z)
{pn_l
(z))-1
+z)
-F,(z)
Fopn-l(Z)
But n-~~
+z>
’
means that I;,(y + z)/I;n(z) is nonincreasing in z, and the failure rate associated with = - d lnF,,(z)/dz. Thus, ~~-1 (z) is nonincreasing in z, and so my
w-l(O)
+ Fcz>
z)
2
(n -
R
1)Pn-l(X)Yn-2(X)
F,(x)
F,,(Z)
is
+ xl ’
Therefore, (6.4) is satisfied with 7J,,,(y) = I;,(y + x)/F,,(x) and en(x) = (n - l)p,-t(x)y”-2(x). As in the proof of Theorem 3.1, this means that H,(y) = 1 - ;(x + y>/F(x> with moments kj (x) = mj (x>/‘(x). Then c,(x)/&t(x)
= cn(x):(x)lmn-t(x)
= (n - l)p,-l(x)y~-2(x)~(x)lm,-l(x),
and using (6.12), one obtains ~en(x) kn-1 (x>
= (n -
l)Pn-l(X)
mn-1
(x)
)&2(x)77(x)
=
yf;y. n
G.E. Willmot/lnsurance:
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Mathematics and Economics 21 (1997) 2542
Also, _ mj(x)
Yn-2cd-(4
cn (X)kj (x) =
Fl--2(X)
h-l(x)
=
Yn-2CX)mjCX) %-2(X)
F(x)
which yields (6.11) upon substitution
’
into (6.6). Finally 00
0J eKYdF(x
+ y)/F(x)
eKYdF(y)/F(x)
= emKx s
s
x
0
0
and the result follows from Theorem 6.1.
As shown by Singh (1989), the 3-IFR class is equivalent to the decreasing variance of residual life or DVRL class introduced by Launer (1984). When n = 3, (6.10) reduces to -1 C(x)
1 - po
qx)
r$-(x)
rF(x)
v(x)
inf Oirx,~;(z)>O
w(z),
l
5 -
_
-+-
4J
e
--KX ,
(6.13)
x > 0,
where
r;(x) = n(x) =
x 2 0.
(6.14)
The df F(y) is n-NWUCX if I;,+t(y + z) > Fn(z)‘,+t(y), i.e. if F’(y) is NWUC (Fagiuoli 1993). This class includes the n-NWU and (n + l)-NWU classes as special cases. Corollary 6.2. Suppose that 4 E (0, 1) satis$es (1.6) and K > 0 satisfies (3.10). ZfF(y) n > 3, then C(x)
5
1-
PO
-
4
i+~n[x)+(J-$f)~}-‘e-KX,
and Pellerey,
is (n - I)-NWUCX where
x>O,
(6.15)
(
where vj (x) is given by (6.7) and n-2 tn(x) =
C
5
j=l
Proof.
' (
Yj(X) -Yn-2(X)p
Since F(y) is (n - l)-NWUCX,
m-1
(0)
F,(Y F(z)
mj (0) m,-2(0)
1
.
(6.16)
thenFn(y + z>2 F,-I (z)~~(Y),implyingthat
El(Y +d , m--1(0) m-2(z)-+z) =m _l(o)Fn_I(z) ~. Fn (Y) n F(z) Fn-l(Z) - m-2(0) F(z) 4-l (0) P ~Yn-2(xFn(Y). n
Therefore, (6.4) is satisfied with 77,,,(y) = F;,(y), i.e. H,(y) = F(y), and c,(x) = m,-l(0)~n-2(x)/m,-2(O). This implies that kj(x) = mi(O). Hence, c,(x)/k,-l(x) = Yn-2(x)/m,-2(O). By (3.10), so” e”YdH,(y) = 4-l and the result follows from Theorem 6.1. 0
G.E. Willmot/lnsurance: Mathematics and Economics 21 (1997) 25-42
40
The 2-NWUCX class includes the increasing variance of residual life or IVRL class which is equivalent to 3-DFR (e.g. Fagiuoli and Pellerey, 1993), and in this case (i.e. it = 3), (6.15) reduces to
L.$E!{~+(~)~]~l
G(x) 5
x20,
epKx,
(6.17)
where r;(x) is given by (6.14). Turning now to lower bounds, we have the following dual to Theorem 6.1. Theorem 6.2. Suppose that 4 E (0, 1) satisfies (1.9), and K > 0 satisjes (3.10). Suppose also that Hx (y) is a df satisfying H,(O) = 0 with moments kj (x) = SOW yj dH,(y), j = 1,2,3, . . . , n - 1, where n 2 3. If; in addition, has (n - 1)th equilibrium df Hn,x(y) = 1 - ??n,x(y) which sutisjies
H,(y)
mn-1
H,,,(Y)
> ~
(0)
G(X)
sup
F;,(Y
05zix,F(z)PO
+z>
F(z)
’
(6.18)
y 2 0,
for some c,,(x) > 0, then
G(x) 2 -1 -
PO
4
I 1
-1
co
_
c,(x) k,lo+w)+-
s
Gl (x>
eKYdH, (y)
b-1 (xl
eKKx, x ? 0,
(6.19)
0
where
lcrn(X)=
‘es
(Ilj(X) -
j=l
Cnb)g)3
(6.20)
.
and Vj (X) =
(6.21)
mj(Z)/F(Z). sup O~ZiX.‘(Z)>O
Proof. The proof is essentially the same as that of Theorem 6.1. As in Theorem 6.1, cr(z, x) < 00, implying that
(6.8) holds. Similarly, (6.19) is trivially true if lo” eKYdH, (y) = 00, and if lo” eKYdH, (y) K 00 then (6.9) holds. Thus, with (6.8), (6.9), and (6.18), the remainder of the proof follows that of Theorem 6.1 with the inequalities reversed. ? ? Simplifications result for many classes of distributions, including the n-DFR and (n - l)-NBUCX classes (e.g. Fagiuoli and Pellerey, 1993). Corollary 6.3. Suppose that 4 E (0, 1) sutisjes then G(x)
2 -1 - PO
is
&t(x) =
rln-2Cx) -
-F(x)
+ 1c/n(x) + s
w--2(x)
4
where vj (x)
1_
(1.9), and
K
>
0 satisfies (3.10). IfF(y)
-’ e-“, n
is n-DFR where n 2 3,
x 2 0,
(6.22)
given by (6.21) and
n25 (rlj(X) - nn-2&&]. j=l
’
(6.23)
41
G.E. Willmot/lnsurance: Mathematics and Economics 2 I (1997) 25-42
Proof. Reverse the inequalities The 3-DFR class is equivalent n = 3, (6.22) reduces to
in Corollary 6.1. to the increasing
0 variance of residual life or IVRL class (Singh, 1989), and when -1 eeKx,
(6.24)
x L 0,
where r;lc(x)
= ql(x)
The n-NBUCX
=
sup Oz~x,F(z)>O
rF(Z),
(6.25)
x 1 0.
class (i.e. Fn(y) is NBUC) contains the n-NBU and (n + I)-NBU classes.
Corollary 6.4. Suppose that 4 E (0, 1) satis$es (1.9) and K > 0 satisjies (3.10). IfF(y)
is (n - l)-NEUCX
where
n 1 3, then
G(x)
1 - PO
l++n~xj+(~)~)p'
1 -
4J
eCKx,
x>O,
(6.26)
(
where qj (x) is given by (6.21) and (6.27)
Proof. Reverse the inequalities For the 2-NBUCX results in
G(x) >
in Corollary 6.2.
class (including
0
the DVRL or equivalently
!$!2(l+(~)~)‘e~Kx,
the 3-IFR class), substitution
of n = 3 into (6.26)
(6.28)
x-0,
where r?(x) is given by (6.25). The results of Theorems 6.1 and 6.2 may be applied to obtain bounds for other higher-order reliability cations (e.g., Fagiuoli and Pellerey, 1993, 1994). The details are straightforward and are omitted here.
classifi-
Acknowledgements This work was supported by a grant from the Natural Sciences and Engineering Thanks also to Xiaodong Lin for helpful comments and suggestions.
Research Council of Canada.
References Abramowitz, M. and I. Stegun (1965). Handbook OfMathematical Functions. Dover, New York. Barlow, R. and F. Proschan (1975). Statistical Theory ofReliability and Life Testing: Probability Models. Holt, Rinehart New York.
and Winston,
G.E. Willmot/Insurance:
42
Mathematics and Economics 21 (1997) 25-42
Fagiuoli, E. and F. Pellerey (1993). New partial orderings and applications. Naval Research Logistics 40, 829-842. Fagiuoli, E. and F. Pellerey (1994). Preservation of certain classes of life distributions under Poisson shock models. Journal of Applied Probability 31,458-465.
Gertsbakh, I. (1989). Statistical Reliability Theory. Marcel Dekker, New York. Hesselager, O., S. Wang and G. Willmot (1997). Exponential and scale mixtures and equilibrium distributions. Scandinavian Actuarial Journal, to appear. Hogg, R. and S. Klugman (1984). Loss Distributions. Wiley, New York. Kaas, R., A. van Heerwaarden and M. Goovaerts (1994). Ordering of Actuarial Risks. Ceuterick, Leuven. Launer, R. (1984). Inequalities for NBUE and NWUE life distributions. Operations Research 32, 660-667. Lin, X. (1996). Tail of compound distributions and excess time. Journal ofApplied Probability 33, 184-195. Massey, W. and W. Wbitt (1993). A probabilistic generalization of Taylor’s theorem. Statistics and Probability Letters 16,5 l-54. Nanda, A., K. Jain and H. Singh (1996). Properties of moments for s-order equilibrium distributions. Journal ofApplied Probability 33, 1108-1111.
Panjer, H. (198 1). Recursive evaluation of a family of compound distributions. Astin Bulletin 12,22-26. Pellerey, F. (1995). On the preservation of some orderings of risks under convolution. Insurance: Mathematics and Economics 16,23-30. Correction ( 1996), 16,8 l-83. Singh, H. (1989). On partial ordering of life distribution. Naval Research Logistics 36, 103-l 10. Tijms, H. (1986). Stochastic Modelling and Analysis: A Computational Approach. Wiley, Chichester, UK. Willmot, G. (1994). Refinements and distributional generalizations of Lundberg’s inequality. Insurance: Mathematics and Economics 15, 49-63. Willmot, G. and X. Lin (1994). Lundberg bounds on the tails of compound distributions. Journal ofApplied Probability 3 1,743-756. Willmot, G. and X. Lin (1997). Simplified bounds on the tails of compound distributions. Journal ofApplied Probability 34, 127-133.
Insurance: Mathematics and Economics 21 (1997) 25-42
Bounds for compound distributions based on mean residual lifetimes and equilibrium distributions Gordon E. Willmot Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ont., Canada N2L 3GI Received March 1997; received in revised form May 1997
Abstract Equilibrium distributions arise naturally in ruin theory and may be characterized in terms of the mean residual lifetime, an analytic and statistical tool for analyzing insurance claim size distributions. Bounds for the right tail of the total claim distribution which refine and generalize previous results are constructed which depend on properties of the equilibrium distribution of the claim size distribution and the mean residual lifetime in particular. Moment based Pareto bounds are considered in some detail. Extensions are considered to higher-order equilibrium distributions. 0 1997 Elsevier Science B.V. Keywords: Lundberg bound; Failure rate; Mean residual lifetime; Ruin theory; s-Equilibrium distribution
1. Introduction Suppose that N is a counting random variable with probability P,, = Pr(N = n),
n = 0, 1,2,.
..,
mass function (1.1)
and tail probabilities cm
a, =
c
pkr
n=0,1,2,...
(1.2)
k=n+l Y2,. . .) be an independent and identically of N with common distribution function (df)
Let VI,
F(y) = 1 - F(y)
= Pr(Yi
5 y),
distributed
y L 0,
sequence of positive random variables independent
(1.3)
and mean E(Y) = &” y dF(y) < 00. Also, let F*“(y) = Pr(Y1 + Y2 + f . . + Y,, > y) be the tail of the n-fold convolution. Define the random sum X = Y1 + Y2 + . . . + Y~v(with X = 0 if N = 0). Define -d(x) = Pr(X > x) = e~,F*~(x),
x > 0.
n=l 0167-6687/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PIISO167-6687(97)00016-4
(1.4)
26
GE. Willmot/Insurance:
Mathematics and Economics 21 (1997) 2.5-42
Upper and lower bounds on G(x) have been obtained by Lin (1996) and others. See Willmot and Lin (1997) and references therein. Suppose 4 E (0, 1) and B(y) = 1 - B(y) is a df satisfying
(1.3
Then bounds for G(x) may be given in terms of B(y) as follows (Lin, 1996). The df B(y) is NWU (NBU) if x(x + y) 1 (c)B(x)B(y) for all x s 0 and y 2 0 (e.g. Barlow and Proschan, 1975). If 4 E (0, 1) satisfies %+I
5
n=0,1,2
4%3
and B(y) is an NWU df satisfying
G(x)
5
1-
PO
-
4
(1.6)
,...) (1 S), then -1
1
inf a(z,x) 05zix,F(z)>O
I
I
x L 0,
(1.7)
where
s 00
a(z, x) =
L
(z(x + y - z)l-’dF(y).
F(z)
Conversely,
(1.8)
Z
if 4 E (0, 1) satisfies n=0,1,2
an+1 ? #a,,
,...)
(1.9)
and B(y) is a NBU df satisfying (1.5), then -1
CC4
1-
> -yjy
PO
a(z,x) sup i o~zix,F(z)>O
1
9
x > 0,
(1.10)
where cr(z, x) is given by (1.8). We remark that 4 is determined by (1.6) or (1.9) and is a parameter associated with the distribution {pn; n = 0, 1, 2, . . .}, and B(y) is subsequently chosen to satisfy (1.5). A detailed discussion of 4 and its relationship to discrete reliability properties may be found in Willmot and Lin (1994). Furthermore, condition (1.6) (( 1.9)) implies that a, I (?)@a~, i.e. that N is stochastically smaller (larger) than a geometric random variable. Therefore, X is stochastically smaller (larger) than a compound geometric with severity df F(y). See, e.g., Kaas et al. (1994, p. 16). Consequently, numerical (rather than analytic) bounds may be obtained by evaluation of the corresponding compound geometric distribution, e.g. using the Panjer recursion (Panjer, 198 1). In this paper we will use these results to construct bounds on G(x) based on properties of the equilibrium df
F,(y) = 1 -‘e(y)
=
y 2
0.
(1.11)
0
Closely related to Fe(y) is the mean residual lifetime
Y > 0.
(1.12)
G.E. Willmot/lnsurance: Mathematics and Economics 21 (1997) 2542
21
Clearly r-~(0) = E(Y), and TF(Y)
=
WWdY) ‘(Y>
(1.13) .
Moreover, the failure rate associated with the df F,(y) is {TF(~)}-’ = F,‘(y)/Fk(y),
F,(Y)
=
so that (1.14)
exp
See, e.g., Gertsbakh (1989). Various reliability classifications of distributions are based on these properties. See, e.g., Pellerey (1995) and Fagiuoli and Pellerey (1993, 1994) for details. A subclass of the NWU (NBU) class is the decreasing (increasing) failure rate or DFR (IFR) class. The df F(y) is DFR (IFR) if F(x + y)/F(x) is nondecreasing (nonincreasing) in x 2 0 for fixed y > 0. A larger class than the DFR (IFR) class is the increasing (decreasing) mean residual lifetime or IMRL (DMRL) class for which TF(Y) is nondecreasing (nonincreasing). A larger class than each of the DFR (IFR), IMRL (DMRL), and NWU (NBU) classes is the new worse (better) than used in convex ordering or NWUC (NBUC) class for which Fk(x + y) > Fe(y)F(x) for all x >_0 and y 2 0. These classifications are very useful from the standpoint of measuring the thickness of the right tail (of great interest to insurers), and the first named class in each case is the heavier tailed of the two mentioned. In this paper we shall generalize and refine previous results by demonstrating that many of these hold for larger classes of distributions. In Section 2 we present some technical preliminaries which are needed subsequently for both upper and lower bounds. In Section 3 we consider upper bounds, and apply these in the case where B(y) is a (moment based) Pareto df in Section 4. Lower bounds are then considered in Section 5. Finally, in Section 6 we consider extensions to higher-order s-equilibrium distributions (see Nanda et al., (1996) and references therein).
2. Some technical preliminaries In this section we present two lemmas which are needed in Sections 3 and 5 in connection with the general upper and lower bounds. In general, we will require that the df B(y) be absolutely continuous with failure rate (2.1)
y ? 0.
cLis(y) = -dlnB(y), dy
The first lemma is an integral identity. Lemma 2.1. Suppose A(y) = 1 - A(y) is a df on (0,~) with mean mA = bM A(y) dy and equilibrium df A,(y) = 1 - x,(y) = sl A(t) dt/mA. Ifthe dfB(y) is twice diferentiable with (B(y)]-’ convex and satisjies co s 0
Iz(x
+ y))-’ dA(y)
swx+
(2.2)
< 00,
Cc
dA(y)
0
Y>
= 1 + mAwE
m
+mA
sasPgx+ ax++ Y)
0
bue(x + VII*Y)
&(Y)
dy.
(2.3)
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Mathematics
and Economics 21 (1997) 25-42
Proof. This identity is effectively a special case of the probabilistic Taylor series expansion of Massey and Whitt (1993) and follows by repeated integration by parts. For notational convenience, let h(y) = (B(x + y))-’ = = PB(X + y) and h”(~)lMy) = L&(X + Y) + IELB@ + u)12. Since NY) exp(.&t+YpLg(f)drl. Then h’(y)lh(y) is nondecreasing, co 05
MY)%Y)
=
m
h(y)dN)
s Y
which implies that lim y,,h(y)A(y)
= 0 since (2.2) holds. Then integration by parts yields
00
cc
h(y)
s
dA(t),
h(t)
5
s Y
d-4(y)
= h(O) + mA
h’(y)
d&(y).
s 0
0
Since (2.2) holds, h(0) = lo” h(O) dA(y) 5 lo” h(y) dA(y) c 00, which implies that lo” h’(y) dAe(y) < co. But h(y) is also convex and so h’(y) is nondecreasing. Thus 00
00 05
h’(y)-;ik(y)
=
h’(y)
d&(t)
h’(f)
I
s Y .
.
.
.
which m turn implies that llmy+,c
s
d&(t),
s Y
= 0. Integration by parts again yields
co d&(y)
= h’(O)
+
~“(Y)&(Y)
dy.
s 0
0
Since h’(y) is nondecreasing, h’(0) = joooh’(O) dAe(y) 5 lam h’(y) dAe(y) < 00, implying that lo” h”(y)&(y) dy < 00. Combining the two integration by parts results yields
The second lemma concerns integrability. Lemma 2.2. Suppose that A(y) = 1 - A(y) is a df on (0, oo), and h(y) 1 0 is difSerentiable with derivative h’(y) L 0 which satisfies Jo” h’(y)A(y) dy < cc. Then so” h(y) dA(y) cl 00 and limy,,h(y)l(y) = 0. Proof. Let t 2 0 and integrate by parts to obtain t
t h’(y)x(y)
s 0
dy + h(O)
= h(t)%t)
+
h(y)
dA(y).
s 0
As t + 00 the left-hand side approaches a finite nonnegative limit and so therefore does the right-hand side. The integrand on the right-hand side is nonnegative, so $ h(y) dA(y) is nondecreasing in t, and therefore has a limit as
29
G.E. Willmot/lnsurance: Mathematics and Economics 21 (1997) 2542 t -+ 00 which must be finite since both terms on the right-hand MY)%Y)
Also, 0 I
=
lim ,,,h(y)A(y)
h(y)
_/y”
dA(w)
I
s?p”
= 0 and the result follows.
0
w 1
h(w)dA(
side are nonnegative,
i.e., &” h(y) dA(y)
since h(y) is nondecreasing,
< co.
which implies that
3. Upper bounds The main upper bound will now be given. In full generality, the conditions of the theorem are quite complicated, but for particular choices of B(y) and F(y) they can and will be simplified. In particular, identification of a df H, (y) whose equilibrium df He,,(y) satisfies (3.1) will be done based on reliability classifications for F(y). Theorem 3.1. Suppose t#~E (0, 1) satisfies (1.6), and the MVUdfB(y) is twice differentiable with {B(y)]-’ convex and satisfies (1 S). If H, (y) = 1 - H,(y) is a dfwith mean m, = so” y d H, (y) satisfying H, (0) = 0 and whose df H,,,(y)
equilibrium He,,(y)
= 1 - H,,,(y)
E(Y)
i
-
mx
= #ox
dt/m,
F,(Y +z1
inf
F(z)
oizix.F(z)>~
y
satisjies (3.1)
1%
-
then -1
cc
5 ~1
C(x)
-
PO
+ y))-’
s 1%
4
,
d&(y)
x
(3.2)
10.
0
Proof. We will use (1.7). Thus assume that 0 5 z F x, F(z) NWU, it follows that
(F(z)&x - z)}-’ (&)I-’ M(y)
a(~, x) i
5
> 0 and consider a(z, x) in (1.8). Since B(Y)
is
W’(z)~(x - z>l-’< 00
using (1.5). Now consider the df A(y) = 1 - A(y) = 1 - -(Y + z)/F(z). Then U(y) = dF(y + z)lF(z), and changing the variable of integration from y to y+z results in cx(z, x) = ~ooo{~(x+y)l-’ dA(y). Sincea(z, x) < co, we may apply Lemma 2.1 with this choice of A(y). In the notation of Lemma 2.1, mA = so” F(y + z) dy/F(z) rF(z) using (1.12). For the associated equilibrium distribution, we have
=
using (1.1 l)-( 1.13). Therefore, from (2.3)
(lI(z,
x>
CISIlgJ
1 + rF(z)KB(x)
=
+
rF(Z)
+
(CLBb
Rx
+
+
Y)12
E;,(Y
+
z> dy.
s 0
Rx)
Y) +
Y)
(3.3)
F,(z)
From (3.1) and (1.13) we have %x(Y)
I
wm4Y m,
+z> F(z)
_
rF
k> mx
Feb’ -e(z)
+
z)
(3.4)
GE. Willmot/Insurance: Mathematics and Economics 21 (1997) 2542
30
Therefore,
s
m&(x+Y) +I/JcLB(x + YN2H ax + Y> 0
s
y&(x + y) +
-
mx
Rx
0
e,x
(Y)
dy
I/JB(X + YN2 I;,(Y + d + Y)
dy < cc
-F,(z)
since (3.3) holds. From Lemma 2.2 with h(y) = PB(X + y)/B(x + y) and A(y) = ??e,x(y) we conclude that + Y>)%(Y) dy < CCL From Lemma 2.2 with h(y) = (B(x + y>)-’ and A(y) = H,(y) we
Jo” (PB(X + Y)/%
conclude that J,“(B(x
s
+ y))-’ dH,(y)
< 00. Then from Lemma 2.1 we obtain
ccl
dHx(y)
0
B(x + y)
From (3.3)-(3.5)
a(z, x) ?
1 + mxm(x) + m x
Rx)
=
s
as&(x +Y) +IPB(X + Y>12-H ax+Y> 0
e,x
(Y> dy.
(3.5)
we have
1 + TF(Z)LLB(X) + m mlL~(X+Y)+b4X+YN2~ x
Rx)
J
Rx + Y)
0
(y)dy
e,x
co
s O” s
1 + md-o(x)
dHx(y>
= 1+ rF(Z)PB(X)+
Rx)
0
= PB(~>~-F(z)- mxl + Rx)
B(x + y)
-
B(x)
dHx(y)
0
B(x + y>.
Using (1.13) and (3.1) with y = 0, we obtain
IF(Z) =
-wFe(z) > m F(z)
-
X’
and so
a(z, x) > m(B(n + y))-’ dHx(y). J 0
Then (3.2) follows from (1.7).
•i
We remark that it is not difficult to replace m, by a more general function of n on the right-hand side of (3. l), but the choice m, is sufficient for our purposes. We now identify suitable choices of Hx (y) for various classes of claim size distributions. Corollary 3.1. Suppose cf~E (0, 1) sati@es convex and satis-es (1 S). Zf F(y)
(1.6), and the hWU df B(y) is twice diferentiable
is IVWUC, then
with {B(y))-’
G.E. Willmot/lnsurance:
Mathematics
C(x)
5 -1
4
PO
1
21 (1997)
31
2542
-1
co
-
and Economics
+ y))-t dF(y)
s 1%~ 0
,
(3.6)
x 2 0.
and (3.1) is satisfied by H,(y)
Proof. Since F(y) is NWUC F,(y + z) 1 F(z)‘e(y) applies. ? ?
= F(y),
and Theorem 3.1
We now turn to the DMRL case. Corollary 3.2. Suppose t$ E (0, 1) satis$es (1.6), and the NWU df B(y) is twice differentiable with (B(y))-’ (1 S). If F(y) is DMRL and F(x) > 0, then
convex and satis$es
-1
co
G(x)
I 1
l-PO5 -F(x) 4
J
x
dF(Y) B(Y)
x > 0.
(3.7)
’
Proof. Using (1.13), one has E(Y);,(y + z)/F(z) = rF(z)Fe(y + z)/F,(z). But (e.g. from (1.14)), F(Y) is DMRL is equivalent to Fe(y) is IFR, i.e. F&y + z>/F~(z) is nonincreasing in z. Thus, for 0 5 z I x one obtains WY)F,(Y
+ z)
F(z) Consider??,(Y)
= rF(z)
E(Y + z) E?(Y +x) 1 rF(x) Te(z) F,(x)
.
= F(x + y)/;(x).
Then m, = rF(x) and as in the proof of Theorem 3.1, one finds that??,,,(y) = Thus, (3.1) is satisfied by ??,(y) = F(x + y)/F(x) and the result follows from Theorem 3.1. 0
F&X + y)/F&).
The next corollary gives a general bound in terms of the mean residual lifetime. Corollary 3.3. Suppose 4 E (0, 1) satisjies (1.6) and the NWZJ df B(y) is twice diferentiable (1.5). IfrF(y) > r > 0, then
with (B(y)}-’
convex and satisfies
C(x) 5 !I/?
(;[,l;:,dy}-‘,
xL0.
(3.8)
Proof. From (1.13) and (1.14), for0 4 z 5 x,
E(Y)l;,(y
+ z)/F(z) = rF(z)F,(y
+ z)ll;,(z)
= rF(z) exp -
s
kAt))-t
z
Thus, (3.1) is satisfied by H,(y)
=
1 - e-J’lr, and Theorem 3.1 applies.
0
dt
32
G.E. Willmot/lnsurance: Mathematics and Economics 21 (1997) 25-42
Lin (1996) used Jensen’s inequality to show that if (B(y)]-’ IS . convex (we remark that his results are slightly more general), then
l-PO4J
G(x) I
--(x
+rj,
x 2 0,
(3.9)
where TF(Y) 2 r > 0. But the df H,(y) = 1 - e-ylr is that of a random variable with mean r, and for {B(y)}-’ convex, Jensen’s inequality yields e-Yl’ 1 -loo dy > r s B(x + Y) B(x + r) ’
0
But this implies that the inequality in (3.8) is tighter than (3.9) and hence a refinement. This is particularly useful when the integral can be easily evaluated, as in the case when B(y) is an exponential tail (in Corollary 3.5) or when B(y) is a Pareto tail (in Corollary 4.2). We remark that a sufficient condition for r-F(y) > r to hold is that - ( d/dy) In F(y) 5 1/r . The new worse than used in expectation or NWUE class is obtained with r = E(Y). The special case where B(y) is an exponential tail is of special interest. That is, when K > 0 satisfies (3.10)
Corollary 3.4. Suppose 4 E (0, 1) satisjies (1.6) Andy > 0 satisfies (3.10). ZfHX(y) = 1-H,(y) is a dfwith mean m, = ir y dZf,(y) satisfying H,(O) = 0 and whose equilibrium df H,,,(y) = 1 - H,,,(y) = # H,(t) dt/m, satisfies (3. l), then (1 - p0) eeKx
(3.11)
Proof. Theorem 3.1 applies with E(X) = eeKX. 0 Willmot and Lin ( 1997) proved (3.7) for the smaller IFR class, and Lin ( 1996) proved (3.6) for the smaller NWU class. Willmot (1994) proved the special cases of Corollaries 3.1-3.3 when B(y) = e+Y. Often no K > 0 satisfies (3.10), however, and a moment based Pareto tail for B(y) is often useful. This special case is now considered.
4. Moment based Pareto bounds For very long tailed claim size distributions, there may only be a finite number of moments (e.g. Hogg and Klugman, 1984). Thus, we assume that moments exist up to order m, i.e. 00 E(Yj)
=
s
yj dF(y)
< co,
Oljlm.
(4.1)
0
In this case choose B(y) = (1 + ~y)-~, the tail of a DFR Pareto distribution. For m > 1, (B(y))-’ is convex and the results of the previous section apply. Then (1 S) becomes
=
(1 +
~y)~
GE. Willmot/lnsurance:
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and Economics
21 (1997) 2542
33
Dci
(1 +~y)~dF(y)
= ;.
(4.2)
s 0
If m is a positive integer,
K
0 may be obtained as the solution to the polynomial
>
(4.3) We now apply results of the previous section. Corollary 4.1. Suppose 4 E (0, 1) satisfies (1.6) and (4.2) holds with
C(x)
Proof.
5
COrOhy
s
0 and m > 1. If F(y) is hWUC,
then
(1 +KX + KYjm dF(y)
,
x > 0.
(4.4)
0
applies with B(y) = (1 +
3.1
>
-1
00
1 - PO 4
K
~y)-~.
Cl
In the special case when m is a positive integer, (4.4) may be expressed as
(4.5)
The incomplete gamma function (e.g. Abramowitz and Stegun, 1965, p. 260) is given for o > 0 by x
s
P(cr, x) =
p-1
,-t
r(a)
0
x 1 0.
dt,
___
(4.6)
We have the following result in terms of the mean residual lifetime. Corollary 4.2. Suppose r$ E (0, 1) satisfies (1.6), and (4.2) holds with
@(Kr)mr(m
where rF(x)
+
,
1)
1: r B 0.
Proof. Corollary 3.3 applies with B(y) = (1 +
~y)-~.
Let
03 t(x) = +
s
(1 +
KX
+
~y)~
eCyi’
dy,
0
and changing variables from y to t = (1 + s(x)
=
(Kr)m
e(‘+KX)IKrr(m
KX
+
+ 1)
Then (4.7) follows by substitution into (3.8).
0
Ky)/(Kr)
results in
K
>
x>O,
0 and m 1 1. Then (4.7)
G.E. Willmot/lnsurance: Mathematics and Economics 21 (1997) 2.5-42
34
As mentioned
in Section 3, the bound (4.7) is a refinement
-d(x) I y(l+KX+KT)m,
of the bound
x 1 0,
obtained by Lin (1996) via Jensen’s inequality. Furthermore, when m is a positive integer simplification results. One has (e.g. Tijms, 1986, p. 18), P(cr, x) = 1 - eAX c,“Id xj/j! for (Y = 1,2,3, . . . Thus, form a positive integer, (4.7) may be expressed as
(4.8)
5. Lower bounds The main lower bound is a dual to Theorem 3.1.
Theorem 5.1. Suppose $J E (0, 1) @s-es and satisfies (1.5). If HI(y) = 1 - H,(y) equiEibrium df H,,, (y) = 1 - p,,,(y)
E(Y)
KdY)
> -
sup mx o~zcx,‘(z)>o
1-
PO
(1.9) and the NBU dfB(y) is twice dzsrentiable with (B(y)}-’ convex is a df with mean m, = &” y dH, (y) satisfying H, (0) = 0 and whose
= s$ ??,(t) _,(Y
+ z)
satisfies
(5.1)
y 10,
’
F(z)
dt/m,
then -I
cc
G(x)
> -
I&x
,
+ y)l-‘d&(y)
(5.2)
x ? 0.
s 0
4
Proof. The proof uses (1.10) and is similar to that of Theorem 3.1. If i,“(B(x is trivially true. Thus, we assume that ~,“(B(x Also, from (5.1), rdz) Fk(y
X,x(Y)
L -
mx
+
+ y))-’ dH,(y)
z)
+ y)}-’ dH,(y) = co then (5.2) < 00 and from Lemma 2.1 relation (3.5) holds.
(5.3)
E(z) .
Thus,
w(z)
s 05~~(X + + s O” &
Rx + Y>
0
Y)
-< mx
0
Since
+ y) + IPB(X + Y>12F,(Y + z)
1 - F&y
_&O” (PUB@ + Y)/%
IPB(X + YN2-
Rx + Y)
+ z)/Fk(z)
‘e(z)
He,x(y)dy
is the equilibrium
+ Y))(F(Y + Z)/‘(Z))
dy
dy
<
00.
df of 1 - F(y + z)/F(z), it follows from Lemma 2.2 that < 00. Again from Lemma 2.2, this means that cr(z, X) =
G.E. Willmot/lnsurance: Mathematics and Economics 21(1997) 2542
3.5
z 1 < oo.Thus,fromLemma2.1,(3.3)holds.Thus,(3.3),(5.3),and(3.5)hold.The + y)l-’ dF(Y +z)/F( remainder of the proof follows by reversing the inequalities in the portion of the proof of Theorem 3.1 following (3.5). 0
/,“I%
It is worth noting that in the special case when B(y) is twice differentiable and IFR, then (B(y))-’ is convex. We have the following corollaries. Corollary 5.1. Suppose 4 E (0, 1) satisfies (1.9), andthe MU&B(y) and satisfies (1.5).
IfF (y) is NBUC, then
G(x)
z -
PO
4
convex
-1
cc
1-
is twice differentiable with {z(y))-’
{x(x
+
y))-’ dF(y)
,
(5.4)
x > 0.
s 0
Proof. Reverse the inequality in Corollary 3.1.
??
Corollary 5.2. Suppose 4 E (0, 1) satisfies (1.9) and the NBV df B(y) is twice differentiable with (??(y))-’ convex and satisfies (1.5). If F (y) is IMRL then
G(x)
l--PO1 7F(x)
(5.5)
Proof. Reverse the inequality in Corollary 3.2.
0
Corollary 5.3. Suppose 4 E (0, 1) satis$es (1.9), and the NBU df B(y) is twice d@erentiabZe with (B(y))-* and satisfies (1.5). Zfr-F(y) 5 r c 00 then
convex
(5.6)
Proof. Reverse the inequality in Corollary 3.3.
??
Simplifications result for the exponential case. Corollary 5.4. Suppose $I E (0, 1) satisfies (1.9) and K > 0 satisjes (3.10). ZfHX(y) = l-2, m, = Jo” y dH,(y) satisfying H,(O) = 0 and whose equilibrium df He,x(y) = 1 - -i?,,,(y) satisjes (5. l), then (1 - po) eSKX ‘(‘)
’ 4 lo” eKYdH, (y) ’
x >_0.
Proof. Theorem 5.1 applies with B(y) = ePKY. 0
(y) is a df with mean = ll pX(t) dt/m,
(5.7)
36
G.E. Willmot/lnsurance: Mathematics and Economics 21 (1997) 2542
Lin (1996) proved (5.4) for the smaller NBU class, as well as in the special case when B(x) = eCKX.Willmot and Lin (1997) proved (5.5) for the smaller DFR class. Lin (1996) also gave the special case of (5.6) when B(x) = eMKX and r = E(Y), i.e. F(y) is new better than used in expectation or NBUE.
6. Higher-order equilibrium distributions Recently, there has been much attention given to higher-order equilibrium distributions. See, e.g., Fagiuoli and Pellerey (1993, 1994) and Nanda et al. (1996) and references therein. Suppose that W = Wt is a nonnegative random variable with mean E( WI) and df A(w) = A 1(w) = 1 - A(w) = 1 - At (w). Then suppose that W2 has the (equilibrium) df AZ(W) = 1 - x2(~) = lowAt(f) dt/E(Wt). In general, let Wj+l have df Ai+l(w) = 1 - xj+t (w) = so” ;Ij(t) dt/E(Wj), which is referred to as the jth equilibrium df of A(w). This requires that E( Wj) -C 00, and by repeated integration by parts (e.g. Hesselager et al., 1997) xj+l(x)
I?(Y
=
-
x)’dA(y)
lo”yj dA(y)
’
(6.1)
x ’ ”
Various reliability classifications are based on properties of the higher-order equilibrium distributions. The approach employed in this paper may be extended to the present situation by employing the probabilistic Taylor series expansion of Massey and Whitt (1993). While the approach is applicable for the general case, for simplicity and due to the importance we shall consider the exponential case B(y) = e- KY.The Massey and Whitt (1993) approach results in
s 00
00
n-l
eKYdA(y) = 1 + c
sE(Wj)
j=l
0
’
eKY&, (y) dy.
+ J
(6.2)
0
The above result can easily be seen to hold if lo” eKYdA(y) < 00 by repeatedly applying the argument used in Lemma 2.1, and if lo” eK-Y&, (y) dy < 00 by repeatedly applying the argument used in Lemma 2.2. For notational convenience, define for x 1 0 and n = 0, 1,2, . . . cc m,(x)
=
s
(Y - x)” dF(y).
(6.3)
Also let Fj(y) = 1 - Fj(y) be the (j - 1)th equilibrium df of F(y) = Fl (y), as described above. We are now in position to give the main results which give upper and lower bounds on the tail of G(x) based on properties of Fn (y). We shall exclude results for IZ= 1 and 2 since these simpler situations have been considered previously and yield simpler bounds. Theorem 6.1. Suppose that C$E (0, 1) satisfies (1.6) and K > 0 satisfies (3.10). Suppose also that H,(y) is a df sati@& H,(O) = 0 with moments kj(x) = lo” yj dH,(y), j = 1,2, . . . , n - 1, where n > 3. ZJ in addition, H,(y) has (n - 1)th equilibrium dfH,,,(y) = 1 - 7f,,,(y) which satisjies (6.4) for some c,,(x) =- 0, then
G.E. Willmot/lnsurance:
Mathematics and Economics 21 (1997) 2542
-1
co
C,(X)
h-1 6)
31
s
e-KX
eKY dff, (y)
,
XLO,
(6.5)
0
where
(6.6) and Yj(x) =
(6.7)
inf mj(Z)/'(Z) Oqix.F(z)zO
Proof. Since (3.10) holds, it follows that for 0 I z 5 x, F(z) > 0, 00
cc s
i
e"("+J'-")dF(y)/F(z)= eKcx-z) s
eKY dF(y)/F(z)
5 e”‘“-“‘/{$‘(z)}
< co.
i
&t(y) < 00 where A(y) = 1 - F(y + z)/‘(z). Since A(y) has jth equilibrium df That is, (.y(z,n) = lo” eK(X+Y) Aj+, (y) = 1 - Fj+l (y + z)/Fj+l (z) as shown in the proof of Theorem 3.1, it follows from (6.2) that cc
s
my eKY
0
But, using (6.1), F,(z) = m,_l(z)/m,_1(0), = 1+
‘?I(z)
dy.
and so cc
n-l
eCKXa(z,X)
+z>
Kj
m.(Z)
C ---_I
j=* j! F(z)
Fn(Y
+
eKY 0
cl
+z1 (2)
dy.
(6.8)
From (6.4),
s
mn-l(O)
eKYHn,,(y)dy I ~
0
s Co
C-2
c,(x)
+z)
.,EdY
0
e
.
dy < cc
F??(z)
since (6.8) holds. Thus, from (6.2),
s 00
eKYdH,(y)
0
n-l
‘kj(x) j=, j!
= 1+ C
Therefore, from (6.4), (6.8), and (6.9),
+ &kn-1
Cx) 7 eKYHn,,Cy) dy. 0
(6.9)
G.E. Willmot/lnsurance:
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Mathematics and Economics 21 (1997) 2542
But from (6.4) with y = 0, cn(x> I m,-t(O)F;,(z)/‘(z)
= ma-l (z)/F(z),
and SO
00
e -KXa(Z,x) z 1 -
s
eKYOK
0
and the result follows from (1.7).
Cl
The df F(y) is n-IFR if F,(y) is IFR. We have the following result. Corollary 6.1. Suppose hat 4 E (0, 1) sutis$es (1.6), and K > 0 satisjies (3.10).
and’(x) G(x)
If F(y) is n-ZFR where n > 3
> 0, then 5
1 -PO
(6.10)
-
4 where yj (x) is given by (6.7) and n-2
mj (X)
Yj(X) - %-2(X)-
&z(x) = cc j=1
(6.11)
*
m-2(x)
i
j!
1
Proof. For notational convenience, let pj (x) be the mean residual lifetime of Fj (x). Then, using (1.13) together
with Nanda et al. (1996), one obtains Fj+l(x)
mj(O) -.-=
Pi(X) =
jmj
(0)
Fj
mjb)
6)
(6.12)
jmj-l(x)'
Then ‘n(Y
m-1
(0)
+z) F(z)
Fn(z) I;,-1 (z)
-.~.
n _l(0)FE1(zr
=m
F(z)
+d Fn(z>
R
Fn(Y
= (n - l)m,_2(0)‘n-1(Z)
-F,(z)
FoPn-l(Z) = (n -
‘n(Y
,)n21_2(Z)
{pn_l
(z))-1
+z)
-F,(z)
Fopn-l(Z)
But n-~~
+z>
’
means that I;,(y + z)/I;n(z) is nonincreasing in z, and the failure rate associated with = - d lnF,,(z)/dz. Thus, ~~-1 (z) is nonincreasing in z, and so my
w-l(O)
+ Fcz>
z)
2
(n -
R
1)Pn-l(X)Yn-2(X)
F,(x)
F,,(Z)
is
+ xl ’
Therefore, (6.4) is satisfied with 7J,,,(y) = I;,(y + x)/F,,(x) and en(x) = (n - l)p,-t(x)y”-2(x). As in the proof of Theorem 3.1, this means that H,(y) = 1 - ;(x + y>/F(x> with moments kj (x) = mj (x>/‘(x). Then c,(x)/&t(x)
= cn(x):(x)lmn-t(x)
= (n - l)p,-l(x)y~-2(x)~(x)lm,-l(x),
and using (6.12), one obtains ~en(x) kn-1 (x>
= (n -
l)Pn-l(X)
mn-1
(x)
)&2(x)77(x)
=
yf;y. n
G.E. Willmot/lnsurance:
39
Mathematics and Economics 21 (1997) 2542
Also, _ mj(x)
Yn-2cd-(4
cn (X)kj (x) =
Fl--2(X)
h-l(x)
=
Yn-2CX)mjCX) %-2(X)
F(x)
which yields (6.11) upon substitution
’
into (6.6). Finally 00
0J eKYdF(x
+ y)/F(x)
eKYdF(y)/F(x)
= emKx s
s
x
0
0
and the result follows from Theorem 6.1.
As shown by Singh (1989), the 3-IFR class is equivalent to the decreasing variance of residual life or DVRL class introduced by Launer (1984). When n = 3, (6.10) reduces to -1 C(x)
1 - po
qx)
r$-(x)
rF(x)
v(x)
inf Oirx,~;(z)>O
w(z),
l
5 -
_
-+-
4J
e
--KX ,
(6.13)
x > 0,
where
r;(x) = n(x) =
x 2 0.
(6.14)
The df F(y) is n-NWUCX if I;,+t(y + z) > Fn(z)‘,+t(y), i.e. if F’(y) is NWUC (Fagiuoli 1993). This class includes the n-NWU and (n + l)-NWU classes as special cases. Corollary 6.2. Suppose that 4 E (0, 1) satis$es (1.6) and K > 0 satisfies (3.10). ZfF(y) n > 3, then C(x)
5
1-
PO
-
4
i+~n[x)+(J-$f)~}-‘e-KX,
and Pellerey,
is (n - I)-NWUCX where
x>O,
(6.15)
(
where vj (x) is given by (6.7) and n-2 tn(x) =
C
5
j=l
Proof.
' (
Yj(X) -Yn-2(X)p
Since F(y) is (n - l)-NWUCX,
m-1
(0)
F,(Y F(z)
mj (0) m,-2(0)
1
.
(6.16)
thenFn(y + z>2 F,-I (z)~~(Y),implyingthat
El(Y +d , m--1(0) m-2(z)-+z) =m _l(o)Fn_I(z) ~. Fn (Y) n F(z) Fn-l(Z) - m-2(0) F(z) 4-l (0) P ~Yn-2(xFn(Y). n
Therefore, (6.4) is satisfied with 77,,,(y) = F;,(y), i.e. H,(y) = F(y), and c,(x) = m,-l(0)~n-2(x)/m,-2(O). This implies that kj(x) = mi(O). Hence, c,(x)/k,-l(x) = Yn-2(x)/m,-2(O). By (3.10), so” e”YdH,(y) = 4-l and the result follows from Theorem 6.1. 0
G.E. Willmot/lnsurance: Mathematics and Economics 21 (1997) 25-42
40
The 2-NWUCX class includes the increasing variance of residual life or IVRL class which is equivalent to 3-DFR (e.g. Fagiuoli and Pellerey, 1993), and in this case (i.e. it = 3), (6.15) reduces to
L.$E!{~+(~)~]~l
G(x) 5
x20,
epKx,
(6.17)
where r;(x) is given by (6.14). Turning now to lower bounds, we have the following dual to Theorem 6.1. Theorem 6.2. Suppose that 4 E (0, 1) satisfies (1.9), and K > 0 satisjes (3.10). Suppose also that Hx (y) is a df satisfying H,(O) = 0 with moments kj (x) = SOW yj dH,(y), j = 1,2,3, . . . , n - 1, where n 2 3. If; in addition, has (n - 1)th equilibrium df Hn,x(y) = 1 - ??n,x(y) which sutisjies
H,(y)
mn-1
H,,,(Y)
> ~
(0)
G(X)
sup
F;,(Y
05zix,F(z)PO
+z>
F(z)
’
(6.18)
y 2 0,
for some c,,(x) > 0, then
G(x) 2 -1 -
PO
4
I 1
-1
co
_
c,(x) k,lo+w)+-
s
Gl (x>
eKYdH, (y)
b-1 (xl
eKKx, x ? 0,
(6.19)
0
where
lcrn(X)=
‘es
(Ilj(X) -
j=l
Cnb)g)3
(6.20)
.
and Vj (X) =
(6.21)
mj(Z)/F(Z). sup O~ZiX.‘(Z)>O
Proof. The proof is essentially the same as that of Theorem 6.1. As in Theorem 6.1, cr(z, x) < 00, implying that
(6.8) holds. Similarly, (6.19) is trivially true if lo” eKYdH, (y) = 00, and if lo” eKYdH, (y) K 00 then (6.9) holds. Thus, with (6.8), (6.9), and (6.18), the remainder of the proof follows that of Theorem 6.1 with the inequalities reversed. ? ? Simplifications result for many classes of distributions, including the n-DFR and (n - l)-NBUCX classes (e.g. Fagiuoli and Pellerey, 1993). Corollary 6.3. Suppose that 4 E (0, 1) sutisjes then G(x)
2 -1 - PO
is
&t(x) =
rln-2Cx) -
-F(x)
+ 1c/n(x) + s
w--2(x)
4
where vj (x)
1_
(1.9), and
K
>
0 satisfies (3.10). IfF(y)
-’ e-“, n
is n-DFR where n 2 3,
x 2 0,
(6.22)
given by (6.21) and
n25 (rlj(X) - nn-2&&]. j=l
’
(6.23)
41
G.E. Willmot/lnsurance: Mathematics and Economics 2 I (1997) 25-42
Proof. Reverse the inequalities The 3-DFR class is equivalent n = 3, (6.22) reduces to
in Corollary 6.1. to the increasing
0 variance of residual life or IVRL class (Singh, 1989), and when -1 eeKx,
(6.24)
x L 0,
where r;lc(x)
= ql(x)
The n-NBUCX
=
sup Oz~x,F(z)>O
rF(Z),
(6.25)
x 1 0.
class (i.e. Fn(y) is NBUC) contains the n-NBU and (n + I)-NBU classes.
Corollary 6.4. Suppose that 4 E (0, 1) satis$es (1.9) and K > 0 satisjies (3.10). IfF(y)
is (n - l)-NEUCX
where
n 1 3, then
G(x)
1 - PO
l++n~xj+(~)~)p'
1 -
4J
eCKx,
x>O,
(6.26)
(
where qj (x) is given by (6.21) and (6.27)
Proof. Reverse the inequalities For the 2-NBUCX results in
G(x) >
in Corollary 6.2.
class (including
0
the DVRL or equivalently
!$!2(l+(~)~)‘e~Kx,
the 3-IFR class), substitution
of n = 3 into (6.26)
(6.28)
x-0,
where r?(x) is given by (6.25). The results of Theorems 6.1 and 6.2 may be applied to obtain bounds for other higher-order reliability cations (e.g., Fagiuoli and Pellerey, 1993, 1994). The details are straightforward and are omitted here.
classifi-
Acknowledgements This work was supported by a grant from the Natural Sciences and Engineering Thanks also to Xiaodong Lin for helpful comments and suggestions.
Research Council of Canada.
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