Department of Mathematical Statistics
University of Umeå
S-901 87 Umeå
Sweden
1980-8
July 1980
SOME PROPERTIES OF THE HNBUE AND HNWUE CLASSES
OF LIFE DISTRIBUTIONS
by
Bengt Klefsjö
American Mathematical Society 1970 subject classification: 60K10, 62N05.
Key words and phrases: Life distribution, survival function, survival
probability, HNBUE, HNWUE, discrete HNBUE, discrete HNWUE, TTT-transform,
Laplace transform, danage model, coherent structure, convolution, mixture.
ABSTRACT
00
f F (x)dx< (>)
t
is studied. We prove
The HNBUE (HNWUE) class of life distributions (i.e. for which
00
< (>) y exp(-t/y)
for
t > 0, where
y = / F(x)dx)
0
that the HNBUE (HNWUE) class is larger than the NBUE (NWUE) class. We also
present some characterizations of the HNBUE (HNWUE) property by using the
Total Time on Test (TTT-) transform and the Laplace transform. Further we
examine whether the HNBUE (HNWUE) property is preserved under the reliabi­
lity operations (1) formation of coherent structure, (2) convolution and
(3) mixture. Some bounds on the moments and on the survival function of a
HNBUE (HNWUE) life distribution are also presented. The class of distribu­
tions with the discrete HNBUE (discrete HNWUE) property (i.e. for which
00
I P. < (>) y(l-l/y)k
j=k J "
is also studied.
00
for
k = 0,1,2
where
y
I p.
i=0 J
00
and
P. =
J
E p, )
k=j+l k
1.
1. Introduction
In many applications (e.g. reliabilityf maintainability, inventory theory
and biometry) different forms of aging are of interest. Therefore, during
the last decades several classes of life distributions based on notions of
aging have been studied. The most well-known of these classes are the IFR,
IFRA, NBU, NBUE and DMRL classes (with duals), the definitions of which
are given in the Appendix.
In this paper we shall mainly study a class of life distributions named
HNBUE (harmonic new better than used in expectation) and its dual class
HNWUE (W = worse).
In Section 2 we shall present the HNBUE and HNWUE classes of life distri­
butions and the corresponding discrete classes.
In Section 3 we shall give some characterizations of the HNBUE (HNWUE)
property using the Total Time on Test (TTT-)transform and the Laplace trans­
form. We shall also study a discrete counterpart to the characterization
by Laplace.trausforas.
In Section 4 we shall eiiamine whether the HNBUE (HNWUE) property is pre­
served under the reliability operations: (1) formation of coherent struc­
ture, (2) convolution and (3) mixture.
In Sections 5 and 6 we shall give some bounds on the moments and on the
survival function of a HNBUE (HNWUE) life distribution.
2. The distribution classes HNBUE and HNWUE
Let
Fj, j » 1, 2, ..., n, denote the life distributions (i.e. distribution
functions with
F.(0 ) = 0) and
J
survival functions of
n
P.sl-F., jsl, 2
J
J
independent units. If
Fg and
the corresponding
F^
denote the
This paper is partially based on previous reports by Klefsjö (1977, 1979).
2.
survival functions of their series and parallel system, respectively, then
n
F (t) = II F.(t)
j-1 J
and
F (t) =
P
H F.(t) ,
j-1 J
where
n
n
iF.(t) = i - n (i-F.(t)).
J
j-1 J
j-1
This means that the mean lifes
y
s
and
y
p
of the series structure and
the parallel structure are given by
(2.1)
y
- / F (x)dx « f { n F.(x)]dj
J
0 s
0 Lj=l J
y
= / F (x)dx = / j li F.(x)ldx .
and
00
(2.2)
P
0 P
r n
Q lj=1 J
J
The integrals in (2.1) and (2.2) are often very difficult to calculate. But
sometimes we can obtain bounds for the mean lifes
yg
and
y^
by using
the following theorem which is a consequence of Theorems 7.3 and 7.4 in
Barlow and Proschan (1975), p. 121.
THEOREM 2.1
Suppose that
tions with finite means
F. and G., j = 1, 2,
n, are survival funcj oo
oo
y. = / F.(x)dx = / G.(x)dx. If
J
0 3
0 J
OO
oo
f F. (x)dx < / G.(x)dx
t J
- t J
then
(2.3)
G.(x)jc
Vs > /
f{
i n G.(x)fdx
" o 4=i
J
J
for
t > 0
and
j = 1, 2,
n,
3.
(2.4)
y
< / < IG.(x)fdx
P - 0
lj=l
J
J
The bounds in (2.3) and (2.4) are simple to calculate if we choose
G^(x) =
00
= exp(-x/y.), x > 0, where y. = / F.(x)dx. This gives rise to the following question:
J
J 0 J
Which life distributions
00
(2.5)
F
00
/ F(x)dx < / G(x)dx
t
have the property that
for
t > 0,
"t
where
00
G(x) = exp(-x/y)
and
y = / F(x)dx ?
0
From Leosna 7#5 on p. 121 in Barlow and Proschan (1975) follows that
(2.5) is true of
F
is IFRA. By using exercise
5
on p. 187 in the
same book it follows that IFRA can be weakend to NBUE. However, as we
shall see later, there are life distributions which are not NBUE but for
which (2.5) is true. Therefore it seems natural to study the larger class
of l ife distributions for which (2.5) holds and hence we make the following
definition.
2.1
Definition of HNBUE (HNWUE) life distributions
DEFINITION 2.1
A life distribution F and its survival function F = 1 - F
oo
with finite mean y = / F(x)dx are said to be harmonic new better than used
0
•in expectation (HNBUE) if
00
(2.6)
/ F(x)dx < y exp(~t/y) for
t
If the reversed inequality is true
t > 0.
F
and
•
F
are said to be harmonic
new worse than used in expectation (HNWUE). This gives a dual class to the
HNBUE class of l ife distributions in the same way as the IFR, IFRA, NBU,
4.
NBUE and DMRL classes have duals (cf. Appendix).
The HNBUE and HNWUE classes were introduced by Rolski (1975). The reason
for the name HNBUE is the following. Suppose for simplicity that
for
t > 0
F(t) > 0
and let
00
e_(t) = f F( x)dx/F(t)
F
t
denote the mean residual life of a unit of age
t. Then the inequality (2.6)
can be written
(2.7)
—
TC
< y for
I
0
t > 0.
{e_,(x)} *dx
F
This inequality says that the integral harmonic mean value of
less than or equal to the integral harmonic mean value of
e (t) is
F
e (0) (i.e. of
a new unit). That (2.6) and (2.7) are equivalent conditions follows frcm
the fact that
/
00
v
00
^-Zn / F(x)dxJ = F(t)// F(x)dx
a.e. .
We also mention that the inequality (2.6) is a special form of the partial
order relation
<
defined in the set of li fe distributions by
00
(2.8)
00
F, <F *> S F.(x)dx < / Fz(x)dx
i - z
~ t
t i
for
t > 0.
This partial order relation was mentioned by Bessler and Veinott Jr.(1966)
and was also used by Stoyan in connection with queuing
problems (see e.g.
Stoyan (1977)). Bessler and Veinott Jr. (1966) also proved that
00
00
f F^.( x)dx < /
t
~ t
for
if and only if
00
00
/ h(x)dF.(x) < / h(x)dF?(x)
0
0
t > 0
5.
for every convex, increasing function
h
for which the integrals exist.
Accordingly
OO
00
/ h(x)dF(x) < / h(x)y *exp(-x/y)dx
0
0
(2.9)
F is HNBUE
for every increasing, convex function
h
for
which the integrals exist.
2.2
Definition of the discrete HNBUE (HNWUE) property
A survival function which is IFR (DFR) is continuous except possibly at
the right (left) hand end point of its interval of su pport (cf. Barlow
and Proschan (1975), p. 77). The corresponding is not true in the HNBUE
(HNWUE) case. There are discrete survival functions which are HNBUE
(HNWUE); for an example see p. 19 (32).
We shall now introduce another aging property which is useful when we
discuss discrete distributions.
Let
p
K
£
be a strictly positive integer valued random variable and let
= P(£>k) denote the corresponding survival probabilities. Let further
1 = QQ > Qi > §2 > ...
denote the corresponding survival probabilities
of a geometric distribution with finite mean
OO
y =
i.e.
00
E Q. = S P . »
j~o J
j-0 J
k
Qk = (1 -1/y) , k = 0, 1,2
Since the discrete counterpart to
the exponential distribution is the geometric distribution it seems na­
tural to say that a discrete distribution with survival probabilities
Q
P
>
> P'2 > ...
is discrete HNBUE if
00
00
Z P. < Z Q. for
j-k J
j=k 1
k =0, 1, 2, ... .
This leads to the following definition.
6.
DEFINITION 2.2
A discrete distribution and its survival probabilities
P^,
00
£ P, are said to be discrete
k=0
harmonic new better than used in expectation (discrete HNBUE) if
k = 0, 1, 2, ..., with finite mean
00
vk
(
IV
I P. < u(l - -j
j=k J " ^
(2.10)
u =
for k = 0, 1, 2, ....
o
If the inequality (2.10) is reversed the distribution and its survival
probabilities are said to be discrete harmonic new worse than used in ex­
pectation (disarete HNWUE).
If
P0, P^,
•••>
are t^le
survival probabilities of a discrete
distribution, the corresponding survival function
(2.11)
F(t) = P^
for
k < t < k + 1
and
F
is given by
k = 0, 1, 2, ....
For this survival function F the HNBUE definition (2.6) is equivalent to that
00
(2.12)
£ P. + (k + 1 - t)P, < y exp(-t/jj)
k "
j=k+l J
for k < t < k + 1
and
k = 0 , 1, 2, ....
There are sequences
—
00
^^^-0
are not
discrete HNBUE but for which
(2.12) holds. An example of such a sequence is given by
P0 = 1,
= I>2 = 1/4
There also are sequences
and
00
P^ = 0
for k > 3.
are
•
discrete HNWUE but for which
the corresponding survival function not is HNWUE. An example of such a se­
quence is given by
PQ - 1, P
- 37/192, P2 = 21/192, ?3 = 5/192, Pk = (l/4)k for k >4.
In fact the condition that a distribution is discrete HNBUE is stronger
than the condition that the corresponding survival function is HNBUE. In
the HNWUE case the situation is just the reverse. These facts are conse­
quences of the following theorem.
THEOREM 2.2
—
Let
00
a secluence
°f survival probabilities of a
discrete distribution with finite mean. Further let
F
be the correspond­
ing survival function given by (2.11).
(a) If
(b) If
^"S ^^screte HNBUE then
F
is HNWUE then
PROOF (a) First notice that if
F
is HNBUE.
*"s discrete HNWUE.
^^^-0 ^aS t^ie ^screte HNBUE property
(2.10) then it follows from the inequality
(2.13)
exp(-l/y) > 1 - (1/y)
that
00
(2.14)
E P. + P < y exp(-k/y) for
k "
j-k+1 3
k = 0, 1, 2,
i.e. that the inequality (2.12) is true for
t = 0, 1, 2, ...• Let
oo
g(t) = V exp(-t/y) - (k+l-t)P, E P.
*
j-k+1 J
for
k < t < k + 1
and
k = 0, 1, 2, ...
Then
g'(t) = —exp(—t/y) + P^ for
If
g'(t) + 0 in
k < t < k + 1
for
k < t < k + 1
and k=0, 1, 2, ..
k = 0, 1, 2, ..., then (2.14) is
sufficient for (2.12) to hold and we are done. Otherwise we have to prove
that
which
g(tfc) > 0, where
g'(tjç)
=
t^ = -y An Pfc
is the value of
t € ]k, k + 1[
for
0•
Since (2.10) holds the proof is complete if
y Pk - (k +
1+
( l\k+1 >
y An Pk)Pk - v( 1 --j
0
for
k = 0, 1, 2, ...
8.
But this condition can be proved by studying the function
(
1^+1
h(u,v) = uv - (k + 1 + u In v)v - ul 1 - — I
on
Π= {(u ,v): k < -u Jin v < k + 1, 0 < v < 1 , u > 1 + kv}.
(b) If
F
is HNWUE then
00
E P. + (k + 1- t)P, > y exp(-t/y) for
j=k+l J
k < t < k + 1
and
k = 0, 1, 2 , . . . .
In particular,
00
I P , > y exp(-k/y)
j-k
"
for
k = 0, 1, 2, ....
By using the inequality (2,13) it follows that
is discrete HNWUE,
•
The situation in the discrete case for the other aging properties men­
tioned in the Appendix vary. For further reference we remark that it can be
proved that
—
00
survival function
^screte
F
^
anc*
is NBUE. In the NWUE case
only if the corresponding
F
is NWUE if (^^=0
is discrete NWUE but the conditions are not equivalent.
2.3
The relations between the HNBUE (HNWUE) and NBUE (NWUE) classes
We shall now prove that the HNBUE (HNWUE) class of life distributions is
larger than the NBUE (NWUE) class and that the same is true in the discrete
case.
THEOREM 2.3
(a) A life distribution which is NBUE (NWUE) is also HNBUE (HNWUE).
(b) A discrete distribution which is discrete NBUE (discrete NWUE) is also
discrete HNBUE (discrete HNWUE).
9.
PROOF (a) If
F
is NBUE we get that
(
_d_
dt
Since
00
g(t) = e
this means that
for
.
00
et/y/ F(x)dxJ = et/y^ f F(x)dx-F(t)J < 0
t > 0.
a.e.
/
/ F(x)dx is absolutely continuous on bounded intervals
t
00
g(t) < g(0) = y for t > 0, i.e. / F(x)dx < y exp(-t/y)
t
(b) The discrete NBUE property that
00
y P^ > Ï
Z P.
P. for k = 0, 1, 2, ..
j-k 2
can be written
00
oo
1 P. <
j=k J *
y
^
I P.
j-k-! J
for
k = 1, 2, 3, ... .
From this the statement follows by using induction on k.
The proof in the HNWUE cases follows by reversing all inequalities. •
An example of a distribution which is discrete HNBUE but not discrete
NBUE is given by
PQ = 1, P
= 1/2, P2 = P3 = 1/8, Pk = 0
for k > 4 .
From Theorem 2.2 and th e comment after this theorem it follows that the
corresponding survival function is HNBUE but not NBUE.
An example of a distribution which is discrete HNWUE but not discrete
NWUE is given by
PQ - 1, Px = 37/192, P2 = 21/192, P3 = 5/192, Pfc = (l/4)k for k >4.
Furthermore
F(t) =
0.55
-t
for 0 < t < 1
for
t > 1
is an example of a s urvival function which is HNWUE but not NWUE,
10.
3. Connections between the HNBUE (HNWUE) property and some
transforms
In this section we shall give connections between the HNBUE and HNWUE
properties and the TTT-transform (defined in Section 3.1). Further we
shall give a characterization of these properties by using the Laplace
transform. In the discrete case we shall study a transform of the sur­
vival probabilities which is analogous to the L aplace transform for
general life distributions.
3.1 Characterization by using the TTT-transform
Barlow and Campo (1975) studied the scaled Total Time on Test (TTT)-
transform
H^Ct)
ipv(t) = —=
for
F
>Ç (i)
0 < t < 1,
-
where
i
F_1(t)_
H (t) =
/
F(x)dx
F
0
and
F-1(t) = inf{x: F(x) > t}.
Some of the aging properties
1.-5. in the Appendix (and their duals) can
be translated to properties of the TTT-transform (see e.g. Barlow (1979) and
Bergman (1979)). For instance
concave (convex) and F
F
is IFR (DFR) if and only if
<Pp(t)
is
is NBUE if and only if ^Pp(t) > t, 0 < t < 1 .
11,
Using the fact that
F
H ''"(l) = / F(x)dx = y
F
0
is a life distribution which is HNBUE then
(3.1)
<PF(fc) Ì.
1
~ exp(-F
1(t)/y)
for
it can be proved that if
0 < t < 1.
00
Furthermore, if the inequality (3.1) holds then / F(s)ds < y exp(-x/y)
for every
X
-1
.
x G {F (t): 0 < t < l}. Particularly, a strictly increasing
life distribution
F
is HNBUE if and only if the condition (3.1) holds.
The situation in the HNWUE case is analogous. However there are
life
distributions which are not HNBUE but for which (3.1) is true. An example
of such a life life distribution is
1
F(x) = - 1/4
0
for
0 < x < 1
for
1 < x < 5
for
x > 5.
The condition (3.1) is weaker than the NBUE condition that
tp„(t) >
t,
—
r
0 < t < 1. Figure 3.1 on p. 12 shows four TTT-transforms (Jh corresponding to
piecewise exponential distributions
but not NBUE.
Fj (see Table 3,1) which are HNBUE
Figure 3.1
The ITr-transform of some piecewise
exponential distributions which are HNBUE but
not NBUE.
TABLE 3.1
The life distributions
F.
J
whose TTT-transforms are illustrated
in Figure 3.1.
e
Fj(x) =
alX
0 < x < A^
cle a2X
< X < A2
3. X
c^e 30
.
. «
A2 < X < A^
a x
c^e~ /
4
» ^
A^
< X
j = 1, 2, 3, 4
0.143
3.600 0.175
3.400
3.460
0.455
96.819
0.359
0.592
1.662
0.600
4.000 0.257
4.200 7.547
0.369
2103.65
0.594
0.806
2.193
0.364
4.033
3.100 9.506
0.228
262.68
0.614
0.964
2.398
0.867
4.000 0.278
5.000 44.501 0.228
181966
1.211
1.423
2.883
0.160
13.
3.2 Characterizations by using the Laplace-transform
Let
(3.2)
oo
$(s) = / e
SXdF(x)
for
s > 0
0
denote the Laplace transform of
F. Further let
/
00
V
lr
SXF(x)dx
(3.3)
for
k = 0, 1,2,...,
ds
and
k+1
a^+^(s) = s
^ar
k ® 0, 1, 2,
(3.4)
aQ(s) = 1 .
Block and Savits (1977) proved that a life distribution
NBU, NBUE, or DMRL (or dual) if and only if
ding discrete property for every
F
^as
is IFR, IFRA,
t*ie corresPon~
s > 0.
The following theorem states that the same is true in the HNBUE and HNWUE
cases.
THEOREM 3.1
00
quence
PROOF
A life distribution
discrete
F
is HNBUE (HNWUE) if and only if the se(discrete HNWUE).
It is sufficient to prove the theorem in the HNBUE case. The HNWUE
case then follows by reversing all inequalities.
The "only if11 part in the HNBUE case was proved by Klefsjö (1977).
Now, take
t > 0
where
s = n/t. From (3.3) and (3.4) it follows that
oo
- Z OL (s) = / G (x) F(x)dx
S k=n+l
0 n
00
(3.5)
and let
14.
This means that
G
is a 0
gamma distribution function with the characteris-
n
tic function
K (u).
(_0°.(x - i-t)n
\s - lu/
\
n /
Since
lim h (u) = eltU ,
n-x» n
converges weakly to the one point distribution
0 x < t
G(x) =
1
x > t.
Accordingly,
00
00
lim / G (x) F(x)dx = / G(x) F(x)dx = / F(x)dx
n^O n
O
t
Fran this result together with (3.5) we now obtain that
(3.6)
<\(s)
lim —
E
rr*» S k=n+l
00
Since
=
^ F(x)dx .
t
*-s discrete HNBUE we get that
(
1 ^
0-1) 1 - TT7TT) »
k-nn °'k(s)" <a<s)" U V1 " 5ÜT;
where
00
(s) =
a
00
E cl (s) =
k=0
1
+ s / F(x)dx =
0
1
+ sy
This gives that
'•7) 7k^(,): 1K- rbj * K1 - TTsf •
(3.
Frcm (3.6), (3.7) and t he fact that
lim u{ 1
ir**> \
j
n+ —7
= y exp(-t/y)
15.
we now have
OO
/ F(x)dx < y exp(-t/y).
t
Let
°
Çj, j = 1, 2, ..., k, be independent randan variables which are
exponentially distributed with
P(£. >x) = exp(-sx), x > 0. Further let g,
^
k
denote the gamma density function of
E Ç, . Then ou (s) can be written
j-i J
^
as
oo
OL (s) = / gv(x) F(x)dx for
0
k = 1, 2, 3, ... .
Now suppose that a device is subjected to some kind of shocks and inter­
pret
as the random damage caused by shock number
j. Further suppose
that the damages accumulate additively and that the device fails when the
accumulated damage exceeds a random threshold with distribution function
Then
k
F.
a (s) is the probability that the device will survive the first
K
shocks. This damage model has been further studied by Esary, Marshall and
Proschan (1973) and Kiefsjö (1977, 198Q1.
3.3 Characterizations by using the generating function
oo
,
a discrete distribution with survival proNow suppose that (P^k-l
OO
00
babilities P, =
E p., k = 0, 1, 2, ... , and finite mean y = S P,.
j-k+1 2
k=0
A discrete counterpart to the Laplace transform for general survival func­
tions is the generating function
0°
k(z) =
If
Ç
(3.8)
OO
I p.z^ = 1 - E z2(1 —z)P. .
J
j=l J
j=0
is a random variable which has a geometric distribution such that
P(€ = j) = z-'(l-z) for
we can write
k(z)
k(z) = 1 —
as
EP(Ç-j)p.
J
j=0
j =0, 1, 2, ...
16.
E, , v = 1, 2, 3, ..., k, are independent and have the distribution (3.8)
V k
then
E E
has a negative binomial distribution, i.e.
If
=(
j )z^(1-z)k
for
j " o, 1, 2, ...
(see Johnson and Kotz (1969), p. 124). This means that a discrete counter­
part to
*-n (3.4) is (with 1-z changed to p and q = 1-p)
00
/k+j-l\
'^<p) * ^0 v
(3.9)
j )pq' 5j
for k = 1, 2, 3,
Q0(p) = 1 •
—
It can be proved that the sequence
p, 0 < p < 1. Therefore we can interpret
oo
^*=0
Q^(p)
decreasing f°r every
as
survival probabilities.
This can also be seen in the following way.
Suppose that a device is subjected to two different types of shocks,
and
B
say. At every discrete time point a shock of type
probability p
a nd a shock of type
B
p(5
v
Further
k
£ £
= j) = pq^
occurs with
occurs with probability q = l - p . If
denotes the number of B-shocks between the A-shocks number
V = 1, 2, 3, ..., then
A
V - 1
and
v,
has the geometric distribution
for
j = 0, 1, 2, ....
represents the number of B-s hocks before A-shock number
v=l
A
k
and
k
/k+j-l\ k
\ ~
1^-0-(7V<
x
If the device has the probability
then
until
for
j
=o,
1, 2, ....
of surviving the first
j
B-shocks
Qk(p) in (3.9) denotes the probability that the device will survive
k
shocks of t ype
A
have occurred.
—
°°
3 )°°
We shall now prove that if (P,),
n
k k=0
has one of the discrete aging proper—
0
0
ties discrete IFR, discrete IFRA etc. then (Q^(p))^=^
for every
p
with
0 < p < 1.
has the same property
17.
THEOREM 3.2
If
discrete IFR (discrete DFR), discrete IFRA
(discrete DFRA), discrete NBU (discrete NWU), discrete NBUE (discrete NWUE),
discrete DMRL (discrete IMRL) or discrete HNBUE (discrete HNWUE) then (Q-. (p) K
K. vJ
defined by (3.9) has the same property for
0 < p < 1.
The proof can be done by using the variation diminishing property of
/k+j-l\ k .
the totally positive kernel K(k,j) = I . I p qJ in k = 1, 2, 3, ...
and
j = 0, 1, 2, ... in a way analogous to that in the proof of Theorem
3.1 in Esary, Marshall and Proschan (1973). For an outline of t otally posi­
tive kernels and the variation diminishing property see Barlow and Proschan
(1975), pp. 92-93. However, we shall give a shorter proof based on the prop00
erties of t he sequence
PROOF OF THEOREM 3.2
Suppose that
defined by (3.4).
r\
is a random variable the distribu­
tion of which is a mixture of Poisson distributions, such that the expected
values
0
of the Poisson distributions vary according to a gamma distribu­
tion with probability density function
r (k) j xk 1exp(-xp/(1 - p)) for x > 0.
gp(x) =
Then
00
j
/k+j-lv . .
P( n = j) = / g (x) eXjrdx = r . Ì p qJ
Q P
J*
\ J /
(cf. Johnson and Kotz (1969), p. 125). This means that, for k > 1 , Q^(p)
can be written as
00
(3.10)
Q,(p) = / g (x) H(x)dx ,
K
0 P
where
00
(3.11)
H(x) =
I e"X
j-0
j
P. .
J
A comparison with (3.3) and (3.4) shows that
same form as
ot^(s)
in (3.4) but with
Q^(p)
^-n (3.10) is of the
s = p/(l-p). From Theorem 1.1 in
—
Block and Savi ts (1977) and Theorem 3.1 above it follows that
00
is discrete IFR (discrete DFR), discrete IFRA (discrete DFRA) etc. if and
only if
H
in (3.11) has the corresponding (nondiscrete) property. But
Theorems 3.1 and 3.2 in Esary, Marshall and Proschan (1973) and Theorem 3.1 in
Kiefsjö (1980) give that
—
H
—
00
is IFR (DFR) IFRA (DFRA), etc. if (Pk)k=0
has the corresponding discrete property. This gives the theorem.
•
We also mention that the conditions in Theorem 3.2 are not necessary
conditions. This follows from the proof above and the fact that
may be IFR (DFR), IFRA (DFRA) etc. although
—
00
does
not
H
in (3.11)
have the
corresponding discrete property (see Esary, Marshall and Proschan (1973),
p. 634).
4.
Preservation of the HNBUE and HNWUE properties under some
rei iability operations
It is known inmost cases whether the distribution classes IFR, IFRA, NBU,
NBUE, and DMRL (with duals) are closed or not under the reliability opera­
tions: (1) formation of coherent structure, (2) convolutions and (3) mixtures
(see Haines (1973) and Barlow and Proschan (1975), pp. 104 and 187). We here
remind of the fact that a structure is coherent if its structure function is
increasing in each argument and each of its components is relevant to the
structure (see Barlow and Proschan (1975), Chapter 1). Simple examples of
coherent structures are seri.es systems, ^ad parallel systems of independent
components.
In this section we will examine whether the HNBUE and HNWUE classes are
closed under the reliability operations mentioned above. The results, which
19.
are the same for the discrete classes too, are summarized in Table 4.1.
TABLE 4.1
A system of independent HNBUE (HNWUE) components is HNBUE (HNWUE)
according to the table.
Formation of c oherent
structures
Convolution of Distri­
butions
Mixture of Dis­
tributions
HNBUE
Not preserved
Preserved
Not preserved
HNWUE
Not preserved
Not preserved
Preserved
Note in particular that a mixture of HNWUE life distributions is HNWUE.
This is interesting since it is (as far as we know) still an open question
whether the NWUE class has this property or not.
We shall now motivate our statements and begin with the HNBUE and HNWUE
classes of life distributions. The discrete properties will be discussed in
Section 4.4.
4.1
Coherent structures
The negative result in the HNBUE case is a consequence of the following
example. Let
F
F(t) =
be the survival function
1
0 < t < 1
1/2
1 < t < 5
0
Then
F
t > 5
is HNBUE. But a series system of tw o independent components with
this survival function has a life distribution which is not HNBUE.
For the HNWUE case we get the negative answer by studying a parallel
structure with two independent components whose life distribution are ex­
ponential with different means. Then the survival function of t he structure
is IFRAandnot exponential (see Barlow and Proschan (1975), p. 83) and there­
fore HNBUE (cf. Appendix) and not HNWUE.
20.
4.2
Convolutions
First, we shall prove that if
and
ables whose life distributions
a distribution function
Let
and
F = F^* F2
E^) = Pj, E(£2) = y2
G2(t) = exp(-t/y2), t > 0
and
£2
are
F2
indepedent random vari­
are HNBUE then
^as
which is also HNBUE.
and l et further
G^t) = expC-t/Uj), t > 0,
G = G^* G2- Then we have to show that
OO
/ F(x)dx < (y1 + y2> exp(-t/(y^+y2)) for
t
t > 0.
From Theorem 4.2 in Marshall and Proschan (1970) it follows that the in­
equalities
OO
OO
/ F.(x)dx < / G(x)dx
t J
t
for
j = 1, 2,
and
t > 0
imply that
oo
oo
/ F(x)dx < / G(x)dx .
t
t
Therefore, it is sufficient to prove that
00
(4.1)
/ G(x)dx < (y +y2) exp(-t/(y1 + y2)) for
t
t > 0,
i.e. that the convolution of two independent exponential distributions is
HNBUE. But this follows from straightforward calculations.
If
and
^2
are
independent and HNWUE then
sarily a HNWUE distribution. If, for example,
^as
not
neces­
and ^ both are ex­
ponentially distributed with expectation
1
then
distribution with the density
t,
t > 0. This distribution has
f(t) = t e
^ ^as
a
gamma
an increasing (non-constant) failure rate and is therefore HNBUE and not
HNWUE.
21.
4.3
Mixtures
If
is
a
family of l ife distributions, where a
tribution function
H(a), the mixture
F
of
is randan with dis­
F^ according to
E
is de­
fined by
(4.2)
F(t) = / Fa(t)dH(a).
Now suppose that every
is an exponential distribution and there­
fore DFR. A mixture of distributions, all of which are DFR, is itself DFR
(see Theorem 4.7 in Barlow and Proschan (1975)). Therefore
F
given by
(4.2) is DFR and also HNWUE. From this it follows that the mixture of HNBU E
distributions
F^
is not necessarily HNBUE. However, if every
the same mean, it is easily seen that the mixture
On the other hand, the mixture F
F
F^
has
is HNBUE.
is HNWUE if every
F^
is HNWUE. To
prove this we have to show that
CO
(4.3)
/ F(x)dx > y exp(-t/y) for
t
t > 0
if
00
(4.4)
/ Fa(x)dx > ya exp(-t/ya) for
t > 0,
where
00
00
y.. = / F (x)dx
a
0
and
y = / F(x)dx = / y dH(a).
a
0
Fran (4.2) and (4.4) we have
/ F(x)dx = /(f F^(x:}.dH(a)i dx = f(f F^(x)dx\lH(a) > / y^exp(-t/y )dH(a).
t
t^
'
x
'
But the function
¥(y) = y exp(-t/y)
according to Jensen's inequality,
is convex for
y > 0 . Therefore,
22.
/ T(ya)dH(a) > ¥(/ i^dHCcx)) = ^(y)
(cf. Rudin (1970), Theorem 3.3) and the proof is complete.
4.4
The discrete properties
The arguments for the classes of distributions which are discrete HNBUE and discrete
HNWUE are analogous to those just given but with the exponential distribution
changed to the geometric distribution. Therefore we only give some comments.
That the discrete HNBUE class not is closed under formation of coherent
structures can be seen by studying
and
P^ = 0 for
—2 00
(P^k-o
^"s
not
, where
Pq = 1, P^, = P
= 1/2
k = 3, 4, 5, .
... This sequence is discrete HNBUE but
discrete HNBUE.
That the convolution of two discrete HNBUE distributions is discrete
HNBUE can be proved in the same manner as for general life distributions.
Using the same method as in Theorem 4.3 by Marshall and Proschan (1970) we
realize that it is sufficient to prove the closure for the geometric dis­
tribution. But if
j = 1, 2, are independent and such that
P(Ç. - k) = Pj^j"1»
k = 1, 2, 3
then it can be proved that the sum
(4.5)
To prove that
^ ^as the survival probabilities
for
—
00
k = 0, 1, 2, .
in (4.5) is discrete HNBUE we have to show
that
for
where
k = 0, 1, 2, ..
23.
•
»
This can be done rather straightforward by using that
is a convex function of
x
for
k.
¥(x) = x , x > 0,
k > 1.
That the class of distributions which is discrete HNWUE not is closed
under convolutions follows from the fact that the negative binomial dis­
tribution has the discrete IFR property with a non-constant discrete fail­
k = 0, l, 2,
ure rate
, and therefore is discrete HNBUE.
A mixture of distributions, all of whic h are discrete HNWUE, is dis­
crete HNWUE. This follows in analogy to the proof in Section 4.3 by using
that
5.
iKx) * x{x- l)/x}k, x > 1 , is a convex function of
x
for
k > 1 .
Moment inequalities
In this section we shall give sane bounds on the moments of a HNBUE (HNWUE)
life distribution. We shall also give the corresponding bounds in the dis­
crete case.
oo
THEOREM 5.1
Ar =
Let
^r/r(r+1)
F
be a life distribution and let
for
r > 0. If
À
< (>)
for
À
> (<) X*
r - 1
for
F
y
= / xrdF(x)
r
0
is HNBUE (HNWUE) then
and
r > 1
<
0 < r < 1.
The bounds are sharp.
PROOF
We prove the theorem in the HNBUE case. The other part follows in
the same way. That the bounds are sharp is seen by studying the exponential
distribution.
The case
r > 1. Since
h(x) = xr
is increasing and convex for
r > 1
it
24.
follows fron (2.9) that
00
M
r
= /
0
00
xrdF(x)
< /
"0
00
xry1
*exp(-x/y,)dx = yì" / trexp(-t)dt =
0
= y^r(r + 1) = xjr(r+ 1).
The case
0 < r < 1.
Since
00
y
r
= r / xr_1 F(x)dx
0
and
00
r / xr 1exp(-x/y1)dx = r yÏT(r) = XÌT(r+l)
0
it is sufficient to prove that
00
00
/ xr ^ F(x) dx > / xr ^exp(-x/y1)dx .
0
"0
But if
F
is HNBUE then
H'(t) « / |i'(x)-exp(-x/y1)|dx > 0
for every
t > 0 . Thus
/ I (x)df(x) > 0
0
where
It(x)
g(x) = x
I""** 1
is the characteristic function of the set
[0,t]. Since
, 0 < r < 1, is decreasing and therefore can be approximated
from below by functions of the form
k
E c.I (x) ,
j=l
j
where
c^ > 0
theorem,
the result follows from the Lebesgue monotone convergence
c
25.
Frcm Theorem 5.1 it follows in particular that
2
y2 1
2y^
if
F
is
HNBUE (HNWUE). Thus for a life distribution which is HNBUE (HNWUE) the co­
efficient of variation
cr/y^ , where
never larger (analler) than
cr
is the standard deviation, is
1.
The same bounds on the moments as in Theorem
5.1 are given by Barlow
and Proschan (1975) in the stronger IFRA (DFRA) case. Moment inequalities
in the NBU (NWU) and NBUE (NWUE) cases are given by Marshall and Proschan
(1972) as
À
< (>) À À
A
r+s
- r s
in the NBU (NWU) case and
X
< (>) X X,
r+1 - r 1
in the NBUE (NWUE) case.
We also remark that if
This is obvious for
is HNBUE then
0 < r < 1
5.1 and the fact that
definition that
F
y^ <
y^ <
00 .
00.
and for
yr <
r > 1
00
for every
r > 0.
it follows from Theorem
In the HNWUE case we know as a part of the
Therefore
yr <
00
for 0<r<l
according to
Theorem 5.1. But we have no guarantee for the finiteness of
yf
for
r > 1.
This is illustrated e.g. by the Pareto distribution with probability den-2-f
sity
f(t) = (l + e)(l+t)
•
Since
, t > 0, with
e > 0
small.
IT
•
y^r (r+l) is the r-th mcment of the exponential distribution
with survival function
G(t) = exp(-t/y^), t > 0, Theorem 5.1 gives compari­
sons between the moments of
F
and
G. We shall now prove that the corre­
sponding comparisons can be done between a distribution which is discrete
HNBUE (discrete HNWUE) and the geometric distribution with the same mean.
The proof is based on the following lemma.
LEMMA 5.2
— o°
Let (P^^g
anc*
—
oo
se<luences
survival probabili­
ties of t wo discrete distributions with the same finite mean
y =
oo
oo
E P. =
E Q..
26,
(a) If
0 < cn < c, < c0 < ...
0 1 2 *
and
E P. < (>) E Q,
k=j
k=j
for
E c.P. < (>) E c.Q.
j=\) J J
j=v -1 ^
for
v = 0, 1, 2, ....
j = 1,2,3,...
then
(b) If
... c„ > c, > c > 0
2 1 0 -
oo
E P
k=j
and
oo
< (>) E Q.
k=j
for
j = 1, 2,3,
then
00
00
E c.P. > (<)
j=0 J j PROOF
E c.Q. .
JJ
j=0
The inequalities follow frcm the fact that
00
r OO
E c.P. = E c A E P, - E 5
P. \>.
=
E (c. - c. ,) Ï P, + c E P.
j—v
j=V Mc=j
k=j+l
j=v+l -1
k=j
k=v
THEOREM 5.3
E p., k = 0, 1, 2, ..., are the survival
00
j-k+1 J
r
probabilities of a discrete distribution with moments y = E j p. . Further
let
Suppose that
P,, =
j"ï .
k
= (1 - 1/y^) , k = 0, 1, 2, ... , be the survival probabilities of a
geometric distribution with moments
If
—
OO
^^^=0
^"s discrete HNBUE
(discrete HNWUE) then
ßr
for
r
>
y > (<) ß
r - r
for
0 < r < 1
1
The bounds are sharp.
PROOF
The inequalities follow immediately by using Lemma 5.2 and the
fact that
oo
U
-
00
r
ì
E jrp = E (j +l)r-jr[ P.
J
J
J
j=l
j=Ol
That the bounds are sharp follows by studying the geometric distribution.
•
27.
From Theorem 5.3 it follows in particular that
P2 < (>)
if
(P )
discrete HNBUE (discrete HNWUE). Accordingly, the coeffi­
cient of variation
6,
- u1
a/y^ < (>) /I - 1/y^
in this case.
Bounds on the survival function
In this section we shall develop bounds on the survival function
F
F
when
is a life distribution which is HNBUE or HNWUE. Such bounds are useful
e.g. in reliability since in a typical situation the only fact known a
priori may be that the time to failure is HNBUE (HNWUE) with expectation
y . First we shall look for bounds in this situation. Then we further
assume that
F(ta|6) is known for some
t^ > 0. From a practical point of
view this assumption may not be too restrictive.
Bounds of similar character in the IFR (DFR), IFRA (DFRA), NBU (NWU),
NBUE (NWTJE) and DMRL (IMRL) cases were given by e.g. Barlow and Marshall
(1964 a, b), Marshall and Proschan (1972), Haines and Singpurwalla (1974)
and Barlow and Proschan (1975). Some of these bounds will be presented
later on.
6.1
Bounds on
F
in the HNWUE case
Our first theorem gives an upper bound on the survival function
F
in the
HNWUE case.
THEOREM 6.1
mean
(6.1)
Suppose that
F
is a life distribution which is HNWUE with
y . Then
F(t) <
(l-exp(-t/y))
for
t > 0 .
28.
REMARK
The value of a bound, such as that in (6.1), at t = 0 is to
be interpreted as the limit when t -* 0+ .
PROOF
The HNWUE definition gives that
t
/ F(x)dx< y(l-exp(-t/y))
0
Since
F
for
t > 0 .
is decreasing
t
f F(x)dx > t F(t)
0
and the theorem follows.
D
Since the NWUE class of life distributions is the largest of the DFR,
DFRA, NWU, NWUE and IMRL classes, all of which are contained in the HNWUE
class (cf. Appendix), it may be of interest to compare our HNWUE bound to
the NWUE bound
(6.2)
F(t) < rr-7-r
~ M
t
for
t > 0
given by Haines and Singpurwalla (1974)• The two bounds are illustrated in
Figure 6,1 on p. 29.
The upper bound in Theorem 6.1 is sharp in the sense that given
and
e > 0
there exists a life distribution
K
with mean
y
t, y
which is
HNWUE and for which
K(t) > ^(l-exp(-t/y)) - e .
For instance, let
F (x) s
-(l-exp(-s/y))
s
for
0 < x < s
exp(-x/y)
for
x > s .
Then it can be proved that
mean
y • Since
Fg is a survival function which is HNWUE with
y(l-exp(-s/y))/s
follows that we can choose
K
as
is a decreasing function of
Ft+fi
s
for sufficiently small
it
6 > 0.
29.
1.0
HNWUE upper bound
cO
>
P
co
0.4
rJO
ed
rO
NWUE upper bound
O
u
ÇU
0.2
J-.u
Time/Mean
Z..U
j.u
4
Figure 6.1 The HNWUE upper bound in (6.1) and the NWUE upper
bound in (6.2)
The only lower bound on
F
we can obtain when
F
is HNWUE is
F(t) > 0. This can be seen in the following way. Let, for 0 < e < 1,
—
Ir+1
F(x) = e
where
b
for
kb < X < (k+l)b
is chosen so that the mean of
F
and
k - 0,1,2,...,
is equal to
y. Then
F
is NWU (see Marshall and Proschan (1972)) and hence HNWUE.
Now suppose that besides
To get upper bounds on
F
y
we also know
F(talc)
for some
t^ > 0.
in this situation (and also for further use)
we need the following lemma, the proof of which is evident.
00
LEMMA 6.2
If
F
is a life distribution with
y
r
for some
r > 1 , then
00
t2
(6.3)
for
r
r-1 ""
= r J x
F(x)dx <
0
t[ F(t1) + (tj-t') F(t2) < r / xr_1 F(x)dx < tj + (t*- t*) F(tx>
0
<
< tj.
We can now prove the following theorem.
THEOREM 6.3
mean
Suppose that
F
is a life distribution which is HNWUE with
y . Then
y(l-exp(-t/y))/t
(6.4)
F(t) < • F(t#)
for
0 < t < t*
for
t* < t 5
(y(l-exp(-t/y)) - t+ FU^l/U-t*) for
where
(6.5)
REMARK
PROOF
A > 0
A
t > A ,
is the solution to
^(t*) » y(l-exp( -A/v))/A.
If
F(t+) » 0 we set A = ».
Lemma 6.2, with
r = 1 , gives that
t
t* FU*) + (t-t*) F(t) < / F(x)dx
O
for
0 < t* < t .
Fran this inequality and the HNWUE definition we get that
F(t) < (y(l-exp(-t/y)) - t* Ht*)}/(t-t+) for 0 < t* < t .
But the inequalities
(6.6)
F(t+) < (y(l-exp(-t/y))- t# F(ts>)}/(t-t„l)
and
F(t*) 1 y(l-exp(-t/y))/t
are equivalent and
the definition of
y(l-exp(-t/y))/t
A
is a decreasing function of
in (6.5) it follows that (6.6) is true for
and th e proof is complete.
t. From
t < A
D
31.
In the NWUE case Haines and Singpurwalla (1974) proved that
y/(y+t)
(6.7)
for 0 < t < t*
F(t) < i P(g
(y—t^F(t^>)/(y+t-t*)
for
t* < t < y(l-F(f+))/F(t*)
for
t >p(l-F(g)/F(g.
The two bounds in (6.4) and (6.7) are illustrated in Figure 6.2.
1.0
The HNWUE upper
bound (6.4)
The HNWUE upper
bound (6.1)
The NWUE upper
bound (6.7)
A
The NWUE upper
bound (6.2)
i
«
i
1.0
2.0
•
3.0
*
'
1
1
4.0
Time/Mean
Figure 6.2 The HNWUE upper bound in (6.4) and the NWUE upper bound in
(6.7). The dotted curves are the HNWUE upper bound in (6.1) and the
NWUE upper bound in (6.2).
32.
The knowledge of
on
F
F(t+)
is not sufficient to get a sharper lower bound
than the obvious one that
FCt^)
for
0 < t < t+
for
t > t+ .
F(t) >
This can be proved by studying the survival function
F (x) =
s
where
s > t^
and
m*)
for
0 < x < s
£•1
for
s+(k-l)b < x < s+kb, k=l,2,3,...,
b
is chosen so that
sufficiently small values on
6.2
Bounds on
F
s
y = s FCt^) + e a/(l-e) . For
it can be proved that
Fg(x) is HNWUE.
in the HNBUE case
We shall now suppose that
F
is a HNBUE life distribution. First we
shall present upper (Theorem 6.4) and lower (Theorem 6.5) bounds which
are useful when only the mean
THEOREM 6.A
with mean
(6.8)
Suppose that
F
y
is a life distribution which is HNBUE
y. Then
for
t < y
for
t > y.
F(t) <
expO^—^-)
PROOF
is known.
Let
t > 0 . By using the HNBUE definition (2.6) we get that
00 _
t
J F( x) dx < jF(x) dx < y exp(-s/y)
s
s
But
J F(x) dx > (t-s) F(t)
since
F
is decreasing. Hence
we obtain that
for every
0 < s < t .
33.
F(t) <
inf
0<s<t
y exp(-s/y)
t-s
=
for
t < y
for
t > y.
,
exp(±~)
We do not know of any analogous upper bounds in the NBUE, NBU and DMRL
cases. However, in the IFRA case Barlow and Proschan (1975) gave the upper
bound
(6.9)
where
1
for
t < y
exp(-wt)
for
t > y ,
F(t) <
w = w(t) >0
is the positive solution to
1 - wy = exp(-wt). Calcu­
lations show, as is expected, that our HNBUE bound is weaker. The two bounds
are illustrated in Figure 6.3.
IFRA
upper bourn
HNBUE
upper bound
ft 0.4
•I
-i
NBUE
lower bound
HNBUE \
• lower bound
1.0
Figure 6.3
2.0
3.0
4.0
Time/Mean
The HNBUE upper bound in (6.8), the IFRA upper bound in
(6.9), the HNBUE lower bound in (6.10) and the NBUE lower bound in (6.13).
34.
The upper bound in Theorem 6.4 is sharp. For
by the one-point distribution at
y . For
t > y
t < y
equality is attained
the sharpness follows by
studying the survival function
F (x) =
s
exp(-x/y)
for
0 < x < s-y
exp((y-s)/y)
for
s-y < x < s
0
for
x > s
in the same way as in the proof of the sharpness of the bound in Theorem 6.1.
THEOREM 6.5
mean
Suppose that
F
is a life distribution which is HNBUE with
y. Then
(6.10)
where
F(t) >
a = a(t)
(6.11)
REMARK
exp(-a/y)
for
0 < t < y
0
for
t > y ,
is the largest non-negative number for which
(a-t+u)exp(-a/y) - y + t = 0 .
Besides
a = 0
there is for
0 < t< y
exactly one positive solu­
tion to the equation (6.11).
PROOF
Let
t ^ 0 . From Lemma 6.2 it follows that
s
J F(x) dx < t + F(t)(s-t)
0
for every
s > t .
Furthermore , the HNBUE definition gives that
s
J F(x) dx > y(l-exp(-s/y)) .
0
Accordingly
(6.12)
F(t) > {y(l-exp(-s/y))-t}/(s-t) for every
i.e.
F(t) > sup {(y(l-exp(-s/y))-rt}/(s-t) .
s > t
,
35.
Standard calculus then gives that for
for
s = a
t < y
the supremum is attained
given by (6.11). Since the right hand side of (6.12) is
negative for every
s > t
if
t > y
the proof is complete.
Table 6.1 gives the lower bound in (6.10) for some values on
t/y
lower
bound
t/y
lower
bound
t/y
lower
bound
0.00
1.000
0.35
0.509
0.70
0.123
0.05
0.903
0.40
0.446
0.75
0.097
•0.10
0.813
0.45
0.334
0.80
0.070
0.15
0.729
0.50
0.285
0.85
0.047
0.20
0.650
0.55
0.239
0.90
0.027
0.25
0.577
0.60
0.198
0.95
0.011
0.30
0.509
0.65
0.161
1.00
0.000
Table 6.1
•
t/y.
Some values of the HNBUE lower bound in (6.10).
The bound in Theorem 6.5 ®ay be compared to the result
(6.13)
F(t) >
y - t
y
for
t < y
o
for
t >
m
given by Marshall and Proschan (1972) in the NBUE case. Figure 6.3 on p. 33
illustrates the two bounds.
The "bound in Theorem 6.5 is sharp, too. For
by the one-point distribution at
y. For
t < y
by
F(x) =
1
for
0 < X < t
exp(-a/y)
for
t < x < ß
0
for
x > ß ,
t > y
this is illustrated
the bound is attained
36.
where
ß = (t-y) exp(a/y) + t
(6.11) it follows that
is chosen so that
/ F(x)dx = y. From
0
and it can be proved that F(t) is HNBUE.
ß > a
Now suppose that in addition to
y
we also know
^(t*)
for some
t* > 0. Then we can improve the bounds in Theorems 6.4 and 6.5.
THEOREM 6.6
mean
Suppose that
F
is a life distribution which is HNBUE with
y. Let
C =
y(l - In F(t*))
if
F(t*) > exp(-t*/y)
t* + y exp(-tj)t/y)/F(t+)
if
F(t+) < expC-t^/y)
and
h(t) = (C-t*)F(t*)/(t-t*)
If
tj^j > y
then
for
^ F(t»)
0 < t <
1 -F(t*)
y - t^ FU*)
exp(-yt/y)
F(t) <
where
(6.14)
If
for
< t < t.
1 -F(tJ
F(0
for
h(t )
for
y = y(t) satisfies
0 <y < 1
t* < t < C
t > C
and
(y-t+yt) exp(-yt/y) - (t*-t) F(t+) = 0.
t* < y
then
F(t) <
1
for
0 < t < t„
^(t^)
for
t+ < t < c
h(t)
for
t > C.
37.
PROOF
Suppose that
t < t* . Then, for every
Ç
with
0 < Ç < 1 , we get that
00
/ F(x)dx > (l-Ç)t F(t) + (t*-t) FU*)
çt
and
/ F(x)dx < y exp(-Çt/y)
Çt
The last inequality follows since
give
F
is HNBUE. These two inequalities
that
F(t) < inf g(Ç) ,
~0C£1
where
y exp(-Çt/y) - (t^-t) F(t*)
g(Ç) (1-Ot
•
Since
g'(Ç) = {(y-t+Çt) exp(-Çt/y) - (t#-t) FU*)}/{(1-Ç)2t}
the numerator in
for
g'(O is an increasing function of
Ç = 1 . Hence the infimum is attained for
g'(0) < 0, i.e. if
t > (y-t* F(t„t))/(1-F(t+))
£ = y
£
which is positive
defined by (6.14) if
and for
Since further (V~t+ F(t+))/(1-F(t*)) < t* if and only if
is complete if
Ç = 0
otherwise.
y < t*
t < t*.
Now suppose that
t > t *. In the same way as before we get that
00
/ F(x)dx > (t*-Çt) F(t*) + (t-t*) F(t) for
€t
and
/ F(x)dx < y exp(-Çt/y).
Çt
Accordingly,
F(t) <
inf k(Ç),
0<Ç<t*/t
0 < Ç < tjt
the proof
38.
where
k(Ç) =
y exp(-Çt/y) - (t^-Çt) F(t*)
t - t*
The infimum is attained for
and for
£ = t^/t
if
Ç = -y Jin F(t+)/t
if
F(t+) > expC-t^/y)
F(t+) < exp(-t+/y). Hence the infimvon is given by
(y-y Jin FU*)- t+) FU*)/(t-t*)
if
FCt*) > expC-t^/y)
y exp(-t+/y)/(t-t+)
if
FCt*) < expC-t^/y) .
and by
But these bounds are sharper than
t > y(l-£n
FCt*)
only when
> t+
if
F(t^) > expC-t^/y)
if
F(t*) < exp(-t*/y) .
and when
t > t* + y exp(-t*/y)/F(t*) > t*
This completes the proof.
For large values on
t
•
the bound in Theorem 6.4 is sharper than
h(t).
This means that we in fact get a sharper bound by changing the function
h(t)
in Theorem 6.6 to
h(t)
for
t< y
for
t > y
h*(t) = «
^min^h(t), exp(~j^)^
We conclude this section with a theorem which in the HNBUE case gives
a lower bound on
THEOREM 6.7
mean
based on the knowledge of
Suppose that
F
min(E, t*),
FCt*)
for some
t* > 0,
is a life distribution which is HNBUE with
y. Let a - a(t) be given by (6.11) and let
D =
where
F(t)
39.
where
E « V(1 - F(t+) + F(t*) In F(t*))/(1 - F(t*))
If
t* < y
then
F(t) >
exp(-a/y) for
0 < t < D
F(t^)
D < t < t*
exp(-6/y) for
0
where
(6.15)
If
6 = <5(t)
for
t* < t <
+ (y-t^J/FCt^)
t > t* + (y-t*)/F(t#) ,
is the positive solution to
(ô-t+y) exp(-6/y) - y + t* + (t-t+) FU*) = 0.
t* > y
then
F(t) >
exp(-a/y) for
0 < t <
F(t*)
for
E < t < t*
for
t > t* .
0
PROOF
for
Suppose that
E
t* < y . From Lemma 6.2 we get for
t* < t < s
that
s
t* + (t-t*) F(t+) + (s-t) F(t) > / F(x)dx > y(1-exp(-s/y)) .
"0
Accordingly,
(6.16)
F(t) > sup{y(1-exp(-s/y)) - t* - (t-t+) F(t#)}/(s-t).
s>t
The problem is now to calculate the right hand side of (6.16). Set
g(s) = {y(1-exp(-s/y)) - t* - (t-t+) F(t+)}/(s-t)
Since
sup g(s) > 0
only when
trivial bound only for
we can prove that
for
s > t.
y - t* - (t-t*) F(t^) >0 we get a non-
t < t+ + (y-t*)/F(t+). For such a value of
t
40.
(i)
gf(s) = {(s-t+y) exp(-s/y) - y+ t*+ (t-t*) FCt^)}/(s-t)2,
(ii)
gf (s) = 0 for at most one value of
(iii)
lim g(s) = 0
s-*»
s > t,
and
(iv)
lim g(s) =
s-*t+
.
Accordingly, sup g(s) = g(ô) , where
For
t < t^
g'(5) « 0 .
we have
F(t) >max(F(t+), exp(-a/y)) ,
where
a = a(t)
tion ©f
t
and
is given by (6.11). Since
F(t+) = exp(-a/y) for
the proof is complete for
The proof when
t*>y
exp(-a/y)
is a decreasing func­
t - ^{l-FCt*) + F(t+)£n F(ti|t)}/(l-FCt^))
t* < y.
is similar to that in the case when
In the NBUE case Haines and Singpurwalla (1974) gave, for
t < t^ < y . •
t+< y ,
the bound
max(F(t*), (y-t)/y)
(6.17)
F(t) > < (y - t* - (t-t*) F(t*))/y
0
They gave no lower bound if
and 6.5 on p.41
t* > y
for
0 < t < t„
for
t* <_ t < t» + (y-t+)/F(t+)
for
t > t* + (y-t*)/F(t*)
and no upper bound at all. Figures 6.4
illustrates the HNBUE bounds in Theorems 6.6 and 6.7 for two
possible combinations of
t*
and
^(t*); one for which
t+/y
< 1
and one for which the opposite inequality holds. In Figure 6.6 the NBUE
bound (6.17) is also illustrated.
0.8.
r—t
cd
>
up
0.6-
co
O
U
HNBUE
upper bound
•h
•8
•§
U
HNBUE «
lower bound
pm
1-°
Figure 6.4
and
Time/Mean^
3*°
The HNBUE bounds in Theorems 6.6 and 6.7 for
t+/p = 2.0
FCt*) =0.3
1.0
HNBUE
upper bound
0.8
0.6
, HNBUE
lower
bound
NBUE
lower
bound
•r<
•S 0.4
0.2
exp(l-t/u)
2.0
1.0
3.0
4.0
Time/Mean
Figure 6.5
The HNBUE bounds in Theorems 6.6 and 6.7 and the NBUE lower
bound in (6.17) for
t^/y = 0.5
and
F(t+) = 0.8.
42.
6.3
A concluding remark
By using Lemma 6.2 and methods similar to those in the proofs of Theorems
6.1, 6.4 and 6.5 it is possible to get bounds on the survival function to a
HNBUE (HNWUE) life distribution containing the mean
y
and another moment
00
y
= JxrdF(x) , with
0
theorem.
r > 1 . For instance we can prove the following
r
THEOREM 6.8
Suppose that
y
y <
r
and with
F
for some
00
is a life distribution which is HNWUE with mean
r > 1 . Then
^(l-exp(-t/y))
(6.18)
where
F(t) <
t^
for
0 < t < tr
for
t > t
00
1
r-1
—(y -rj X
exp(-x/y) dx)
tr r t
,
is the positive solution to
00
—£-(yr~r f x r 1exp(-x/y) dx) = ^-(l-exp(-tr/y)).
t
t
r
r
r
The upper bound on
on p. 43
when
r = 2
F
and
in Theorem 6.8 is illustrated in Figure 6.6
y2/y 2 = 2.1.
However, some of the bounds based on
y
and
y^
which we have obtained
contain rather unpleasant and cumbersome expressions and seem at the pre­
sent stage to be of rather limited practical value. Therefore our research
in this case will continue .
1.0.
0.8-
cd
.S oM
>
u
3
co
The HNWUE upper
bound in
m (6.1h
Co.IK
^
fx
<4-1
O
ft 0.4.
:
The HNWUE upper
V
bound in (6.18)
'S
•8
u
* 0.2
1-°
Figure 6.6
....i
Time/Mean 2"°
The HNWUE upper bound in (6.18) when
3"°
r•2
and
V^/u
2
= 2.1. The dotted curve is the HNWUE upper bound in (6.1).
Acknowledgements
The author is very grateful to Dr Bo Bergman, Professor Gunnar Kulldorff
and Dr Kerstin Vännman for valuable comments.
=
APPENDIX
A life distribution
F (i.e a distribution function with
and its survival function
and
(i)
_
F » 1 - F
„lOjit)
p(t)
t
for
x > 0;
increasing failure rate in average (IFRA) if
-in F(t)
t
is increasing on
S;
(iii) new better than used (NBU) if
F(x)F(y) > F(x+y)
for
x > 0
and
y > 0;
new better than used in expectation (NBUE) if
F(x) f F(y)dy > / F(x+y)dy
0
"0
for
(v)
00 —
F(x)dx
increasing failure rate (IFR) if the conditional survival function
is a decreasing function of
(iv)
y =
S « {t: F(t) > 0} are said to be (or to have)
t
(ii)
with finite mean
F(0 ) = 0)
x > 0;
decreasing mean residual life (DMRL) if
co
t « _2— / F(x)dx
F(t) t
is decreasing on
S;
45.
(vi)
harmonio neu better than used in expectation (HNBUE) if
f F( x)dx < y exp(-t/y)
t
for
t > 0.
By reversing the inequalities and changing decreasing (increasing) to
increasing (decreasing) we get the dual classes DFR, DFRA, NWU, NWUE, IMRL
and HNWUE. Here D = decreasing, I = increasing
and
W = worse.
The relations between the different classes are illustrated by the
following figure.
IFRA
x.
Y
(DFRA)
NBU
(NWU)
X
NBUE
(NWUE)
—
X
y
HNBUE
(HNWUE)
DMRL
(IMRL)
A discrete life distribution and its survival probabilities
00
j=k+l Pj » ^
=
^
0,1,2,..., with finite mean
00
—
y = ^_q P^
and
=
—
S = {k: P^ > 0}
are said to be (or to have)
(i)
discrete increasing failure rate (discrete IFR) if (P^/P^_^)^=1
is decreasing on
(ii)
S;
discrete increasing failure rate in average (discrete IFRA) if
(P^k)^=l
is decreasing;
(iii) discrete new better than used (discrete NBU) if
j,k = 0,1,2,...;
P.P, > P. .
j k - j+k
for
46.
(iv)
diserete new better than used in expectation (discrete ÏÏBUE) if
P,
ft P. < E? , P.
k E?
J=0
j - j=k j
(v)
k = 0,1,2,...;
>» »
»
discrete decreasing mean residual life (discrete DMRL) if
00
—
—
(£j_k Pj)/P^
(vi)
for
is decreasing in
k
on
S;
discrete harmonic new better than used in expectation (discrete HNBUE)
if
Z"
J
P. < y(l-l/y)k
J ""
for k =0,1,2,....
By reversing the inequalities and changing decreasing (increasing) to
increasing (decreasing) we get definitions of the corresponding dual proper­
ties.
The relations between these classes of discrete life distributions are the
same as in the general case.
Different properties of the classes IFR, IFRA, NBU, NBUE and DMRL
and their duals are discussed e.g. by Bryson and Siddiqui (1969), Marshall
and Proschan (1972), Haines (1973) and Barlow and Proschan (1975).
47.
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