A^VÇÚO 1 32 ò
1 5 Ï 2016 c 10
Chinese Journal of Applied Probability and Statistics
Oct., 2016, Vol. 32, No. 5, pp. 519-529
doi: 10.3969/j.issn.1001-4268.2016.05.007
Comparing MRL for Increasing Concave Function of Two
Random Variables ∗
LI Ping
(School of Business, Hebei Normal University, Shijiazhuang, 050024, China )
LING Xiaoliang
(College of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018, China )
Abstract:
The mean residual life (MRL) function plays a very important role in the area
of reliability engineering, survival analysis, and many other fields. In this paper, we introduce and
study a new stochastic order which gives stochastic comparison for mean residual life of strictly
increasing concave function of two random variables. We show that this new stochastic order lies
between the hazard rate and mean residual life orders. The preservation properties under mixtures
are presented here. Finally, we give some applications of this new order in reliability theory.
Keywords:
mean residual life order; mixture; preservation properties; renewal processes
2010 Mathematics Subject Classification:
§1.
Primary 60E15; Secondary 60K10, 62N05
Introduction
Mean residual life (MRL) function is a commonly used measure in describing the
lifetime of items in reliability engineer, survival analysis and actuarial science. Let X be
a lifetime random variable with distribution function F and survival function F ≡ 1 − F .
MRL function of X is defined by
mX (t) = E[X − t | X > t] =
hZ
∞
i
F (x)dx F (t),
t > 0.
t
The lifetime variable X should be smaller than another lifetime variable in some
stochastic sense if the MRL function of X is smaller than that of another lifetime variable.
This statement provides the mean residual life order which has been defined and studied
in the literature. Let Y be another lifetime random variable which has survival function
G. Then X is said to be smaller than Y in the mean residual life order (denoted by
X 6mrl Y ) if mX (t) 6 mY (t) for all t > 0, or equivalently, X 6mrl Y iff
hZ ∞
i.h Z ∞
i
G(x)dx
F (x)dx is increasing in t > 0.
t
∗
t
The project was supported by the National Natural Science Foundation of China (11501162, 71231001, 7142
0107023) and the Natural Science Foundation of Hebei Province (A2014208133).
Received January 27, 2015. Revised November 10, 2015.
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Chinese Journal of Applied Probability and Statistics
Vol. 32
The hazard rate (HR) function of X is defined as hX (t) = f (t)/ F (t), where f is the
density function of X. Then random variable X is said to be smaller than Y in the hazard
rate order (denoted by X 6hr Y ) if G(t)/ F (t) is increasing in t > 0. Readers are referred
to [1] and [2] for a general reference about above two stochastic orders. The dual concept
of the mean residual life is called mean inactivity time or reversed mean residual life, and
it is given by (see [3])
eX (t) = E[t − X | X 6 t] =
hZ
t
F (x)dx
i
F (t),
t > 0.
0
Based on the comparison of mean inactivity times of eX 2 (t) and eY 2 (t), Kayid and
Izadkhah [4] propose a new stochastic order called strong mean inactivity time order which
lies between the reversed hazard rate and the mean inactivity time orders. They provided
some properties and applications of strong mean inactivity time order. Motivated by this,
in this paper, we introduce a new stochastic order which lies between the hazard rate and
mean residual life orders.
Definition 1
Let φ be a nonnegative decreasing function. X is said to be smaller
than Y in the mean residual life order with respect to φ (denoted by X 6φmrl Y ) if
hZ
∞
φ(x)F (x)dx
i
∞
hZ
F (t) 6
t
φ(x)G(x)dx
i
G(t)
for all t > 0,
t
or equivalently, X 6φmrl Y if and only if
hZ
∞
φ(x)G(x)dx
t
i.h Z
∞
i
φ(x)F (x)dx is increasing in t > 0.
t
Let ϕ be a strictly increasing concave function, and be differentiable, one can easily
check that ϕ(X) 6mrl ϕ(Y ) is equivalent to X 6φmrl Y , where φ(x) = dϕ(x)/dx is a strictly
decreasing function. It shows that the order 6φmrl gives a comparison of mean residual life
functions of ϕ(X) and ϕ(Y ).
The purpose of this paper is to introduce and study the order 6φmrl . In Section 2, we
present some implications with respect to the order 6φmrl , and build some of its preservation
properties under mixtures. In Section 3, we provide some applications of this new order
in reliability theory.
Throughout this paper, the terms increasing and decreasing mean non-decreasing and
non-increasing, respectively. All integral and expectation are implicity assumed to exist
whenever they are written.
No. 5
LI P., LING X. L.: Comparing MRL for Increasing Concave Function of Two Random Variables
§2.
521
Implications and Preservations
Before proceeding to state our main results, we firstly recall the following two lemmas
which are essentially due to [5] and [6], respectively.
Lemma 2
Assume that W (x) is Lebesgue-Stieltjes measure, not necessarily positive.
R∞
R∞
If h(x) is non-negative and increasing, and t dW (x) > 0, for all t > 0, then 0 h(x)dW (x)
> 0.
A non-negative function ψ(θ, x) is said to be TP2 in (θ, x) ∈ R × R if ψ(θ1 , x1 )ψ(θ2 ,
x2 ) > ψ(θ1 , x2 )ψ(θ2 , x1 ) for all θi ∈ R and xi ∈ R such that θ1 6 θ2 and x1 6 x2 .
Lemma 3
Let ψ(θ, x) be an TP2 function in (θ, x) ∈ R × R, and hi (θ) be TP2 in
(i, θ) ∈ {1, 2} × R, where hi (θ) is a probability density function in θ for each i. Then φi (x) =
R
R ψ(θ, x)hi (θ)dθ is TP2 in (i, x) ∈ {1, 2} × R.
The next result shows that the order 6φmrl lies between 6hr and 6mrl orders.
Theorem 4
Let X and Y be two nonnegative random variables, and let φ be a
nonnegative decreasing function. Then
(i) X 6hr Y implies that X 6φmrl Y ;
(ii) X 6φmrl Y implies that X 6mrl Y .
Proof
(i) Note that X 6hr Y implies that G(x)/ F (x) is increasing in x, or equivalently
h F (x) G(x) i
6 0,
for all x > t.
−
F (t)
G(t)
Then we immediately have
Z
∞
t
h F (x) G(x) i
φ(x)
dx 6 0,
−
F (t)
G(t)
which states that X 6φmrl Y .
(ii) Note that for any fixed t > 0, we have
Z ∞
1 h φ(x)G(x) φ(x)F (x) i
mY (t) − mX (t) =
−
dx
φ(x)
G(t)
F (t)
t
Z ∞
1 h φ(x)G(x) φ(x)F (x) i
−
=
I[x > t]dx,
φ(x)
G(t)
F (t)
0
where I(x > t) stands for the indicator function of the set [x > t]. For all 0 6 s 6 t,
we have
Z ∞h
s
φ(x)G(x) φ(x)F (x) i
−
I[x > t]dx =
G(t)
F (t)
Z
t
∞h
φ(x)G(x) φ(x)F (x) i
−
dx > 0.
G(t)
F (t)
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Chinese Journal of Applied Probability and Statistics
Vol. 32
For all s > t, we have
hZ ∞
i.h Z ∞
i
φ(x)G(x)dx
φ(x)F (x)dx
s
s
hZ ∞
i.h Z ∞
i G(t)
φ(x)G(x)dx
φ(x)F (x)dx >
>
> 0.
F (t)
t
t
Hence, X 6φmrl Y implies
Z ∞h
φ(x)G(x) φ(x)F (x) i
−
dx > 0
G(t)
F (t)
s
for all s, t > 0.
On the other hand, 1/φ(x) is an increasing function. Finally, applying Lemma 2, we
obtain that
Z
mY (t) − mX (t) =
t
That is X 6mrl Y .
∞
1 h φ(x)G(x) φ(x)F (x) i
−
dx > 0.
φ(x)
G(t)
F (t)
In general, the order 6φmrl does not imply the order 6hr . However, according to
Theorem 2.A.7 of [1], we have X 6hr Y if, and only if, X 6φmrl Y holds for all s > 0, where
φ(x) = e−sx , x > 0.
Residual life at random time (RLRT) is defined by XY = [X − Y | X > Y ], see [7] for
more detail stochastic properties of RLRT. X is said to be smaller than Y in the Laplace
R∞
R∞
transform order (denoted by X 6Lt Y ) if 0 e−sx F (x)dx 6 0 e−sx G(x)dx for all s >
0; X is said to be smaller than Y in the moment generating function order (denoted by
R∞
R∞
X 6mg Y ) if 0 esx F (x)dx 6 0 esx G(x)dx for all s > 0; X is said to be smaller than Y in
the mean inactivity time order (denoted by X 6MIT Y ) if E[t−X | X 6 t] > E[t−Y | Y 6 t]
for all t > 0; X is said to be smaller than Y in the strong mean inactivity time order
(denoted by X 6SMIT Y ) if
hZ t
i
hZ t
i
xF (x)dx /F (t) >
xG(x)dx /G(t)
0
for all t > 0.
0
Let Y be independent of X, some authors have proven that XY 6Lt(mg,MIT,SMIT) X if and
only if Xt 6Lt(mg,MIT,SMIT) X for all t > 0, see [4, 8–10] for more details. We give the
corresponding result for 6φmrl order.
Theorem 5
XY 6φmrl X for any Y that is independent of X if and only if Xt 6φmrl X
for all t > 0.
Proof
Let Xt 6φmrl X for all t > 0. It follows that
Z
Z ∞
F (t + s) ∞
φ(x)F (x)dx.
φ(x)F (t + x)dx 6
F (s)
s
s
No. 5
LI P., LING X. L.: Comparing MRL for Increasing Concave Function of Two Random Variables
Note that, the survival function of XY is given by
i.h Z
hZ ∞
F (x + t)dG(t)
P{XY > x} =
∞
523
i
F (t)dG(t) .
0
0
Then, for any s > 0, we have
i
hZ ∞
φ(x)P{XY > x}dx /P{XY > s}
s
hZ ∞
nZ ∞
i o.h Z ∞
i
φ(x)
=
F (x + t)dG(t) dx
F (s + t)dG(t)
0
s
0
nZ ∞Z ∞
o.h Z ∞
i
φ(x)F (x + t)dx dG(t)
=
F (s + t)dG(t)
0
s
0
i
o.h Z ∞
i
n Z ∞ h F (t + s) Z ∞
φ(x)F (x)dx dG(t)
F (s + t)dG(t)
6
F (s)
s
0
0
Z ∞
hZ ∞
i.h Z ∞
i
=
F (s + t)dG(t)
φ(x)F (x)dx
F (s + t)dG(t)F (s)
0
s
0
i
hZ ∞
=
φ(x)F (x)dx / F (s).
s
Then we get XY 6mrl X.
On the other hand, if XY 6φmrl X for any random variable Y . By taking Y as a
degenerate variable, then we have Xt 6φmrl X for all t > 0.
The following theorem shows that the order
6φmrl
may be preserved under mixture
under a stronger condition.
Theorem 6
Let X , Y and Θ be random variables such that [X | Θ = θ] 6φmrl [Y | Θ =
θ0 ] for all θ and θ0 in the support of Θ. Then X 6φmrl Y .
Proof
Select θ and θ0 in the support of Θ. Let F (·|θ), G(·|θ), F (·|θ0 ), G(·|θ0 ) be the
survival function of [X | Θ = θ], [Y | Θ = θ], [X | Θ = θ0 ], and [Y | Θ = θ0 ], respectively. It
is sufficient to show that for α ∈ (0, 1), we have
Z ∞
i
h Z ∞
α
φ(u)F (u|θ)du + (1 − α)
φ(u)F (u|θ0 )du αF (t|θ) + (1 − α)F (t|θ0 )
t
Zt ∞
h Z ∞
i
6 α
φ(u)G(u|θ)du + (1 − α)
φ(u)G(u|θ0 )du αG(t|θ) + (1 − α)G(t|θ0 ) .
t
t
By the assumptions of the theorem, they satisfy
i
hZ ∞
i
hZ ∞
φ(u)G(u|θ)du / G(t|θ),
φ(u)F (u|θ)du / F (t|θ) 6
t
t
hZ ∞
i
hZ ∞
i
φ(u)G(u|θ0 )du / G(t|θ0 ),
φ(u)F (u|θ)du / F (t|θ) 6
t
t
hZ ∞
i
hZ ∞
i
φ(u)G(u|θ)du / G(t|θ),
φ(u)F (u|θ0 )du / F (t|θ0 ) 6
t
t
524
Chinese Journal of Applied Probability and Statistics
hZ
∞
∞
i
hZ
φ(u)F (u|θ0 )du / F (t|θ0 ) 6
t
Vol. 32
i
φ(u)G(u|θ0 )du / G(t|θ0 ).
t
It is easy to verify that the latter four inequalities imply the former one, and then we get
the result.
Consider a family of distribution functions {Gθ , θ ∈ R+ }. Let now X(θ) be a random
variable having distribution function Gθ , and let Θi be a random variables with distribution
functions Fi , for i = 1, 2 and supports in R+ . Let Y1 and Y2 be two random variables such
that Yi =st X(Θi ), i = 1, 2, that is, suppose that the distribution function of Yi is given
by
∞
Z
y ∈ R+ , i = 1, 2.
Gθ (y)dFi (θ),
Hi (y) =
0
The following result is a closure of the order 6φmrl under mixture.
Theorem 7
If Θ1 6lr Θ2 , and if X(θ) 6φmrl X(θ0 ) whenever θ 6 θ0 , then Y1 6φmrl Y2 .
Proof For i = 1, 2, let fi be the density function of Θi . Y1 6φmrl Y2 means that
nZ ∞
hZ ∞
i o.n Z ∞
hZ ∞
i o
φ(u)
Gθ (u)f2 (θ)dθ du
φ(u)
Gθ (u)f1 (θ)dθ du
t
0
t
0
is increasing in t, which is equivalent to say that
Z ∞
Z ∞hZ
hZ ∞
i
φ(u)
Gθ (u)fi (θ)dθ du =
t
0
0
∞
i
φ(u)Gθ (u)du fi (θ)dθ
t
is TP2 in (i, t) ∈ {1, 2} × R+ . Assumption Θ1 6lr Θ2 means that fi (θ) as a function of
R∞
i ∈ {1, 2} and θ is TP2 . The assumption X(θ) 6φmrl X(θ0 ) means that t φ(u)Gθ (u)du,
as a function of t and θ is TP2 . By Lemma 3, we have
Z ∞
Z ∞hZ ∞
hZ ∞
i
i
φ(u)
Gθ (u)fi (θ)dθ du =
φ(u)Gθ (u)du fi (θ)dθ
t
0
0
t
is TP2 in (i, t) ∈ {1, 2} × R+ . This completes the proof.
§3.
Reliability Applications
In this section, we discuss some relevant applications in reliability theory involving
the order 6φmrl .
Let the lifetimes of n components X1 , X2 , . . . , Xn be independent and not necessarily
identically distributed random variables with distribution functions F1 , F2 , . . . , Fn , respectively. Let X1:n 6 X2:n 6 · · · 6 Xn:n be the order statistics corresponding to the random
variables X1 , X2 , . . . , Xn and let X1:n 6 X2:n 6 · · · 6 Xn−1:n−1 be the order statistics
No. 5
LI P., LING X. L.: Comparing MRL for Increasing Concave Function of Two Random Variables
525
corresponding to the random variables X1 , X2 , . . . , Xn−1 . Boland et al. [11] proved that
if Xi 6hr Xn , i = 1, 2, . . . , n − 1, then Xk:n−1 6hr Xk+1:n , k = 1, 2, . . . , n − 1. Hu et
al. [12] proved that if Xi 6mrl Xn , i = 1, 2, . . . , n − 1, then Xn−1:n−1 6mrl Xn:n . In the
following theorem we show that if the n-th component with lifetime Xn is stronger than
other components in the mean residual life order with respect φ, then the parallel system
with lifetime Xn−1:n−1 is smaller than the parallel system with lifetime Xn:n in the mean
residual life order with respect φ.
If Xi 6φmrl Xn , i = 1, 2, . . . , n − 1, then Xn−1:n−1 6φmrl Xn:n .
Theorem 8
Proof
Note that the following equation holds,
1−
n
Q
Fi (x) =
i=1
where
0
Q
i=1
φ(x)
n−1
P
F i (x)
i=1
s
n−1
P
∞
φ(x)
⇐⇒
i=1
i−1
Q
F i (s)
i−1
Q
n−1
P
F i (x)
i=1
n−1
P
φ(x)
n
P
Fj (s)
i−1
Q
φ(x)
x > 0,
i−1
Q
Z
n−1
P
F i (x)
i−1
Q
6
F n (s)
Fj (s)
Fj (s)
j=1
n−1
Q
φ(x)F n (x)
s
Fj (x)dx
j=1
∞
Fj (x)dx
i−1
Q
F i (x)
F i (s)
Fj (x)dx
j=1
n−1
Q
Fj (s)
j=1
j=1
i=1
s
n
P
i=1
Z
⇐⇒
Fj (x),
i=1
s
6
j=1
F i (s)
i=1
∞
∞
Fj (x)dx
j=1
s
Z
i−1
Q
j=1
Z
j=1
i=1
Z
F i (x)
≡ 1. Then Xn−1:n−1 6φmrl Xn:n is equivalent to, for all s > 0,
∞
Z
n
P
i−1
Q
∞
φ(x)F n (x)
Fj (x)dx 6
j=1
s
F n (s)
n−1
Q
Fj (x)dx
j=1
n−1
Q
n−1
P
F i (s)
i=1
Fj (s)
i−1
Q
Fj (s).
j=1
j=1
For i = 1, 2, . . . , n − 1, Xi 6φmrl Xn means that, for all t > s > 0,
Z ∞
nh Z ∞
i
o
φ(x)F i (x)dx > 0.
φ(x)F n (x)dx / F n (s) F i (s) −
t
Note that
i−1
Q
t
Fj (x) is an increasing function of x, applying Lemma 2, we have
j=1
nh Z
∞
φ(x)F n (x)
i−1
Q
Z
i
o
Fj (x)dx / F n (s) F i (s) −
j=1
s
s
∞
φ(x)F i (x)
i−1
Q
Fj (x)dx > 0.
j=1
Then we have, for i = 1, 2, . . . , n − 1,
nh Z ∞
i.h
io
n−1
n−1
i−1
Q
Q
Q
Fj (x)dx
F n (s)
Fj (s) F i (s)
Fj (s)
φ(x)F n (x)
s
j=1
j=1
j=1
526
Chinese Journal of Applied Probability and Statistics
=
∞
nh Z
φ(x)F n (x)
>
i.h
io
n−1
Q
Fj (x)dx
F n (s)
Fj (s) F i (s)
j=1
s
∞
nh Z
n−1
Q
φ(x)F n (x)
i−1
Q
j=i
∞
Z
o
Fj (x)dx / F n (s) F i (s) >
i
j=1
s
Vol. 32
φ(x)F i (x)
i−1
Q
Fj (x)dx.
j=1
s
This implies, for all s > 0,
Z
∞
Z
φ(x)
n−1
P
F i (x)
i=1
s
i−1
Q
∞
φ(x)F n (x)
Fj (x)dx 6
s
j=1
F n (s)
n−1
Q
Fj (x)dx
j=1
n−1
Q
Fj (s)
n−1
P
F i (s)
i=1
i−1
Q
Fj (s).
j=1
j=1
Therefore Xn−1:n−1 6φmrl Xn:n .
Now let X1 , X2 , . . . , Xn be independent and identically distributed (i.i.d.) lifetime
random variables from F and let Y1 , Y2 , . . . , Yn be also i.i.d. lifetime random variables
from G. Some authors studied which stochastic orders for series system are inherited for
the components, see [13] and [14] for details. The following result shows that if the lifetimes
of two series systems with i.i.d. components are 6φmrl ordered, then their components are
also 6φmrl ordered.
If X1:n 6φmrl Y1:n , then Xi 6φmrl Yi for all i = 1, 2, . . . , n.
Theorem 9
Proof
X1:n 6φmrl Y1:n implies that for all s > t > 0,
Z ∞
n
n
n
n
φ(x)(F (t)G (x) − F (x)G (t))dx > 0.
s
Note that
n
n
n
n
F (t)G (x) − F (x)G (t) = (F (t)G(x) − F (x)G(t))
n
P
(F (t)G(x))n−i (F (x)G(t))i−1 .
i=1
Since
n
P
(F (t)G(x))n−i (F (x)G(t))i−1
−1
is increasing in x, and applying Lemma 2, we
i=1
obtain that
Z ∞h
Zt ∞
n
P
(F (t)G(x))n−i (F (x)G(t))i−1
i−1
n
n
n
n
φ(x) F (t)G (x) − F (x)G (t) dx
i=1
φ(x)(F (t)G(x) − F (x)G(t))dx > 0.
=
t
That is Xi 6φmrl Yi for all i = 1, 2, . . . , n.
For a sequence of mutually i.i.d. non-negative random variable, {Xn , n = 1, 2, . . .},
n
P
having common distribution function F , with F (0) = 0. Define Sn =
Xi as the
i=1
No. 5
LI P., LING X. L.: Comparing MRL for Increasing Concave Function of Two Random Variables
527
time of the n-th arrival, for all n = 1, 2, . . ., with S0 = 0. Let N (t) = sup{n, Sn 6 t}
represents the number of arrivals during the interval [0, t]. Then N = {N (t), t > 0} is a
renewal process with underlying distribution F . The excess lifetime at time t is defined by
γ(t) = SN (t)+1 − t. Several results have been given in the context of renewal processes for
new better than used in the sense of some stochastic orders, see for example [10, 15–17].
We give a corresponding result for the order 6φmrl .
Theorem 10
Proof
If γ(t) 6φmrl γ(s) whenever 0 6 s 6 t, then Xt 6φmrl X for all t > 0.
By conditional probability, it follows that
Z t
P(γ(t) > x) = F (t + x) +
P(γ(t − y) > x)dF (y),
0
see [18; p. 193]. Integrating both sides gives, for all s > 0,
Z ∞
φ(x)P(γ(t) > x)dx
s
Z ∞
Z ∞hZ t
i
=
φ(x)F (t + x)dx +
φ(x)P(γ(t − y) > x)dF (y) dx
s
s
0
Z ∞
Z thZ ∞
i
=
φ(x)F (t + x)dx +
φ(x)P(γ(t − y) > x)dx dF (y)
s
0
Zs ∞
#
Z t"
Z ∞
φ(x)P(γ(t) > x)dx
s
>
φ(x)F (t + x)dx +
P(γ(t − y) > s) dF (y)
P(γ(t) > s)
0
s
Z ∞
Z ∞
Z
i
φ(x)P(γ(t) > x) h t
=
φ(x)F (t + x)dx +
P(γ(t − y) > s)dF (y) dx
P(γ(t) > s)
0
Zs ∞
Zs ∞
φ(x)P(γ(t) > x)
=
φ(x)F (t + x)dx +
[P(γ(t) > s) − F (t + s)]dx
P(γ(t) > s)
s
s
Z ∞
Z ∞
Z ∞
φ(x)P(γ(t) > x)
=
φ(x)F (t + x)dx +
φ(x)P(γ(t) > x)dx − F (t + s)
dx,
P(γ(t) > s)
s
s
s
where the inequality holds due to γ(t) 6φmrl γ(t − y). Thus,
hZ ∞
i
hZ ∞
i
φ(x)F (t + x)dx / F (t + s) 6
φ(x)P(γ(t) > x)dx /P(γ(t) > s).
s
Note that
s
γ(t) 6φmrl
hZ ∞
γ(0) =st X, we then have, for all s > 0 and t > 0,
i
hZ ∞
i
φ(x)F (x)dx / F (s).
φ(x)F (t + x)dx / F (t + s) 6
s
That is
Xt 6φmrl
X for all t > 0.
s
The renewal function M (t) is defined as the expected number of renewals in [0, t],
M (t) = E[N (t)]. For all t > 0 and x > 0, the following equation also holds,
Z t
F (t − u + x)dM (u).
P(γ(t) > x) = F (t + x) +
0
528
Chinese Journal of Applied Probability and Statistics
Vol. 32
In the literature, several results deal with the ageing property F and stochastic comparison
of excess time in a renewal process, see [17] and references therein. The following result
gives an investigation on the behavior of the excess lifetime of a renewal process in terms
of the order 6φmrl property of the interarrival times.
Theorem 11
Proof
Z
If Xt 6φmrl X for all t > 0, then γ(t) 6φmrl γ(0) for all t > 0.
Note that Xt 6φmrl X for all t > 0 means that
hZ ∞
i
∞
φ(x)F (x)dx / F (s)
F (t + x)dx 6 F (t + s)
s
for all t > 0,
s
then we have
Z ∞
φ(x)P(γ(t) > x)dx
s
Z ∞
Z ∞
hZ t
i
=
φ(x)F (t + x)dx +
φ(x)
F (t − u + x)dM (u) dx
s
s
0
Z thZ ∞
Z ∞
i
φ(x)F (t − u + x)dx dM (u)
=
φ(x)F (t + x)dx +
0
s
s
Z
Z ∞
Z th
i
F (t − u + s) ∞
φ(x)F (x)dx dM (u)
6
φ(x)F (t + x)dx +
F (s)
s
s
0
Z ∞
nh Z ∞
i
oZ t
=
φ(x)F (t + x)dx +
φ(x)F (x)dx / F (s)
F (t − u + s)dM (u)
s
s
0
Z ∞
nh Z ∞
i
o
=
φ(x)F (t + x)dx +
φ(x)F (x)dx / F (s) [P(γ(t) > s) − F (t + s)]
s
s
Z
nh ∞
i
o
nhZ ∞
i
o
6
φ(x)F (x)dx / F (s) F (t + s)+
φ(x)F (x)dx / F (s) [P(γ(t) > s)−F (t + s)]
s
s
nhZ ∞
i
o
=
φ(x)F (x)dx /F (s) P(γ(t) > s).
s
Therefore,
hZ
∞
i
hZ
φ(x)P{γ(t) > x}dx /P(γ(t) > s) 6
s
That is γ(t) 6φmrl γ(0) for all t > 0.
∞
i
φ(x)F (x)dx / F (s).
s
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