Journal of Inequalities in Pure and
Applied Mathematics
SOME INEQUALITIES AND BOUNDS FOR WEIGHTED
RELIABILITY MEASURES
volume 3, issue 4, article 60,
2002.
BRODERICK O. OLUYEDE
Department of Mathematics and Computer Science
Georgia Southern University
Statesboro, GA 30460.
EMail: boluyede@gasou.edu
Received 22 October, 2001;
accepted 17 July, 2002.
Communicated by: N.S. Barnett
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
071-01
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Abstract
Weighted distributions occur naturally in a wide variety of settings with applications in reliability, forestry, ecology, bio-medicine, and many other areas. In
this note, bounds and stability results on the distance between weighted reliability functions, residual life distributions, equilibrum distributions with monotone weight functions and the exponential counterpart in the class of distribution functions with increasing or decreasing hazard rate and mean residual
life functions are established. The problem of selection of experiments from
the weighted distributions as opposed to the original distributions is addressed.
The reliability inequalities are applied to repairable systems.
2000 Mathematics Subject Classification: 39C05.
Key words: Reliability inequalities, Stochastic Order, Weighted distribution functions,
Integrable function.
The author expresses his gratitude to the editor and referee for many useful comments and suggestions.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Some Basic Results and Utility Notions . . . . . . . . . . . . . . . . . . 5
3
Inequalities for Weighted Reliability Measures . . . . . . . . . . . . 8
4
Inequalities for Repairable Systems . . . . . . . . . . . . . . . . . . . . . 16
5
Selection of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
References
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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1.
Introduction
When data is unknowingly sampled from a weighted distribution as opposed to
the parent distribution, the survival function, hazard function, and mean residual
life function (MRLF) may be under or overestimated depending on the weight
function. For size-biased sampling, the analyst will usually give an over optimistic estimate of the survival function and mean residual life functions. It is
well known that the size-biased distribution of an increasing failure rate (IFR)
distribution is always IFR. The converse is not true. Also, if the weight function is monotone increasing and concave, then the weighted distribution of an
IFR distribution is an IFR distribution. Similarly, the size-biased distribution of
a decreasing mean residual (DMRL) distribution has decreasing mean residual
life. The residual life at age t, is a weighted distribution, with survival function
given by
(1.1)
F t (x) =
F (x + t)
,
F (t)
for x ≥ 0. The weight function is W (x) = f (x + t)/f (x), where f (u) =
dF (u)/du, the hazard function and mean residual life functions are λFt (x) =
λF (x + t) and δFt (x) = δF (x + t). It is clear that if F is IFR, (DMRL) distribution, then Ft is IFR, (DMRL) distribution. The hazard function λF (x)
and meanR residual life function δF (x) are given by λF (x) = f (x)/F (x), and
∞
δF (x) = x F (u)du/F (x) respectively. The functions λF (x), δF (x), and F (x)
are equivalent ([6]). The purpose of this article is to establish bounds and stability results on the distance between weighted reliability functions, residual life
distributions, equilibrium distributions with monotone weight functions and the
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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exponential counterpart in the class of life distributions with increasing or decreasing hazard rate and mean residual life functions. We also present results
on increasing hazard rate average (IHRA) weighted distributions and obtain inequalities for the weighted mean residual life function for large values of the
response. In Section 2 some basic results and utility notions are presented.
Section 3 contains stochastic inequalities for reliability measures under distributions with monotone weight functions as well as inequalities for IHRA and
mean residual life weighted distributions. In Section 4 inequalities and stability results for repairable systems are established. Section 5 contains results on
selection of experiments under weighted distributions as opposed to the parent
distributions.
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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2.
Some Basic Results and Utility Notions
Let X be a nonnegative random variable with distribution function F (x) and
probability density function (pdf) f (x). Let W (x) be a positive weight function
such that 0 < E(W (X)) < ∞. The weighted distribution of X is given by
(2.1)
F W (x) =
F (x){W (x) + MF (x)}
,
E(W (X))
where
R∞
MF (x) =
x
0
F (t)W (t)dt
F (x)
,
assuming W (x)F (x) −→ 0 as x −→ ∞. The corresponding pdf of the weighted
random variable XW is
(2.2)
fW (x) =
W (x)f (x)
,
E(W (X))
x ≥ 0, where 0 < E(W (X) < ∞. We now give some basic and important
definitions.
Definition 2.1. Let X and Y be two random variables with distribution functions F and G respectively. We say F <st G, stochastically ordered, if F (x) ≤
G(x), for x ≥ 0 or equivalently, for any increasing function Φ(x),
(2.3)
E(Φ(X)) ≤ E(Φ(Y )).
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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Definition 2.2. A distribution function F is an increasing hazard rate (IHR)
distribution if F (x + t)/F (t) is decreasing in 0 < t < ∞ for each x ≥ 0. Similarly, a distribution function F is an decreasing hazard rate (DHR) distribution
if F (x + t)/F (t) is increasing in 0 < t < ∞ for each x ≥ 0. It is well known
that IHR (DHR) implies DMRL (IMRL).
Definition 2.3. Let F be a right-continous distribution such that F (0+) = 0. F
is said to be an increasing hazard rate average (IHRA) distribution if and only
if for all 0 ≤ α ≤ 1, and x ≥ 0,
(2.4)
α
F (αx) ≥ F (x).
It is well known that F is an IHRA distribution if and only if
Z
1/α
Z
α x
(2.5)
g(x)dF (x) ≤
g
dF (x)
,
α
for all 0 < α < 1 and all nonnegative non-decreasing functions g.
Consider a renewal process with life distribution F (x) and weighted distribution function FW (x), with weight function W (x) > 0. Let Xt denote the
residual life of the unit functioning at time t. Then as t −→ ∞, Xt has the
limiting reliability function given by
Z ∞
−1
F e (x) = µ
F (y)dy,
(2.6)
x
x ≥ 0. The corresponding limiting pdf is
(2.7)
fe (x) =
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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F (x)
,
µF
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x ≥ 0. The weighted equilibrium reliability or survival function is
Z ∞
−1
(2.8)
F We (x) = µFW
F W (y)dy,
x
where FW (x) is given by (2.1). Note that F We (x) can be expressed as
(2.9)
F We (x) = µ−1
FW F W (x)δFW (x),
x ≥ 0, and the corresponding hazard function is
(2.10)
λFWe (x) =
fWe (x)
= (δFWe (x))−1 ,
F We (x)
x ≥ 0, where
(2.11)
x ≥ 0.
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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δFWe (x) = (δF (x)F (x))−1
Z
∞
F (y)δF (y)dy,
x
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3.
Inequalities for Weighted Reliability Measures
In this section, we derive inequalities for reliability measures for weighted distributions. Bounds and stability results on the distance between the equilibrium
reliability functions, weighted reliability functions and the size-biased equilibrium exponential distributions are established. These results are given in the
context of life distributions with monotone hazard and mean residual life functions.
Theorem 3.1. ([1]). If F has DMRL, then Sk (x) ≤ Sk (0)e−x/µ , k = 1, 2, . . . ,
and Sk (x) ≥ µSk−1 (0)e−x/µ − µSk−1 (0) + Sk (0), k = 2, 3, . . . , where

F (x)
if k = 0,



R∞
Sk (x) =
F (x + t)tk−1 dt


 0
if k = 1, 2, . . . ,
(k − 1)!
is a sequence of decreasing functions for which F possess moments of order J,
that is µk = E(X k ) exists, k = 1, 2, . . . , J.
We let S−1 (x) = f (x) be the pdf of F if it exists. Then Sk (0) = µk /k!,
and Sk0 (x) = −Sk−1 (x), k = 0, 1, 2, . . . , J. The ratio Sk−1 (x)/Sk (x) is a hazard function of a distribution function with survival function Sk (x)/Sk (0). The
inequalities in Theorem 3.1 are reversed if F has increasing mean residual life
(IMRL).
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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Theorem 3.2. Let F We (x) be an IHR weighted equilibrium reliability function
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with increasing weight function. Then
Z ∞
2 µ2 −x/µ
.
(3.1)
|F We (x) − xe
|dx ≤ 2 µ −
2µ 0
Proof. Let A = {x|F We ≤ xe−x/µ }. Then we have for x > 0,
Z ∞
Z
−x/µ
|F We (x) − xe
|dx ≤ 2 (xe−x/µ − F We (x))dx
0
ZA∞
≤2
(xe−x/µ − F e (x))dx
Z0 ∞ S1 (x)
−x/µ
=2
xe
−
dx
µ
0
S2 (0)
2
=2 µ −
µ
µ2
2
(3.2)
=2 µ −
.
2µ
The first inequality is trivial and the second inequality is due to the stochastic
order between F We (x) and F e (x) when W (x) is increasing in x ≥ 0.
Theorem 3.3. Let F We be the weighted equilibrium reliability function with
decreasing hazard rate(DHR). If the weight function W (x) is increasing, then
Z ∞
|F We (x) − xe−x/µ |dx ≥ 2e−η/µ max{0, |µ − ηµ − 1|},
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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0
for η ≥ µ.
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Proof. Let F We be DHR reliability function, then there exist η ≥ µ such that
F We (x) ≤ xe−x/µ or F We (x) ≥ xe−x/µ as x ≤ η or x ≥ η. Now,
Z ∞
Z ∞
−x/µ
|F We (x) − xe
|dx = 2
(F We (x) − xe−x/µ )dx
0
Zη ∞
≥2
(F e (x) − xe−x/µ )dx
η
Z ∞
2
=
(S1 (x) − µe−x/µ )dx
µ
η
2
=
(S2 (η) − {µ2 ηe−η/µ + µe−η/µ })
µ
−η/µ
≥ 2S1 (η) − 2e
(3.3)
Broderick O. Oluyede
(µη + 1)
−η/µ
≥ 2µS0 (η) − 2e
Some Inequalities and Bounds
for Weighted Reliability
Measures
(µη + 1)
= 2e−η/µ (µ − ηµ − 1).
The first inequality is due to the fact that W (x) is increasing in x, so that F We (x)
and F e (x) are stochastically ordered. The last two inequalities are due to the
that Sk (x) ≥ µSk−1 (x) for all x ≥ 0, k = 1, 2, . . . , where Sk (x) =
Rfact
∞
Sk−1 (y)dy.
x
Theorem 3.4. Let F and F W be the parent and weighted survival functions
respectively. Suppose W (x) is increasing and F has DMRL, then
2W (x)
µ2
2
µ2
−x/µ
F W (x) ≥ e
max 0,
+
− −
,
(3.4)
α
αµ
α 2µ
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where 0 < α = E(W (X)) < ∞.
Proof. Note that
Z ∞
0
F W (x) = α
W (x)F (x) +
F (y)W (y)dy
x
R∞
S2 (y)dy
−1 W (x)S2 (x)
≥α
+ x
S2 (0)
S2 (0)
−1
= (αS2 (0)) {W (x)S2 (0) + S3 (x)}
−1
≥ (αS2 (0))−1 {W (x)[µS1 (0)e−x/µ − µS1 (x)]
+ µS2 (0)e−x/µ − µS2 (0)}
−1
(3.5)
Broderick O. Oluyede
−x/µ
= (αS2 (0)) {µ(e
[W (x)S1 (0) + S2 (0)] − S1 (0) − S2 (0))}
n
µ2 µ2 o
= 2(αµ)−1 e−x/µ W (x)µ +
−µ−
2
2
2W
(x)
µ
2
µ
2
2
= e−x/µ
+
− −
.
α
αµ
α αµ
The inequalities follows from the fact that W (x) is increasing and from the
application of Theorem 3.1.
Theorem 3.5. Let FW be a weighted distribution function with increasing weight
function W (x) ≥ 0, then δF (x) ≤ δFW (x) ≤ (λFW (x))−1 for all x > 0. Furthermore, if the hazard rate λFW (x) is such that
(3.6)
Some Inequalities and Bounds
for Weighted Reliability
Measures
c
λFW (x) ≥
x
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for x ≥ x0 , where c is a positive real number, then
Z t
1
(3.7)
{λF (x)}−k dx ≥ c∗ t−k
t
0
for t > 0, k > 1, where c∗ is a real number.
Proof. Let W (x) be increasing in x, then λFW (x) ≤ λF (x) and δF (x) ≤
δFW (x) for all x > 0. Note that,
R∞
Z ∞
F W (y)dy
x
(3.8)
δFW (x) =
≤
e−λFW (x)y dy = (λFW (x))−1 ,
F (x)
0
for all x > 0. Consequently,
Z t
Z t
1
1
−k
{δFW (x)} dx ≥
{λFW (x)}k dx
t
t
0
Z0 t 1
c k
≥
dx
t
x
0
(3.9)
= c∗ t−k
for t > 0.
Theorem 3.6. Let FW be a weigthed distribution function with increasing weight
function W (x) ≥ δ ∗ > 0. If F is an IHRA distribution, then FW is an IHRA
distribution.
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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Proof. Let F be an IHR distribution, and W (x) ≥ δ ∗ > 0 be increasing. We
α
show that F W (αx) ≥ F W (x) for all 0 < α ≤ 1, and x ≥ 0.
Clearly,
F W (αx) ≥ F (αx)
for all 0 ≤ α ≤ 1, and x ≥ 0. This follows from the fact that FW and F are
stochastically ordered. For all 0 ≤ α ≤ 1, and x ≥ 0, set
α
α
h(x) = F (x) − F W (x),
∗
and δ = E(W (X)). Simple computation yeilds
W (x)f (x)
α−1
α−1
0
h (x) = αF W (x)
− αF
(x)f (x)
E(W (X))
W (x)
α−1
(3.10)
≥ αF
(x)f (x)
−1 .
E(W (X))
α−1
Since αF
(x)f (x) ≥ 0, W (x) ≥ δ ∗ and h(0) = 0, thus h(x) is increasing
and h(x) ≥ 0. Using the fact that F is an IHRA distribution we get
(3.11)
α
F W (αx) ≥ F (αx) ≥ F (x)
for all 0 ≤ α ≤ 1, and x ≥ 0. Since h(x) ≥ 0 for all x ≥ 0, it follows therefore
that
α
F W (αx) ≥ F W (x),
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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for all 0 ≤ α ≤ 1, and x ≥ 0. The proof is complete.
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Theorem 3.7. Let GW and HW be two weighted distribution functions with
increasing weight functions. Suppose the conditions of the previous theorem
are satisfied, then the convolution of GW and HW , GW ∗ HW , is an IHRA
distribution.
Proof. The proof follows directly from [2].
Theorem 3.8. Let FW be a weighted distribution function with increasing weight
function W (x) and pdf fW (x) > 0 for x ≥ x0 . Suppose the hazard function
λFW (x) is such that λFW (x) ≥ xc for x ≥ x0 , where c is a real positive number.
If X is the original random variable then
Some Inequalities and Bounds
for Weighted Reliability
Measures
P (X − x ≤ xt|X > t) ≤ 1 − (1 + t)−c
Broderick O. Oluyede
for all t > 0 and x ≥ x0 .
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Proof. Let W (x) be increasing in x, then the hazard function of the distribution
function F of X satisfies the inequality
c
λF (x) ≥ λFW (x) ≥
x
for x ≥ x0 . Now for t > 0,
Z (1+t)x
Z
λF (y)dy ≥
x
(3.12)
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(1+t)x
λFW (y)dy
x
Z (1+t)x 1
≥c
dy ≥ 1 − (1 + t)−1 ,
y
x
for t > 0, using the fact that ln(a) ≥ 1 − a−1 for a > 0.
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The last result provides simple inequalities for the lower bound of the residual life time distributions for large values of x from the use of the information
about the hazard function of the weighted distribution function.
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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4.
Inequalities for Repairable Systems
In this section we obtain useful inequalities for repairable systems under weighted
distributions. Let {Xi }∞
i=1 be a sequence of operating times from a repairable
system that start functioning at time t = 0. The sequence of times {Xi }∞
i=1 form
a renewal-type stochastic point process. Following [4], if a system has virtual
age Tm−1 = t immediately after the (m − 1)th repair, then the length of the mth
cycle Xm has the distribution
(4.1)
Ft (x) = P (Xm ≤ x|Tm−1 = t) =
{F (x + t) − F (t)}
,
F (t)
x ≥ h0, where Fi (x) = 1−F (x) is the reliability function of a new system. When
Pj
t=
i=1 Xi , j = 1, 2, . . . , m − 1, minimal repair is performed, keeping the
virtual age intact and when t = 0 we have perfect repair. The virtual age of
the system is equal to its operating time for the case of minimal repair. The
reliability function corresponding to (4.1) is given by
(4.2)
F t (x) =
F (x + t)
,
F (t)
x ≥ 0. The weighted reliability function corresponding to (4.2) is given by
(4.3)
x ≥ 0.
F W (x + t)
F Wt (x) =
,
F W (t)
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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Theorem 4.1. If F Wt (x) is an IHR reliability function with increasing weight
function. Then
Z ∞
µ
2
−x/µ
|dx ≤ 2µ 1 − 2 .
(4.4)
|F Wt (x) − e
2µ
0
Proof. Let A = {x|F Wt ≤ e−x/µ }. Then we have for x ≥ 0,
Z ∞
Z
−x/µ
|F Wt (x) − e
|dx ≤ 2 (e−x/µ − F Wt (x))dx
0
ZA∞
≤2
(e−x/µ − F Wt (x))dx
0Z ∞
−x/µ
≤2
(e
− F W (x + t))dx
Z0 ∞
−x/µ
≤2
(e
− F (x + t))dx
0
Z ∞
S1 (x + t)
−x/µ
≤2
e
−
dx
µ
0
µ2
=2 µ−
2µ
µ2
(4.5)
= 2µ 1 − 2 .
2µ
The first two inequalities are straightforward,the third inequality follows from
the fact that W (x) is increasing, so that F W and F are stochastically ordered.
The fourth and fifth inequalities follow from Theorem 3.1.
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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Theorem 4.2. If F Wt (x) is an DHR reliability function,
Z ∞
(4.6)
|F Wt (x) − e−x/µ |dx ≥ 2µe−/µ |e−t/µ − 1|,
0
provided W (x) is an increasing weight function.
Proof. Let F Wt be a DHR survival function, then there exist ≥ µ such that
F Wt ≤ e−x/µ or F Wt ≥ e−x/µ as x ≤ or x ≥ . Now,
Z ∞
Z ∞
−x/µ
|F Wt (x) − e
|dx = 2
(F Wt (x) − e−x/µ )dx
0
Z ∞
≥2
(F W (x + t) − e−x/µ )dx
Z ∞
≥2
(F (x + t) − e−x/µ )dx
= 2{S1 ( + t) − µe−/µ }
≥ 2µS0 ( + t) − µe−/µ
(4.7)
= 2µe−/µ (e−/µ − 1).
The first inequality is trivial. The second inequality follows from the fact that
W (x) is increasing, so that F W (y) ≥ F (y) for all y ≥ 0. The last inequality
follow from Theorem 3.1.
Theorem 4.3. If F We (x) is an IHR reliability function with increasing weight
function. Then
Z ∞
µ
2
−x/µ
(4.8)
|F We (x) − e
|dx ≤ 2µ 1 − 2 .
2µ
0
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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Proof. Let D = {x|F We ≤ e−x/µ }. Then we have for x ≥ 0,
Z ∞
Z
−x/µ
|dx ≤ 2 (e−x/µ − F We (x))dx
|F We (x) − e
0
ZD∞
≤2
(e−x/µ − F We (x))dx
Z0 ∞ Z ∞
−1
−x/µ
=2
F W (y))dy dx
e
− µF W
0
x
Z ∞
Z ∞
−1
−x/µ
≤2
e
− µF W
F (y))dy dx
0
(4.9)
x
2S2 (0)
= 2µ −
µ
µ2
= 2µ −
µ
µ2
= 2µ 1 − 2 .
2µ
The first two inequalities are straightforward,the third inequality follows from
the fact that W (x) is increasing, so that F We and F e are stochastically ordered.
Theorem 4.4. If F We (x) is an DHR reliability function,
2 Z ∞
µ2 −x/µ
−/µ µ
(4.10)
|F We (x) − e
|dx ≥ 2µe
µ2 − µ ,
0
provided W (x) is an increasing weight function.
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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Proof. Using the fact that F We have DHR survival function, we have, for ≥ µ
Z ∞
Z ∞
−x/µ
|dx = 2
(F We (x) − e−x/µ )dx
|F We (x) − e
0
Z ∞ Z ∞
−1
−x/µ
=2
F (y)dy − e
dx
µF W
x
Z ∞
Z ∞
−1
−x/µ
F (y)dy − e
dx
≥2
µF W
x
Z ∞
2
(S1 (x) − µFW e−x/µ )dx
=
µF W 2
=
S2 () − 2µFW (µe−/µ )
µF W
2µ
≥
S1 () − 2µFW (µe−/µ )
µF W
2
2µ
≥
S0 () − 2µFW (µe−/µ )
µF W
µ2 −/µ
3 −/µ
= 2µ e
−2
e
µ
2
µ2
µ
−/µ
(4.11)
−
,
= 2µe
µ2
µ
R∞
where µFW = 0 F W (x)dx. The inequalities follows from the fact that W (x)
is increasing, so that F W (y) ≥ F (y) for all y ≥ 0, and Sk (x) ≥ Sk−1 (x) for all
x ≥ 0, k ≥ 1.
Some Inequalities and Bounds
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5.
Selection of Experiments
In this section, we deal with the problem of sampling and selection of experiments from weighted distribution as opposed to the original distribution. The
results also extends to any general competing distribution functions F and G
with probability density functions f and g respectively. For this purpose we
consider two probability spaces (Ω, Ψ, ν1 ) and (Ω, Ψ, ν2 ) such that the probability measures ν1 and ν2 are absolutely continous with respect to each other.
Let λ be a probability measure defined on Ψ and equivalent to ν1 and ν2 . Suppose f (x) and g(x) are Radon-Nikodym derivatives of ν1 and ν2 with respect to
λ.
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
Theorem 5.1. Let
1. B1 (x; η) = min
f (x)
gW (x)
− η, 0 ,
gW (x)
f (x)
− η, 0 ,
and
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2. B2 (x; c) = min
and suppose f (x) and gW (x) satisfy,
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Pf (x : gW (x) = 0) = PgW (x : f (x) = 0),
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B(ηf, gW ) ≤ B(f, ηgW )
Page 21 of 27
if and only if
EgW {B1 (x, ; η)} ≤ Ef {B2 (x; η)},
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where Ef and EgW are expectations with respect to the probability density functions f and gW , respectively, gW (x) = W (x)f (x)/E(W (X)), W (x) is increasing in x, and
Z
B(f, gW ) = max{f (x), gW (x)}dλ(x).
Proof. Note that
Z
B(f, ηgW ) =
Z
f (x) dλ(x) +
{x:ηgW (x)≤f (x)}
ηgW (x) dλ(x).
{x:ηgW (x)>f (x)}
Similarly,
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
Z
Z
B(ηf, gW ) =
ηf (x) dλ(x) +
{x:ηf (x)>gW (x)}
gW (x) dλ(x).
{x:ηf (x)≤gW (x)}
Note that,
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Z
B(ηf, gW ) − B(f, ηgW ) =
{ηgW (x) − f (x)} λ(x)
{x:ηgW (x)>f (x)}
Z
−
{ηf (x) − gW (x)} dλ(x)
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{x:ηf (x)>gW (x)}
Z
{(f (x)/gW (x)) − η}gW (x) dλ(x)
=
{x:ηgW (x)>f (x)}
Z
−
{(gW (x)/f (x)) − η}f (x)dλ(x)
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{x:ηf (x)>gW (x)}
(5.1)
= EgW {B1 (x, η)} − Ef {B2 (x, η)}.
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Consequently,
B(ηf, gW ) ≤ B(f, ηgW )
if and only if
EgW {B1 (x, η)} ≤ Ef {B2 (x, η)}.
Now let us consider the case in which one of the two hypothesis is true. The
hypothesis are H0 : X ∼ f and Y ∼ gW , and H1 : X ∼ gW and Y ∼ f .
Let RX(Y ) (πi ) be the Bayes risk when X(Y ) is performed and πi the prior
probability that Hi is true, i = 0, 1. We obtain the following comparisons of gW
and f.
Theorem 5.2. B(ηf, gW ) ≤ B(f, ηgW ) if and only if RX (π0 ) ≤ RY (π0 ), where
η = α0 π0 /α1 π1 , π1 > 0, and αi > 0 is the loss if Hi is true and is not accepted.
Proof. Let D = {x : (α0 π0 /α1 π1 )f (x) ≤ gW (x)} and di the Bayes decision to
accept Hi , then
RX (π0 ) = α0 π0 P (d1 |H0 ) + α1 π1 P (d0 |H1 )
Z
Z
f (x)dλ(x)
= α0 π0
gW (x)dλ(x) + α1 π1
D
(5.2)
= B(α0 π0 gW , α1 π1 f )
= α1 π1 B(gW , ηf ).
Dc
Some Inequalities and Bounds
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Also,
RY (π0 ) = α1 π1 B(f, ηgW ),
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so that
B(ηf, gW ) ≤ B(f, ηgW )
if and only if
RX (π0 ) ≤ RY (π0 ).
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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6.
Applications
Example 6.1. Let
 α−1 α
x β −x/β


e
if x > 0, α > 0, β > 0

Γ(α)
f (x; α, β) =



0
otherwise.
Then µ = αβ and µ2 = α(α + 1)β 2 and the hazard rate is increasing for α ≥ 1
and decreasing for α ≤ 1. If W (x) = x, then the weighted pdf is given by
 α α+1
x β


e−x/β if x > 0, α > 0, β > 0

Γ(α + 1)
fW (x; α, β) =



0
otherwise.
Applying Theorem 4.1, for α > 1, we have
Z ∞
α(α
+
1)
−x/αβ
= β|α − 1|.
(6.1)
|F Wt (x) − e
|dx ≤ 2αβ 1 −
2α2
0
With β = 1/2,
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Z
(6.2)
Some Inequalities and Bounds
for Weighted Reliability
Measures
C(α) =
0
∞
|F Wt (x) − e−2x/α |dx ≤
|α − 1|
.
2
Similarly, for α < 1 and β = 1/2, we have
Z ∞
|α − 1|
(6.3)
C(α) =
|F We (x) − e−2x/α |dx ≤
.
2
0
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Note that µFW = β(α + 1). Consequently, for α < 1 and β = 1/2, we get
Z ∞
1
−2x/α
−2/α |dx ≥ 2αe
|F Wt (x) − e
2(1 + α) .
0
Example 6.2. Let
f (x; β) =
 −1/2 −1 −x/β
β e
if x > 0
 2π

0
otherwise.
The corresponding weighted pdf with W (x) = x is given by

 2xβ −2 e−x/β if x > 0
gW (x; β) =

0
otherwise.
Note that, by Theorem 5.2 and for β ≥ η ≥ 1, B(ηf, gW ) ≤ B(f, ηgW ) and
RX (π0 ) ≤ RY (π0 ). Consequently, the experiment with lower Bayes risk is selected.
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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References
[1] R. BARLOW, A.W. MARSHALL AND F. PROSCHAN, Properties of probability distributions with monotone hazard rate, Ann. Math. Statist., 34
(1963), 375–389.
[2] H.W. BLOCK AND T.H. SAVIT, The IFRA problem, Annals of Probab.,
4(6) (1976), 1030–1032.
[3] R.C. GUPTA AND J.P. KEATING, Relations for reliability measures under
length biased sampling, Scan. J. Statist., 13 (1985), 49–56.
[4] M. KAJIMA, Some results for repairable systems with general repair, J.
Appl. Probab., 26 (1989), 89–102.
[5] G.P. PATIL AND C.R. RAO,Weighted distributions and size-biased sampling with applications to wildlife and human families, Biometrics, 34
(1978), 179–189.
[6] S.M. ROSS, Stochastic Processes, Bell System Wiley, New York, (1983).
Some Inequalities and Bounds
for Weighted Reliability
Measures
Broderick O. Oluyede
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