Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 3, Issue 4, Article 60, 2002
SOME INEQUALITIES AND BOUNDS FOR WEIGHTED RELIABILITY
MEASURES
BRODERICK O. OLUYEDE
D EPARTMENT OF M ATHEMATICS AND C OMPUTER S CIENCE
G EORGIA S OUTHERN U NIVERSITY
S TATESBORO , GA 30460.
boluyede@gasou.edu
Received 22 October, 2001; accepted 17 July, 2002
Communicated by N.S. Barnett
A BSTRACT. 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.
Key words and phrases: Reliability inequalities, Stochastic Order, Weighted distribution functions, Integrable function.
2000 Mathematics Subject Classification. 39C05.
1. I NTRODUCTION
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)
ISSN (electronic): 1443-5756
c 2002 Victoria University. All rights reserved.
The author expresses his gratitude to the editor and referee for many useful comments and suggestions.
071-01
2
B RODERICK O. O LUYEDE
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 λ
R F∞(x) and mean 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 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.
2. S OME BASIC R ESULTS AND U TILITY N OTIONS
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
F (t)W 0 (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 )).
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)
J. Inequal. Pure and Appl. Math., 3(4) Art. 60, 2002
α
F (αx) ≥ F (x).
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W EIGHTED R ELIABILITY M EASURES
3
It is well known that F is an IHRA distribution if and only if
1/α
Z
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
(2.6)
F e (x) = µ
F (y)dy,
x
x ≥ 0. The corresponding limiting pdf is
F (x)
,
µF
fe (x) =
(2.7)
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)
−1
δFWe (x) = (δF (x)F (x))
Z
∞
F (y)δF (y)dy,
x
x ≥ 0.
3. I NEQUALITIES FOR W EIGHTED R ELIABILITY M EASURES
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.
J. Inequal. Pure and Appl. Math., 3(4) Art. 60, 2002
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B RODERICK O. O LUYEDE
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).
Theorem 3.2. Let F We (x) be an IHR weighted equilibrium reliability function with increasing
weight function. Then
Z ∞
2 µ2 −x/µ
.
(3.1)
|dx ≤ 2 µ −
|F We (x) − xe
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|},
0
for η ≥ µ.
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
η
Z ∞
≥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
≥ 2S1 (η) − 2e−η/µ (µη + 1)
≥ 2µS0 (η) − 2e−η/µ (µη + 1)
(3.3)
J. Inequal. Pure and Appl. Math., 3(4) Art. 60, 2002
= 2e−η/µ (µ − ηµ − 1).
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W EIGHTED R ELIABILITY M EASURES
5
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 fact that Sk (x) ≥ µSk−1 (x) for
R∞
all x ≥ 0, k = 1, 2, . . . , where Sk (x) = x Sk−1 (y)dy.
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
(3.4)
F W (x) ≥ e
max 0,
+
− −
,
α
αµ
α 2µ
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)}
(3.5)
= (αS2 (0))−1 {µ(e−x/µ [W (x)S1 (0) + S2 (0)] − S1 (0) − S2 (0))}
n
µ2 µ2 o
−µ−
= 2(αµ)−1 e−x/µ W (x)µ +
2
2
2W (x)
µ2
2
µ2
= 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)
λFW (x) ≥
c
x
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
J. Inequal. Pure and Appl. Math., 3(4) Art. 60, 2002
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B RODERICK O. O LUYEDE
for all x > 0. Consequently,
Z t
Z t
1
1
−k
{δFW (x)} dx ≥
{λFW (x)}k dx
t
t
0
0
Z t 1
c k
≥
dx
t
x
0
= c∗ t−k
(3.9)
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.
Proof. Let F be an IHR distribution, and W (x) ≥ δ ∗ > 0 be increasing. We show that
α
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
(x)f (x)
h (x) = αF W (x)
− αF
E(W (X))
W (x)
α−1
(3.10)
≥ αF
(x)f (x)
−1 .
E(W (X))
α−1
(x)f (x) ≥ 0, W (x) ≥ δ ∗ and h(0) = 0, thus h(x) is increasing and h(x) ≥ 0.
Since αF
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),
for all 0 ≤ α ≤ 1, and x ≥ 0. The proof is complete.
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
P (X − x ≤ xt|X > t) ≤ 1 − (1 + t)−c
for all t > 0 and x ≥ x0 .
J. Inequal. Pure and Appl. Math., 3(4) Art. 60, 2002
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W EIGHTED R ELIABILITY M EASURES
7
Proof. Let W (x) be increasing in x, then the hazard function of the distribution function F of
X satisfies the inequality
c
x
λF (x) ≥ λFW (x) ≥
for x ≥ x0 . Now for t > 0,
Z
(1+t)x
Z
(1+t)x
λF (y)dy ≥
(3.12)
x
Z
(1+t)x
λFW (y)dy ≥ c
x
x
1
dy ≥ 1 − (1 + t)−1 ,
y
for t > 0, using the fact that ln(a) ≥ 1 − a−1 for a > 0.
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.
4. I NEQUALITIES FOR R EPAIRABLE S YSTEMS
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
F (x) = 1 − F (x) is the reliability function of a new system. When t =
hP≥ 0, where
i
j
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
F t (x) =
(4.2)
F (x + t)
,
F (t)
x ≥ 0. The weighted reliability function corresponding to (4.2) is given by
F Wt (x) =
(4.3)
F W (x + t)
,
F W (t)
x ≥ 0.
Theorem 4.1. If F Wt (x) is an IHR reliability function with increasing weight function. Then
Z
∞
−x/µ
|F Wt (x) − e
(4.4)
0
J. Inequal. Pure and Appl. Math., 3(4) Art. 60, 2002
µ2 |dx ≤ 2µ 1 − 2 .
2µ
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B RODERICK O. O LUYEDE
Proof. Let A = {x|F Wt ≤ e−x/µ }. Then we have for x ≥ 0,
Z ∞
Z
−x/µ
|dx ≤ 2 (e−x/µ − F Wt (x))dx
|F Wt (x) − e
0
ZA∞
≤2
(e−x/µ − F Wt (x))dx
0Z ∞
−x/µ
≤2
(e
− F W (x + t))dx
0
Z ∞
−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.
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 increas
ing, 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 ∞
µ2 −x/µ
(4.8)
|F We (x) − e
|dx ≤ 2µ 1 − 2 .
2µ
0
J. Inequal. Pure and Appl. Math., 3(4) Art. 60, 2002
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W EIGHTED R ELIABILITY M EASURES
9
Proof. Let D = {x|F We ≤ e−x/µ }. Then we have for x ≥ 0,
Z ∞
Z
−x/µ
|F We (x) − e
|dx ≤ 2 (e−x/µ − F We (x))dx
0
ZD∞
≤2
(e−x/µ − F We (x))dx
Z0 ∞ Z ∞
−1
−x/µ
=2
e
− µF W
F W (y))dy dx
0
x
Z ∞
Z ∞
−1
−x/µ
≤2
F (y))dy dx
e
− µF W
0
x
2S2 (0)
µ
µ2
= 2µ −
µ
µ2
= 2µ 1 − 2 .
2µ
= 2µ −
(4.9)
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.
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
µF W
F (y)dy − e
dx
x
Z ∞
Z ∞
−1
−x/µ
≥2
µF W
F (y)dy − e
dx
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
µ
µ2
−/µ
= 2µe
(4.11)
−
,
µ2
µ
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10
B RODERICK O. O LUYEDE
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.
5. S ELECTION OF E XPERIMENTS
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.
Theorem 5.1. Let
f (x)
(1) B1 (x; η) = min gW (x) − η, 0 ,
and
gW (x)
(2) B2 (x; c) = min f (x) − η, 0 ,
and suppose f (x) and gW (x) satisfy,
Pf (x : gW (x) = 0) = PgW (x : f (x) = 0),
then
B(ηf, gW ) ≤ B(f, ηgW )
if and only if
EgW {B1 (x, ; η)} ≤ Ef {B2 (x; η)},
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
Z
B(f, ηgW ) =
f (x) dλ(x) +
{x:ηgW (x)≤f (x)}
ηgW (x) dλ(x).
{x:ηgW (x)>f (x)}
Similarly,
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,
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)
{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)
{x:ηf (x)>gW (x)}
(5.1)
= EgW {B1 (x, η)} − Ef {B2 (x, η)}.
J. Inequal. Pure and Appl. Math., 3(4) Art. 60, 2002
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W EIGHTED R ELIABILITY M EASURES
11
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
= α0 π0
gW (x)dλ(x) + α1 π1
f (x)dλ(x)
D
Dc
= B(α0 π0 gW , α1 π1 f )
(5.2)
= α1 π1 B(gW , ηf ).
Also,
RY (π0 ) = α1 π1 B(f, ηgW ),
so that
B(ηf, gW ) ≤ B(f, ηgW )
if and only if
RX (π0 ) ≤ RY (π0 ).
6. A PPLICATIONS
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
(6.1)
|F Wt (x) − e
|dx ≤ 2αβ 1 −
= β|α − 1|.
2α2
0
J. Inequal. Pure and Appl. Math., 3(4) Art. 60, 2002
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12
B RODERICK O. O LUYEDE
With β = 1/2,
Z
(6.2)
∞
C(α) =
|F Wt (x) − e−2x/α |dx ≤
0
|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
Note that µFW = β(α + 1). Consequently, for α < 1 and β = 1/2, we get
Z ∞
1
−2x/α
−2/α |F Wt (x) − e
|dx ≥ 2αe
2(1 + α) .
0
Example 6.2. Let
f (x; β) =

 2π −1/2 β −1 e−x/β if x > 0
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.
R EFERENCES
[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).
J. Inequal. Pure and Appl. Math., 3(4) Art. 60, 2002
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