1. INTRODUCTION
As is well known, lifetime distributions can be classified by
stochastic ordering. For various definitions of stochastic orderings, we
refer the reader to Shaked and Shanthikumar [13]. For some definitions of classes of life
distributions, see Barlow and Proschan [1]. Based on the increasing concave ordering,
Deshpande, Kochar, and Singh [5]
suggested the NBU(2) (new better than used of second order) class. Based
on the increasing convex ordering, Cao and Wang [4] proposed the NBUC (new better than used in
convex ordering) class. These two classes are extensions of the NBU (new
better than used) class.
Properties of these two classes of lifetime distributions have been
studied by various authors. For details, see Li, Li, and Jing [9], Li and Kochar [8], Franco, Ruiz, and Ruiz [6], Hu and Xie [7], and Pellerey and Petakos [12] and references therein. In those articles,
preservations of these two classes of lifetime distributions under various
operations have been studied. In this article, we give some integral
inequality characterizations of the NBUC and NBU(2) classes. These
characterizations are, in a sense, similar to that given in Block and
Savits [3] for the IFRA (increasing
failure rate average) class. These characterizations might be useful in
various situations. In Block, Li, and Savits [2], inequalities of this type are used to obtain
preservation under mixture results analogous to Lynch [11] for certain classes of lifetime
distributions. In Li [10], a certain
kind of integral inequality characterization of NBU(2) is used to prove
that the NBU(2) class is closed under the formation of parallel systems.
Here, again, we give some preservation under mixture results for the NBUC
and NBU(2) classes as an application of the new characterizations. The
preservation results are then used to provide new proofs for the closure
under convolution property of the NBUC and NBU(2) classes that were first
obtained in Hu and Xie [7].
2. MAIN RESULTS
Generally, we will use F to denote the cumulative
distribution function of a lifetime random variable. Then F = 1
− F is its survival function. The NBUC class and NBU(2)
class are defined as follows.
Definition 2.1: F is said to belong to the NBUC class if
for all a, b ≥ 0, we have
Definition 2.2: F is said to belong to the NBU(2) class
if for all a, b ≥ 0, we have
Consider a family of lifetime distributions
{F(t|θ) : θ ≥ 0}. Let M
be the mixing distribution and
FM(t) =
∫F(t|θ) dM(θ) be the
mixed distribution. We will first present our new characterizations of the
two classes and then apply them to obtain preservation under mixture
results for the NBUC and NBU(2) classes. As an application of the
preservation results, we will rederive the known result that these two
classes are closed under convolution.
2.1. NBUC
The new characterization of the NBUC property is given by the
following lemma.
Lemma 2.1: F is NBUC iff
holds for all nonnegative, nondecreasing functions g, any
nonnegative, nondecreasing convex function h, and any measurable
function
.
Proof: First, we show the sufficiency. Let
,
and take α(t) = (a/t) ∧
(a/(a + b)). Note that
α(t)t ≥ a iff t ≥ a
+ b, and, in this case, we have h((1 −
α(t))t) = t − a −
b. Therefore,
On the other hand, the right-hand side is
Therefore, we obtain the inequality
Replace a with a + ε in the above inequality and
let ε → 0+. This yields
that is, F is NBUC.
Next, we show the necessity. Suppose that F is NBUC. We first
consider the case
.
Among all the functions
,
the function defined by α0(t) =
(a/t) ∧ (a/(a +
b)) will make the integrand
g(α(t))h((1 −
α(t))t) the largest possible since it is not zero
iff α(t)t ≥ a and (1 −
α(t))t > b, and in this case, to make
the value the largest possible, α(t) should be as small as
possible. From the proof of the sufficiency part, it is easy to see
that
for any
.
Thus, if we let
where gi ≥ 0, 0 ≤
a1 ≤ ··· ≤
an < +∞,
hi ≥ 0, and 0 ≤ b1
≤ ··· ≤ bm <
+∞, then it follows that
Thus, we have shown that the result is correct for functions of the
above form. Since any nonnegative, nondecreasing function is a
nondecreasing limit of functions like g and any nonnegative,
nondecreasing convex function is a nondecreasing limit of functions like
h, it follows from the Lebesgue monotone convergence theorem that
the inequality holds. █
Corollary 2.1: If F is NBUC, then
holds for all nonnegative, nondecreasing functions g, any
nonnegative, nondecreasing convex function h, and any constant
α ∈ [0,1].
Next, we consider the problem of preservation under mixture of the
NBUC class. Analogous to Lynch [11],
we require two conditions:
for all a,b,θ ≥ 0 and some
measurable function α : [0,+∞) × [0,+∞)
× [0,+∞) → [0,1].
(NBUC2) The mixing distribution M(θ) is
NBUC.
Remark: Note, by (NBUC1),
that is, F(t|θ) is an NBUC
distribution for each θ ≥ 0.
Theorem 2.1: Let
{F(t|θ),θ ≥ 0} and M
satisfy (NBUC1) and (NBUC2). Then
FM is NBUC. Conversely, if
FM is NBUC whenever
{F(t|θ),θ ≥ 0} satisfies
(NBUC1), then M is NBUC.
Proof: The converse part follows by taking
F(t|θ) =
I[0,θ+ε)(t) with ε > 0.
Then F(t|θ) is increasing in
is convex in θ, and
where α(a,b,θ) = a ∧
θ/θ. Therefore, F(t|θ)
satisfies (NBUC1). Since FM is NBUC, it
follows that M(t) =
FM(t + ε) satisfies
for all a ≥ ε (if we define M(t) =
1 for t < 0, this is true for all a ≥ 0). By
letting ε → 0+, we conclude by the Lebesgue dominated convergence
theorem that
Since
,
we have
Now, replace the a above with a + ε and let
ε → 0+. Thus,
that is, M is NBUC.
The other part of the proof is a direct application of Lemma 2.1. In
fact,
In the second inequality, we applied Lemma 2.1 since
M(θ) is NBUC,
is nonnegative, increasing, and convex in θ and
F(t|θ) is increasing in θ for each
t. Thus, FM is NBUC. █
As an application of the above mixture preservation result, we
rederive the preservation of NBUC under the convolution result of Hu and
Xie [7].
Corollary 2.2: Suppose that F1 and
F2 are two NBUC lifetime distributions. Then their
convolution is also NBUC.
Proof: The survival function F of the convolution
is given by
Then, by Fubini's theorem,
By the NBUC property of F1, for s <
a,
Let α(a,s) = (s ∧
a)/s. Then, for all s ≥ 0, we have
Note that F1(t) is decreasing in
t, and, therefore,
is an increasing, convex function of s. The desired result
now follows as an application of Theorem 2.1. █
2.2. NBU(2)
The new characterization of the NBU(2) property is given by the
following lemma.
Lemma 2.2: F is NBU(2) iff
for all nonnegative, nondecreasing functions g, any nonnegative,
nondecreasing concave function h, and any measurable function
.
Proof: First, we show the sufficiency. Let
g(t) =
1[a,+∞)(t) and
h(t) =
,
and take α(t) = (a ∧
t)/t. We note that α(t)t ≥
a iff t ≥ a, and in this case, we have
h((1 − α(t))t) = (t −
a) ∧ b. Then
On the other hand,
Therefore, we must have
Replace a with a + ε in the above inequality and
let ε → 0+ to conclude that
that is, F is NBU(2).
Next, we show the necessity. Suppose that F is NBU(2). First,
we consider the case
.
Among all the functions
,
the function defined by α0(t) = (a ∧
t)/t will make the integrand
g(α(t))h((1 −
α(t))t) for the above-defined g and
h the largest possible since it is not zero iff
α(t)t ≥ a, and in this case, to make the
value the largest possible, α(t) should be as small as
possible. Then from the proof of the sufficiency part, it is easy to see
that
for any
.
Thus, if we let
where gi ≥ 0, 0 ≤
a1 ≤ ··· ≤
an < +∞, and
hi ≥ 0, 0 ≤ b1 ≤
··· ≤ bm <
+∞, then it follows that
Thus we have shown that the result is true for functions of the above
form. Since any nonnegative, nondecreasing function is a nondecreasing
limit of functions like g and any nonnegative, nondecreasing
concave function is a nondecreasing limit of functions like h, it
follows from the Lebesgue monotone convergence theorem that the result is
true. █
Corollary 2.3: If F is NBU(2), then
for all nonnegative, nondecreasing functions g, any nonnegative,
nondecreasing concave function h, and any constant α ∈
[0,1].
Next, we consider the mixture preservation problem of NBU(2). For the
NBU(2) preservation result, we require the following:
for all a,b,θ ≥ 0 and some
measurable function α : [0,+∞) × [0,+∞)
× [0,+∞) → [0,1].
(NBU(2)2) The mixing distribution M(θ) is
NBU(2).
Remark: By (NBU(2)1), it is easy to see that
that is, F(t|θ) is an NBU(2)
distribution for each θ ≥ 0.
Theorem 2.2: Let
{F(t|θ),θ ≥ 0} and M satisfy
(NBU(2)1) and (NBU(2)2). Then FM is NBU(2).
Conversely, if FM is NBU(2) whenever
{F(t|θ),θ ≥ 0} satisfies
(NBU(2)1), then M is NBU(2).
Proof: The converse part follows by taking
F(t|θ) =
1[0,θ+ε)(t). Then,
F(t|θ) is increasing in θ for each
t and
is concave in θ. Moreover, for α(a,b,θ) =
a ∧ θ/θ,
Therefore, F(t|θ) satisfies (NBU(2)1).
By our assumption, FM is NBU(2). Since
M(t) = FM(t +
ε) in this case, we conclude that for a ≥ ε,
By letting ε → 0+, we have
Since
,
we have
Now, replace a in the above inequality with a +
ε and let ε → 0+. Thus,
that is, M is NBU(2).
The other part of the proof is a direct application of Lemma 2.2. In
fact,
In the second inequality, we have used Lemma 2.2, noting that
F(t|θ) is nondecreasing in θ,
M(θ) is NBU(2), and
is nonnegative, nondecreasing, and concave in θ. Thus,
FM is NBU(2). █
As an application of the above mixture preservation result, we give
another proof of the preservation of NBU(2) under the convolution result.
This result appeared previously in Hu and Xie [7].
Corollary 2.4: Suppose that F1 and
F2 are two NBU(2) lifetime distributions. Then their
convolution is also NBU(2).
Proof: Analogous to the proof of Corollary 2.2, we let
α(a,s) = (s ∧
a)/s. Consider
Since F1 is NBU(2), we have
Now, note that F1(t) is decreasing in
t and that
is an increasing, concave function of s. The result then
follows as a direct application of Theorem 2.2. █
