Stochastic Orders and Ageing Classes

Stochastic Orders and Ageing Classes

Abstract This article covers the knowledge of the recently used stochastic orders, age- ing classes, and relations between them. Definitions and relations between particular stochastic orders are presented here. The paper also contains definitions (especially using stochastic orders), relations and closure properties of ageing classes. Closure properties are investigated here under five useful reliability operations: convolutions, mixtures, formation of monotonic systems, maxima, and minima.

2010 Mathematics Subject Classification: 60–02, 62–02, 60E15, 62N05.

Key words and phrases: stochastic orders, aging classes, closure properties, convolu- tion, mixture, monotonic system.

1. Introduction. Reliability theory describes the random times of working of devices which are susceptible to failures. For this type of research, specific concepts are developed, such as the residual life distribution function, the failure rate function, and the mean residual life function. Their definitions are given in the section 2. It also contains the basic definitions and facts about monotonic systems. Monotonic systems create a very simple and convenient model. It describes operating of devices composed of arranged parts. It is a rapidly developing theory, being used in many areas of science. For more information, see e.g. [2], [13].

Various stochastic orders, that is the orders on the sets of probability distribu- tions, are useful in many fields of probability and mathematical statistics. Many about the orders can be learned from the book [14]. In the section 3 of this article, only these orders that are most helpful in working with the ageing classes are defined and compared. It is not a suprise that this criterium is fulfilled most of all by the simple, classic orders. Relations between the introduced orders are summarized in Figure 1.

In the section 4, some of the ageing classes (both those known for decades and

some of those introduced relatively recently) are described. Emphasis was placed on

two issues: the relationship between classes and the closedness of each class under

the most frequently used reliability operations: convolutions, mixtures, and creating

monotonic systems. Classes closed under the maxima and minima are also mentioned

there. The results are summarized in Figure 2 and in Table 2.

Everyone interested in applying presented concepts will surely welcome the ac- cessible and reliable book [10].

2. Preliminaries.

2.1. Basic notations and terminology. Let X and Y be nonnegative random variables with probability distribution functions F and G, respectively. We identify variables with their distribution functions. Denote by S _F the support of F , by F = 1 − F the tail, by F ⁻¹ (p) = inf{x : F (x) ­ p}, p ∈ (0, 1), the quantile function, by F ⁻¹ (0) and F ⁻¹ (1) the lower and upper bounds of S _F , respectively, and put I _F := 0, F ⁻¹ (1)
; for G analogously. Denote the density functions of F and G by f and g, respectively, if they exist. We assume that F (0) = G(0) = 0 and the expectation values of X and Y (denoted by µ F and µ G , respectively) are finite.

We denote by ψφ the composition of functions φ and ψ, i.e. ψφ(x) = ψ φ(x)
. In this paper ’increasing’ means ’nondecreasing’ and ’decreasing’ means ’non increas- ing’.

Definition 2.1 The residual life distribution function of F is given by F (x|u) = F (x + u)

F (u) for u ∈ I _F , x ∈ [0, ∞).

Definition 2.2 Assume that F is absolutely continuous. The failure rate function of F is given by r _F (x) = ^{f (x)}

F (x) for x ∈ 0, F ⁻¹ (1)
. Absolute continuity is not necessary to define r F (0) = lim x→0 − ln F (x)

x .

Definition 2.3 The mean residual life function of F is given by m F (x) =

R ∞ x F (t)dt

F (x) for x ∈ I F .

We omit the subscripts of µ, r(x) and m(x) whenever it is obvious which distribution function we think of. Functional relations between F (x), F (x|u), r(x) and m(x) are presented in Table 1. We assume there differentiability if it is needed.

Note that F (x|0) = F (x) and m(0) = µ.

There is one more handy function. It is based on the distribution function instead of the tail function.

Definition 2.4 Assume that F is absolutely continuous. The reverse failure rate function of F is given by ˘ r F (x) = _{F (x)} ^{f (x)} = ln F (x)
0

for x ∈ S F .

We will also use four well-known operations: convolution, mixture, maximum, and

minimum. We recall their definitions and basic properties.

as function of F (x) as function of F (x|u)

F (x) = F (x) F (x|0)

F (x|u) = ^{F (x+u)}

F (u) F (x|u)

r(x) = − ln F (x)
0

− ln F (x|0)
0

m(x) = R ∞

0

F (t+x)

F (x) dt R ∞

0 F (t|x)dt as function of r(x) as function of m(x) F (x) = e ⁻

R

0

r(t)dt µ

m(x) e ⁻ R

0 dt m(t)

F (x|u) = e ⁻ R

u

r(t)dt m(u) m(x+u) e ⁻

R

dt m(t)

r(x) = r(x) ^m _m(x)

^(x)+1

m(x) = R ∞ 0 e ⁻

R

r(u)du

dt m(x)

Table 1: Relations between F (x), F (x|u), r(x) and m(x)

Definition 2.5 Assume that X and Y are independent. The distribution function of random variable X + Y is called convolution of F and G and denoted by F ∗ G.

It is clear that F ∗ G = G ∗ F and (F ∗ G)(x) =

Z x 0

F (x − t)dG(t).

If F and G are absolutely continuous then F ∗ G is also absolutely continuous with the density function denoted by f ∗ g and given by

(f ∗ g)(x) = Z x

0

f (x − t)g(t)dt.

Definition 2.6 Let F = {F θ : θ ∈ Θ} be a family of distributions indexed by a parameter θ which takes values in a set Θ. Let H be a distribution function with support S H = Θ. The distribution function given by

F (x) = Z

Θ

F θ (x)dH(θ)

is called the mixture of F with respect to H and H is called the mixing distribution.

In this paper we consider only the mixtures of families of distributions having the same support.

Assume that X and Y are independent. Denote the distribution function of max{X, Y } and min{X, Y } by F ∨ G and F ∧ G, respectively. It is evident that F ∨ G = G ∨ F and

(F ∨ G)(x) = F (x)G(x). (1)

It is also obvious that F ∧ G = G ∧ F and

F ∧ G(x) = F (x)G(x). (2)

The principal distribution in studying life distributions is the exponential distribu- tion M λ (x) = 1 − e ^−λx , λ > 0. Nice properties of this distribution are gathered in the following lemma.

Lemma 2.7

• M (x) = e ^−λx , M ⁻¹ (p) = − _λ ¹ ln(1 − p), µ M = _λ ¹ ,

• M is absolutely continuous with density function x 7→ λe ^−λx ,

• M (x|u) = M (x), r M (x) = λ, m M (x) = λ, ˘ r _M (x) = _e

^λ −1 ,

• M λ ∗ M λ ∼ Gamma(2, λ), M λ

∧ M λ

= M λ

+λ

.

2.2. Monotonic systems. There is a rich literature on monotonic systems (see e.g. [2], [13]). In this article we just introduce the necessary mathematical definitions and facts, do not describe the ideas which hide behind them.

Fix n ∈ N.

Definition 2.8 Function ϕ : {0, 1} ⁿ → {0, 1} is called a monotonic system if i) ϕ(0, . . . , 0) = 0,

ii) ϕ(1, . . . , 1) = 1,

iii) ϕ is increasing in each of its arguments.

Theorem 2.9 Every monotonic system {0, 1} ⁿ 3 (c ₁ , . . . , c _n ) 7→ ϕ(c ₁ , . . . , c _n ) is a polynomial of c 1 , . . . , c n with integer coefficients.

Definition 2.10 Let X 1 , . . . , X n be pairwise independent nonnegative random variables with distributions F 1 , . . . , F _n , respectively. The reliability F of a mono- tonic system ϕ with components’ distributions F ₁ , . . . , F _n is given for x ∈ [0, ∞) by

F (x) = P 

ϕ I _[0,X

₎ (x), . . . , I _[0,X

₎ (x)
= 1  , where I [0,X

) is an indicator of set [0, X i ), i = 1, . . . , n.

Theorem 2.11 Every reliability of a monotonic system with components’ distribu- tions F 1 , . . . , F n is a polynomial of F 1 , . . . , F n with integer coefficients.

Theorem 2.12 Every reliability of a monotonic system is a tail function.

There are two very simple and useful monotonic systems.

Example 2.13 The monotonic system given by

ϕ(c 1 , . . . , c n ) = max{c 1 , . . . , c n } is called the parallel system. For this system

F (x) = 1 − F (x) = P 

ϕ I _[0,X

₎ (x), . . . , I _[0,X

₎ (x)
= 0 

=

= P I _[0,X

₎ (x) = 0, . . . , I _[0,X

₎ (x) = 0
=

= P I _[0,X

₎ (x) = 0
· . . . · P I [0,X

) (x) = 0
=

= F 1 (x) · . . . · F n (x), that is F = F ₁ ∨ . . . ∨ F _n .

Example 2.14 The monotonic system given by

ϕ(c 1 , . . . , c n ) = min{c 1 , . . . , c n } is called the series system. For this system

F (x) = P 

ϕ I _[0,X

₎ (x), . . . , I _[0,X

₎ (x)
= 1 

=

= P I _[0,X

₎ (x) = 1, . . . , I _[0,X

₎ (x) = 1
=

= P I _[0,X

₎ (x) = 1
· . . . · P I [0,X

) (x) = 1
=

= F 1 (x) · . . . · F n (x), that is F = F 1 ∧ . . . ∧ F n .

We see that if components have distributions F and G then F ∨ G and F ∧ G are reliabilities of two-element parallel and series system, respectively.

3. Some stochastic orders.

3.1. Definitions. There are a lot of various stochastic orders invented. In this paper we consider some of them which are useful in the investigation of ageing classes. All the following definitions are taken from [14].

Definition 3.1 F is said to be smaller than G in the likehood ratio order (denoted as F ¬ lr G) if

P (X ∈ A)P (Y ∈ B) ­ P (X ∈ B)P (Y ∈ A) for all measurable sets A, B ⊂ [0, ∞) such that

x ∈ A, y ∈ B ⇒ x ¬ y.

If F and G are absolutely continuous then, equivalently, function x 7→ g(x)

f (x)

is increasing on S F ∪ S G (here a/0 is taken to be equal to ∞ whenever a ∈ (0, ∞)).

Definition 3.2 F is said to be smaller than G in the hazard rate order (denoted as F ¬ hr G) if function

x 7→ G(x) F (x)

is increasing on I _F ∪ I _G (here a/0 is taken to be equal to ∞ whenever a ∈ (0, ∞)).

Equivalently,

F (x)G(y) ­ F (y)G(x) for all x, y ∈ [0, ∞), x ¬ y, or, assuming absolute continuity,

r F (x) ­ r G (x)

for all x ∈ I _F ∪ I _G (here r _F (x) is taken to be equal to ∞ whenever x ­ F ⁻¹ (1); for r _G analogously).

Definition 3.3 F is said to be smaller than G in the mean residual life order (denoted as F ¬ mrl G) if function

x 7→

R ∞ x G(t)dt R ∞

x F (t)dt

is increasing on I F ∪ I G (here a/0 is taken to be equal to ∞ whenever a ∈ (0, ∞)).

Equivalently,

Z ∞ x

F (t)dt Z ∞

y

G(t)dt ­ Z ∞

y

F (t)dt Z ∞

x

G(t)dt for all x, y ∈ [0, ∞), x ¬ y, or

m _F (x) ¬ m _G (x)

for all x ∈ I F ∪ I G (here m F (x) is taken to be equal to 0 whenever x ­ F ⁻¹ (1); for m G analogously).

Definition 3.4 F is said to be smaller than G in the usual stochastic order (de- noted as F ¬ st G) if

F (x) ¬ G(x) for all x ∈ [0, ∞). Equivalently,

F ⁻¹ (p) ¬ G ⁻¹ (p) for all p ∈ (0, 1), or

Eφ(X) ¬ Eφ(Y )

for all increasing functions φ : R → R for which the expectations exist.

Note the obvious fact.

Theorem 3.5 If F ¬ ^st G and F ­ st G then F = G.

Definition 3.6 F is said to be smaller than G in the dispersive order (denoted as F ¬ disp G) if function

p 7→ G ⁻¹ (p) − F ⁻¹ (p) is increasing on (0, 1).

If S F is a possibly infinite interval then, equivalently, function x 7→ G ⁻¹ F (x) − x

is increasing on S _F .

Definition 3.7 F is said to be smaller than G in the convex order (denoted as F ¬ cx G) if µ F = µ G and

Z ∞ x

F (t)dt ¬ Z ∞

x

G(t)dt

for all x ∈ [0, ∞). Equivalently,

Eφ(X) ¬ Eφ(Y )

for all convex functions φ : R → R for which the expectations exist.

Definition 3.8 F is said to be smaller than G in the increasing convex order (denoted as F ¬ icx G) if

Z ∞ x

F (t)dt ¬ Z ∞

x

G(t)dt

for all x ∈ [0, ∞). Equivalently,

Eφ(X) ¬ Eφ(Y )

for all increasing convex functions φ : R → R for which the expectations exist.

Definition 3.9 F is said to be smaller than G in the increasing concave order (denoted as F ¬ icv G) if

Z x 0

F (t)dt ¬ Z x

0

G(t)dt for all x ∈ [0, ∞). Equivalently,

Eφ(X) ¬ Eφ(Y )

for all increasing concave functions φ : R → R for which the expectations exist.

Note that all the foregoing orders can be defined in the same way for the arbitrary

(not necessarily nonnegative) random variables whereas the following cannot.

Definition 3.10 Assume that S ^F is a possibly infinite interval and S G = I G . F is said to be smaller than G in the convex transform order (denoted as F ¬ c G) if function

x 7→ G ⁻¹ F (x) is convex on S F .

Definition 3.11 Assume that S F is a possibly infinite interval and S G = I G . F is said to be smaller than G in the star order (denoted as F ¬ ∗ G) if function

x 7→ G ⁻¹ F (x) x is increasing on S F \{0}. Equivalently, function

p 7→ G ⁻¹ (p) F ⁻¹ (p) is increasing on (0, 1).

Definition 3.12 Assume that S F is a possibly infinite interval and S G = I G . F is said to be smaller than G in the superadditive order (denoted as F ¬ su G) if

G ⁻¹ F (x + y) ­ G ⁻¹ F (x) + G ⁻¹ F (y) for all x, y ∈ [0, ∞).

Definition 3.13 F is said to be smaller than G in the Laplace transform order (denoted as F ¬ Lt G) if Ee ^−sX ­ Ee ^−sY , i.e.

Z ∞ 0

e ^−st F (t)dt ¬ Z ∞

0

e ^−st G(t)dt for all s ∈ [0, ∞).

3.2. Relations. We can expect some connotations between orders. In fact, they are collected in Graph 1.

Relations F ¬ _c G ⇒ F ¬ _∗ G ⇒ F ¬ _su G are an immediate consequence of the following Lemma (see [4, p. 1207]).

Lemma 3.14 Let φ be a nonnegative continuous function which vanishes at the ori- gin. If φ is convex then φ is starshaped. If φ is starshaped then φ is superadditive.

The dashed arrows are a graphical presentation of the following obvious fact.

Theorem 3.15 If F ¬ ^cx G then F ¬ icx G. If F ­ cx G then F ¬ icv G.

The dotted arrows are a graphical presentation of implications which occur under

some additional assumptions (see [14, p. 214, 154]).

F ¬ c G  F ¬ _∗ G



F ¬ su G



F ¬ lr G



F ¬ disp G



F ¬ hr G

s{ 

F ¬ _cx G

 #+

F ¬ _st G

s{ 

F ¬ _mrl G

F ¬ icx G F ¬ icv G



F ¬ Lt G

Figure 1: Relations between orders

Theorem 3.16 Let F ¬ su G. If F ¬ _st G or lim

x→0

G

F (x)

x ­ 1 then F ¬ _disp G.

Theorem 3.17 Let F ¬ disp G. If F ⁻¹ (0) = G ⁻¹ (0) > −∞ then F ¬ st G. If F ⁻¹ (1) = G ⁻¹ (1) < ∞ then F ­ st G.

The other connotations in Graph 1 are clear.

It is interesting that Theorems 3.5 and 3.17 imply that if F ¬ disp G and F ⁻¹ (0) = G ⁻¹ (0) > −∞, F ⁻¹ (1) = G ⁻¹ (1) < ∞ then F = G.

4. Some ageing classes.

4.1. Definitions.

So far, a lot of ageing classes have been invented and the new ones are still being introduced. In this class we present the ILR and DLR classes (see [13, pp. 98–

103]) and the classes which can be found in [11]. They can be defined in numerous methods, especially by using stochastic orders. In the definitions we list the ways taken from [11], [13] and [14]. We do not present ageing classes in historical order, but from the ’strongest’ to the ’weakest’. Assumptions and conditions for the dual classes are written in brackets.

We denote by F (·|u) the distribution function of random variable [X − u|X > u],

i.e. F (x|u) = P (X − u ¬ x|X > u). Clearly 1 − F (x|u) = F (x|u).

Definition 4.1 Assume that S ^F is a possibly infinite interval [S F = [0, ∞)]. F is said to be ILR [DLR] (increasing [decreasing] likelihood ratio) if

F (·|u) ¬ _lr [­ _lr ]F (·|v) for all u, v ∈ I _F , u ­ v, or, equivalently,

F (·|u) ¬ lr [­ lr ]F for all u ∈ I F .

If F is absolutely continuous then, equivalently, function x 7→ ln f (x) is concave [convex] on S F .

Definition 4.2 Assume that S F is a possibly infinite interval [S F = [0, ∞)]. F is said to be IFR [DFR] (increasing [decreasing] failure rate) if it meets one of the following equivalent conditions:

• F ¬ c [­ c ]M λ for any λ > 0,

• F (·|u) ¬ hr [­ hr ]F (·|v) for all u, v ∈ I F , u ­ v,

• F (·|u) ¬ disp [­ disp ]F (·|v) for all u, v ∈ I F , u ­ v,

• F (·|u) ¬ st [­ st ]F (·|v) for all u, v ∈ I F , u ­ v,

• F (·|u) ¬ _hr [­ _hr ]F for all u ∈ I _F ,

• function x 7→ − ln F (x) is convex [concave] on S F ,

• function u 7→ F (x|u) is decreasing [increasing] on I F for all x ∈ [0, ∞),

• ^{F (x+u)}

F (x) ­ [¬] ^{F (y+u)}

F (y) for every u ­ 0 and y ­ x ­ 0 such that y + u ∈ S F . If S F = [0, ∞) then, equivalently, F (·|u) ¬ disp [­ disp ]F for all u ∈ S F .

If F is absolutely continuous then, equivalently, function r F is increasing [decreasing]

on S F .

Definition 4.3 Assume that S ^F is a possibly infinite interval [S F = [0, ∞)]. F is said to be IFRA [DFRA] (increasing [decreasing] failure rate average) if it meets one of the following equivalent conditions:

• F ¬ _∗ [­ _∗ ]M for any λ > 0,

• function x 7→ − ln F (x)/x is increasing [decreasing] on S F \{0},

• function x 7→ F (x)
^1/x

is decreasing [increasing] on S _F \{0},

• (F (x)
^α

¬ [­]F (αx) for all α ∈ (0, 1), x ∈ [0, ∞).

If F is absolutely continuous then, equivalently, function x 7→ R x

0 r _F (t)dt/x is in-

creasing [decreasing] on S F .

Definition 4.4 Assume that S ^F is a possibly infinite interval [S F = [0, ∞)]. F is said to be NBU [NWU] (new better [worse] than used) if it meets one of the following equivalent conditions:

• F ¬ _su [­ _su ]M _λ for any λ > 0,

• F (·|u) ¬ st [­ st ]F for all u ∈ I F ,

• F (x|u) ¬ [­]F (x) for all u ∈ I _F , x ∈ [0, ∞),

• F (x + y) ¬ [­]F (x)F (y) for every x, y ­ 0.

Definition 4.5 [Assume that S F = [0, ∞)]. F is said to be DMRL [IMRL] (de- creasing [increasing] mean residual life) if it meets one of the following equivalent conditions:

• F (·|u) ¬ mrl [­ mrl ]F (·|v) for all u, v ∈ I F , u ­ v,

• F (·|u) ¬ icx [­ icx ]F (·|v) for all u, v ∈ I F , u ­ v,

• F (·|u) ¬ mrl [­ mrl ]F for all u ∈ I F ,

• function m _F is decreasing [increasing] on I _F .

Definition 4.6 [Assume that F ⁻¹ (0) = 0 and F ⁻¹ (1) = ∞]. F is said to be NBUC [NWUC] (new better [worse] than used in convex order) if F (·|u) ¬ icx [­ icx ]F for all u ∈ S F , or, equivalently,

Z ∞ x

F (t|u)dt ¬ [­]

Z ∞ x

F (t)dt

for all u ∈ I F , x ∈ [0, ∞).

Definition 4.7 [Assume that F ⁻¹ (0) = 0 and F ⁻¹ (1) = ∞]. F is said to be NBUE [NWUE] (new better [worse] than used in expectation) if

m _F (x) ¬ [­]µ F

for all x ∈ I _F .

Definition 4.8 [Assume that F ⁻¹ (0) = 0 and F ⁻¹ (1) = ∞]. F is said to be HNBUE [HNWUE] (harmonically new better [worse] than used in expectation) if it meets one of the following equivalent conditions:

• F ¬ _cx [­ _cx ]M _λ for ¹ _λ = µ _F ,

• R ∞

x F (t)dt ¬ [­]µ _F e ⁻

for all x ∈ [0, ∞),

• 1/ ¹ _x R x

0 dt/m F (t)
¬ [­]µ F for all x ∈ (0, ∞).

Definition 4.9 [Assume that F ⁻¹ (0) = 0 and F ⁻¹ (1) = ∞]. F is said to be in L- class [L-class] ([dual] Laplace class of distributions) if F ­ Lt [¬ Lt ]M λ for _λ ¹ = µ F , or, equivalently,

Z ∞ 0

e ^−st F (t)dt ­ [¬] µ F

1 + µ F s for all s ∈ [0, ∞).

Definition 4.10 Assume that F is absolutely continuous [and S F = [0, ∞)]. F is said to be NBUFR [NWUFR] (new better [worse] than used in failure rate) if

r _F (x) ­ [¬]r _F (0) for all x ∈ I F .

Definition 4.11 Assume that F is absolutely continuous [and S ^F = [0, ∞)]. F is said to be NBUFRA [NWUFRA] (new better [worse] than used in failure rate average) if − ln F (x) ­ [¬]r F (0)x for all x ∈ I F , or, equivalently,

Z x 0

r _F (t)dt ­ [¬]r _F (0)x for all x ∈ I F .

Definition 4.12 Assume that S F is a possibly infinite interval and F ⁻¹ (0) = 0 [F ⁻¹ (1) = ∞]. F is said to be BT [UBT] ([upside-down] bathtub shape) if there exists x ₀ ∈ F ⁻¹ (0), F ⁻¹ (1)
such that function

x 7→ − ln F (x)

is concave [convex] on F ⁻¹ (0), x 0
and convex [concave] on x 0 , F ⁻¹ (1)
, or, equiv- alently, function r _F is decreasing [increasing] on F ⁻¹ (0), x ₀

and increasing [de- creasing] on x 0 , F ⁻¹ (1)
.

Definition 4.13 Assume that F ⁻¹ (0) = 0 [F ⁻¹ (1) = ∞]. F is said to be IDMRL [DIMRL] (increasing then decreasing [decreasing then increasing] mean residual life) if there exists x 0 ∈ F ⁻¹ (0), F ⁻¹ (1)
such that function m F is increasing [decreasing]

on F ⁻¹ (0), x ₀
and decreasing [increasing] on x 0 , F ⁻¹ (1)
.

Definition 4.14 Assume that F ⁻¹ (0) = 0 [F ⁻¹ (1) = ∞]. F is said to be NWBUE

[NBWUE] (new worse then better [better then worse] than used in expectation) if

there exists x 0 ∈ F ⁻¹ (0), F ⁻¹ (1)
such that m F (x) ­ [¬]µ F for all x ∈ F ⁻¹ (0), x 0

and m F (x) ¬ [­]µ F for all x ∈ x 0 , F ⁻¹ (1)
.

Definition 4.15 Assume that S ^F is a possibly infinite interval. F is said to be DRFR (decreasing reverse failure rate) if function

x 7→ ln F (x) is concave on S F , or, equivalently,

F (x + u)

F (x) ­ F (y + u) F (y)

for every u ­ 0 and y ­ x > F ⁻¹ (0) such that y + u ∈ S F .

If F is absolute continuous then, equivalently, function ˘ r F is decreasing on S F . Note that, by analogy, one can consider the IRFR (increasing reverse failure rate) class, but there is no IRFR distribution satysfying our assumptions because inequal- ity

F (x + u)

F (x) ¬ F (y + u) F (y)

with x → 0 implies that F (0) > 0. Therefore we omit that class in this paper.

4.2. Relations. Connotations between the introduced ageing classes are shown in Graph 2. We use abbreviations there — e.g. ’IFR’ means ’F is IFR’.

The implications ILR ⇒ DRFR, NBU ⇒ NBUFR, NWU ⇒ NWUFR are held under the assumption of absolute continuity of F (see [13, p. 101] and [11, p. 31]).

DFR ⇒ DRFR because for any u ­ 0 and y ­ x ­ 0 we have F (x + u)

F (x) ¬ F (y + u)

F (y) ⇐⇒ F (x) − F (x + u)

F (x) ­ F (y) − F (y + u) F (y)

⇐⇒ F (x + u) − F (x)

F (x) ­ F (y + u) − F (y) F (y)

⇒ F (x + u) − F (x) ­ F (y + u) − F (y)

⇒ F (y)

F (x) ­ F (y + u) − F (y) F (x + u) − F (x)

⇐⇒ F (x + u) − F (x)

F (x) ­ F (y + u) − F (y) F (y)

⇐⇒ F (x + u)

F (x) ­ F (y + u) F (y) .

The dashed arrows are a graphical presentation of the following facts.

Theorem 4.16 Let F be absolutely continuous.

• If r(0)µ ¬ 1 then BT ⇒ DMRL.

• If r(0)µ > 1 then BT ⇒ IDMRL.

• If r(0)µ ­ 1 then UBT ⇒ IMRL.

• If r(0)µ < 1 then UBT ⇒ DIMRL.

Proof of Theorem 4.16 can be found in [11, p. 118].

I LR  +3D R F R D LR

 I F R

s{

 B T

 { U B T

 # D F R ck

 #+ I F R A  D F R A

 N B U

 #+ D M R L

 I D M R L

 #+ D I M R L

 s{ I M R L  N W U  s{ N B U F R  N B U C

 N W U C

 N W U F R

 N B U F R A N B U E

 N W B U E N B W U E N W U E

 N W U F R A

H N B U E

 H N W U E

 L L

Figure 2: Relations b et w een ageing classes

Theorem 4.17 Let F be IDMRL [DIMRL] with the distinguished point x ⁰ ∈ F ⁻¹ (0), F ⁻¹ (1)
. If there exists x 1 ∈ x 0 , F ⁻¹ (1)

such that m(x 1 ) = µ then F is NWBUE [NBWUE]. If there is no such x 1 then F is NWUE [NBUE].

Proof Assume that F is IDMRL with the distinguished point x 0 . Since m(0) = µ and m is increasing on F ⁻¹ (1), x ₀
, it is obvious that m(x) ­ µ for any x ∈

F ⁻¹ (0), x 0
. Because m is decreasing on x 0 , F ⁻¹ (1)
, function x 7→ m(x) − µ changes the sign at most once (from ’+’ to ’−’) on x 0 , F ⁻¹ (1)
. If it does, then F is NWBUE, if it does not then F is NWUE. For the dual classes we proceed

analogously. _

In [11, p. 116] it is written that if S _F = [0, ∞) then IDMRL ⇒ NWBUE and DIMRL

⇒ NBWUE. This is not true, as shown by the following examples.

Example 4.18 Let

F (x) =



 

 

1

(x+1)

, x ∈ [0, 2],

1

54 (8 − x), x ∈ [2, 4],

2

27 e ^−(x−4)/2 , x ∈ [4, ∞).

Then

m(x) =







x + 1, x ∈ [0, 2], (8 − x)/2, x ∈ [2, 4],

2, x ∈ [4, ∞),

so F is clearly IDMRL and NWUE, but not NWBUE.

Example 4.19 Let

F (x) =



 

 

6−x

6 , x ∈ [0, 4],

1

3(x−3)

, x ∈ [4, 5],

1

12 e ^−(x−5)/2 , x ∈ [5, ∞).

Then

m(x) =







(6 − x)/2, x ∈ [0, 4], x − 3, x ∈ [4, 5],

2, x ∈ [5, ∞),

so F is clearly DIMRL and NBUE, but not NBWUE.

The other connotations in Graph 2 are a result of respective relations between stochastic orders or are evident.

4.3. Closure properties. The closure of ageing classes under some operations is a subject of intense research. In this paper we focus on five types of operations:

convolutions, mixtures (with respect to the arbitrary mixing distribution), formation of monotonic systems, maxima and minima. The closure properties of the discussed ageing classes are collected in Table 2. Note the following fact.

Theorem 4.20 For any ageing class except DRFR the exponential distribution M is the unique distribution belonging to this class and its dual class simultaneously.

M is also DRFR.

convol. mixture maximum minimum mon. sys.

ILR closed not closed not closed not closed not closed IFR closed not closed not closed closed not closed

IFRA closed not closed closed closed closed

NBU closed not closed closed closed closed

DMRL not closed not closed not closed not closed not closed NBUC closed not closed closed not closed not closed NBUE closed not closed closed not closed not closed HNBUE closed not closed not closed not closed not closed L closed not closed not closed not closed not closed

NBUFR closed not closed closed closed closed

NBUFRA closed not closed closed closed closed

BT not closed not closed not closed not closed not closed IDMRL not closed not closed not closed not closed not closed NWBUE not closed not closed not closed not closed not closed convol. mixture maximum minimum mon. sys.

DLR not closed closed not closed closed not closed DFR not closed closed not closed closed not closed DFRA not closed closed not closed closed not closed NWU not closed not closed not closed closed not closed IMRL not closed closed not closed not closed not closed NWUC not closed not closed not closed not closed not closed NWUE not closed not closed not closed not closed not closed HNWUE not closed closed not closed not closed not closed L not closed closed not closed not closed not closed NWUFR not closed not closed not closed closed not closed NWUFRA not closed closed not closed closed not closed UBT not closed not closed not closed not closed not closed DIMRL not closed not closed not closed not closed not closed NBWUE not closed not closed not closed not closed not closed DRFR closed not closed closed not closed not closed

Table 2: Closure properties of ageing classes

Closedness under convolutions is mentioned: ILR and DRFR — in [13, p. 99, 179]

(absolute continuity required), IFR and IFRA — in [2, p. 83–84], NBU and NBUE

— in [2, p. 155–156], NBUC and HNBUE — in [11, p. 36], L — in [9], NBUFR and NBUFRA — in [1]. Furthermore, we know from [1] that the convolution of any two NBUFRA distributions is NBUFR.

In [11, p. 38] it is stated that DMRL is not closed under convolutions. The following simple example shows that the other classes are not closed either.

Example 4.21 Let

F = G = M 1 .

F ∗ G ∼ Gamma(2, 1) has the density (f ∗ g)(x) = xe ^−x , the logarithm of which is strictly concave. Therefore F ∗ G is ILR. Hence, due to Fact 4.20, F ∗ G is neither

’from’ DLR ’to’ NWUFRA nor in L (but it is DRFR). Moreover, r _{F ∗G} is strictly increasing and m F ∗G is strictly decreasing, so F ∗G can be neither BT, UBT, IDMRL, DIMRL, NWBUE, nor NBWUE.

Closedness under mixtures is mentioned: DLR — in [13, p. 100] (absolute continuity required), DFR and DFRA — in [2, p. 86], IMRL and HNWUE — in [11, p. 38], L

— in [9], NWUFRA — in [1].

In [11, p. 36, 38] it is stated that NWU, NWUC and NWUE are not closed under mixtures. There is a sophisticated counterexample for NWUFR in [1]. The following example shows that all the other introduced classes except DRFR are not closed either.

Example 4.22 Let

Θ = {1, 2}, F ₁ = M ₁ , F ₂ = M ₂ ; dH(1) = dH(2) = 1/2.

F (x) = 1 − (e ^−x + e ^−2x )/2 has the density f (x) = ¹ ₂ e ^−x (1 + 2e ^−x ) the logarithm of which is strictly convex. Therefore F is DLR. Hence, due to Fact 4.20, F is neither ’from’ ILR ’to’ NBUFRA nor in L (but it is DRFR). Moreover, r F is strictly decreasing and m F is strictly increasing, so F can be neither BT, UBT, IDMRL, DIMRL, NWBUE, nor NBWUE.

To show that DRFR is not closed under mixtures, we need another example.

Example 4.23 Let

Θ = {1, 2}, S F

= S F

= [0, 1], F 1 (x) = x, F 2 (x) = x ⁵ and

dH(1) = dH(2) = 1/2.

F 1 and F 2 are clearly DRFR whereas d ²

dx ² ln F ₁ (x) + F ₂ (x)

2 = −5x ⁸ + 10x ⁴ − 1 (x ⁴ + 1) ² x ² has a zero in (1 − 0.4 √

5) ^1/4 ≈ 0.57, so F = (F 1 + F 2 )/2 is not DRFR.

As we know, distributions of maximum and minimum are examples of the reliability of monotonic systems. Hence, if an ageing class is closed under the formation of monotonic systems then it is also closed under maxima and minima. And conversely

— if an ageing class is not closed under maxima or under minima then it cannot be closed under the formation of the monotonic systems. This observation helps to fill out Table 2.

Closednesses under formation of monotonic systems are mentioned: IFRA and NBU — in [2, p. 71], NBUFR — in [11, p. 37], NBUFRA — in [12].

Closedness under maxima of NBUC and NBUE is proved in [5]. DRFR is clearly closed under maxima due to (1).

HNBUE and L are not closed under maxima, as shown by the following examples.

Example 4.24 Let

F (x) =







0, x ∈ [0, 2], 19/20, x ∈ [2, 9], 1, x ∈ [9, ∞).

F is HNBUE, but F ∨ F is not HNBUE (however it is in L).

Example 4.25 Let

F (x) =







0, x ∈ [0, 2], 19/20, x ∈ [2, 13], 1, x ∈ [13, ∞).

F is in L (but not HNBUE) and F ∨ F is not in L.

The next example shows that the other classes except UBT and DIMRL are not closed under maxima.

Example 4.26 Let

F = M ₁ , G = M ₂ .

F ∨ G(x) = 1 − e ^−x − e ^−2x + e ^−3x is UBT, so it is neither BT, IFR, DFR, ILR, nor DLR. It is also IFRA, so due to Fact 4.20 it can be neither ’from’ DFRA ’to’

NWUFRA nor in L. It is NBUE as well, therefore it is neither NWBUE nor NBWUE.

Finally, it is DIMRL, therefore it can be neither DMRL, IMRL, nor IDMRL.

The example showing that UBT and DIMRL are not closed under maxima is a little more complicated.

Example 4.27 Let

F (x) =



 

 

(2−x)

4 , x ∈ [0, 1],

1

4x

, x ∈ [1, 2],

1

16 e ^2−x , x ∈ [2, ∞) and

G(x) =



 

 

 

 

e ^−x , x ∈ [0, 2],

(4−x)

4e

, x ∈ [2, 3],

1

4e

(x−2)

, x ∈ [3, 4],

1

16 e ^2−x , x ∈ [4, ∞).

F and G are both UBT and DIMRL, but r F ∨G and m F ∨G have five and four intervals of monotonicity, respectively.

IFR, DFR, DFRA, NWU, NWUFR and NWUFRA are clearly closed under the minima due to (2). If we assume absolute continuity, we can prove closedness of DLR as well.

Theorem 4.28 Let S F = S G = [0, ∞). Let F and G be absolutely continuous. If F and G are DLR then F ∧ G is also DLR.

Proof We know that DLR is closed under mixtures. We use an idea from [ 13, p.

93]: we express F ∧ G as a mixture of tails of DLR distributions.

We have

F ∧ G(x) = F (x)G(x) = Z ∞

x

F (t)g(t)dt + Z ∞

x

G(t)f (t)dt =

= ρH 1 (x) + (1 − ρ)H 2 (x), where

ρ = Z ∞

0

F (t)g(t)dt, 1 − ρ = Z ∞

0

G(t)f (t)dt and

H 1 (x) = 1 ρ

Z ∞ x

F (t)g(t)dt, H 2 (x) = 1 1 − ρ

Z ∞ x

G(t)f (t)dt.

It is clear that ρ ∈ (0, 1) and H 1 and H 2 are absolutely continuous tails with supports S H

= S H

= [0, ∞). All we need is to check that H 1 and H 2 are DLR.

Denote by h 1 the density function of H 1 . F is DLR, so it is also DFR, i.e. function x 7→ ln F (x) is convex. Thus the function

ln h 1 (x) = − ln ρ + ln F (x) + ln g(x)

is convex as a sum of the convex functions. For H 2 we proceed analogously. _ In [11, p. 36] it is stated that NBUC and NWUC are both not closed under minima.

There is simple counterexample for DMRL in [6], for NBUE in [13, p. 175], for HNBUE in [8, p. 19] and for L in [9]. One can also find absolutely continuous counterexample for ILR and DRFR in [7]. The following example (taken from [3]) shows that IMRL, NWUE, HNWUE and L are not closed under minima either.

Example 4.29 Let

F (x) =



 

 

e ^−x , x ∈ [0, 1], e ^−4x+3 , x ∈ [1, 2], e −(x+498)/100 , x ∈ [2, ∞).

F is UBT and IMRL, so NWUE, HNWUE and in L as well, but F ∧ F is not in L, so neither HNWUE, NWUE, nor IMRL.

In Example 4.27 distributions F and G are both UBT, DIMRL and NBWUE, but r _{F ∧G} and m _{F ∧G} have both four intervals of monotonicity and function x 7→

m F ∧G (x) − µ F ∧G changes the sign three times. There is a similar example for BT,

IDMRL and NWBUE.

Example 4.30 Let

F (x) =



 

 

1

(x+1)

, x ∈ [0, 1],

1

16 (3 − x) ² , x ∈ [1, 2],

1

16 e ^4−2x , x ∈ [2, ∞) and

G(x) =



 

 

 

 

e ^−2x , x ∈ [0, 2],

1

e

(x−1)

, x ∈ [2, 3],

(5−x)

16e

, x ∈ [3, 4],

1

16 e ^4−2x , x ∈ [4, ∞).

F and G are both BT, IDMRL and NWBUE, but — as in Example 4.27 — r F ∧G and m F ∧G have both four intervals of monotonicity and function x 7→ m F ∧G (x) − µ F ∧G

changes the sign three times.

References

[1] A. M. Abouammoh and A. N. Ahmed, The new better than used failure rate class of life distribution, Adv. Appl. Prob. 20 (1988), 237–240.

[2] R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart and Winston, Inc., New York, 1975.

[3] M. Brown, Further monotonicity properties for specialized renewal processes, Ann. Proba- bility 9 (1981), 891–895.

[4] A. M. Br¨ uckner and E. Ostrow, Some function classes related to the class of convex func- tions, Pacific J. Math. 12 (1962), 1203–1215.

[5] J. Cai and Y. Wu, A note on the preservation of the NBUC class under formation of parallel systems with dissimilar components, Microelectron. Reliab. 37 (1997), 359–360.

[6] M. Franco, J. M. Ruiz and M. C. Ruiz, On closure of the IFR(2) and NBU(2) classes, J.

Appl. Prob. 38 (2001), 235–241.

[7] M. Franco, M. C. Ruiz and J. M. Ruiz, A note on closure of the ILR and DLR classes under formation of coherent systems, Statistical Papers 44 (2003), 279–288.

[8] B. Klefsj¨ o, Some Properties of the HNBUE and HNWUE Classes of Life Distributions, Report 1980-8 (1980), Univ. Ume˚ a.

[9] B. Klefsj¨ o, A useful ageing property based on the Laplace transform, J. Appl. Prob. 20 (1983), 615–626.

[10] B. Kopociński, Zarys teorii odnowy i niezawodności, PWN, Warszawa 1973.

[11] C.-D. Lai and M. Xie, Stochastic Ageing and Dependence for Reliability, Springer, New York 2006.

[12] W.-Y. Loh, A new generalization of the class of NBU distributions, IEEE Trans. Rel. 33 (1984), 419–422.

[13] A. W. Marshall and I. Olkin, Life Distributions, Springer, New York, 2007.

[14] M. Shaked and J. G. Shantikumar, Stochastic Orders, Springer, New York, 2007.

Porządki stochastyczne i klasy rozkładów czasów życia

Streszczenie. Teoria niezawodności opisuje losowe czasy działania urządzeń po- datnych na awarie. Na potrzeby tego typu badań opracowano specyficzne branżowe pojęcia, m.in. resztowy czas życia, intensywność awarii, czy średni resztowy czas życia. Ich definicje zostały podane w Rozdziale 2. Znalazły się w nim także pod- stawowe definicje i fakty dotyczące systemów monotonicznych. Systemy monoton- iczne tworzą bardzo prosty i wygodny model, opisujący działanie urządzeń składa- jących się z odpowiednio ułożonych względem siebie części. Jest to dynamicznie rozwijająca się teoria, znajdująca zastosowania w wielu dziedzinach nauki. Więcej na ten temat można dowiedzieć się z książek [2] i [13].

Rozmaite porządki stochastyczne, czyli porządki na zbiorach rozkładów prawdo- podobieństwa, mają zastosowanie w wielu dziedzinach rachunku prawdopodobieństwa i statystyki matematycznej. Wiele o porządkach można nauczyć się z książki [14].

W Rozdziale 3 niniejszego artykułu zostały zdefiniowane i porównane ze sobą tylko te porządki, które najbardziej przydają się w pracy z klasami rozkładów czasu ży- cia. Nie powinno dziwić, że takie kryterium wyboru spełniają przede wszystkim porządki, które prostotą definicji i wszechstronnym zastosowaniem zasłużyły sobie na miano klasycznych. Relacje między wprowadzonymi porządkami zostały pod- sumowane na Rysunku 1.

W Rozdziale 4 opisane zostały wybrane klasy rozkładów czasu życia – zarówno te znane od kilkudziesięciu lat, jak i niektóre spośród wprowadzonych stosunkowo niedawno. Nacisk został położony na dwa zagadnienia: relacje między poszczegól- nymi klasami oraz domkniętości poszczególnych klas względem najczęściej używanych operacji niezawodnościowych: splotów, mieszanek i tworzenia systemów monoton- icznych. Zostały też wyróżnione klasy domknięte ze względu na maksima i minima.

Wyniki zostały podsumowane na Rysunku 2 i w Tabeli 2.

Wszystkim zainteresowanym zastosowaniem przedstawionych pojęć na pewno przypadnie do gustu przystępnie i rzetelnie napisana książka [10].

Słowa kluczowe: porządki stochastyczne, klasy rozkładów czasów życia, domknię- tość ze względu na operacje niezawodnościowe, splot, mieszanka, system monoton- iczny.

Patryk Miziuła was born in Bydgoszcz in 1987. He received the M.Sc.

degree in mathematics in 2011 from the Univeristy of Nicolaus Coper- nicus in Torun. Currently he is Ph.D. student of this university. His research interests include reliability theory, order statistics, stochastic orders, ageing classes, design criteria for regression models.

Patryk Miziuła urodził się w Bydgoszczy w 1987 r. W 2011 r. ukończył matematykę na Uniwersytecie Mikołaja Kopernika w Toruniu, otrzymując tytuł magistra. Aktualnie jest słuchaczem studiów doktoranckich na tejże uczelni. Jego zainteresowania naukowe obejmują m.in. teorię niezawodności, statystyki porządkowe, porządki stochastyczne, klasy rozkładów czasu życia, kryteria planowania doświadczeń w modelach regresji.

Patryk Miziuła

Uniwersytet Mikołaja Kopernika w Toruniu

Wydział Matematyki i Informatyki, ul. Chopina 12/18, 87-100 Toruń, Polska E-mail: bua@mat.umk.pl

(Received: 15th of May 2012)