Introduction and Summary -...

CHAPTER 1

Introduction and Summary

1.1 Introduction

This thesis is concerned with the properties and applications of certain classes of

lifetime distributions in reliability modeling and analysis, when lifetime is measured in

discrete units. The present century is marked with a greater demand for high quality

and reliability. Buyer’s expectations with regard to the ability of a product or system to

perform its intended function are usually very high. Simple consumer items such as

small electronic items, small appliances, mechanical devices etc are expected to work

without fail. For more complex items such as computers, communications devices,

A part of this chapter have appeared in the paper Jose and Jayamol (2006)

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CHAPTER 1. INTRODUCTION AND SUMMARY

automobiles etc most consumers are not ready to tolerate very few failures. For highly

sophisticated systems such as nuclear power plants, rocket propulsion systems etc

failures can be disastrous and hence very high reliability is mandatory. With the strong

concern for manufacturing best quality products, the importance of reliability theory

has increased significantly with the innovations in recent technology. Reliability theory

has been applied extensively in various areas such as industry, engineering, defence,

actuaries, biomedical sciences, space technology etc. It also finds applications in the

study of shock models, metal wear, corrosion models, fatigue crack growth etc.

1.2 Review of Literature

The notion of ageing plays an important role in reliability and maintenance theory.

Several classes of lifetime distributions have been proposed in order to model dif-

ferent aspects of ageing. In the study of various classes of lifetime distributions,

stochastic ordering plays an important role. Various types of stochastic orderings

are described in Shaked and Shanthikumar (1994, 2007), Szekli (1995) and Muller

and Stoyan (2002).

Some of the important classes of lifetime distributions described in the literature

are Increasing Failure Rate (IFR), Increasing Failure Rate in Average (IFRA), De-

creasing Mean Residual Life (DMRL), New Better than Used (NBU ), New Better

than Used in Expectation (NBUE), and Harmonically New Better than Used in Ex-

pectation (HNBUE), and their respective duals Decreasing Failure Rate (DFR), De-

creasing Failure Rate in Average (DFRA), Increasing Mean Residual Life (IMRL),

New Worse than Used (NWU ), New Worse than Used in Expectation (NWUE), Har-

monically New Worse than Used in Expectation (HNWUE). In addition to the above

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classes of distributions, researchers have recently introduced some more classes

such as L-class, M-class, LM-class, L(α)-class, LD-class etc.

The vast majority of literature on the various criteria for ageing treats lifetime

as continuous with only occasional references to the discrete case. Recently many

researchers have shown interest for reliability analysis in the discrete time domain,

see for example Xekalaki (1983), Gupta and Richard (1997), Roy and Gupta (1999),

Shaked et al (1995), Baracquemond and Gaudoin (2002) and Kemp (2004).

Xekalaki (1983), points out that limitations of measuring devices and the fact that

discrete models provide good approximations to their continuous counterparts, ne-

cessitates assessment of reliability in discrete time. Further discrete models do occur

in a natural way as in fatigue studies, the time to failure is measured in terms of the

number of cycles to failure, number of shocks, number of copies etc., which are ob-

viously integer valued. Actuaries and Biostatisticians are interested in the lifetimes

of persons or organisms, measured in months, weeks, or days. Accordingly elabo-

ration of various concepts analogous to those in the continuous case has become

necessary to distinguish classes of discrete lifetime distributions based on the notion

of ageing. Discrete lifetimes have important applications. For reliability engineers,

‘time’ can also be the number of times that a piece of equipment is operated, or the

number of miles that a plane is flown. The definitions and a number of results relat-

ing to IFR, IFRA, DMRL,NBU, NBUE are given in Esary et al (1973), Block et al

(2003), Kemp (2004), Golifourshni and Asadi (2008).

As a discrete analogue to L - class Klefsjo (1983) introduced G - class and Mohan

and Ravi (2002) extended this to obtained G(α) - class. Klar and Muller (2003) intro-

duced M - class closely related to L - class, which is obtained by replacing Laplace

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transform ordering by moment generating function ordering. They gave another class

of distributions called LM - class. A random variable X is said to be in LM - class if

it is in the L - class and in the M - class. For details see Klefsjo (1983), Mohan and

Ravi (2002, 2003), Klar and Muller (2003), Jose and Jayamol (2006), Jayamol and

Jose (2006), Jayamol and Jose (2007a,b), Jayamol and Jose (2008a,b). The problem

of testing the null hypothesis of exponentiality against alternatives belonging to a hi-

erarchy of ageing classes has been treated extensively in the literature. An excellent

survey of the tests of exponentiality versus the various life distributions belonging to

various ageing classes is given in Hollander and Proschan (1972). For recent ap-

proaches, see Chaudhuri (1997), Baringhaus and Henze (1991), Klar (2002), Klefsjo

and Westberg (1996) and Pandit and Anuradha (2007).

1.3 Summary of the Present Work

The main objectives of the present study are the following.

1. To introduce ageing classes when lifetime is measured in discrete units.

2. To study the properties and obtain characterizations for the classes of life time

distributions introduced.

3. To obtain limits of moments and other functionals that have many uses in relia-

bility and maintenance.

4. To develop test statistics for testing whether the lifetime distribution is geometric

against the developed class of distributions.

5. To explore applications of the results in real life contexts such as shock models,

cumulative damage models etc.

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The thesis is organized into 6 chapters as follows. Chapter 1 gives a summary

of the thesis as well as a brief review of literature on the topic. Second chapter of

this thesis deals with alternate probability generating function (a.p.g.f.) ordering, a

discrete analogue of Laplace transform ordering. Then we make a detailed study of

the properties of this ordering. After that a.p.g.f. ratio ordering and its properties are

discussed. Using a.p.g.f. ordering we developed a new class of lifetime distributions

called G∗ class and studied its properties. A sub class called G∗D class is also intro-

duced and studied. Then it is generalized to obtain G∗(α) and G∗(α, γ) classes. The

dual classes are also introduced. Also we obtain the moments and coefficient of vari-

ation of the random variables whose distributions belong to these classes. For details

see Jayamol and Jose (2006, 2008a).

In the third chapter, we analyze the relation between a.p.g.f.s and moments of

the parent and derived distributions. Using these relations we characterize G∗ class

and G∗D class. Also we give some characterizations of geometric law using coefficient

of variations and length biased random variable. From the definition of G∗ and G∗

classes, it can be seen that, geometric distribution forms the boundary between the

two classes. Accordingly it is important to ascertain the conditions under which the

geometric law emerges within these classes. Our characterization provides a simple

property that meets this objective. Also using this result one can construct suitable

statistics to test whether an equipment whose life is measured in discrete time units

failed due to uncontrollable factors alone, against the alternative that it degrades over

time in the sense of G∗-ageing. For details see Jayamol and Jose (2007a,b), Jose and

Jayamol (2008).

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In the fourth chapter we discuss the relation between infinite divisibility and the

G∗ and G∗(α) classes. Using infinitely divisible distributions we give a new character-

ization of geometric law within G∗ and G∗ classes. Compound geometric distributions

arise naturally in many contexts of applied probability such as queuing theory, relia-

bility and stress-strength modeling. Also we discuss geometric compounds and their

properties. We proved that G∗ class of distributions is closed under geometric com-

pounding. More details are available in Jayamol and Jose (2008c).

In Chapter 5 of this thesis, we propose tests for geometricity against G∗ class as

a discrete counter part for the test of exponentiality versus L class. The procedures

are based on empirical a.p.g.f.. The asymptotic properties are studied. Empirical

quantiles and empirical powers are calculated for various alternatives by using the

programming language R. For a real data set, it is verified that the newly developed

test is consistent. For details see Jayamol and Jose (2008d).

In Chapter 6 we study some shock models leading to G∗ class such as Poisson

shock models. The life distribution H(t) of a device subject to shocks governed by

a Poisson process is considered as a function of the probabilities Pk of surviving the

first k shocks. Various properties of the discrete failure distribution Pk are shown to be

reflected in corresponding properties of the continuous life distribution H(t). A certain

cumulative damage model also is investigated. Various applications of these models

in reliability modeling are also considered, see Jayamol and Jose (2008b).

Thus the thesis mainly deals with a.p.g.f. ordering, G∗ class of life distributions and

their properties. We also study their applications in the context of reliability modeling,

shock models, cumulative damage models, testing for geometricity and characteriza-

tions of various classes of lifetime distributions.

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1.4 Preliminary Concepts

1.4.1 Important Reliability Concepts

In this section we consider some important reliability concepts and their inner relation-

ships.

(i) Reliability

Let X (X > 0), denote the failure time of a unit, with cumulative distribution

function F (t) = P (X < t) and probability density function f(t)(0 < t < ∞), i.e.,

f(t) = dF (t)dt

and F (t) =∫ t

0f(u)du. In reliability theory they are called failure time

distribution and failure density function. Then

R(t) = P (X > t) = 1− F (t) =

∫ ∞

t

f(u)du = F (t),

is called the reliability function or survival function. The mean µ = E(X) =∫∞0

tf(t)dt =∫∞

0R(t)dt, if it exist, is called the mean time to failure (MTTF ) or

mean lifetime. If the device under consideration is renewed through maintenance

and repair, E(X) is also known as the mean time between failure (MTBF ).

(ii) Failure rate

The instantaneous failure rate function h(t) is

h(t) =f(t)

F (t), F (t) < 1.

It is known by different names such as failure rate, hazard rate, risk rate, force of

mortality, etc. (Badenius (1970)). Then H(t) =∫ t

0h(u)du, is called the cumulative

hazard function, and it satisfies the relation

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R(t) = exp[−∫ t

0

h(u)du]

= exp(−H(t)).

In addition we have the inequalities

H(t)

1 + H(t)≤ F (t) ≤ H(t) ≤ F (t)

F (t).

Let

Ft(x) = P (X > t + x/X > t) (1.4.1)

=F (x + t)

F (t),

denotes the survival function of the residual lifetime after time t. It is the condi-

tional probability that a unit (or a system) of age t will survive for an additional x

units of time.

Let Xt denotes the random variable with survival function Ft(x), then the

function m(t) = E(Xt) is called the mean residual function. That is, the mean

residual lifetime of a unit of age t is defined as

m(t) = E(X − t/X > t).

=

∫ ∞

0

Ft(x)dx.

The function defined in (1.4.1) highlights different aspects of survival and residual

life distributions. The hazard rate and the mean residual life function are related

by

h(t) =1 + m′(t)

m(t).

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It is well known that h(t) determines the distribution function uniquely and hence,

m(t) also characterizes the distribution. Also, F (t) and m(t) are related by

F (t) =m(0)

m(t)exp

{−

∫ t

0

dx

m(x)

}.

(iii) Equilibrium distribution

Let F be a distribution function with finite mean µ such that F (x) = 0 for X < 0,

and let

f1(x) =F (x)

µ=

F (x)∫∞0

F (z)dz, x ≥ 0, (1.4.2)

= 0, x < 0.

The density function f1 is called the equilibrium distribution or the first derived

distribution. For any distribution F such that F (0) = 0, f1(0) = 1µ

and, F can be

obtained from f1. Clearly every equilibrium density f1 is decreasing. The above

distribution arises as the limiting distribution of the forward recurrence time in a

renewal process (see Feller (1971)).

It is straight forward to show that the hazard rate h1 of the equilibrium distri-

bution is the reciprocal of the mean residual life, that is, h1(t) = 1m(t)

.

(iv) Laplace transform

For a non-negative random variable X with distribution function F and survival

function F = 1− F,

LF (s) =

∫ ∞

0

e−sxdF(x) and LF(s) =

∫ ∞

0

e−sxF(x)dx,

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are called Laplace Stieltjes transforms of F and F , respectively. Then

LF (s) =1− LF (s)

sfor all s > 0.

(v) Starshaped function

A function f(x) defined on [0 ∞) is said to be starshaped (antistarshaped) iff(x)

xis increasing (decreasing) in x > 0 and f(0) = 0, or equivalently,

f(ax) ≤ (≥)af(x) for 0 ≤ a ≤ 1, x ≥ 0.

1.4.2 Stochastic Orders

The objectives of a reliability study are understanding the failure phenomena, estimat-

ing and predicting reliability, optimization etc. In order to study the failure phenomena

of the systems or units we consider failure time distribution. Since the failure time as

well as the residual lifetime are random variables, several types of stochastic orders

have been developed by various researchers (see Shaked and Shanthikumar (1994),

Klefsjo (1983)).

Mann and Whitney (1947) were the pioneers who introduced the concept of stochas-

tic order for the first time. It was used by Birnbaum (1948) in a study of peakedness

and the notion of ordering distributions was brought into clear focus by Lehmann

(1955) and by Bickel and Lehmann (1975). The introduction of an ordering to rep-

resent the idea that one distribution has more of some characteristic than another

requires careful consideration of the characteristic nature. A very rich literature exists

concerning the topics of stochastic orderings, dependence and ageing properties and

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related applications in reliability, life testing, and survival analysis. For more details,

see Shaked and Shanthikumar (1994, 2007), Szekli (1995) and Muller and Stoyan

(2002).

The ordering≤ of distributions is said to be (i) Reflexive if F ≤ F, for all distribution

functions F ; and (ii) Transitive if F ≤ G and G ≤ H implies F ≤ H. Orderings with

these two properties are called preorders: if in addition they satisfy the condition that

F ≤ G and G ≤ F implies F = G, then the orders are called partial orders.

(i) Stochastic Order

Let X and Y be two random variables such that P [X > u] ≤ P [Y > u] for all

u ∈ (−∞,∞). Then X is said to be smaller than Y in the usual stochastic order

and is denoted by X ≤st Y.

(ii) Hazard Rate Order

Let X and Y be two non-negative random variables with absolutely continuous

distribution functions and hazard rate functions r1(·) and r2(·) respectively, such

that r1(t) ≥ r2(t), t ≥ 0. Then X is said to be smaller than Y in the hazard

rate order and is denoted by X ≤hr Y. It may be noted that if X and Y are two

random variables such that X ≤hr Y, then X ≤st Y . It is clear that the hazard rate

ordering would be appropriate for comparing the life lengths of identical devices,

one operating in a more hazardous environment than the other, since the hazard

rate can be interpreted as the probability of failure in the next instant of time given

that the system has survived up to a specific time.

(iii) Star Order

Suppose X has distribution function F and Y has distribution function G. Then

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X is said to be smaller than Y in star order, denoted by X ≤∗ Y , if G−1(F ) is star

shaped in x, that is,G−1(F (x))

xis increasing in x ≥ 0.

(iv) Supperadditive Order

Suppose that X has distribution function F and Y has distribution function G.

Then, X is said to be smaller than Y in the supperadditive order if G−1(F ) =

G−1(F ) is supperadditive in x, that is, G−1(F (x + y)) ≥ G−1(F (x)) + G−1(F (y)),

x, y ≥ 0.

(v) Convex Order

Let X and Y be two random variables such that E(φ(X)) ≤ E(φ(Y )) for all

convex functions φ : R → R, provided the expectations exist. Then X is said to

be smaller than Y in the convex order and is denoted by X ≤cx Y .

(vi) Laplace Transform Order

Let X and Y be two non-negative random variables such that,

E(e−sX) ≤ E(e−sY ) for all s > 0. (1.4.3)

Then X is said to be smaller than Y in the Laplace transform order and is denoted

by X ≤L Y. The above ordering concepts among probability distributions based

on comparison of their Laplace transforms are taken from Klefsjo (1983). It differs

from the definition used by Stoyan (1983), where the reversed inequality is used

in (1.4.3). In the sequel of our work we use the definition given by Klefsjo (1983).

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1.4.3 Ageing Classes

The notion of ageing was first introduced in Barlow and Marshall (1964). Prior to that

the physics of failure mainly focused on the properties of specially chosen families of

failure distributions. Barlow and Proschan (1975) generalized the main assumptions

of the theory of ageing distributions. Important contributions to this theory were made

by Bryson and Siddiqui (1969), Proschan and Hollander (1984), Cox and Oakes etc.

Definition of ageing classes

Various nonparametric families of distributions have been studied in the context of

reliability theory. The theory developed for these families has thus involved the notion

of components and systems, which might be mechanical, electrical, or hydraulic. But

the same ideas can often be applied to biological systems also. In this section we

review the definitions of the existing ageing criteria and the ordering of the classes of

ageing distributions related to one another.

Definition 1.4.1 The distribution function F (x) is an IFR(DFR) distribution if Ft(x)

is decreasing (an increasing) function of t for all x ∈ R+.

IFR distributions appear when it is apriori known that the failure rate of the units

increases as the operating time increases.

DFR distributions are used to model various practical cases, such as, metals

whose strength increases with annealing, systems characterized by infant mortality

have a failure rate which decreases in the initial time interval. A mixture of exponential

distributions is a DFR distribution.

The Weibull distribution with distribution function F (x) = 1−exp(−(λx)α), λ, x ≥ 0

has IFR for α > 1, DFR for 0 < α < 1, and constant failure rate for α = 1.

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Definition 1.4.2 The distribution function F (t) is an IFRA (DFRA) distribution if − ln F (t)t

is an increasing (a decreasing) function of t ∈ R+.

IFRA is the minimal class of distributions that describe the life of a monotone

system consisting of components with IFR life distributions (Barlow and Proschan

(1975)).

Definition 1.4.3 The distribution function F (x) is an NBU(NWU) distribution if F (x+

y) ≤ (≥)F (x)F (y), x, y ∈ R+.

The above group of distributions has been invoked in order to develop a scientific

approach to technical maintenance with the objectives to reduce the number of fail-

ures during operation, to increase the time to failure and to achieve these goals while

reducing the total maintenance and on-site repair costs.

Definition 1.4.4 The distribution function F (x) is an NBUE(NWUE) distribution if

∫ ∞

t

F (x)dx ≤ (≥) µF (t), t ≥ 0. (1.4.4)

It can be seen that∫∞

tF (x)

F (t)dx represents the conditional mean of the remaining life

of a unit of age t. Hence (1.4.4) implies that a used unit of age t has smaller mean

remaining life than a new unit if F is NBUE.

NBUE (NWUE) distributions arise in connection with replacement of compo-

nents and are used for the development of mathematical models of maintenance of

complex systems. NBUE condition is the condition that a used device of any age has

a mean residual life smaller than the mean residual life of a new device of the same

kind. Inequality (1.4.4) implies that the distribution function F1 with density f1 given by

(1.4.2) is stochastically less than F .

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Definition 1.4.5 The distribution function F (x) with finite mean is DMRL(IMRL) if

m(t) =∫∞

0Ft(x)dx is decreasing (increasing) in t ≥ 0.

Definition 1.4.6 The distribution function F (x) is HNBUE(HNWUE) if∫∞

tF (x)dx ≤

µ exp(−t/µ), for t > 0.

Definition 1.4.7 The distribution function F (x) is GHNBUE(GHNWUE) if η(F ) ≤

(≥)1, where η(F ) = 1 if µF = 0 and η(F ) = σµF

if µF > 0, where σ2 and µF are

variance and the mean of the distribution.

1.4.4 Discrete Lifetime Distributions

The vast majority of literature on the various criteria for ageing treats lifetime as con-

tinuous with only occasional references to the discrete case. Recently there is active

interest for reliability modeling and analysis in the discrete time domain.

Let X denote a discrete lifetime random variable whose probability mass function

and cumulative distribution function are given by,

pk = P [X = k], k = 0, 1, 2, . . . and Pk = P [X ≤ k].

Then the failure rate function is defined as hk = pk

Pk−1, where Pk = P [X > k]. It

gives the conditional probability of the failure of the device at time k, given that it has

not failed by time k − 1. The failure rate function uniquely determines the distribu-

tion. Shaked and Shanthikumar (1994) proved the following necessary and sufficient

conditions for a sequence {hk, k ≥ 1} to be a failure rate function.

1. For all k < m, hk < 1 and hm = 1, the distribution is defined over {1, 2, . . . ,m.}

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2. For all k ∈ N+ = {1, 2, . . .} , 0 ≤ hk ≤ 1 and∞∑i=1

hi = ∞. The distribution is

defined over k ∈ N+.

The mean residual life (or mean remaining life) at time k is defined as µk =∞∑

n=k+1

Pn

Pk

.

The mean lifetime µ = E[X] =∞∑

k=0

Pk < ∞.

The classes of discrete ageing distributions have not been much discussed in the

literature. The definitions and some results relating to IFR, IFRA, DMRL,NBU,

NBUE are given in Esary et al (1973).

Now we shall consider the definitions of discrete version of the above mentioned

ageing classes. For further properties see Esary et al (1973), Klefsjo (1982).

Definition 1.4.8 A discrete survival probability Pk =∞∑

j=k+1

pj, with support on {0, 1, . . .}

probability mass function pk = Pk−1 − Pk for k = 1, 2, 3, . . . and p0 = 1 − P0 is said to

be

(i) IFR if Pk+1

Pkis decreasing in k.

For IFR distributions, Salvia and Bollinger (1982) shows that Pk−1 ≤ (1−h0)k ≈

exp(−hk0) and that µ ≤ (1−h0)

h0.

(ii) IFRA if ¯(Pk)−1/k

is decreasing in k.

The class of distributions distinguished by IFRA property was introduced for

continuous random variables by Birnbaum et al (1966) in an attempt to find a

new class of life distributions that reflect the phenomenon of wear-out. Klefsjo

(1982) has considered the discrete IFRA class, preferring to define it in terms

of the behavior of [F (x)]1/x where F (x) = P [X > x], as in the continuous case.

Consider the lifetime distribution H(t) of a device subject to shocks governed

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by a Poisson process with probability Pk of surviving k shocks. If the discrete

distributions formed by Pk is IFRA, then H(t) is also IFRA (Esary et al (1973)).

(iii) DMRL if∞∑

j=k

Pj

Pk

is decreasing in k.

Ebrahimi (1986) showed that X is DMRL if there exists a non-increasing se-

quence {an} > 0 such that for all n,

hi = 1− ai

1 + ai + 1for i = 0, 1, 2, . . . .

(iv) NBU if Pk+j ≤ PkPj.

The quantity Pk+j

Pkrepresents the survival function of a unit of age k or the condi-

tional probability that a unit of age k will survive for an additional j units of time.

At k = 0,Pk+j

Pk= Pj is the survival function of a new unit and accordingly the age-

ing of the device can be studied by comparing Pk+j

Pkand Pj. Thus Pk+j ≤ PkPj if

and only if the older system that has aged has no better chance of surviving for a

duration of j than does a new system. In other words, the new unit is better than

the used one or NBU.

(v) NBUE if∞∑

j=k

Pj ≤ Pk

∞∑j=0

Pj. This condition states that the expected remaining life

of a unit surviving age k is not larger than the expected life of a new unit so that

the new unit is better than the unit of age k in terms of the expected life length.

(vi) HNBUE if µ =∞∑

j=0

Pj is finite and∞∑

j=k

Pj ≤ µ(1− 1/µ)k, k = 0, 1, 2, . . ..

It is obvious from the definition that NBUE class is contained in the HNBUE

class.

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Corresponding dual classes of distributions are also defined by reversing the di-

rection of monotonicity or inequality in the above definitions.

1.4.5 L-class and M-class

In addition to the classes of lifetime distributions discussed above, recently various

researchers have introduced and studied some more classes such as L-class, M-

class, LM-class, L(α)-class, LD-class etc.

Based on the Laplace transform order between a random life and an exponential

life, Klefsjo (1983) proposed the class L as a larger class which includes most of

the previously known classes, such as DMRL,NBUE, HNBUE etc. Lin (1998)

introduced a more general family of classes of life distributions which includes L-

class. This family is denoted by L(α), 0 < α < ∞. Klefsjo (1983) also introduced a

discrete analogue of the L - class and it is denoted by G.

A sub set LD of L-class was introduced in Mitra et al (1995). Mohan and Ravi

(2002) studied the properties of the family of classes of distributions L(α). They define

a sub class of the family L(α), denoted by LD(α). A discrete analogue called G(α)

of the family L(α) is introduced by Mohan and Ravi (2002). Dual properties of each

class are also discussed in the literature.

Given below are the important definitions and properties of various classes.

Definition 1.4.9 (Klefsjo (1983)) Let F be a life distribution with survival function

F = 1− F and finite mean µ =∫∞

0F (t)dt. The class of life distributions for which

∫ ∞

0

e−stF (t)dt ≥ µ

1 + sµfor all s ≥ 0, (1.4.5)

is called the L - class.

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Equivalently, a life distribution F belongs to L class if and only if its Laplace (Stielt-

jes) transform L(s) satisfies the relation LF (s) ≡ E(e−sX) ≤ 11+sµ

≡ E(e−sYµ) ≡ LGµ(s)

in which Yµ is a random variable obeying the exponential distribution Gµ with mean

µ = µF and G0 = F0, the degenerate distribution at zero.

The dual class of L- class denoted by L, is defined as follows.

Definition 1.4.10 (Klefsjo (1983)) A life distribution F (ie, a distribution function with

F(0-)=0) with survival function F =1-F and finite mean µ =∫∞

0F (t)dt belongs to L

class if ∫ ∞

0

e−stF (t)dt ≤ µ

1 + sµfor s ≥ 0.

This definition is closely related to the partial order relation ≤(L) defined on the

set of distributions with F(0) = 0, given by (1.4.3). The importance of the classes

L and L of life distributions lies in the fact that in many situations it is difficult or

even impossible to get an explicit expression of the survival function but possible to

determine the Laplace transform of F . The inequality (1.4.5) can be interpreted in

different ways, which are interesting from a reliability point of view.

First, suppose that three independent components C1, C2 and C3 have the ran-

dom times to failure ξ1, ξ2 and ξ3 respectively. Here ξ1 is supposed to have the survival

function F , and ξ2 and ξ3 to have exponential distributions with means∫∞

0F (t)dt and

1/s, respectively. Then the inequality (1.4.5) means that the expectation of min(ξ1, ξ3)

is at least as large as the expectation of min(ξ2, ξ3). This in turn means an inequality

between the mean times to failure of the corresponding series systems, see Klefsjo

(1983) for details.

Another interpretation of (1.4.5) is the following. Suppose that F is the survival

19


function for a certain component in a certain environment. Now add to this envi-

ronment the possibility of a catastrophe which will destroy the component and let

the time to the catastrophe have an exponential distribution with mean 1/s. Then∫∞0

se−stF (t)dt denotes the probability that a catastrophe takes place during the life-

time of the component. Therefore the inequality (1.4.5) means that this probability is

at least as large as the probability of a catastrophe during the lifetime of a component

with exponential time to failure with the same mean as F .

A third interpretation is the following. Suppose that a producer with survival func-

tion F (t) produces c units per unit time. Then cF (t)dt (approximately) represents the

amount of output during (t, t+ dt) and cF (t)e−stdt the present value of this output, as-

suming a discount factor s. Accordingly c∫∞

0F (t)e−stdt represents the present value

of the total production of the producer. Thus (1.4.5) is a comparison of the present

value of the output of two producers. See Alzaid et al (1991) for more details.

The family denoted by L(α), 0 < α < ∞ is defined as follows.

Definition 1.4.11 (Lin (1998), Mohan and Ravi (2002)) Let Gα,β be the gamma distri-

bution with density function

gα,β(x) =1

Γ(α)βαxα−1e−

xβ for x ≥ 0, α > 0, β > 0. (1.4.6)

Then for each α > 0,L(α) - class of life distributions is given by

L(α) =⋃β≥0

{F : µ(F ) = αβ, F ≤L Gα,β}

=⋃β≥0

{F : µ(F ) = αβ,LF (s) ≤ (1 + βs)−α for s ≥ 0}

= {F : LF (s) ≤ 1

(1 + µ(F )α s)α

, s ≥ 0}.

20


in which µ(F ) stands for the mean of F.

The dual class L(α) is defined by reversing the inequality. Thus L(α) contains all

those distribution functions not greater than a gamma distribution function with mean

µ(F ) in the Laplace transform order and L(1) ≡ L, as defined by Klefsjo (1983). So

L(α) class is some times known as L -like class.

Klar (2002) shows that the failure rate of a random life in the class L may go to 0

as time goes to infinity so that the class L may not be a reasonable notion of positive

ageing. As a result, Klar and Muller (2003) defined the class M of life distributions by

replacing the Laplace transform order by the moment generating function order and

showed that this class gives a more reasonable model of the positive ageing process,

and it is large enough in the sense that the class M also includes HNBUE as a

reasonable subclass.

Klar and Muller (2003) introduced M and LM classes of lifetime distributions,

defined as follows.

Definition 1.4.12 (a) A non-negative random variable X with mean µ = E(X) > 0

and distribution function F is said to be in the M -class if

E(etX) =

∫ ∞

0

etxdF (x) ≤ 1

1− µtfor 0 ≤ t ≤ 1/µ.

Here, M(t, 1/µ) = (1 − µt)−1 is the moment generating function of an exponential

distribution with mean µ.

(b) X is said to be in the LM class, if it is in the L-class and in the M class. For the

LM class the inequality must hold for all −∞ < t < 1/µ.

As a discrete analogue of L-class, Klefsjo (1983) also introduced a class called

21


G-class as follows. Let ξ be a strictly positive integer-valued random variable and let

Pk = P [ξ > k], k = 0, 1, 2, . . . denote the corresponding survival probabilities. Further

let 1 = Q0 ≥ Q1 ≥ . . . denote the corresponding survival probabilities of a geometric

distribution with mean µ =∑∞

k=0 Qk =∑∞

k=0 Pk where Qk = (1− 1µ)k for k = 0, 1, 2, . . . .

Definition 1.4.13 (Klefsjo (1983)) A discrete life distribution and its survival proba-

bilities Pk, k = 0, 1, 2, . . . with mean µ =∑∞

k=0 Pk, are said to belong to the G class

if∞∑

k=0

Pkpk ≥ µ

p + (1− p)µ, 0 ≤ p ≤ 1. (1.4.7)

Its dual class G is obtained by reversing the inequality (Klefsjo (1983)).

As a discrete analogue to L(α)-class, Mohan and Ravi (2002) introduced the G(α)-

class defined as follows.

Definition 1.4.14 (Mohan and Ravi (2002)) Let f denote the probability mass func-

tion (p.m.f) of a positive integer valued random variable with probability generating

function (p.g.f) gs, defined by

gf (s) =∞∑

n=1

snf(n), 0 ≤ s ≤ 1.

Let µk(f) =∑∞

n=1 nkf(n) denote the kth moment of f for k=1,2,3,. . . , we assume

µ1(F ) < ∞. Then for any α, with 0 < α < ∞

G(α) = {f : gf (s) ≤pαs

(1− qs)α, 0 ≤ s ≤ 1},

where p = αµ1(f)−1+α

which ensures that the mean of the bounding negative binomial

is equal to that of f . The dual G(α) is defined by reversing the inequality.

22


It may be noted that G(1) is not the class G introduced by Klefsjo (1983).

Mohan and Ravi (2002) introduced a family of classes of distributions LD(α) as

an extension of the class LD introduced by Mitra et al (1995). They define LD(α) as

follows.

Definition 1.4.15

LD(α) = {F : LF (s) ≤ 1

(1 + µ1(F )sα

)α, LF1(s) ≤

1

(1 + µ1(F1)sα

)α, s ≥ 0},

where LF1(s) is the Laplace transform of f1 given by (1.4.2).

Its dual LD(α) is obtained by reversing the inequalities, provided µ1(F1) < ∞. It

may be noted that F is in LD(α) if and only if F is in L(α) and F1 is in L(α). LD(1) is

the class LD introduced by Mitra et al (1995).

Mohan and Ravi (2002) define GD(α) as a discrete analogue to LD(α).

Definition 1.4.16

GD(α) = {f : gf (s) ≤pαs

(1− qs)α, gf1(s) ≤

pαs

(1− q1s)α, 0 < s ≤ 1},

where p1 = αµ1(f1)−1+α

, q1 = 1 − p1. GD(α), the dual is obtained by reversing the

inequalities provided that µ1(f1) < ∞.

GD(1) is the class GD introduced by Bhattacharjee and Basu (2002), see also

Mohan and Ravi (2002).

1.4.6 Poisson Processes

Definition 1.4.17 Suppose events are occurring successively in time, with the inter-

vals between successive events independently and identically distributed according

23


to an exponential distribution Gλ(t) = 1− e−λt. Let N(t) denote the number of events

during [0, t]. Then the stochastic process {N(t), t ≥ 0} is called a Poisson process

with mean rate (or intensity) λ.

Examples of the Poisson Processes in Life Testing and Reliability

The Poisson processes arise quite naturally in life testing situations when the under-

lying life distribution is exponential. The following are commonly occurring examples.

1. Maintained Unit. A unit is put into operation at time 0. Each time failure occurs,

the failed unit is replaced by a fresh unit of the same type. Assume life lengths

are independently and identically distributed according to an exponential distri-

bution Gλ(t) = 1−e−λt. Let N(t) denotes the number of failures observed during

[0, t]. Then {N(t), t ≥ 0} is a Poisson process with mean rate (or intensity) λ.

2. Sampling with replacement. A sample of n units is randomly selected from a

population having life distributions Gλ(t) = 1−e−λt. The failure rate λ is unknown

and is to be estimated from data collected during the life testing experiment. The

n units are put on test at time 0, each time a failure occurs, it is replaced by a

new, randomly selected units. If N(t) denotes the number of failures observed

during [0, t], then {N(t), t ≥ 0} is a Poisson process with mean rate nλ.

The Poisson process has many applications in reliability theory, especially in the study

of shock models.

1.4.7 Characterizations of Probability Models

Except for distribution free methods, all of the statistical approaches depend upon well

defined probability models. Since the assumption of an inappropriate model can lead

24


to erroneous conclusion, in some applications the assumption of a model involves little

or no risk. For example, the central limit theorem may make the normal distribution a

clear choice. In many applications the appropriate model is not at all clear, but what

constitutes an appropriate model clearly depends upon how the model is used and

what is expected of it.

In a number of applications, parametric models have played a prominent historical

role in statistical theory, and were long studied under the older method of curve fitting.

Some of the best known early work is that of Pearson (1895), who created a set of

curves or frequency distributions that might be suitable for fitting to data arising in a

variety of contexts. Elderton and Johnson (1969) indicate that “The advantages of any

system of curves depend on the simplicity of the formulae and the number of classes

of observations that can dealt with satisfactory . . .”.

There was an underlying assumption that a distribution provides a reasonable fit

to data is sufficient. But what is reasonable for one purpose may not be reasonable

for another. This concern was captured by Kingman (1978), who noted that “Although

it is often possible to justify the use of a distribution empirically, simply because it

appears to fit the data, it is more satisfactory if the structure of the distribution reflects

plausible features of the underlying mechanism”.

The study of characterization of probability distributions appears to have begun

with the work of Gauss. A full fledged development of this area, as part of mathe-

matical statistics began taking shape only in the late fifties of 20th century. Consistent

with the emphasis placed on normal distribution in the early stages of development of

statistical theory, initially the work on characterization theorems also were concerned

primarily with normal models. Although in 1923 Polya characterized the normal law,

25


by the identical distribution of two linear statistics, a real spurt in this direction be-

gan only with the conjecture of Levy, that for independent random variables X and

Y , X + Y is normal if and only if X and Y also follow the same law. Cramer (1946)

proved Levy’s conjecture and in the following year Raikov established a similar result

for Poisson variables. Modest activities in the forties due to the world war gave way

to rapid growth in the next decade, with the review paper by Lukacs (1956) and the

monograph by Lukacs and Laha (1964), which established a new line of thought. The

first authoritative book on the tools employed in providing characterizations along with

a large collection of results covering most probability distributions was published by

Kagan et al (1973). This along with the books by Galambos and Kotz (1977), Mathai

and Pederzoli (1977), Azlarov and Volodin (1986) and Patil et al (1975) contain most

of the literature on the subject in recent times.

1.4.8 Tests for Exponentiality

The exponential hypothesis has attracted significant attention of researchers because

of its important implications in reliability theory as well as in stochastic modeling ap-

plications. When one is dealing with failure times of items such as transistors, fuses,

air monitors etc., where failure is brought about by sudden shocks rather than wear

and tear, the exponential assumption is particularly justified. There are numerous

characterizations of exponential distribution and many goodness of fit tests for ex-

ponentiality have been suggested by various authors. Ascher (1990), D’Agostino and

Stephens (1986), Doksum and Yandell (1984) provide overviews over this subject. For

recent approaches, see Ahmad and Alwasel (1999), Castillo and Puig (1999), Chaud-

huri (1997), Baringhaus and Henze (1991), Gupta and Richard (1997), Klar (2002),

Klefsjo and Westberg (1996), Niktin (1996) and Sen and Srivastava (1991).

26


The problem of testing the null hypothesis of exponentiality against alternatives

belonging to a hierarchy of ageing classes such as IFR, IFRA, DMRL, NBU ,

NBUE, HNBUE has been treated extensively in the literature. An excellent survey

of the tests of exponentiality versus the various life distribution belonging to the above

mentioned ageing classes is given in Hollander and Proschan (1984). Test procedures

for IFR and IFRA alternatives have been dealt within Bickel and Doksum (1969), Bickel

(1969) and Proschan and Pyke(1967), NBU and NBUE alternatives have been found

in Hollander and Proschan (1972), Koul (1978) and De Souza Borges et al (1984).

HNBUE alternatives have been dealt with by Klefsjo (1986). An approach involving

linear functions of order statistics to test exponentiality against HNBUE alternatives

was used by Klefsjo (1986).

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35

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