variable depending on speci¬ed r. If r is also a random variable, then con-

sider T as the associated cumulative time-to-failure for random r. The density

function for T can be expressed as a sum of conditional terms. We have

γ rv trv’1 ’γt

fTr (t) = fT (t|r) = e

“(rv)

and

∞

fT (t) = fT (t|r)P (R = r),

r=1

where P (R = r) is the probability that a randomly selected data record includes

r failures.

Consider a speci¬c observed data set with N data records. For each of the N

data records, r and Trj are known. The set of cumulative operating times {Trj }

form a non-homogeneous population that can be modelled using a mixture

model. The Trj variables are associated with di¬erent gamma distributions

depending on r with the same scale parameter. Mixture models are gener-

ally used when there is a nonhomogeneous population composed of distinct

subpopulations. The most common situation is when the proportion of each

subpopulation is known or can be estimated, but it is unknown which mem-

bers belong to which subpopulations. The probability density function for a

non-homogeneous population can be expressed as weighted sum of the respec-

tive probability density functions. The weights represent the probability that

7.3 Illustrative examples 135

a randomly selected member of the population is from a particular subpopula-

tion. For our problem, the subpopulations are characterized by the number of

failures within the merged data set. Denote nr the number of data records with

exactly r failures. The weights are the probabilities that a randomly selected

Trj from the N data records has r failures. This probability is nr . Denote

N

n = (n1 , n2 , ..., nm ) where m = max{r|nr > 0} is the maximum number of

m

failures for any data record within a data set and N = r=1 nr . Thus we have

m

nr γ rv trv’1 ’γt

1

fT (t|n) = e. (7.7)

N “(rv)

r=1

• Maximum likelihood estimate

A likelihood function for γ and v is based on the observed data and relation

(7). The likelihood function l(v, γ) is expressed as a product of fT (t|n) for the

N data records

nr

m m kv’1

nk γ kv Trj

1

e’γTrj .

l(v, γ) =

N “(kv)

r=1 j=1 k=1

Estimates of v and γ can be obtain using a Newton search to minimize l(v, γ).

An alternative and more preferred likelihood function can be developed by

exploiting the observation that the subpopulations are clearly identi¬ed within

the overall population. The alternative likelihood function can be expressed as

the product of the conditional density functions for Trj and fT (t|r) as follows

nr

m

l2 (v, γ) = fT (v, γ|Trj )

r=1 j=1

and

nr

m m

¯

ln l2 (v, γ) = ’γM t + M v ln γ + (vr ’ 1) ln Trj ’ nr ln “(rv), (7.8)

r=1 j=1 r=1

m

where M = r=1 rnr denotes total number of failures associated with all N

m nr

1

¯

data records and t = M r=1 j=1 Trj is average time-to-failure.

Taking partial derivatives of the log-likelihood function ln l2 gives the equations

136 7 The exact LR test of the scale in the gamma family

v

ˆ

γ=

ˆ (7.9)

¯

t

and

nr

m m

v

r ln Trj ’

M ln( ) + rnr ψ(rv) = 0, (7.10)

¯

t r=1 j=1 r=1

where ψ is Euler™s digamma function. The digamma function is in Coit and

Jint (2000) approximated by the equation

1 1 1 1

ψ(x) = ln x ’ ’ ’

+ + ... (7.11)

2x 12x2 120x4 252x6

Now v can be solved from the equations (10) and (11) using a Newton-Raphson

ˆ

or bisection search, v is then substituted into equation (9) to determine γ . The

ˆ ˆ

resulting gamma estimates should be tested for goodness-of-¬t.

• Testing the scale of the failure process

Suppose, that the shape parameter is v and consider the test of the hypothe-

sis (2):

H0 : γ = γ0 versus H1 : γ = γ0

about the scale of the failure process. Such testing problem is usual in analysis

of failure processes. Suppose that we have the only one observation Tr1 which

is typical in the case of unsystematic rebooting of some device. Then Tr1

has Gamma(γ, rv) distribution and the optimal test is the exact LR test of

the scale in the gamma family. We use the quantlet etgammaexacttest in

the form etgammaexacttest(x, ±, γ0 , rv) where x = {Tr1 } is data vector. The

power function can be evaluated with the help of the following theorem:

Theorem 2 The exact power p(γ, ±) of the LR test based on the Wilks statis-

tics of the hypothesis (2) of the Gamma(v, γ) distribution on the level ± at the

point γ of the alternative has the form

γ γ

c±,N c±,N

W’1 (’e’1’ 2vN )}+FvN {’vN W0 (’e’1’ 2vN )},

p(γ, ±) = 1’FvN {’vN

γ0 γ0

7.4 Implementation to the XploRe 137

where c±,N denotes the critical value of the exact test of the hypothesis (2) on

the level ±.