H0 : γ = 11.1992 versus H1 : γ = 11.1992. (7.5)

7.3 Illustrative examples 131

The relevant idea is to test the hypothesis (5) exactly. The asymptotical Wilks

test is oversized and the power is smaller than the exact one. The signi¬cance

of this e¬ect can be clearly seen from the Table 3 in Appendix Oversizing of the

asymptotics. The value of the Wilks statistics of the LR test of the hypothesis

(6) is ’2 ln » = 0.14981 and the corresponding p-value is 0.69873. Therefore

the null hypothesis cannot be rejected at the 0.05 signi¬cance level. Much

more detailed discussion can be found in Stehl´ (2002), where the exact power

±k

function is derived (in Theorem 1):

The exact power p(γ, ±) of the LR test based on the Wilks statistics of the

hypothesis (2) of the Erlang(k, 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’ 2kN )) + FkN (’kN W0 (’e’1’ 2kN )),(7.6)

p(γ, ±) = 1 ’ FkN (’kN

γ0 γ0

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

the level ±.

We obtain the exact power p(10.0938, 0.05) = 0.9855. Since the power of our

test is high at the point corresponding to the penalization time, the probability

of penalization is small. The power function p(γ, 0.05) of the LR test of the

hypothesis (5) is for γ ∈ (7, 15) displayed in the Figure 1.

Time to complete after breakdown

Consider that a job was being processed for the past u hours when the system

breaks down. When the breakdown is ¬xed, the company would like to estimate

how much longer the processing will take. Denote p(t) the probability P {X ¤

u+t|X > u} (X denotes the processing time) that once the breakdown is ¬xed,

the processing will continue for less than t hours given that the breakdown

occurred u hours after the processing started. It is easy to see that

k’1 (γ(u+t))i

’γt i=0 i!

p(t) = 1 ’ e .

k’1 (γu)i

i=0 i!

Consider the processing times given from the Table 7.2. Then the processing

time has Erlang (16, 11.1992) distribution. If a system breakdown occurs 1.5

hour after processing starts, then the probability p(0.5) that the processing will

132 7 The exact LR test of the scale in the gamma family

Table 7.3: Airplane indicator light reliability data

Failures Cumulative operating time(hours)

2 T21 = 51000

9 T91 = 194900

8 T81 = 45300

8 T82 = 112400

6 T61 = 104000

5 T51 = 44800

be complete within 30 minutes after the breakdown is ¬xed equals to 0.83079.

The application of the exact LR test of the scale would be similar like in the

part Promising a due date.

7.3.2 Estimation with missing time-to-failure information

For the assessment of component reliability, ¬eld data has many distinct ad-

vantages (Coit and Jint, 2000). For all of the advantages of the ¬eld data,

there are also disadvantages, including incomplete or inaccurate data report-

ing and others. Several of these disadvantages are described in more detail by

Coit, Dey and Jint (1986). The disadvantage to be addressed in Coit and Jint

(2000) is the fact that the individual times-to-failure are often missing. The

data is often only available in the form of r collective failures observed Trj

cumulative hours with no further delineation or detail available (Anon, 1991)

and (Anon, 1997). Quantities r and Trj are known but the individual failure

times are not. Analysts may have many of these merged data records available

for the same component. Table 7.4 presents a data set of this type. Here Trj

is the jth cumulative operating time with r failures, i.e.

Trj = X1 + ... + Xr ,

where Xi is the ith time-to-failure.

There has been other research concerned with the use of data with missing at-

tributes. Dey (1982) has developed a simulation model to observe the behavior

of grouped data and test an exponential distribution assumption. Coit and

Dey (1999) have developed and demonstrated a hypothesis test to evaluate an

7.3 Illustrative examples 133

exponential distribution assumption when there is missing time-to-failure data.

The grouped exponential data was modelled using a k-Erlang distribution (Coit

and Jint, 2000). The hypothesis test was demonstrated to successfully reject

the exponential distribution when it was not appropriate even without detailed

knowledge of component time-to-failure. Coit and Jint (2000) made following

assumptions:

• Component times-to-failure are iid

• Component time-to-failure are gamma distributed

• Repair times are insigni¬cant compared to operating time

• System repair does not degrade or otherwise a¬ect the reliability of the

unfailed components.

Time to failure distributions

For standard data sets, there are well-known techniques to estimate parameters

for many distributions and types of censoring, and to objectively evaluate the

applicability of these distributions. However, the analysis of reliability data

with missing time-to-failure is a nonstandard problem and it is not addressed

in these references.

Without individual time-to-failure data, it is impossible to ¬t the data to most

popular distributions (e.g. gamma, Weibull, lognormal) using standard tech-

niques such as MLE. However, the MLE for the exponential distribution with

Type I or II censoring only depends on the number of failures and the cumu-

lative times.

This is a product of the memoryless property associated with the exponential

distribution. The limitations of the available ¬eld data and the simplicity of

the exponential MLE have been used to rationalize the exponential distribution

in applications where it would seemingly be a poor choice. the constant hazard

function associated with the exponential distribution is not intuitively appro-

priate for most failure mechanisms which can be attributed to the accumulation

of stress, such as fracture, fatigue, corrosion and wear mechanisms.

Incorrect assumptions of the underlying distribution can have dire consequences.

For many ¬‚edging companies, major decisions are made with limited data be-

ing used as rationale. When an incorrect distribution is assumed, particularly

for reasons of convenience, it is particularly dangerous.

134 7 The exact LR test of the scale in the gamma family

The gamma distribution is ¬‚exible distribution that can model many particu-

lar component failure mechanisms. Methods to determine MLE estimates are

available (Lawless, 1982) for Type I, Type II and chronologically grouped fail-

ure data. However, none of the available MLE estimators pertain to the data

with missing failure times.

Gamma-distribution maximum-likelihood estimates

• Distribution of cumulative time-to-failure

Coit and Jint (2000) determine gamma distribution MLEs with missing failure

times. The likelihood function is based on the distribution parameters (γ, v)

and observed r and Trj values rather than times-to-failure and censor times

which are unknown. When time-to-failure Xi is distributed according to the

r

gamma distribution Gamma(γ, v) then Trj = i=1 Xi is distributed in ac-