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discussed below.
The same example of a log normal distribution in time can be used, as shown in figure 7. Taking
the log of time, μ is the mean and σ2 is the variance of a normal distribution. Log t is expressed
as being affected by contributing factors, and they can be expressed in a linear relationship, as
shown in equation 7. The contributing factors in this study were aircraft model (x1), operators
12
(x2) and others. μ is the explainable portion of the equation, and ε is the unexplainable/error/
residual portion. Since log t is distributed normally with mean μ and the varianceσ2, after
removing μ, ε is distributed normally with mean 0 and the variance σ2. Index i=1,…,n is the
units under evaluation.
( 2 )
1 2 log 0 1 2 ... , ~ 0, i i i i i i i t=μ +ε =β +xβ + x β + +ε ε N σ (7)
Again, estimating the parameters μ, σ2, β0, β1, and β2 is achieved by using the maximum
likelihood principle. The maximum likelihood is simply the probability of seeing the observed
data as shown in equation 8. It finds a parameter set that most likely generated the observed
data. The likelihood function for a combination of n independent units is
0 1 2 0 1 2 1 2
1
1
1
( , , ,..., ) ( , , ,..., ; , ,...)
{ log( ) }{ log( ) log( )
{1 log( ) }
n
i
i
n
nor i i li nor i i nor i i di
i
nor i i ri
L L x x
t t t
t
β β β σ β β β σ
μ μ
σ σ σ
μ
σ
=
−
=
=
= Φ ⎢⎣⎡ − ⎥⎦⎤ Φ ⎢⎣⎡ − ⎥⎦⎤−Φ ⎢⎣⎡
− Φ ⎡⎢⎣ − ⎤⎥⎦
Π
Π }
μ − ⎤⎥⎦
(8)
where μ = β 0 + x1β 1 + x2β 2 + ... i and li, di, and ri are indicator variables for left-censored, exactor
interval-censored, and right-censored observations. Similarly, the terms in the brackets are the
probabilities of seeing the data for each unit, which could be left-censored, exact- or intervalcensored,
and right-censored. The parameters μ, σ2, β
0, β
1, and β
2 are estimated by maximizing
function (equation 8) and are denoted as ˆ, ˆ , ˆ , ˆ , and ˆ . 0 1 2
μ σ 2 β β β
Diagnostic checking is a critical part of the modeling process. To ensure that the curve fits the
data well, residuals are often computed and checked for patterns.
ˆ log ˆ ~ (0,1)
ˆ
i i
i
i
t N
μ
ε
σ
−
= 􀀅 (9)
The residuals can be obtained and plotted against a standard normal distribution or against the
fitted values for check of nonlinearity.
In the statistical literature, maximum likelihood estimators give good statistical properties.
Commercial software packages such as SAS, S-PLUS, and MINITAB are readily available. In
this study, S-PLUS [9] was used. Readers are referred to the textbooks by Lehmann and
Casella [10] and Pawitan [11].
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4. RESULTS.
4.1 RISK ANALYSIS.
The in-service data received varied in quality. Some records provided only the numbers of
orders (to reflect maximum numbers of removals). Other records included detailed shop
procedures and information on subpart replacements. In some cases, complete repair records for
all years were provided. In other cases, several years of records were missing. In general, in
areas where in-depth analysis was desired, more detailed data was available.
During the analysis, if exact failure estimations were not possible, a simple bounding approach
was used. In other words, if a risk estimation of all failure modes was available, then the risk
estimation of one specific failure mode was less than the overall risk. For the most part, this
occurred for reliable components for which the number of failures was scarce. A simple
bounding approach was quite adequate.
During the review of records, most critical failure modes were prioritized. For example,
attention was given to symptoms of jamming for the servocontrol actuators. Attention was given
to the loss of electrical signals for both channels of the travel limitation units, but not much
attention was given to the loss of one channel. For the most part, most critical failure modes
were recorded.
The results were organized to discuss how the designs of the units had been upgraded or
　
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