Remaining Life of a Pipeline

Maximun Likelihood parameters estimation
To completely define eq. (iii), it is necessary to find the values of the parameters “ν” and “σ E ”. This is
possible by applying the method of Maximum Likelihood Estimator, as follows:
ln( likelihood)
= Λ =
n
∑
i=
1
⎡ 1 ⎡ Ei
Ni
ln⎢
φ ⎢
⎣σ
EEi
⎣
− E0e
σ
E
−υt
⎤⎤
⎥⎥
⎦
⎦
(iv)
The parameters estimate “υˆ ” and “ σˆ ”, will be found by solving for all the evidence equation (iv),
so that:
E
∂Λ
∂υ
= 0
∂Λ
= 0
∂σ E
(v)
(vi)
Once the estimators “υˆ ” and “ σˆ ” have been found based on the evidence, the equation (iv) is
E
totally defined for any value of t i , and it represents the distribution function of the stress.
pdf
E(
t))
= f ( E,
t)
=
ˆ σ
1
e
2π
1 ⎡
− ⎢
2 ⎢⎣
( E(
t )
−E
e
0
2
ˆ σ E
(
E
) ⎤
⎥
⎥⎦
− ˆ υt
2
(vii)
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