IEC 61710:2013

IEC 61710 ®
Edition 2.0 2013-05
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
colour
inside
Power law model – Goodness-of-fit tests and estimation methods

Modèle de loi en puissance – Essais d'adéquation et méthodes d'estimation
des paramètres
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IEC 61710 ®
Edition 2.0 2013-05
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
colour
inside
Power law model – Goodness-of-fit tests and estimation methods

Modèle de loi en puissance – Essais d'adéquation et méthodes d'estimation

des paramètres
INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
COMMISSION
ELECTROTECHNIQUE
PRICE CODE
INTERNATIONALE
CODE PRIX XA
ICS 03.120.01; 03.120.30 ISBN 978-2-83220-797-0

– 2 – 61710 © IEC:2013
CONTENTS
FOREWORD . 5
INTRODUCTION . 7
1 Scope . 8
2 Normative references . 8
3 Terms and definitions . 8
4 Symbols and abbreviations . 8
5 Power law model . 9
6 Data requirements . 10
6.1 General . 10
6.1.1 Case 1 – Time data for every relevant failure for one or more copies
from the same population . 10
6.1.2 Case 1a) – One repairable item . 10
6.1.3 Case 1b) – Multiple items of the same kind of repairable item
observed for the same length of time . 11
6.1.4 Case 1c) – Multiple repairable items of the same kind observed for
different lengths of time . 11
6.2 Case 2 – Time data for groups of relevant failures for one or more repairable
items from the same population . 12
6.3 Case 3 – Time data for every relevant failure for more than one repairable
item from different populations . 12
7 Statistical estimation and test procedures . 13
7.1 Overview . 13
7.2 Point estimation . 13
7.2.1 Case 1a) and 1b) – Time data for every relevant failure . 13
7.2.2 Case 1c) – Time data for every relevant failure . 14
7.2.3 Case 2 – Time data for groups of relevant failures . 15
7.3 Goodness-of-fit tests . 16
7.3.1 Case 1 – Time data for every relevant failure. 16
7.3.2 Case 2 – Time data for groups of relevant failures . 17
7.4 Confidence intervals for the shape parameter . 18
7.4.1 Case 1 – Time data for every relevant failure. 18
7.4.2 Case 2 – Time data for groups of relevant failures . 19
7.5 Confidence intervals for the failure intensity . 20
7.5.1 Case 1 – Time data for every relevant failure. 20
7.5.2 Case 2 – Time data for groups of relevant failures . 20
7.6 Prediction intervals for the length of time to future failures of a single item . 21
7.6.1 Prediction interval for length of time to next failure for case 1 – Time
data for every relevant failure . 21
7.6.2 Prediction interval for length of time to Rth future failure for case 1 –
Time data for every relevant failure . 22
7.7 Test for the equality of the shape parameters β ,β , ., β . 23
1 2 k
7.7.1 Case 3 – Time data for every relevant failure for two items from
different populations . 23
7.7.2 Case 3 – Time data for every relevant failure for three or more items
from different populations . 24
Annex A (informative) The power law model – Background information . 30
Annex B (informative) Numerical examples . 31

61710 © IEC:2013 – 3 –
Annex C (informative) Bayesian estimation for the power law model . 41
Bibliography . 56

Figure 1 – One repairable item . 10
Figure 2 – Multiple items of the same kind of repairable item observed for same length
of time . 11
Figure 3 – Multiple repairable items of the same kind observed for different lengths of
time . 12
Figure B.1 – Accumulated number of failures against accumulated time for software
system . 32
Figure B.2 – Expected against observed accumulated times to failure for software
system . 32
Figure B.3 – Accumulated number of failures against accumulated time for five copies
of a system . 35
Figure B.4 – Accumulated number of failures against accumulated time for an OEM
product from vendors A and B . 37
Figure B.5 – Accumulated number of failures against time for generators . 38
Figure B.6 – Expected against observed accumulated number of failures for
generators . 39
Figure C.1 – Plot of fitted Gamma prior (6,7956, 0,0448) . 47
for the shape parameter of the power law model . 47
Figure C.2 – Plot of fitted Gamma prior (17,756 6, 1447,408) for the expected number
of failures parameter of the power law model . 47
Figure C.3 – Subjective distribution of number of failures. 51
Figure C.4 – Plot of the posterior probability distribution for the number of future
failures, M . 54
Figure C.5 – Plot of the posterior cumulative distribution for the number of future
failures, M . 55

Table 1 – Critical values for Cramer-von-Mises goodness-of-fit test at 10 % level of
significance. 25
Table 2 – Fractiles of the Chi-square distribution . 26
Table 3 – Multipliers for two-sided 90 % confidence intervals for intensity function for
time terminated data . 27
Table 4 – Multipliers for two-sided 90 % confidence intervals for intensity function for
failure terminated data . 28
Table 5 – 0,95 fractiles of the F distribution . 29
Table B.1 – All relevant failures and accumulated times for software system . 31
Table B.2 – Calculation of expected accumulated times to failure for Figure B.2 . 33
Table B.3 – Accumulated times for all relevant failures for five copies of a system
(labelled A, B, C, D, E) . 34
Table B.4 – Combined accumulated times for multiple items of the same kind of a
system . 34
Table B.5 – Accumulated operating hours to failure for OEM product from vendors A
and B . 36
Table B.6 – Grouped failure data for generators . 38
Table B.7 – Calculation of expected numbers of failures for Figure B.6 . 40
Table C.1 – Strengths and weakness of classical and Bayesian estimation . 42

– 4 – 61710 © IEC:2013
Table C.2 – Grid for eliciting subjective distribution for shape parameter β . 46
Table C.3 – Grid for eliciting subjective distribution for expected number of failures
parameter η . 46
Table C.4 – Comparison of fitted Gamma and subjective distribution for shape
parameter β . 48
Table C.5 – Comparison of fitted Gamma and subjective distribution for expected
number of failures by time T = 20 000 h parameter η . 48
Table C.6 – Times to failure data collected on system test . 49
Table C.7 – Summary of estimates of power law model parameters . 50
Table C.8 – Time to failure data for operational system . 53

61710 © IEC:2013 – 5 –
INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
POWER LAW MODEL –
GOODNESS-OF-FIT TESTS
AND ESTIMATION METHODS
FOREWORD
1) The International Electrotechnical Commission (IEC) is a worldwide organization for standardization comprising
all national electrotechnical committees (IEC National Committees). The object of IEC is to promote
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2) The formal decisions or agreements of IEC on technical matters express, as nearly as possible, an international
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6) All users should ensure that they have the latest edition of this publication.
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8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is
indispensable for the correct application of this publication.
9) Attention is drawn to the possibility that some of the elements of this IEC Publication may be the subject of
patent rights. IEC shall not be held responsible for identifying any or all such patent rights.
International Standard IEC 61710 has been prepared by IEC technical committee 56:
Dependability.
This second edition cancels and replaces the first edition, published in 2000, and constitutes
a technical revision.
The main changes with respect to the previous edition are listed below:
– the inclusion of an additional Annex C on Bayesian estimation for the power law model.
The text of this standard is based on the following documents:
FDIS Report on voting
56/1500/FDIS 56/1508/RVD
Full information on the voting for the approval of this standard can be found in the report on
voting indicated in the above table.

– 6 – 61710 © IEC:2013
This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.
The committee has decided that the contents of this publication will remain unchanged until
the stability date indicated on the IEC web site under "http://webstore.iec.ch" in the data
related to the specific publication. At this date, the publication will be
• reconfirmed,
• withdrawn,
• replaced by a revised edition, or
• amended.
IMPORTANT – The 'colour inside' logo on the cover page of this publication indicates
that it contains colours which are considered to be useful for the correct
understanding of its contents. Users should therefore print this document using a
colour printer.
61710 © IEC:2013 – 7 –
INTRODUCTION
This International Standard describes the power law model and gives step-by-step directions
for its use. There are various models for describing the reliability of repairable items, the
power law model being one of the most widely used. This standard provides procedures to
estimate the parameters of the power law model and to test the goodness-of-fit of the power
law model to data, to provide confidence intervals for the failure intensity and prediction
intervals for the length of time to future failures. An input is required consisting of a data set
of times at which relevant failures occurred, or were observed, for a repairable item or a set of
copies of the same item, and the time at which observation of the item was terminated, if
different from the time of final failure. All output results correspond to the item type under
consideration.
Some of the procedures can require computer programs, but these are not unduly complex.
This standard presents algorithms from which computer programs should be easy to
construct.
– 8 – 61710 © IEC:2013
POWER LAW MODEL –
GOODNESS-OF-FIT TESTS
AND ESTIMATION METHODS
1 Scope
This International Standard specifies procedures to estimate the parameters of the power law
model, to provide confidence intervals for the failure intensity, to provide prediction intervals
for the times to future failures, and to test the goodness-of-fit of the power law model to data
from repairable items. It is assumed that the time to failure data have been collected from an
item, or some identical items operating under the same conditions (e.g. environment and
load).
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and
are indispensable for its application. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any
amendments) applies.
IEC 60050-191:1990, International Electrotechnical Vocabulary (IEV) – Chapter 191:
Dependability and quality of service
3 Terms and definitions
For the purposes of this document, the terms and definitions of IEC 60050-191 apply.
4 Symbols and abbreviations
The following symbols and abbreviations apply:
β shape parameter of the power law model
ˆ
β estimated shape parameter of the power law model
β ,β lower, upper confidence limits for β
LB UB
C Cramer-von-Mises goodness-of-fit test statistic
C ()M critical value for the Cramer-von-Mises goodness-of-fit test statistic at γ level of
1−γ
significance
χ Chi-square goodness-of-fit test statistic
2 2
χ ()υ γ th fractile of the χ distribution with υ degrees of freedom
γ
d number of intervals for groups of failures
E[]N()t expected accumulated number of failures up to time t
E[t ] expected accumulated time to jth failure
j
61710 © IEC:2013 – 9 –
ˆ
E[N[t(i)]] estimated expected accumulated number of failures up to t(i)
ˆ
E[t ] estimated expected accumulated time to jth failure
j
F (ν ,ν ) γ th fractile for the F distribution with (ν ,ν ) degrees of freedom
γ 1 2 1 2
i general purpose indicator
j general purpose indicator
k number of items
L, U multipliers used in calculation of confidence intervals for failure intensity
λ scale parameter of the power law model

λ estimated scale parameter of the power law model
M parameter for Cramer-von-Mises statistical test
N number of relevant failures
number of failures for jth item
N
j
N (t) accumulated number of failures up to time t
N[t(i)] accumulated number of failures up to time t(i)
R difference between the order number of future (predicted) failure and order
number of last (observed) failure
T accumulated relevant time
*
total accumulated relevant time for time terminated test
T
T total accumulated relevant time for jth item
j
T ,T lower, upper prediction limits for the length of time to the Rth future failure
RL RU

T estimated median time to (N+1)th failure
N +1
t accumulated relevant time to the ith failure
i
t ith failure time for jth item
i j
t total accumulated relevant time for failure terminated test
N
t total accumulated relevant time to Nth failure of jth item
N j
t(i − 1), t(i) endpoints of ith interval of time for grouped failures
z(t) failure intensity at time t
ˆ
z(t) estimated failure intensity at time t
z , z lower, upper confidence limits for failure intensity
LB UB
5 Power law model
The statistical procedures for the power law model use the relevant failure and time data from
the test or field studies. The basic equations for the power law model are given in this clause.
Background information on the model is given in Annex A and examples of its application are
given in Annex B.
The expected accumulated number of failures up to test time t is given by:
β
[ ( )] with
E N t = λt λ > 0,β > 0, t > 0
– 10 – 61710 © IEC:2013
where
λ is the scale parameter;
β is the shape parameter ( 0 < β < 1 corresponds to a decreasing failure intensity; β = 1
corresponds to a constant failure intensity; β > 1 corresponds to an increasing failure
intensity).
The failure intensity at time t is given by:
d
β −1
z(t) = E[N(t)] = λβ t with t > 0
dt
Thus the parameters λ and β both affect the failure intensity in a given time.
and .
Methods are given in 7.2 for maximum likelihood estimation of the parameters of λ β
Subclause 7.3 gives goodness-of-fit tests for the model and 7.4 and 7.5 give confidence
interval procedures. Subclause 7.6 gives prediction interval procedures and 7.7 gives tests for
the equality of the shape parameters. The model is simple to evaluate. However when β < 1,
theoretically z(0) = ∞ (i.e. z(t) tends to infinity as t tends to zero) and z(∞) = 0 (i.e. z(t) tends
to zero as t tends to infinity); but this theoretical limitation does not generally affect its
practical use.
6 Data requirements
6.1 General
6.1.1 Case 1 – Time data for every relevant failure for one or more copies
from the same population
The normal evaluation methods assume the observed times to be exact times of failure of a
single repairable item or a set of copies of the same repairable item. The figures below
illustrate how the failure times are calculated for three general cases.
6.1.2 Case 1a) – One repairable item
For one repairable item observed from time 0 to time T, the relevant failure time, t , is the
i
elapsed operating time (that is, excluding repair and other down times) until the occurrence of
the i-th failure as shown in Figure 1.
A
B
B
0 1 2 3 T
IEC  996/13
Key
A operating time, B down time
Figure 1 – One repairable item
*
Time terminated data are observed to T , which is not a failure time, and failure terminated
data are observed to t , which is the time of the Nth failure. Time terminated and failure
N
terminated data use slightly different formulae.

61710 © IEC:2013 – 11 –
6.1.3 Case 1b) – Multiple items of the same kind of repairable item observed for the
same length of time
It is assumed there are k items, which all represent the same population. That is, they are
nominally identical items operating under the same conditions (e.g. environment and load).
*
When all items are observed to time T , which is not a failure time (i.e. time terminated data),
then the failure time data are combined by superimposing failure times (t , i = 1, 2.,N) for all k
i
items on the same time line as shown in Figure 2.

A
B
C
D
t t t t t T*
0 1 N-1 N
2 3
IEC  997/13
Key
A item 1
B item 2
C item k
D superimposed process
Figure 2 – Multiple items of the same kind of repairable item
observed for same length of time
6.1.4 Case 1c) – Multiple repairable items of the same kind observed for different
lengths of time
When all items do not operate for the same period of time, then the time at which observation
of the jth item is terminated T ( j = 1,2,.,k), where T < T < . < T , is noted. The failure data
j 1 2 k
are combined by superimposing all the failure times for all k items on the same time line as
shown in Figure 3. The times to failure are t , i = 1, 2,.,N , where N = the total number of
i
failures observed accumulated over the k items.

– 12 – 61710 © IEC:2013
T
A
T
B
T
C 3
T
k
D
0 t t t t t t t
1 2 3 4 5 6
IEC  998/13
Key
A item 1
B item 2
C item 3
D item k
t time
Figure 3 – Multiple repairable items of the same kind observed
for different lengths of time
If each item is a software system then the repair action should be done to the other systems
which did not fail at that time.
6.2 Case 2 – Time data for groups of relevant failures for one or more repairable
items from the same population
This alternative method is used when there is at least one copy of an item and the data
consist of known time intervals, each containing a known number of failures.
The observation period is over the interval and is partitioned into d intervals at times
(0, T)
. The ith interval is the time period between and , where
0 < t(1) < t(2) < . < t(d) t(i − 1) t(i)
i 1, 2,.,dt, (0) 0 and td( ) = T . It is important to note that the interval lengths and the
number of failures per interval need not be the same.
6.3 Case 3 – Time data for every relevant failure for more than one repairable item
from different populations
It is assumed there are k items which do not represent the same population and are to be
compared. It should be noted that if each item is to be considered individually then it is
appropriate to use case 1a) in 6.1.2.
If direct comparisons of the items are to be made then as an extension of 6.1 the following
notation is used:
t denotes the ith failure time for the process corresponding to the jth item;
ij
denotes the number of failures observed for the jth item;
N
j
t is the time of the Nth failure for the jth item;
N
j
where i = 0, 1, 2, … N and j = 1, 2, …k.
j
==
61710 © IEC:2013 – 13 –
7 Statistical estimation and test procedures
7.1 Overview
In case 1 – time data for every relevant failure – the formulae given for failure terminated data
assume one repairable item, that is . All output results correspond to that item. The
k = 1
formulae given for time terminated data assume k copies of the item observed for the same
length of time. If there is only one repairable item then k = 1. The point estimation procedures
for all the aforementioned cases are given in 7.2.1. The appropriate procedures for the case
when all copies are observed for different lengths of time are given in 7.2.2. Procedures for
the case of time data for groups of relevant failures are given in 7.2.3.
An appropriate goodness-of-fit test, as described in 7.3 shall be performed after the
parameter estimation procedures of 7.2. Note that these tests, and the procedures given in
7.4 to 7.7 for constructing interval estimates and carrying out statistical tests, distinguish only
between the cases of time data for every relevant failure (i.e. all instances of case 1 data –
1a), 1b) and 1c)) and time data for groups of relevant failures (i.e. case 2)).
The inference procedures that follow provide approximate estimates in some circumstances
and so caution is required if they are to be applied if the number of observed failures is less
than 10.
7.2 Point estimation
7.2.1 Case 1a) and 1b) – Time data for every relevant failure
This method applies only when the time of failure has been logged for every failure as
described in 6.1.2 and 6.1.3.
Step 1: Calculate the summation:
N *
 
T
 
S = ln (time terminated)
1 ∑
 
t
j
j =1  
N
 
t
N
 
S = ln (failure terminated)
2 ∑
 
t
j
 
j =1
Step 2: Calculate the (unbiased) estimate of the shape parameter from the formula:
β
∧
N −1
β = (time terminated)
S
∧
N − 2
(failure terminated)
β =
S
Step 3: Calculate the estimate of the scale parameter λ from the formula:
∧
N
λ = (time terminated)
∧
β
*
k(T )
– 14 – 61710 © IEC:2013
∧
N
λ= (failure terminated)
∧
β
k (t )
N
Step 4: Calculate the estimate of the failure intensity z(t), for any time t > 0 , from the formula:
∧
∧ ∧ ∧
β −1
z(t) = λ β t
∧
z(t) estimates the current failure intensity for t over the range represented by the data.
"Extrapolated" estimates for a future t may be obtained similarly, but should be used with the
usual caution associated with extrapolation.
Step 5: Given N observed failures the last of which occurred at t , the median time to the
N
(N+1)th failure can be estimated from the formula:
 
−1
 
N +1
 
0,5 −1
ˆ
T = t exp (time terminated)
 
N +1 N
 
ˆ
Nβ
 
(N −1)
 
 
 
−1
 
N +1
 
0,5 − 1
ˆ
T = t exp (failure terminated)
 
N +1 N
 
ˆ
Nβ
 
(N − 2)
 
 
7.2.2 Case 1c) – Time data for every relevant failure
This method applies only when the time of failure has been logged for every failure as
described in 6.1.4.
Step 1: Assemble the data into the times to failure, t , i=1,2,.,N, where N is the total number
i
of failures over the k copies and T , j=1,2,.,k, is the end of the observation period for the jth
j
copy.
ˆ
Step 2: The maximum likelihood estimate of the shape parameter β is the value of β which
satisfies the formula:
k
ˆ
β
N T lnT
∑ j j
N
N
j=1
+ ln t − = 0
∑ i
k
ˆ
ˆ
β β
i=1
T
∑ j
j=1
ˆ
An iterative method shall be used to solve the formula for β .
Step 3: Calculate the estimate of the scale parameter λ from the formula:

61710 © IEC:2013 – 15 –
N
ˆ
λ =
k
ˆ
β
T
∑
j
j=1
Step 4: Calculate the estimate of the failure intensity z(t), for any time t > 0 , from the formula:
∧
∧ ∧ ∧
β −1
z(t) = λ β t
∧
z(t)estimates the current failure intensity for t over the range represented by the data.
"Extrapolated" estimates for a future t may be obtained similarly, but should be used with the
usual caution associated with extrapolation.
7.2.3 Case 2 – Time data for groups of relevant failures
This method applies when the data set consists of known time intervals, each containing a
known number of failures as described in 6.2.
Step 1: Assemble into a data set the number of relevant failures N recorded in the ith
i
interval [t(i − 1),t(i)], i = 1, 2,., d . The total number of relevant failures is
d
N = N
∑ i
i =1
ˆ
Step 2: The maximum likelihood estimate of the shape parameter β is the value of β which
satisfies the formula:
∧ ∧
 
d
β β
[t(i)] ln t(i) − [t(i −1)] ln t(i −1) 
N − ln t(d ) = 0
i
∑  
∧ ∧
 β β 
i =1
[t(i)] − [t(i −1)]
 
∧ ∧
β β
[ ]
Note that t(0) = 0 and [t(0)] lnt(0) = 0. All t(.) terms may be normalized with respect to t(d )
disappears. An iterative method shall be used to solve the
and then the final term ln td( )

ˆ
formula for β .
Step 3: Calculate the estimate of the scale parameter λ from the formula:
∧
N
λ =
∧
β
( )
t d
Step 4: Calculate the estimate of the failure intensity z(t), for any test time t > 0, from the
formula:
∧
∧ ∧ ∧
β −1
z(t) = λ β t
– 16 – 61710 © IEC:2013
∧
z(t) estimates the current failure intensity for t over the range represented by the data.
"Extrapolated" estimates for a future t may be obtained similarly, but should be used with the
usual caution associated with extrapolation.
7.3 Goodness-of-fit tests
7.3.1 Case 1 – Time data for every relevant failure
7.3.1.1 Cramer-von-Mises test
ˆ
Step 1: Calculate β from step 2 in 7.2.1 or step 2 in 7.2.2.
Step 2: Calculate the Cramer-von-Mises goodness-of-fit test statistic given by the formula:
∧
 
β
M
 
 t 
1 j  2 j −1 
 
C = +  − 
 
∑
 
12M T 2M
 
 
 
j =1
 
 
where
*
M = N and  (time terminated)
T = T
M = N − 1 and T = t (failure terminated)
N
Step 3: Select the critical value C (M ) for the Cramer-von-Mises test corresponding to M
0,90
from Table 1, which gives critical values at a 10 % significance level.
Step 4: If:
2 2
C > C (M )
0,90
then the hypothesis that the power law model fits the data cannot be accepted. Otherwise, on
the basis of the data analysed, the power law model can be used as a working hypothesis.
7.3.1.2 Graphical procedure
When the failure times are known, the graphical procedure described below may be used to
obtain additional information about the correspondence between the model and the data. This
involves plotting the expected time to the jth failure, E(t ) , against the observed time to the
j
jth failure. Further details about the approach are given in Annexes A and B.
ˆ ˆ
Step 1: Calculate from step 2 in 7.2.1 and from step 3 in 7.2.1.
β λ
Step 2: Calculate the estimate of the expected time to the jth failure, j=1,2,.,N, from the
formula:
∧
∧
 j 
  β
E(t ) =
j
 
ˆ
 λ 
61710 © IEC:2013 – 17 –
∧
Step 3: Plot E(t ) against t on identical linear scales. The visual agreement of these points
j
j
with a line of 45 ° through the origin is a subjective measure of the applicability of the model.
7.3.2 Case 2 – Time data for groups of relevant failures
7.3.2.1 Chi-square test
ˆ 
Step 1: Calculate β from step 2 in 7.2.3 and λ from step 3 in 7.2.3.
Step 2: Calculate the expected number of failures in the time interval [t(i − 1),t(i)] which is
approximated by:
ˆ ˆ
 
β β
ˆ
e = λ [t(i)] − [t(i −1)]
i  
 
Step 3: For each interval, e shall not be less than 5, and if necessary, adjacent intervals
i
should be combined before the test. For d intervals (after combination if necessary) and with
N the same as in 7.2.3, calculate the statistic:
i
d
(N − e )
i i
χ =
∑
e
i
i =1
Step 4: Select the critical value from a χ distribution with (d − 2) degrees of freedom and a
10 % significance level from Table 2, i.e. χ (d − 2).
0,90
2 2
Step 5: If the test statistic χ exceeds the critical value χ (d − 2) then the hypothesis that
0,90
the power law model fits the data cannot be accepted. Otherwise, on the basis of the data
analysed, the power law model can be used as a working hypothesis.
The Chi-square test is a large sample test and so will need large data sets to detect
deviations from the power law model that are practically important.
7.3.2.2 Graphical procedure
When the data set consists of known time intervals, each containing a known number of
failures, the graphical procedure described below may be used to obtain additional
information about the correspondence between the model and the data. This involves plotting
the expected number of failures against those observed at each endpoint. Further details of
the approach are given in B.5.
Step 1: For each endpoint t(i), calculate the observed number of failures from 0 to t(i) from
the formula:
i
N[t(i)] = N
∑ j
j =1
Step 2: Calculate the estimate of the corresponding expected number of failures E[N[t(i)]]
from the formula:
– 18 – 61710 © IEC:2013
ˆ
β
ˆ ˆ
E[N[t(i)]] = λt(i)
ˆ
Step 3: Plot E[N[t(i)]] against N[t(i)]on identical linear scales. The visual agreement of these
points with a line of 45 ° through the origin is a subjective measure of the applicability of the
model.
7.4 Confidence intervals for the shape parameter
7.4.1 Case 1 – Time data for every relevant failure
The shape parameter β in the power law model determines if the failure intensity changes
with time. If 0 < β < 1, there is decreasing failure intensity; if β = 1, there is a constant failure
intensity; if β > 1, there is an increasing failure intensity.
For a two-sided confidence interval for β when individual failure times are available, follow
the steps below as appropriate for time and failure terminated data.
Two-sided 90 % confidence interval for – Time terminated data
β
ˆ
Step 1: Calculate from step 2 in 7.2.1 or from step 2 in 7.2.2.
β
Step 2: Calculate:
χ (2N)
0,05
D =
L
2(N −1)
χ (2N)
0,95
D =
U
2(N −1)
where the fractiles of the χ distribution are given in Table 2.
Step 3: Calculate the lower confidence limit for β from the formula:
∧
β = D β
LB L
and the upper confidence limit for β from the formula:
∧
β = D β
UB U
Step 4: The two-sided 90 % confidence interval for β is given by (β ,β ).
LB UB
NOTE One-sided 95 % lower and upper limits for β are β and β , respectively.
LB UB
Two-sided 90 % confidence interval for β – Failure terminated data
ˆ
Step 1: Calculate β from step 2 in 7.2.1.

61710 © IEC:2013 – 19 –
Step 2: Calculate:
χ (2(N −1))
0,05
D =
L
2(N − 2)
χ (2(N −1))
0,95
D =
U
2(N − 2)
where the fractiles of the χ distribution are given in Table 2.
Step 3: Calculate the lower confidence limit for β from the formula:
∧
β = D β
LB L
and the upper confidence limit for β from the formula:
∧
β = D β
UB U
Step 4: The two-sided 90 % confidence interval for β is given by (β , β ).
LB UB
NOTE One-sided 95 % lower and upper limits for β are β and β , respectively.
LB UB
7.4.2 Case 2 – Time data for groups of relevant failures
ˆ
Step 1: Calculate β from step 2 in 7.2.3.
Step 2: Calculate:
t(i)
P(i) = with i = 1, 2,.,d
t(d )
Step 3: Calculate the expression:
∧ ∧ ∧ ∧
 
β β β β
[P(i)] ln[P(i)] − [P(i − 1)] ln[P(i − 1)] 
d
 
 
A =
∑
∧ ∧
 
i =1
β β
[P(i)] − [P(i − 1)] 
 
 
Step 4: Calculate:
C =
A
Step 5: For an approximate two-sided 90 % confidence interval for β , calculate:

– 20 – 61710 © IEC:2013
1,64C
S =
N
where N is the total number of failures.
Step 6: Calculate the lower confidence limit for β from the formula:
∧
β = β (1− S)
LB
and the upper confidence limit for β from the formula:
∧
β = β (1+ S)
UB
Step 7: The two-sided 90 % confidence interval for β is given by (β ,β ).
LB UB
NOTE One-sided 95 % lower and upper limits for β are β and β , respectively.
LB UB
7.5 Confidence intervals for the failure intensity
7.5.1 Case 1 – Time data for every relevant failure
∧
Step 1: Calculate z(t) from step 4 in 7.2.1 or step 4 in 7.2.2.
Step 2: For a two-sided 90 % confidence interval refer to Table 3 (time terminated) and
Table 4 (failure terminated) and locate values of L and U for the appropriate sample size N.
Step 3: Calculate the lower confidence limit for z(t) from the formula:
∧
z(t)
z =
LB
U
and the upper confidence limit for z(t) from the formula:
∧
z(t)
z =
UB
L
Step 4: The two-sided 90 % confidence interval for z(t)
...