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BS EN 61710:2013

BS EN 61710:2013

$198.66

Power law model. Goodness-of-fit tests and estimation methods

Published By Publication Date Number of Pages
BSI 2013 60
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IEC 61710:2013 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). 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. Keywords: power law model, Bayesian estimation, reliability of repairable items

PDF Catalog

PDF Pages PDF Title
6 English
CONTENTS
9 INTRODUCTION
10 1 Scope
2 Normative references
3 Terms and definitions
4 Symbols and abbreviations
11 5 Power law model
12 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
6.1.2 Case 1a) – One repairable item
Figures
Figure 1 – One repairable item
13 6.1.3 Case 1b) – Multiple items of the same kind of repairable item observed for the same length of time
6.1.4 Case 1c) – Multiple repairable items of the same kind observed for different lengths of time
Figure 2 – Multiple items of the same kind of repairable item observed for same length of time
14 6.2 Case 2 – Time data for groups of relevant failures for one or more repairable items from the same population
6.3 Case 3 – Time data for every relevant failure for more than one repairable item from different populations
Figure 3 – Multiple repairable items of the same kind observedfor different lengths of time
15 7 Statistical estimation and test procedures
7.1 Overview
7.2 Point estimation
7.2.1 Case 1a) and 1b) – Time data for every relevant failure
16 7.2.2 Case 1c) – Time data for every relevant failure
17 7.2.3 Case 2 – Time data for groups of relevant failures
18 7.3 Goodness-of-fit tests
7.3.1 Case 1 – Time data for every relevant failure
19 7.3.2 Case 2 – Time data for groups of relevant failures
20 7.4 Confidence intervals for the shape parameter
7.4.1 Case 1 – Time data for every relevant failure
21 7.4.2 Case 2 – Time data for groups of relevant failures
22 7.5 Confidence intervals for the failure intensity
7.5.1 Case 1 – Time data for every relevant failure
7.5.2 Case 2 – Time data for groups of relevant failures
23 7.6 Prediction intervals for the length of time to future failures of a single item
7.6.1 Prediction interval for length of time to next failure for case 1 – Time data for every relevant failure
24 7.6.2 Prediction interval for length of time to Rth future failure for case 1 – Time data for every relevant failure
25 7.7 Test for the equality of the shape parameters β1,β 2, …, β k
7.7.1 Case 3 – Time data for every relevant failure for two items from different populations
26 7.7.2 Case 3 – Time data for every relevant failure for three or more items from different populations
27 Tables
Table 1 – Critical values for Cramer-von-Mises goodness-of-fit testat 10 % level of significance
28 Table 2 – Fractiles of the Chi-square distribution
29 Table 3 – Multipliers for two-sided 90 % confidence intervals for intensity function for time terminated data
30 Table 4 – Multipliers for two-sided 90 % confidence intervals for intensity function for failure terminated data
31 Table 5 – 0,95 fractiles of the F distribution
32 Annex A (informative) The power law model – Background information
33 Annex B (informative) Numerical examples
Table B.1 – All relevant failures and accumulated times for software system
34 Figure B.1 – Accumulated number of failures against accumulated timefor software system
Figure B.2 – Expected against observed accumulated times to failurefor software system
35 Table B.2 – Calculation of expected accumulated times to failure for Figure B.2
36 Table B.3 – Accumulated times for all relevant failuresfor five copies of a system (labelled A, B, C, D, E)
Table B.4 – Combined accumulated times for multiple items of the same kind of a system
37 Figure B.3 – Accumulated number of failures against accumulated timefor five copies of a system
38 Table B.5 – Accumulated operating hours to failure for OEM product from vendors A and B
39 Figure B.4 – Accumulated number of failures against accumulated time for an OEM product from vendors A and B
40 Figure B.5 – Accumulated number of failures against time for generators
Table B.6 – Grouped failure data for generators
41 Figure B.6 – Expected against observed accumulated number of failures for generators
42 Table B.7 – Calculation of expected numbers of failures for Figure B.6
43 Annex C (informative) Bayesian estimation for the power law model
44 Table C.1 – Strengths and weakness of classical and Bayesian estimation
48 Table C.2 – Grid for eliciting subjective distribution for shape parameter β
Table C.3 – Grid for eliciting subjective distribution for expected number of failures parameter η
49 Figure C.1 – Plot of fitted Gamma prior (6,7956, 0,0448) for the shape parameter of the power law model
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
50 Table C.4 – Comparison of fitted Gamma and subjective distributionfor shape parameter β
Table C.5 – Comparison of fitted Gamma and subjective distribution for expected number of failures by time T = 20 000 h parameter η
51 Table C.6 – Times to failure data collected on system test
52 Table C.7 – Summary of estimates of power law model parameters
53 Figure C.3 – Subjective distribution of number of failures
55 Table C.8 – Time to failure data for operational system
56 Figure C.4 – Plot of the posterior probability distribution for the number of future failures, M
57 Figure C.5 – Plot of the posterior cumulative distribution for the number of future failures, M
58 Bibliography

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BS EN 61710:2013
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