BS IEC 60605-4-2001 Equipment reliability testing - Statistical procedures for exponential distribution - Point estimates confidence intervals prediction intervals and tolerance in.pdf

1、BRITISH STANDARD BS IEC 60605-4:2001 Equipment reliability testing Part 4: Statistical procedures for exponential distribution Point estimates, confidence intervals, prediction intervals and tolerance intervals ICS 29.020; 03.120.01 NO COPYING WITHOUT BSI PERMISSION EXCEPT AS PERMITTED BY COPYRIGHT

2、LAWBS IEC 60605-4:2001 This British Standard, having been prepared under the direction of the Management Systems Sector Policy and Strategy Committee, was published under the authority of the Standards Policy and Strategy Committee on 31 August 2001 BSI 31 August 2001 ISBN 0 580 38414 4 National for

3、eword This British Standard reproduces verbatim IEC 60605-4:2001 and implements it as the UK national standard. The UK participation in its preparation was entrusted to Technical Committee DS/1, Dependability and terotechnology, which has the responsibility to: A list of organizations represented on

4、 this committee can be obtained on request to its secretary. From 1 January 1997, all IEC publications have the number 60000 added to the old number. For instance, IEC 27-1 has been renumbered as IEC 60027-1. For a period of time during the change over from one numbering system to the other, publica

5、tions may contain identifiers from both systems. Cross-references The British Standards which implement international or European publications referred to in this document may be found in the BSI Standards Catalogue under the section entitled “International Standards Correspondence Index”, or by usi

6、ng the “Find” facility of the BSI Standards Electronic Catalogue. A British Standard does not purport to include all the necessary provisions of a contract. Users of British Standards are responsible for their correct application. Compliance with a British Standard does not of itself confer immunity

7、 from legal obligations. aid enquirers to understand the text; present to the responsible international/European committee any enquiries on the interpretation, or proposals for change, and keep the UK interests informed; monitor related international and European developments and promulgate them in

8、the UK. Summary of pages This document comprises a front cover, an inside front cover, the IEC title page, pages 2 to 33 and a back cover. The BSI copyright date displayed in this document indicates when the document was last issued. Amendments issued since publication Amd. No. Date CommentsNORME IN

9、TERNATIONALE CEI IEC INTERNATIONAL STANDARD 60605-4 Deuxime dition Second edition 2001-08 Essai de fiabilit des quipements Partie 4: Mthodes statistiques de distribution exponentielle Estimateurs ponctuels, intervalles de confiance, intervalles de prdiction et intervalles de tolrance Equipment relia

10、bility testing Part 4: Statistical procedures for exponential distribution Point estimates, confidence intervals, prediction intervals and tolerance intervals Numro de rfrence Reference number CEI/IEC 60605-4:2001 BSIEC606054:2001 60605-4 IEC:2001 2 CONTENTS FOREWORD 4 1 Scope 6 2 Normative referenc

11、es. 6 3 Definitions and symbols . 7 3.1 Definitions 7 3.2 Symbols 7 4 Assumptions and requirements 8 4.1 Assumptions and information required for point estimates and confidence intervals . 9 4.2 Assumptions and requirements for prediction intervals 9 4.3 Assumptions and requirements for tolerance in

12、tervals . 9 5 Procedure for calculating point estimates and confidence intervals 10 5.1 Time terminated tests 11 5.1.1 Point estimates 11 5.1.2 Confidence limits11 5.2 Analytical procedure Failure terminated tests.15 5.2.1 Point estimate15 5.2.2 Confidence intervals.15 6 Prediction intervals for the

13、 number of failures in a future period.17 6.1 Two-sided prediction intervals L2 r and U2 r .17 6.2 One-sided prediction interval17 7 Procedure for assigning tolerance intervals .17 7.1 Upper Poisson tolerance bound17 7.2 Lower Poisson tolerance bound18 Annex A (informative) Examples.19 A.1 Point est

14、imate of MTTF.19 A.2 Application of lower one-sided 90 % confidence limit on mean time to failure (MTTF).19 A.3 Application of two-sided 90 % confidence limits on MTTF 19 A.4 Application of a two-sided 90 % prediction interval.20 A.5 Application of upper 90 % tolerance bound at 95 % confidence20 A.6

15、 Application of lower 90 % tolerance bound at 95 % confidence 20 Annex B (informative) Relation between confidence, prediction and tolerance intervals .22 B.1 Confidence intervals.22 B.2 Prediction intervals.22 B.3 Tolerance intervals.23 BSIEC606054:2001 BSI31August2001 2 BSIEC606054:2001 BSI31Augus

16、t2001 2 BSIEC606054:2001 BSI31August2001 260605-4 IEC:2001 3 Annex C (normative) Computation of accumulated test time * T 24 C.1 Case 1, one repaired item with constant failure intensity .24 C.2 Case 2, more than one repaired item with identical constant failure intensities 25 C.3 Case 3, non-repair

17、ed items 26 Annex D (normative) Tables of fractiles of the chi-squared distribution: ) ( 2 27 Annex E (normative) Integral of the chi-squared distribution and cumulative Poisson distribution .29 Annex F (normative) 0,95 fractiles of the F distribution32 Bibliography 33 BSIEC606054:2001 BSI31August20

18、01 3 BSIEC606054:2001 BSI31August2001 3 BSIEC606054:2001 BSI31August2001 360605-4 IEC:2001 4 INTERNATIONAL ELECTROTECHNICAL COMMISSION _ EQUIPMENT RELIABILITY TESTING Part 4: Statistical procedures for exponential distribution Point estimates, confidence intervals, prediction intervals and tolerance

19、 intervals FOREWORD 1) The IEC (International Electrotechnical Commission) is a worldwide organization for standardization comprising all national electrotechnical committees (IEC National Committees). The object of the IEC is to promote international co-operation on all questions concerning standar

20、dization in the electrical and electronic fields. To this end and in addition to other activities, the IEC publishes International Standards. Their preparation is entrusted to technical committees; any IEC National Committee interested in the subject dealt with may participate in this preparatory wo

21、rk. International, governmental and non-governmental organizations liaising with the IEC also participate in this preparation. The IEC collaborates closely with the International Organization for Standardization (ISO) in accordance with conditions determined by agreement between the two organization

22、s. 2) The formal decisions or agreements of the IEC on technical matters express, as nearly as possible, an international consensus of opinion on the relevant subjects since each technical committee has representation from all interested National Committees. 3) The documents produced have the form o

23、f recommendations for international use and are published in the form of standards, technical specifications, technical reports or guides and they are accepted by the National Committees in that sense. 4) In order to promote international unification, IEC National Committees undertake to apply IEC I

24、nternational Standards transparently to the maximum extent possible in their national and regional standards. Any divergence between the IEC Standard and the corresponding national or regional standard shall be clearly indicated in the latter. 5) The IEC provides no marking procedure to indicate its

25、 approval and cannot be rendered responsible for any equipment declared to be in conformity with one of its standards. 6) Attention is drawn to the possibility that some of the elements of this International Standard may be the subject of patent rights. The IEC shall not be held responsible for iden

26、tifying any or all such patent rights. International Standard IEC 60605-4 has been prepared by IEC technical committee 56: Dependability. This second edition cancels and replaces the first edition published in 1986 and its amendment 1 (1989), of which it constitutes a technical revision. The text of

27、 this standard is based on the following documents: FDIS Report on voting 56/737/FDIS 56/756/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. This publication has been drafted in accordance with the ISO/IEC Direct

28、ives, Part 3. Annexes A and B are for information only. Annexes C, D, E and F form an integral part of this standard. BSIEC606054:2001 BSI31August2001 4 BSIEC606054:2001 BSI31August2001 4 BSIEC606054:2001 BSI31August2001 460605-4 IEC:2001 5 The committee has decided that the contents of this publica

29、tion will remain unchanged until 2006. At this date, the publication will be reconfirmed; withdrawn; replaced by a revised edition, or amended. BSIEC606054:2001 BSI31August2001 5 BSIEC606054:2001 BSI31August2001 5 BSIEC606054:2001 BSI31August2001 560605-4 IEC:2001 6 EQUIPMENT RELIABILITY TESTING Par

30、t 4: Statistical procedures for exponential distribution Point estimates, confidence intervals, prediction intervals and tolerance intervals 1 Scope This International Standard provides statistical methods for evaluating point estimates, confidence intervals, prediction intervals and tolerance inter

31、vals for the failure rate of items, whose time to failure follows an exponential distribution. This implies that the failure rate (see IEV 191-12-02) is constant with respect to time. It should be noted that although reference is made above to failure rate, the numerical methods to be described are

32、equally applicable to other event rates provided the times to the occurrence of the event follow an exponential distribution. Thus, with this qualification, the numerical methods will apply, for example, to constant failure intensities and constant repair rates (see IEV 191-12-04 and IEV 191-13-02 r

33、espectively). For convenience, however, and to avoid unnecessary repetition, reference will be made in what follows only to failures and failure rates. Use of the procedures in this standard should be supported by tests to confirm the validity of the assumption of constant failure rate or failure in

34、tensity (see IEC 60605-6). This standard is also applicable whenever a random sample of items is subjected to a test of times to failure for the purpose of estimating measures of reliability performance. 2 Normative references The following normative documents contain provisions which, through refer

35、ence in this text, constitute provisions of this part of IEC 60605. For dated references, subsequent amendments to, or revisions of, any of these publications do not apply. However, parties to agreements based on this part of IEC 60605 are encouraged to investigate the possibility of applying the mo

36、st recent editions of the normative documents indicated below. For undated references, the latest edition of the normative document referred to applies. Members of IEC and ISO maintain registers of currently valid International Standards. IEC 60050(191):1990, International Electrotechnical Vocabular

37、y (IEV) Chapter 191: Depend- ability and quality of service IEC 60605-6:1997, Equipment reliability testing Part 6: Tests for the validity of the constant failure rate or constant failure intensity assumptions ISO 3534-1:1993, Statistics Vocabulary and symbols Part 1: Probability and general statist

38、ical terms ISO 3534-2:1993, Statistics Vocabulary and symbols Part 2: Statistical quality control BSIEC606054:2001 BSI31August2001 6 BSIEC606054:2001 BSI31August2001 6 BSIEC606054:2001 BSI31August2001 660605-4 IEC:2001 7 3 Definitions and symbols For the purposes of this International Standard, the

39、terms and definitions of IEC 60050(191), ISO 3534-1 and ISO 3534-2 apply, together with the following definitions and symbols. Additional constants and functions are defined in the text. 3.1 Definitions 3.1.1 two-sided confidence interval (see ISO 3534-1, 2.57) when L2 m and U2 m are two functions o

40、f the observed values such that, m being a population parameter to be estimated, the probability ) ( Pr 2 U 2 L m m m is at least equal to ) 1 ( (where ) 1 ( is a fixed number, positive and less than 1), the interval between 2 L m and 2 U m is a two- sided ) 1 ( confidence interval for m 3.1.2 one-s

41、ided confidence interval (see ISO 3534-1, 2.58) when 1 U m(or L1 m ) is a function of the observed values such that, m being a population parameter to be estimated, the probability ) ( Pr 1 U m m (or the probability ) ( Pr 1 L m m ) is at least equal to ) 1 ( (where ) 1 ( is a fixed number, positive

42、 and less than 1), the interval from the smallest possible value of m up to 1 U m (or the interval from L1 mup to the largest possible value of m) is a one-sided ) 1 ( confidence interval for m NOTE See annex B for further discussion. 3.1.3 tolerance limits (see ISO 3534-2, 1.4.3) two limits L2 and

43、U2 on the observed number of failures between which at least a proportion P of the population may be asserted (with probability 1 ) to lie NOTE The interval L2 to U2 is known as the tolerance interval. 3.1.4 (see ISO 3534-2, 1.4.4) tolerance difference between the upper and lower tolerance limits. N

44、OTE More descriptive definitions of the above quantities and their relationship to one another are given in annex B. 3.2 Symbols cl Lower or upper one-sided confidence limit on a dependability parameter, such as MTTF (mean time to failure), failure rate, reliability, etc. Significance level: ) 1 ( 1

45、00 % is the confidence level at which confidence intervals and limits are calculated L1, U1 Lower and upper one-sided values of a ) 1 ( 100 % tolerance interval L2, U2 Lower and upper two-sided values of a ) 2 / 1 ( 100 % tolerance interval True constant failure rate BSIEC606054:2001 BSI31August2001

46、 7 BSIEC606054:2001 BSI31August2001 7 BSIEC606054:2001 BSI31August2001 760605-4 IEC:2001 8 Estimated constant failure rate U1 1 L , Lower and upper one-sided confidence limit on true failure rate U2 2 L , Lower and upper two-sided confidence limit on true failure rate m True mean time to failure m E

47、stimated mean time to failure U1 1 L ,m m Lower and upper one-sided confidence limit on true mean time to failure U2 2 L ,m m Lower and upper two-sided confidence limit on true mean time to failure n Total number of items under test P Proportion of a future population (used for the purpose of assign

48、ing tolerance intervals) r Number of failures observed r L1 ,r U1 Lower and upper one-sided values of a % ) 1 ( 100 prediction interval r L2 ,r U2 Lower and upper two-sided values of a % ) 2 / 1 ( 100 prediction interval t Calendar time at which the test was terminated * T Accumulated test time: the total number of item-time units, e.g. item-hours, obtained at the termination of the test (see annex C) f w Length of future periods (used for the purpose of assigning prediction intervals) p w Length of past periods (used for the purpose of ass