5404:
2311:
5390:
1514:
875:
5428:
5416:
280:, it is not actually a probability because it can exceed 1. Erroneous expression of the failure rate in % could result in incorrect perception of the measure, especially if it would be measured from repairable systems and multiple systems with non-constant failure rates or different operation times. It can be defined with the aid of the
2294:
75:, which is also called the "useful life period". Because of this, it is incorrect to extrapolate MTBF to give an estimate of the service lifetime of a component, which will typically be much less than suggested by the MTBF due to the much higher failure rates in the "end-of-life wearout" part of the "bathtub curve".
2175:), the failure rate for a complex system is simply the sum of the individual failure rates of its components, as long as the units are consistent, e.g. failures per million hours. This permits testing of individual components or subsystems, whose failure rates are then added to obtain the total system failure rate.
165:
Mechanical movement is the predominant failure mechanism causing mechanical and electromechanical devices to wear out. For many devices, the wear-out failure point is measured by the number of cycles performed before the device fails, and can be discovered by cycle testing. In cycle testing, a device
151:
Time lag is one of the serious drawbacks of all failure rate estimations. Often by the time the failure rate data are available, the devices under study have become obsolete. Due to this drawback, failure-rate prediction methods have been developed. These methods may be used on newly designed devices
2108:
Increasing failure rate is an intuitive concept caused by components wearing out. Decreasing failure rate describes a system which improves with age. Decreasing failure rates have been found in the lifetimes of spacecraft, Baker and Baker commenting that "those spacecraft that last, last on and on."
865:
Hazard rate and ROCOF (rate of occurrence of failures) are often incorrectly seen as the same and equal to the failure rate. To clarify; the more promptly items are repaired, the sooner they will break again, so the higher the ROCOF. The hazard rate is however independent of the time to repair and
55:
The failure rate of a system usually depends on time, with the rate varying over the life cycle of the system. For example, an automobile's failure rate in its fifth year of service may be many times greater than its failure rate during its first year of service. One does not expect to replace an
202:
Given a component database calibrated with field failure data that is reasonably accurate , the method can predict product level failure rate and failure mode data for a given application. The predictions have been shown to be more accurate than field warranty return analysis or even typical field
2191:
Suppose it is desired to estimate the failure rate of a certain component. A test can be performed to estimate its failure rate. Ten identical components are each tested until they either fail or reach 1000 hours, at which time the test is terminated for that component. (The level of statistical
123:
From field failure rate reports, statistical analysis techniques can be used to estimate failure rates. For accurate failure rates the analyst must have a good understanding of equipment operation, procedures for data collection, the key environmental variables impacting failure rates, how the
2067:
A decreasing failure rate (DFR) describes a phenomenon where the probability of an event in a fixed time interval in the future decreases over time. A decreasing failure rate can describe a period of "infant mortality" where earlier failures are eliminated or corrected and corresponds to the
2125:
is ⩟ 1, and when the failure rate is increasing the coefficient of variation is ⩜ 1. Note that this result only holds when the failure rate is defined for all t ⩟ 0 and that the converse result (coefficient of variation determining nature of failure rate) does not hold.
130:
Many organizations maintain internal databases of failure information on the devices or systems that they produce, which can be used to calculate failure rates for those devices or systems. For new devices or systems, the historical data for similar devices or systems can serve as a useful
67:(MTBF, 1/λ) is often reported instead of the failure rate. This is valid and useful if the failure rate may be assumed constant â often used for complex units / systems, electronics â and is a general agreement in some reliability standards (Military and Aerospace). It does in this case
1146:
659:
2099:
with DFR renewal function, inter-renewal times are concave. Brown conjectured the converse, that DFR is also necessary for the inter-renewal times to be concave, however it has been shown that this conjecture holds neither in the discrete case nor in the continuous case.
2156:) rate of a device is the number of failures that can be expected in one billion (10) device-hours of operation. (E.g. 1000 devices for 1 million hours, or 1 million devices for 1000 hours each, or some other combination.) This term is used particularly by the
2182:
improves the mission failure rate, but makes the series failure rate (also called the logistics failure rate) worseâthe extra components improve the mean time between critical failures (MTBCF), even though the mean time before something fails is worse.
2205:
1766:
1639:
158:
The most accurate source of data is to test samples of the actual devices or systems in order to generate failure data. This is often prohibitively expensive or impractical, so that the previous data sources are often used
1927:
1281:
145:
that provides failure rate data for many military electronic components. Several failure rate data sources are available commercially that focus on commercial components, including some non-electronic components.
1508:
989:
2057:
463:
78:
The reason for the preferred use for MTBF numbers is that the use of large positive numbers (such as 2000 hours) is more intuitive and easier to remember than very small numbers (such as 0.0005 per hour).
90:, where the time to recover from failure can be neglected and the likelihood of failure remains constant with respect to time, the failure rate is simply the multiplicative inverse of the MTBF (1/λ).
1394:
2979:
421:
108:
Failure rates are important factors in the insurance, finance, commerce and regulatory industries and fundamental to the design of safe systems in a wide variety of applications.
2289:{\displaystyle {\frac {6{\text{ failures}}}{7502{\text{ hours}}}}=0.0007998\,{\frac {\text{failures}}{\text{hour}}}=799.8\times 10^{-6}\,{\frac {\text{failures}}{\text{hour}}},}
2420:. Proceedings of the 2014 International Symposium - BEYOND REGULATORY COMPLIANCE, MAKING SAFETY SECOND NATURE, Hilton College Station-Conference Center, College Station, Texas.
733:
1654:
332:
254:
981:
687:
3108:
2345:
807:
760:
172:
1959:
1182:
954:
905:
860:
450:
4525:
1306:
82:
The MTBF is an important system parameter in systems where failure rate needs to be managed, in particular for safety systems. The MTBF appears frequently in the
2171:
Under certain engineering assumptions (e.g. besides the above assumptions for a constant failure rate, the assumption that the considered system has no relevant
5030:
831:
780:
352:
278:
175:(FMEDA) is a systematic analysis technique to obtain subsystem / product level failure rates, failure modes and design strength. The FMEDA technique considers:
1542:
5180:
4804:
166:
is cycled as rapidly as practical until it fails. When a collection of these devices are tested, the test will run until 10% of the units fail dangerously.
3445:
2370:
4578:
5017:
1198:
2459:
3440:
3140:
1141:{\displaystyle h(t)=\lim _{\Delta t\to 0}{\frac {R(t)-R(t+\Delta t)}{\Delta t\cdot R(t)}}=-{\frac {\mathrm {d} }{\mathrm {d} t}}\ln R(t).}
1816:
203:
failure analysis given that these methods depend on reports that typically do not have sufficient detail information in failure records.
4044:
3192:
219:, divided by the total time expended by that population, during a particular measurement interval under stated conditions. (MacDiarmid,
5432:
1405:
654:{\displaystyle \lambda (t)={\frac {R(t_{1})-R(t_{2})}{(t_{2}-t_{1})\cdot R(t_{1})}}={\frac {R(t)-R(t+\Delta t)}{\Delta t\cdot R(t)}}\!}
2805:
1967:
4827:
4719:
2988:
2739:
1528:
3093:
5464:
5005:
4879:
1771:
Thus, for an exponential failure distribution, the hazard rate is a constant with respect to time (that is, the distribution is "
1185:
5063:
4724:
4469:
3840:
3430:
2138:
is the most common unit in practice. Other units, such as miles, revolutions, etc., can also be used in place of "time" units.
142:
4054:
5114:
4326:
4133:
4022:
3980:
3035:
2942:
2918:
2899:
2865:
3219:
1329:
137:
Handbooks of failure rate data for various components are available from government and commercial sources. MIL-HDBK-217F,
5459:
5357:
4316:
2084:
1784:
4366:
3103:
4908:
4857:
4842:
4832:
4701:
4573:
4540:
4321:
4151:
2849:
1309:
152:
to predict the device's failure rates and failure modes. Two approaches have become well known, Cycle
Testing and FMEDA.
86:
design requirements, and governs frequency of required system maintenance and inspections. In special processes called
4977:
4278:
5479:
5252:
5053:
4032:
3701:
3165:
362:
5137:
5104:
2145:
as failures per million, or 10, especially for individual components, since their failure rates are often very low.
5469:
5454:
5109:
4852:
4611:
4517:
4497:
4405:
4116:
3934:
3417:
3289:
1788:
64:
4283:
4049:
3907:
4869:
4637:
4358:
4212:
4141:
4061:
3919:
3900:
3608:
3329:
2969:(Commercial Practices ed.). Rome, New York: Reliability Analysis Center and Rome Laboratory. pp. 35â39.
908:
4982:
5352:
5119:
4667:
4632:
4596:
4381:
3823:
3732:
3691:
3603:
3294:
3133:
2172:
2122:
2110:
2080:
1645:
1522:
57:
4389:
4373:
2418:
Combining field failure data with new instrument design margins to predict failure rates for SIS Verification
5484:
5261:
4874:
4814:
4751:
4111:
3973:
3963:
3813:
3727:
2360:
2355:
2325:
2179:
1780:
810:
49:
5022:
4959:
2999:
1517:
Exponential failure density functions. Each of these has a (different) constant hazard function (see text).
5299:
5229:
4714:
4601:
3598:
3495:
3402:
3281:
3180:
2927:
216:
5420:
4298:
5324:
5266:
5209:
5035:
4928:
4837:
4563:
4447:
4306:
4188:
4180:
3995:
3891:
3869:
3828:
3793:
3760:
3706:
3681:
3636:
3575:
3535:
3337:
3160:
5403:
4293:
2310:
5474:
5247:
4822:
4771:
4747:
4709:
4627:
4606:
4558:
4437:
4415:
4384:
4170:
4121:
4039:
4012:
3968:
3924:
3686:
3462:
3342:
2696:
2380:
2142:
1776:
281:
1761:{\displaystyle h(t)={\frac {f(t)}{R(t)}}={\frac {\lambda e^{-\lambda t}}{e^{-\lambda t}}}=\lambda .}
5394:
5319:
5242:
4923:
4687:
4680:
4642:
4550:
4530:
4502:
4235:
4101:
4096:
4086:
4078:
3896:
3857:
3747:
3737:
3646:
3425:
3381:
3299:
3224:
3126:
2687:
Baker, J. C.; Baker, G. A. S. . (1980). "Impact of the space environment on spacecraft lifetimes".
2193:
2073:
1800:
692:
38:
4969:
5408:
5219:
5073:
4918:
4794:
4691:
4675:
4652:
4429:
4163:
4146:
4106:
4017:
3912:
3874:
3845:
3805:
3765:
3711:
3628:
3314:
3309:
3058:
3050:
2759:
2669:
2631:
2613:
2586:
2545:
2501:
2365:
2350:
2316:
287:
230:
124:
equipment is used at the system level, and how the failure data will be used by system designers.
94:
5314:
5284:
5276:
5096:
5087:
5012:
4943:
4799:
4784:
4759:
4647:
4588:
4454:
4442:
4068:
3985:
3929:
3852:
3696:
3618:
3397:
3271:
3031:
2984:
2974:
2938:
2914:
2895:
2821:
2801:
2735:
2455:
2375:
963:
669:
5339:
5294:
5058:
5045:
4938:
4913:
4847:
4779:
4657:
4265:
4158:
4091:
4004:
3951:
3770:
3641:
3435:
3234:
3201:
3014:
2778:
2727:
2704:
2661:
2623:
2576:
2535:
2491:
2447:
2430:
W. M. Goble, "Field
Failure Data â the Good, the Bad and the Ugly," exida, Sellersville, PA
2833:
2722:
Saleh, Joseph Homer; Castet, Jean-François (2011). "On Time, Reliability, and
Spacecraft".
785:
738:
5256:
5000:
4862:
4789:
4464:
4338:
4311:
4288:
4257:
3884:
3879:
3833:
3563:
3214:
3112:
3097:
3070:
2853:
2480:"Bounds, Inequalities, and Monotonicity Properties for Some Specialized Renewal Processes"
2096:
1935:
1158:
930:
881:
836:
426:
87:
41:
or component fails, expressed in failures per unit of time. It is usually denoted by the
1289:
2763:
2700:
2109:
The reliability of aircraft air conditioning systems were individually found to have an
1634:{\displaystyle F(t)=\int _{0}^{t}\lambda e^{-\lambda \tau }\,d\tau =1-e^{-\lambda t},\!}
5489:
5205:
5200:
3663:
3593:
3239:
3090:
1772:
816:
765:
337:
263:
42:
2782:
2524:"DFR Property of First-Passage Times and its Preservation Under Geometric Compounding"
1308:
is the failure time. The failure distribution function is the integral of the failure
5448:
5362:
5329:
5192:
5153:
4964:
4933:
4397:
4351:
3956:
3658:
3485:
3249:
3244:
2822:"A Realistic Evaluation of Memory Hardware Errors and Software System Susceptibility"
2157:
1796:
1787:, the hazard function may not be constant with respect to time. For some such as the
72:
3515:
3079:
Military
Handbook, âReliability Prediction of Electronic Equipment, MIL-HDBK-217F, 2
2652:
Proschan, F. (1963). "Theoretical
Explanation of Observed Decreasing Failure Rate".
2635:
5304:
5237:
5214:
4459:
3755:
3653:
3588:
3530:
3452:
3407:
3000:"Approximation of Mean Time Between Failure When a System has Periodic Maintenance"
2755:
2665:
2340:
2163:
The relationship of FIT to MTBF may be expressed as: MTBF = 1,000,000,000 x 1/FIT.
2431:
5347:
5309:
4992:
4893:
4755:
4568:
4535:
4027:
3944:
3939:
3583:
3540:
3520:
3500:
3490:
3259:
2894:(Fourth ed.). Englewood Cliffs, New Jersey: Prentice-Hall. pp. 26â32.
2604:
Yu, Y. (2011). "Concave renewal functions do not imply DFR interrenewal times".
2451:
452:
is the time to (first) failure distribution (i.e. the failure density function).
257:
102:
101:
and trucking is "mean distance between failures", a variation which attempts to
83:
915:
Calculating the failure rate for ever smaller intervals of time results in the
17:
4193:
3673:
3373:
3304:
3254:
3229:
3149:
2952:
Knowles, D. I. (1995). "Should We Move Away From 'Acceptable
Failure Rate'?".
2731:
2306:
1513:
1188:
that describes the probability of failure (at least) up to and including time
116:
Failure rate data can be obtained in several ways. The most common means are:
2846:
2627:
2581:
2564:
2540:
2523:
2496:
2479:
4346:
4198:
3818:
3613:
3525:
3510:
3505:
3470:
3018:
1792:
874:
34:
260:
that a failure occurs in a specified interval given no failure before time
2764:"A note on comparing response times in the M/GI/1/FB and M/GI/1/PS queues"
3862:
3480:
3357:
3352:
3347:
3319:
3028:
System
Reliability Theory; Models, Statistical methods, and Applications
1276:{\displaystyle \operatorname {P} (T\leq t)=F(t)=1-R(t),\quad t\geq 0.\!}
5367:
5068:
2673:
2590:
2549:
2505:
2335:
2330:
1804:
189:
The effect of each component failure mode on the product functionality,
98:
5289:
4270:
4244:
4224:
3475:
3266:
2883:
Safety
Instrumented System Design: Techniques and Design Verification
2708:
105:
actual loaded distances to similar reliability needs and practices.
2565:"Further Monotonicity Properties for Specialized Renewal Processes"
1922:{\displaystyle h(t)={\frac {f(t)}{1-F(t)}}={\frac {F'(t)}{1-F(t)}}}
2618:
1512:
873:
813:, where the condition is that no failure has occurred before time
2885:, Research Triangle Park, NC: International Society of Automation
2866:"Mission Reliability and Logistics Reliability: A Design Paradox"
1503:{\displaystyle h(t)={\frac {f(t)}{1-F(t)}}={\frac {f(t)}{R(t)}}.}
3209:
2954:
Communications in
Reliability Maintainability and Supportability
2196:
is not considered in this example.) The results are as follows:
2052:{\displaystyle F(t)=1-\exp {\left(-\int _{0}^{t}h(t)dt\right)}.}
45:
5178:
4745:
4492:
3791:
3561:
3178:
3122:
192:
The ability of any automatic diagnostics to detect the failure,
2911:
An
Introduction to Reliability and Maintainability Engineering
2134:
Failure rates can be expressed using any measure of time, but
127:
Historical data about the device or system under consideration
3118:
2446:. Springer Series in Reliability Engineering. pp. 1â84.
2980:
Reliability Engineering and Risk Analysis: A Practical Guide
27:
Frequency with which an engineered system or component fails
2965:
MacDiarmid, Preston; Morris, Seymour; et al. (n.d.).
2402:
Electrical & Mechanical Component Reliability Handbook
2299:
or 799.8 failures for every million hours of operation.
1151:
A continuous failure rate depends on the existence of a
2847:"Calculating Failure Rates of Series/Parallel Networks"
2400:
198:
The operational profile (environmental stress factors).
3047:
The Customer Needs A Maintenance-Free Operating Period
2208:
1970:
1938:
1819:
1657:
1545:
1408:
1389:{\displaystyle F(t)=\int _{0}^{t}f(\tau )\,d\tau .\!}
1332:
1292:
1201:
1161:
992:
966:
933:
884:
839:
819:
788:
768:
741:
695:
672:
466:
429:
365:
340:
290:
266:
233:
5031:
Autoregressive conditional heteroskedasticity (ARCH)
3115:
by Hubert Kirrmann, ABB Research Center, Switzerland
960:
failure rate or we say instantaneous hazard rate as
5338:
5275:
5228:
5191:
5146:
5128:
5095:
5086:
5044:
4991:
4952:
4901:
4892:
4813:
4770:
4700:
4666:
4620:
4587:
4549:
4516:
4428:
4337:
4256:
4211:
4179:
4132:
4077:
4003:
3994:
3804:
3746:
3720:
3672:
3627:
3574:
3461:
3416:
3390:
3372:
3328:
3280:
3200:
3191:
195:
The design strength (de-rating, safety factors) and
2937:. New York: John Wiley & Sons. pp. 8â30.
2288:
2051:
1953:
1921:
1760:
1633:
1502:
1388:
1300:
1275:
1176:
1140:
975:
948:
899:
854:
825:
801:
774:
754:
727:
681:
653:
444:
415:
346:
326:
272:
248:
211:The failure rate can be defined as the following:
3105:Fault Tolerant Computing in Industrial Automation
2820:Xin Li; Michael C. Huang; Kai Shen; Lingkun Chu.
1630:
1385:
1272:
650:
56:exhaust pipe, overhaul the brakes, or have major
2079:Mixtures of DFR variables are DFR. Mixtures of
1009:
416:{\displaystyle \lambda (t)={\frac {f(t)}{R(t)}}}
4579:Multivariate adaptive regression splines (MARS)
2724:Spacecraft Reliability and Multi-State Failures
2444:Failure Rate Modelling for Reliability and Risk
2346:Failure modes, effects, and diagnostic analysis
1525:can be used to model the failure distribution (
173:Failure modes, effects, and diagnostic analysis
2517:
2515:
139:Reliability Prediction of Electronic Equipment
3134:
3045:Turner, T.; Hockley, C.; Burdaky, R. (1997).
2178:Adding "redundant" components to eliminate a
8:
2798:Analysis of Queues: Methods and Applications
2416:Goble, William M.; Iwan van Beurden (2014).
334:, the probability of no failure before time
215:The total number of failures within an item
2473:
2471:
2442:Finkelstein, Maxim (2008). "Introduction".
2113:, and thus in the pooled population a DFR.
1529:List of important probability distributions
134:Government and commercial failure rate data
5188:
5175:
5092:
4898:
4767:
4742:
4513:
4489:
4217:
4000:
3801:
3788:
3571:
3558:
3197:
3188:
3175:
3141:
3127:
3119:
2960:(1). International RMS Committee, USA: 23.
2647:
2645:
1399:The hazard function can be defined now as
2617:
2580:
2539:
2495:
2371:Reliability theory of aging and longevity
2273:
2272:
2263:
2240:
2239:
2225:
2215:
2209:
2207:
2016:
2011:
1998:
1969:
1937:
1876:
1835:
1818:
1803:it is monotonic decreasing (analogous to
1735:
1718:
1708:
1673:
1656:
1615:
1595:
1583:
1570:
1565:
1544:
1465:
1424:
1407:
1375:
1357:
1352:
1331:
1293:
1291:
1200:
1160:
1106:
1100:
1098:
1027:
1012:
991:
965:
932:
883:
838:
818:
793:
787:
767:
746:
740:
716:
703:
694:
671:
585:
570:
548:
535:
517:
495:
482:
465:
428:
381:
364:
339:
289:
265:
232:
2121:When the failure rate is decreasing the
1775:"). For other distributions, such as a
1648:. The hazard rate function for this is:
2933:Kapur, K. C.; Lamberson, L. R. (1977).
2913:. Boston: McGraw-Hill. pp. 23â32.
2392:
1807:), while for many it is not monotonic.
5105:KaplanâMeier estimator (product limit)
3066:
3056:
2977:; Kaminskiy, M.; Krivtsov, V. (2010).
2141:Failure rates are often expressed in
284:, also called the survival function,
7:
5415:
5115:Accelerated failure time (AFT) model
2892:Logistics Engineering and Management
870:Failure rate in the continuous sense
186:The failure modes of each component,
183:The functionality of each component,
5427:
4710:Analysis of variance (ANOVA, anova)
3077:U.S. Department of Defense, (1991)
3030:. New York: John Wiley & Sons.
1810:Solving the differential equation
4805:CochranâMantelâHaenszel statistics
3431:Pearson product-moment correlation
1202:
1107:
1101:
1068:
1057:
1013:
967:
673:
626:
615:
207:Failure rate in the discrete sense
25:
3026:Rausand, M.; Hoyland, A. (2004).
2998:Mondro, Mitchell J. (June 2002).
2935:Reliability in Engineering Design
2689:Journal of Spacecraft and Rockets
71:relate to the flat region of the
5426:
5414:
5402:
5389:
5388:
3007:IEEE Transactions on Reliability
2309:
1534:exponential failure distribution
1186:cumulative distribution function
5064:Least-squares spectral analysis
2890:Blanchard, Benjamin S. (1992).
1262:
4045:Mean-unbiased minimum-variance
2666:10.1080/00401706.1963.10490105
2606:Journal of Applied Probability
2085:hyperexponentially distributed
2031:
2025:
1980:
1974:
1948:
1942:
1913:
1907:
1893:
1887:
1867:
1861:
1847:
1841:
1829:
1823:
1699:
1693:
1685:
1679:
1667:
1661:
1555:
1549:
1491:
1485:
1477:
1471:
1456:
1450:
1436:
1430:
1418:
1412:
1372:
1366:
1342:
1336:
1256:
1250:
1235:
1229:
1220:
1208:
1171:
1165:
1132:
1126:
1086:
1080:
1063:
1048:
1039:
1033:
1019:
1002:
996:
943:
937:
894:
888:
849:
843:
722:
696:
644:
638:
621:
606:
597:
591:
576:
563:
554:
528:
523:
510:
501:
488:
476:
470:
439:
433:
407:
401:
393:
387:
375:
369:
321:
315:
300:
294:
243:
237:
48:(lambda) and is often used in
1:
5358:Geographic information system
4574:Simultaneous equations models
2783:10.1016/S0167-6377(03)00061-0
728:{\displaystyle (t_{2}-t_{1})}
256:, is often thought of as the
4541:Coefficient of determination
4152:Uniformly most powerful test
2909:Ebeling, Charles E. (1997).
2522:Shanthikumar, J. G. (1988).
1646:exponential density function
866:of the logistic delay time.
93:A similar ratio used in the
5110:Proportional hazards models
5054:Spectral density estimation
5036:Vector autoregression (VAR)
4470:Maximum posterior estimator
3702:Randomized controlled trial
3049:. Leatherhead, Surrey, UK:
2983:(2nd ed.). CRC Press.
2771:Operations Research Letters
2452:10.1007/978-1-84800-986-8_1
907:plotted for a selection of
327:{\displaystyle R(t)=1-F(t)}
249:{\displaystyle \lambda (t)}
227:Although the failure rate,
180:All components of a design,
60:problems in a new vehicle.
5506:
4870:Multivariate distributions
3290:Average absolute deviation
2881:Goble, William M. (2018),
2800:. CRC Press. p. 703.
2796:Gautam, Natarajan (2012).
2199:Estimated failure rate is
1799:), for others such as the
1789:deterministic distribution
1785:hypertabastic distribution
909:log-logistic distributions
65:mean time between failures
5384:
5187:
5174:
4858:Structural equation model
4766:
4741:
4512:
4488:
4220:
4194:Score/Lagrange multiplier
3800:
3787:
3609:Sample size determination
3570:
3557:
3187:
3174:
3156:
2732:10.1002/9781119994077.ch1
2569:The Annals of Probability
2528:The Annals of Probability
2484:The Annals of Probability
2081:exponentially distributed
1795:increasing (analogous to
1532:). A common model is the
1523:probability distributions
5353:Environmental statistics
4875:Elliptical distributions
4668:Generalized linear model
4597:Simple linear regression
4367:HodgesâLehmann estimator
3824:Probability distribution
3733:Stochastic approximation
3295:Coefficient of variation
2123:coefficient of variation
2117:Coefficient of variation
2111:exponential distribution
976:{\displaystyle \Delta t}
682:{\displaystyle \Delta t}
5465:Reliability engineering
5013:Cross-correlation (XCF)
4621:Non-standard predictors
4055:LehmannâScheffĂ© theorem
3728:Adaptive clinical trial
3019:10.1109/TR.2002.1011521
2361:Reliability engineering
2356:Frequency of exceedance
2326:Annualized failure rate
2180:single point of failure
2063:Decreasing failure rate
1961:, it can be shown that
1781:log-normal distribution
811:conditional probability
809:. Note that this is a
50:reliability engineering
5409:Mathematics portal
5230:Engineering statistics
5138:NelsonâAalen estimator
4715:Analysis of covariance
4602:Ordinary least squares
4526:Pearson product-moment
3930:Statistical functional
3841:Empirical distribution
3674:Controlled experiments
3403:Frequency distribution
3181:Descriptive statistics
2928:Federal Standard 1037C
2628:10.1239/jap/1308662647
2582:10.1214/aop/1176994317
2541:10.1214/aop/1176991910
2497:10.1214/aop/1176994773
2290:
2053:
1955:
1923:
1762:
1644:which is based on the
1635:
1518:
1504:
1390:
1302:
1277:
1178:
1142:
977:
950:
912:
901:
856:
827:
803:
776:
756:
729:
683:
655:
446:
417:
348:
328:
274:
250:
5325:Population statistics
5267:System identification
5001:Autocorrelation (ACF)
4929:Exponential smoothing
4843:Discriminant analysis
4838:Canonical correlation
4702:Partition of variance
4564:Regression validation
4408:(JonckheereâTerpstra)
4307:Likelihood-ratio test
3996:Frequentist inference
3908:Locationâscale family
3829:Sampling distribution
3794:Statistical inference
3761:Cross-sectional study
3748:Observational studies
3707:Randomized experiment
3536:Stem-and-leaf display
3338:Central limit theorem
2291:
2083:random variables are
2054:
1956:
1924:
1763:
1636:
1516:
1505:
1391:
1303:
1278:
1179:
1143:
978:
951:
902:
877:
857:
828:
804:
802:{\displaystyle t_{2}}
777:
757:
755:{\displaystyle t_{1}}
730:
684:
666:over a time interval
656:
447:
418:
349:
329:
275:
251:
5460:Engineering failures
5248:Probabilistic design
4833:Principal components
4676:Exponential families
4628:Nonlinear regression
4607:General linear model
4569:Mixed effects models
4559:Errors and residuals
4536:Confounding variable
4438:Bayesian probability
4416:Van der Waerden test
4406:Ordered alternative
4171:Multiple comparisons
4050:RaoâBlackwellization
4013:Estimating equations
3969:Statistical distance
3687:Factorial experiment
3220:Arithmetic-Geometric
3091:Bathtub curve issues
2834:"Reliability Basics"
2381:Weibull distribution
2206:
2143:engineering notation
1968:
1954:{\displaystyle F(t)}
1936:
1817:
1777:Weibull distribution
1655:
1543:
1406:
1330:
1290:
1199:
1177:{\displaystyle F(t)}
1159:
1153:failure distribution
990:
983:approaches to zero:
964:
949:{\displaystyle h(t)}
931:
900:{\displaystyle h(t)}
882:
862:in the denominator.
855:{\displaystyle R(t)}
837:
817:
786:
766:
739:
693:
670:
464:
445:{\displaystyle f(t)}
427:
363:
338:
288:
282:reliability function
264:
231:
95:transport industries
5320:Official statistics
5243:Methods engineering
4924:Seasonal adjustment
4692:Poisson regressions
4612:Bayesian regression
4551:Regression analysis
4531:Partial correlation
4503:Regression analysis
4102:Prediction interval
4097:Likelihood interval
4087:Confidence interval
4079:Interval estimation
4040:Unbiased estimators
3858:Model specification
3738:Up-and-down designs
3426:Partial correlation
3382:Index of dispersion
3300:Interquartile range
2967:Reliability Toolkit
2701:1980JSpRo..17..479B
2074:decreasing function
2021:
1801:Pareto distribution
1575:
1362:
1301:{\displaystyle {T}}
956:. This becomes the
5480:Statistical ratios
5340:Spatial statistics
5220:Medical statistics
5120:First hitting time
5074:Whittle likelihood
4725:Degrees of freedom
4720:Multivariate ANOVA
4653:Heteroscedasticity
4465:Bayesian estimator
4430:Bayesian inference
4279:KolmogorovâSmirnov
4164:Randomization test
4134:Testing hypotheses
4107:Tolerance interval
4018:Maximum likelihood
3913:Exponential family
3846:Density estimation
3806:Statistical theory
3766:Natural experiment
3712:Scientific control
3629:Survey methodology
3315:Standard deviation
3111:2014-03-26 at the
3096:2014-11-29 at the
3051:ERA Technology Ltd
2852:2016-03-03 at the
2760:Harchol-Balter, M.
2563:Brown, M. (1981).
2478:Brown, M. (1980).
2366:Reliability theory
2351:Force of mortality
2317:Mathematics portal
2286:
2068:situation where λ(
2049:
2007:
1951:
1919:
1758:
1631:
1561:
1519:
1500:
1386:
1348:
1298:
1273:
1174:
1138:
1026:
973:
946:
913:
897:
852:
823:
799:
772:
752:
725:
679:
651:
442:
413:
344:
324:
270:
246:
5470:Survival analysis
5455:Actuarial science
5442:
5441:
5380:
5379:
5376:
5375:
5315:National accounts
5285:Actuarial science
5277:Social statistics
5170:
5169:
5166:
5165:
5162:
5161:
5097:Survival function
5082:
5081:
4944:Granger causality
4785:Contingency table
4760:Survival analysis
4737:
4736:
4733:
4732:
4589:Linear regression
4484:
4483:
4480:
4479:
4455:Credible interval
4424:
4423:
4207:
4206:
4023:Method of moments
3892:Parametric family
3853:Statistical model
3783:
3782:
3779:
3778:
3697:Random assignment
3619:Statistical power
3553:
3552:
3549:
3548:
3398:Contingency table
3368:
3367:
3235:Generalized/power
2461:978-1-84800-985-1
2376:Survival analysis
2281:
2280:
2277:
2248:
2247:
2244:
2231:
2228:
2218:
2091:Renewal processes
1917:
1871:
1747:
1703:
1495:
1460:
1115:
1090:
1008:
826:{\displaystyle t}
775:{\displaystyle t}
648:
580:
411:
347:{\displaystyle t}
273:{\displaystyle t}
143:military standard
112:Failure rate data
88:renewal processes
63:In practice, the
39:engineered system
16:(Redirected from
5497:
5430:
5429:
5418:
5417:
5407:
5406:
5392:
5391:
5295:Crime statistics
5189:
5176:
5093:
5059:Fourier analysis
5046:Frequency domain
5026:
4973:
4939:Structural break
4899:
4848:Cluster analysis
4795:Log-linear model
4768:
4743:
4684:
4658:Homoscedasticity
4514:
4490:
4409:
4401:
4393:
4392:(KruskalâWallis)
4377:
4362:
4317:Cross validation
4302:
4284:AndersonâDarling
4231:
4218:
4189:Likelihood-ratio
4181:Parametric tests
4159:Permutation test
4142:1- & 2-tails
4033:Minimum distance
4005:Point estimation
4001:
3952:Optimal decision
3903:
3802:
3789:
3771:Quasi-experiment
3721:Adaptive designs
3572:
3559:
3436:Rank correlation
3198:
3189:
3176:
3143:
3136:
3129:
3120:
3074:
3068:
3064:
3062:
3054:
3041:
3022:
3004:
2994:
2970:
2961:
2948:
2924:
2905:
2886:
2869:
2863:
2857:
2843:
2837:
2831:
2825:
2818:
2812:
2811:
2793:
2787:
2786:
2768:
2752:
2746:
2745:
2719:
2713:
2712:
2684:
2678:
2677:
2649:
2640:
2639:
2621:
2601:
2595:
2594:
2584:
2560:
2554:
2553:
2543:
2519:
2510:
2509:
2499:
2475:
2466:
2465:
2439:
2433:
2428:
2422:
2421:
2413:
2407:
2406:
2397:
2319:
2314:
2313:
2295:
2293:
2292:
2287:
2282:
2278:
2275:
2274:
2271:
2270:
2249:
2245:
2242:
2241:
2232:
2230:
2229:
2226:
2220:
2219:
2216:
2210:
2150:Failures In Time
2058:
2056:
2055:
2050:
2045:
2044:
2040:
2020:
2015:
1960:
1958:
1957:
1952:
1928:
1926:
1925:
1920:
1918:
1916:
1896:
1886:
1877:
1872:
1870:
1850:
1836:
1767:
1765:
1764:
1759:
1748:
1746:
1745:
1730:
1729:
1728:
1709:
1704:
1702:
1688:
1674:
1640:
1638:
1637:
1632:
1626:
1625:
1594:
1593:
1574:
1569:
1509:
1507:
1506:
1501:
1496:
1494:
1480:
1466:
1461:
1459:
1439:
1425:
1395:
1393:
1392:
1387:
1361:
1356:
1307:
1305:
1304:
1299:
1297:
1282:
1280:
1279:
1274:
1183:
1181:
1180:
1175:
1147:
1145:
1144:
1139:
1116:
1114:
1110:
1104:
1099:
1091:
1089:
1066:
1028:
1025:
982:
980:
979:
974:
955:
953:
952:
947:
921:
920:
906:
904:
903:
898:
878:Hazard function
861:
859:
858:
853:
832:
830:
829:
824:
808:
806:
805:
800:
798:
797:
781:
779:
778:
773:
761:
759:
758:
753:
751:
750:
734:
732:
731:
726:
721:
720:
708:
707:
688:
686:
685:
680:
660:
658:
657:
652:
649:
647:
624:
586:
581:
579:
575:
574:
553:
552:
540:
539:
526:
522:
521:
500:
499:
483:
451:
449:
448:
443:
422:
420:
419:
414:
412:
410:
396:
382:
353:
351:
350:
345:
333:
331:
330:
325:
279:
277:
276:
271:
255:
253:
252:
247:
97:, especially in
21:
5505:
5504:
5500:
5499:
5498:
5496:
5495:
5494:
5445:
5444:
5443:
5438:
5401:
5372:
5334:
5271:
5257:quality control
5224:
5206:Clinical trials
5183:
5158:
5142:
5130:Hazard function
5124:
5078:
5040:
5024:
4987:
4983:BreuschâGodfrey
4971:
4948:
4888:
4863:Factor analysis
4809:
4790:Graphical model
4762:
4729:
4696:
4682:
4662:
4616:
4583:
4545:
4508:
4507:
4476:
4420:
4407:
4399:
4391:
4375:
4360:
4339:Rank statistics
4333:
4312:Model selection
4300:
4258:Goodness of fit
4252:
4229:
4203:
4175:
4128:
4073:
4062:Median unbiased
3990:
3901:
3834:Order statistic
3796:
3775:
3742:
3716:
3668:
3623:
3566:
3564:Data collection
3545:
3457:
3412:
3386:
3364:
3324:
3276:
3193:Continuous data
3183:
3170:
3152:
3147:
3113:Wayback Machine
3098:Wayback Machine
3087:
3065:
3055:
3044:
3038:
3025:
3002:
2997:
2991:
2973:
2964:
2951:
2945:
2932:
2921:
2908:
2902:
2889:
2880:
2877:
2875:Further reading
2872:
2864:
2860:
2854:Wayback Machine
2844:
2840:
2832:
2828:
2819:
2815:
2808:
2795:
2794:
2790:
2766:
2754:
2753:
2749:
2742:
2721:
2720:
2716:
2709:10.2514/3.28040
2686:
2685:
2681:
2651:
2650:
2643:
2603:
2602:
2598:
2562:
2561:
2557:
2521:
2520:
2513:
2477:
2476:
2469:
2462:
2441:
2440:
2436:
2429:
2425:
2415:
2414:
2410:
2399:
2398:
2394:
2390:
2385:
2315:
2308:
2305:
2259:
2221:
2211:
2204:
2203:
2189:
2169:
2132:
2119:
2106:
2097:renewal process
2093:
2065:
2003:
1999:
1966:
1965:
1934:
1933:
1897:
1879:
1878:
1851:
1837:
1815:
1814:
1731:
1714:
1710:
1689:
1675:
1653:
1652:
1611:
1579:
1541:
1540:
1481:
1467:
1440:
1426:
1404:
1403:
1328:
1327:
1288:
1287:
1197:
1196:
1157:
1156:
1105:
1067:
1029:
988:
987:
962:
961:
929:
928:
919:hazard function
918:
917:
880:
879:
872:
835:
834:
815:
814:
789:
784:
783:
764:
763:
742:
737:
736:
712:
699:
691:
690:
668:
667:
625:
587:
566:
544:
531:
527:
513:
491:
484:
462:
461:
425:
424:
397:
383:
361:
360:
336:
335:
286:
285:
262:
261:
229:
228:
209:
114:
28:
23:
22:
18:Hazard function
15:
12:
11:
5:
5503:
5501:
5493:
5492:
5487:
5485:Error measures
5482:
5477:
5472:
5467:
5462:
5457:
5447:
5446:
5440:
5439:
5437:
5436:
5424:
5412:
5398:
5385:
5382:
5381:
5378:
5377:
5374:
5373:
5371:
5370:
5365:
5360:
5355:
5350:
5344:
5342:
5336:
5335:
5333:
5332:
5327:
5322:
5317:
5312:
5307:
5302:
5297:
5292:
5287:
5281:
5279:
5273:
5272:
5270:
5269:
5264:
5259:
5250:
5245:
5240:
5234:
5232:
5226:
5225:
5223:
5222:
5217:
5212:
5203:
5201:Bioinformatics
5197:
5195:
5185:
5184:
5179:
5172:
5171:
5168:
5167:
5164:
5163:
5160:
5159:
5157:
5156:
5150:
5148:
5144:
5143:
5141:
5140:
5134:
5132:
5126:
5125:
5123:
5122:
5117:
5112:
5107:
5101:
5099:
5090:
5084:
5083:
5080:
5079:
5077:
5076:
5071:
5066:
5061:
5056:
5050:
5048:
5042:
5041:
5039:
5038:
5033:
5028:
5020:
5015:
5010:
5009:
5008:
5006:partial (PACF)
4997:
4995:
4989:
4988:
4986:
4985:
4980:
4975:
4967:
4962:
4956:
4954:
4953:Specific tests
4950:
4949:
4947:
4946:
4941:
4936:
4931:
4926:
4921:
4916:
4911:
4905:
4903:
4896:
4890:
4889:
4887:
4886:
4885:
4884:
4883:
4882:
4867:
4866:
4865:
4855:
4853:Classification
4850:
4845:
4840:
4835:
4830:
4825:
4819:
4817:
4811:
4810:
4808:
4807:
4802:
4800:McNemar's test
4797:
4792:
4787:
4782:
4776:
4774:
4764:
4763:
4746:
4739:
4738:
4735:
4734:
4731:
4730:
4728:
4727:
4722:
4717:
4712:
4706:
4704:
4698:
4697:
4695:
4694:
4678:
4672:
4670:
4664:
4663:
4661:
4660:
4655:
4650:
4645:
4640:
4638:Semiparametric
4635:
4630:
4624:
4622:
4618:
4617:
4615:
4614:
4609:
4604:
4599:
4593:
4591:
4585:
4584:
4582:
4581:
4576:
4571:
4566:
4561:
4555:
4553:
4547:
4546:
4544:
4543:
4538:
4533:
4528:
4522:
4520:
4510:
4509:
4506:
4505:
4500:
4494:
4493:
4486:
4485:
4482:
4481:
4478:
4477:
4475:
4474:
4473:
4472:
4462:
4457:
4452:
4451:
4450:
4445:
4434:
4432:
4426:
4425:
4422:
4421:
4419:
4418:
4413:
4412:
4411:
4403:
4395:
4379:
4376:(MannâWhitney)
4371:
4370:
4369:
4356:
4355:
4354:
4343:
4341:
4335:
4334:
4332:
4331:
4330:
4329:
4324:
4319:
4309:
4304:
4301:(ShapiroâWilk)
4296:
4291:
4286:
4281:
4276:
4268:
4262:
4260:
4254:
4253:
4251:
4250:
4242:
4233:
4221:
4215:
4213:Specific tests
4209:
4208:
4205:
4204:
4202:
4201:
4196:
4191:
4185:
4183:
4177:
4176:
4174:
4173:
4168:
4167:
4166:
4156:
4155:
4154:
4144:
4138:
4136:
4130:
4129:
4127:
4126:
4125:
4124:
4119:
4109:
4104:
4099:
4094:
4089:
4083:
4081:
4075:
4074:
4072:
4071:
4066:
4065:
4064:
4059:
4058:
4057:
4052:
4037:
4036:
4035:
4030:
4025:
4020:
4009:
4007:
3998:
3992:
3991:
3989:
3988:
3983:
3978:
3977:
3976:
3966:
3961:
3960:
3959:
3949:
3948:
3947:
3942:
3937:
3927:
3922:
3917:
3916:
3915:
3910:
3905:
3889:
3888:
3887:
3882:
3877:
3867:
3866:
3865:
3860:
3850:
3849:
3848:
3838:
3837:
3836:
3826:
3821:
3816:
3810:
3808:
3798:
3797:
3792:
3785:
3784:
3781:
3780:
3777:
3776:
3774:
3773:
3768:
3763:
3758:
3752:
3750:
3744:
3743:
3741:
3740:
3735:
3730:
3724:
3722:
3718:
3717:
3715:
3714:
3709:
3704:
3699:
3694:
3689:
3684:
3678:
3676:
3670:
3669:
3667:
3666:
3664:Standard error
3661:
3656:
3651:
3650:
3649:
3644:
3633:
3631:
3625:
3624:
3622:
3621:
3616:
3611:
3606:
3601:
3596:
3594:Optimal design
3591:
3586:
3580:
3578:
3568:
3567:
3562:
3555:
3554:
3551:
3550:
3547:
3546:
3544:
3543:
3538:
3533:
3528:
3523:
3518:
3513:
3508:
3503:
3498:
3493:
3488:
3483:
3478:
3473:
3467:
3465:
3459:
3458:
3456:
3455:
3450:
3449:
3448:
3443:
3433:
3428:
3422:
3420:
3414:
3413:
3411:
3410:
3405:
3400:
3394:
3392:
3391:Summary tables
3388:
3387:
3385:
3384:
3378:
3376:
3370:
3369:
3366:
3365:
3363:
3362:
3361:
3360:
3355:
3350:
3340:
3334:
3332:
3326:
3325:
3323:
3322:
3317:
3312:
3307:
3302:
3297:
3292:
3286:
3284:
3278:
3277:
3275:
3274:
3269:
3264:
3263:
3262:
3257:
3252:
3247:
3242:
3237:
3232:
3227:
3225:Contraharmonic
3222:
3217:
3206:
3204:
3195:
3185:
3184:
3179:
3172:
3171:
3169:
3168:
3163:
3157:
3154:
3153:
3148:
3146:
3145:
3138:
3131:
3123:
3117:
3116:
3101:
3086:
3085:External links
3083:
3082:
3081:
3075:
3042:
3036:
3023:
3013:(2): 166â167.
2995:
2989:
2971:
2962:
2949:
2943:
2930:
2925:
2919:
2906:
2900:
2887:
2876:
2873:
2871:
2870:
2858:
2838:
2826:
2813:
2807:978-1439806586
2806:
2788:
2758:; Bansal, N.;
2747:
2740:
2714:
2679:
2660:(3): 375â383.
2641:
2612:(2): 583â588.
2596:
2575:(5): 891â895.
2555:
2534:(1): 397â406.
2511:
2490:(2): 227â240.
2467:
2460:
2434:
2423:
2408:
2405:. exida. 2006.
2391:
2389:
2386:
2384:
2383:
2378:
2373:
2368:
2363:
2358:
2353:
2348:
2343:
2338:
2333:
2328:
2322:
2321:
2320:
2304:
2301:
2297:
2296:
2285:
2269:
2266:
2262:
2258:
2255:
2252:
2238:
2235:
2224:
2217: failures
2214:
2188:
2185:
2168:
2165:
2131:
2128:
2118:
2115:
2105:
2102:
2092:
2089:
2064:
2061:
2060:
2059:
2048:
2043:
2039:
2036:
2033:
2030:
2027:
2024:
2019:
2014:
2010:
2006:
2002:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1950:
1947:
1944:
1941:
1930:
1929:
1915:
1912:
1909:
1906:
1903:
1900:
1895:
1892:
1889:
1885:
1882:
1875:
1869:
1866:
1863:
1860:
1857:
1854:
1849:
1846:
1843:
1840:
1834:
1831:
1828:
1825:
1822:
1769:
1768:
1757:
1754:
1751:
1744:
1741:
1738:
1734:
1727:
1724:
1721:
1717:
1713:
1707:
1701:
1698:
1695:
1692:
1687:
1684:
1681:
1678:
1672:
1669:
1666:
1663:
1660:
1642:
1641:
1629:
1624:
1621:
1618:
1614:
1610:
1607:
1604:
1601:
1598:
1592:
1589:
1586:
1582:
1578:
1573:
1568:
1564:
1560:
1557:
1554:
1551:
1548:
1511:
1510:
1499:
1493:
1490:
1487:
1484:
1479:
1476:
1473:
1470:
1464:
1458:
1455:
1452:
1449:
1446:
1443:
1438:
1435:
1432:
1429:
1423:
1420:
1417:
1414:
1411:
1397:
1396:
1384:
1381:
1378:
1374:
1371:
1368:
1365:
1360:
1355:
1351:
1347:
1344:
1341:
1338:
1335:
1296:
1284:
1283:
1271:
1268:
1265:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1173:
1170:
1167:
1164:
1149:
1148:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1113:
1109:
1103:
1097:
1094:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1024:
1021:
1018:
1015:
1011:
1007:
1004:
1001:
998:
995:
972:
969:
945:
942:
939:
936:
896:
893:
890:
887:
871:
868:
851:
848:
845:
842:
822:
796:
792:
771:
749:
745:
724:
719:
715:
711:
706:
702:
698:
678:
675:
664:
663:
662:
661:
646:
643:
640:
637:
634:
631:
628:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
584:
578:
573:
569:
565:
562:
559:
556:
551:
547:
543:
538:
534:
530:
525:
520:
516:
512:
509:
506:
503:
498:
494:
490:
487:
481:
478:
475:
472:
469:
456:
455:
454:
453:
441:
438:
435:
432:
409:
406:
403:
400:
395:
392:
389:
386:
380:
377:
374:
371:
368:
343:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
269:
245:
242:
239:
236:
225:
224:
208:
205:
200:
199:
196:
193:
190:
187:
184:
181:
177:
176:
170:
167:
163:
160:
156:
153:
149:
146:
135:
132:
128:
125:
121:
113:
110:
37:with which an
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5502:
5491:
5488:
5486:
5483:
5481:
5478:
5476:
5473:
5471:
5468:
5466:
5463:
5461:
5458:
5456:
5453:
5452:
5450:
5435:
5434:
5425:
5423:
5422:
5413:
5411:
5410:
5405:
5399:
5397:
5396:
5387:
5386:
5383:
5369:
5366:
5364:
5363:Geostatistics
5361:
5359:
5356:
5354:
5351:
5349:
5346:
5345:
5343:
5341:
5337:
5331:
5330:Psychometrics
5328:
5326:
5323:
5321:
5318:
5316:
5313:
5311:
5308:
5306:
5303:
5301:
5298:
5296:
5293:
5291:
5288:
5286:
5283:
5282:
5280:
5278:
5274:
5268:
5265:
5263:
5260:
5258:
5254:
5251:
5249:
5246:
5244:
5241:
5239:
5236:
5235:
5233:
5231:
5227:
5221:
5218:
5216:
5213:
5211:
5207:
5204:
5202:
5199:
5198:
5196:
5194:
5193:Biostatistics
5190:
5186:
5182:
5177:
5173:
5155:
5154:Log-rank test
5152:
5151:
5149:
5145:
5139:
5136:
5135:
5133:
5131:
5127:
5121:
5118:
5116:
5113:
5111:
5108:
5106:
5103:
5102:
5100:
5098:
5094:
5091:
5089:
5085:
5075:
5072:
5070:
5067:
5065:
5062:
5060:
5057:
5055:
5052:
5051:
5049:
5047:
5043:
5037:
5034:
5032:
5029:
5027:
5025:(BoxâJenkins)
5021:
5019:
5016:
5014:
5011:
5007:
5004:
5003:
5002:
4999:
4998:
4996:
4994:
4990:
4984:
4981:
4979:
4978:DurbinâWatson
4976:
4974:
4968:
4966:
4963:
4961:
4960:DickeyâFuller
4958:
4957:
4955:
4951:
4945:
4942:
4940:
4937:
4935:
4934:Cointegration
4932:
4930:
4927:
4925:
4922:
4920:
4917:
4915:
4912:
4910:
4909:Decomposition
4907:
4906:
4904:
4900:
4897:
4895:
4891:
4881:
4878:
4877:
4876:
4873:
4872:
4871:
4868:
4864:
4861:
4860:
4859:
4856:
4854:
4851:
4849:
4846:
4844:
4841:
4839:
4836:
4834:
4831:
4829:
4826:
4824:
4821:
4820:
4818:
4816:
4812:
4806:
4803:
4801:
4798:
4796:
4793:
4791:
4788:
4786:
4783:
4781:
4780:Cohen's kappa
4778:
4777:
4775:
4773:
4769:
4765:
4761:
4757:
4753:
4749:
4744:
4740:
4726:
4723:
4721:
4718:
4716:
4713:
4711:
4708:
4707:
4705:
4703:
4699:
4693:
4689:
4685:
4679:
4677:
4674:
4673:
4671:
4669:
4665:
4659:
4656:
4654:
4651:
4649:
4646:
4644:
4641:
4639:
4636:
4634:
4633:Nonparametric
4631:
4629:
4626:
4625:
4623:
4619:
4613:
4610:
4608:
4605:
4603:
4600:
4598:
4595:
4594:
4592:
4590:
4586:
4580:
4577:
4575:
4572:
4570:
4567:
4565:
4562:
4560:
4557:
4556:
4554:
4552:
4548:
4542:
4539:
4537:
4534:
4532:
4529:
4527:
4524:
4523:
4521:
4519:
4515:
4511:
4504:
4501:
4499:
4496:
4495:
4491:
4487:
4471:
4468:
4467:
4466:
4463:
4461:
4458:
4456:
4453:
4449:
4446:
4444:
4441:
4440:
4439:
4436:
4435:
4433:
4431:
4427:
4417:
4414:
4410:
4404:
4402:
4396:
4394:
4388:
4387:
4386:
4383:
4382:Nonparametric
4380:
4378:
4372:
4368:
4365:
4364:
4363:
4357:
4353:
4352:Sample median
4350:
4349:
4348:
4345:
4344:
4342:
4340:
4336:
4328:
4325:
4323:
4320:
4318:
4315:
4314:
4313:
4310:
4308:
4305:
4303:
4297:
4295:
4292:
4290:
4287:
4285:
4282:
4280:
4277:
4275:
4273:
4269:
4267:
4264:
4263:
4261:
4259:
4255:
4249:
4247:
4243:
4241:
4239:
4234:
4232:
4227:
4223:
4222:
4219:
4216:
4214:
4210:
4200:
4197:
4195:
4192:
4190:
4187:
4186:
4184:
4182:
4178:
4172:
4169:
4165:
4162:
4161:
4160:
4157:
4153:
4150:
4149:
4148:
4145:
4143:
4140:
4139:
4137:
4135:
4131:
4123:
4120:
4118:
4115:
4114:
4113:
4110:
4108:
4105:
4103:
4100:
4098:
4095:
4093:
4090:
4088:
4085:
4084:
4082:
4080:
4076:
4070:
4067:
4063:
4060:
4056:
4053:
4051:
4048:
4047:
4046:
4043:
4042:
4041:
4038:
4034:
4031:
4029:
4026:
4024:
4021:
4019:
4016:
4015:
4014:
4011:
4010:
4008:
4006:
4002:
3999:
3997:
3993:
3987:
3984:
3982:
3979:
3975:
3972:
3971:
3970:
3967:
3965:
3962:
3958:
3957:loss function
3955:
3954:
3953:
3950:
3946:
3943:
3941:
3938:
3936:
3933:
3932:
3931:
3928:
3926:
3923:
3921:
3918:
3914:
3911:
3909:
3906:
3904:
3898:
3895:
3894:
3893:
3890:
3886:
3883:
3881:
3878:
3876:
3873:
3872:
3871:
3868:
3864:
3861:
3859:
3856:
3855:
3854:
3851:
3847:
3844:
3843:
3842:
3839:
3835:
3832:
3831:
3830:
3827:
3825:
3822:
3820:
3817:
3815:
3812:
3811:
3809:
3807:
3803:
3799:
3795:
3790:
3786:
3772:
3769:
3767:
3764:
3762:
3759:
3757:
3754:
3753:
3751:
3749:
3745:
3739:
3736:
3734:
3731:
3729:
3726:
3725:
3723:
3719:
3713:
3710:
3708:
3705:
3703:
3700:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3679:
3677:
3675:
3671:
3665:
3662:
3660:
3659:Questionnaire
3657:
3655:
3652:
3648:
3645:
3643:
3640:
3639:
3638:
3635:
3634:
3632:
3630:
3626:
3620:
3617:
3615:
3612:
3610:
3607:
3605:
3602:
3600:
3597:
3595:
3592:
3590:
3587:
3585:
3582:
3581:
3579:
3577:
3573:
3569:
3565:
3560:
3556:
3542:
3539:
3537:
3534:
3532:
3529:
3527:
3524:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3487:
3486:Control chart
3484:
3482:
3479:
3477:
3474:
3472:
3469:
3468:
3466:
3464:
3460:
3454:
3451:
3447:
3444:
3442:
3439:
3438:
3437:
3434:
3432:
3429:
3427:
3424:
3423:
3421:
3419:
3415:
3409:
3406:
3404:
3401:
3399:
3396:
3395:
3393:
3389:
3383:
3380:
3379:
3377:
3375:
3371:
3359:
3356:
3354:
3351:
3349:
3346:
3345:
3344:
3341:
3339:
3336:
3335:
3333:
3331:
3327:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3296:
3293:
3291:
3288:
3287:
3285:
3283:
3279:
3273:
3270:
3268:
3265:
3261:
3258:
3256:
3253:
3251:
3248:
3246:
3243:
3241:
3238:
3236:
3233:
3231:
3228:
3226:
3223:
3221:
3218:
3216:
3213:
3212:
3211:
3208:
3207:
3205:
3203:
3199:
3196:
3194:
3190:
3186:
3182:
3177:
3173:
3167:
3164:
3162:
3159:
3158:
3155:
3151:
3144:
3139:
3137:
3132:
3130:
3125:
3124:
3121:
3114:
3110:
3107:
3106:
3102:
3099:
3095:
3092:
3089:
3088:
3084:
3080:
3076:
3072:
3060:
3052:
3048:
3043:
3039:
3033:
3029:
3024:
3020:
3016:
3012:
3008:
3001:
2996:
2992:
2990:9780849392474
2986:
2982:
2981:
2976:
2972:
2968:
2963:
2959:
2955:
2950:
2946:
2940:
2936:
2931:
2929:
2926:
2922:
2916:
2912:
2907:
2903:
2897:
2893:
2888:
2884:
2879:
2878:
2874:
2867:
2862:
2859:
2855:
2851:
2848:
2845:Vita Faraci.
2842:
2839:
2835:
2830:
2827:
2824:. 2010. p. 6.
2823:
2817:
2814:
2809:
2803:
2799:
2792:
2789:
2784:
2780:
2776:
2772:
2765:
2761:
2757:
2751:
2748:
2743:
2741:9781119994077
2737:
2733:
2729:
2726:. p. 1.
2725:
2718:
2715:
2710:
2706:
2702:
2698:
2694:
2690:
2683:
2680:
2675:
2671:
2667:
2663:
2659:
2655:
2654:Technometrics
2648:
2646:
2642:
2637:
2633:
2629:
2625:
2620:
2615:
2611:
2607:
2600:
2597:
2592:
2588:
2583:
2578:
2574:
2570:
2566:
2559:
2556:
2551:
2547:
2542:
2537:
2533:
2529:
2525:
2518:
2516:
2512:
2507:
2503:
2498:
2493:
2489:
2485:
2481:
2474:
2472:
2468:
2463:
2457:
2453:
2449:
2445:
2438:
2435:
2432:
2427:
2424:
2419:
2412:
2409:
2404:
2403:
2396:
2393:
2387:
2382:
2379:
2377:
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2332:
2329:
2327:
2324:
2323:
2318:
2312:
2307:
2302:
2300:
2283:
2267:
2264:
2260:
2256:
2253:
2250:
2236:
2233:
2222:
2212:
2202:
2201:
2200:
2197:
2195:
2186:
2184:
2181:
2176:
2174:
2166:
2164:
2161:
2159:
2158:semiconductor
2155:
2151:
2146:
2144:
2139:
2137:
2129:
2127:
2124:
2116:
2114:
2112:
2103:
2101:
2098:
2090:
2088:
2086:
2082:
2077:
2075:
2071:
2062:
2046:
2041:
2037:
2034:
2028:
2022:
2017:
2012:
2008:
2004:
2000:
1995:
1992:
1989:
1986:
1983:
1977:
1971:
1964:
1963:
1962:
1945:
1939:
1910:
1904:
1901:
1898:
1890:
1883:
1880:
1873:
1864:
1858:
1855:
1852:
1844:
1838:
1832:
1826:
1820:
1813:
1812:
1811:
1808:
1806:
1802:
1798:
1797:"wearing out"
1794:
1790:
1786:
1782:
1778:
1774:
1755:
1752:
1749:
1742:
1739:
1736:
1732:
1725:
1722:
1719:
1715:
1711:
1705:
1696:
1690:
1682:
1676:
1670:
1664:
1658:
1651:
1650:
1649:
1647:
1627:
1622:
1619:
1616:
1612:
1608:
1605:
1602:
1599:
1596:
1590:
1587:
1584:
1580:
1576:
1571:
1566:
1562:
1558:
1552:
1546:
1539:
1538:
1537:
1535:
1531:
1530:
1524:
1515:
1497:
1488:
1482:
1474:
1468:
1462:
1453:
1447:
1444:
1441:
1433:
1427:
1421:
1415:
1409:
1402:
1401:
1400:
1382:
1379:
1376:
1369:
1363:
1358:
1353:
1349:
1345:
1339:
1333:
1326:
1325:
1324:
1322:
1318:
1314:
1312:
1294:
1269:
1266:
1263:
1259:
1253:
1247:
1244:
1241:
1238:
1232:
1226:
1223:
1217:
1214:
1211:
1205:
1195:
1194:
1193:
1191:
1187:
1184:, which is a
1168:
1162:
1154:
1135:
1129:
1123:
1120:
1117:
1111:
1095:
1092:
1083:
1077:
1074:
1071:
1060:
1054:
1051:
1045:
1042:
1036:
1030:
1022:
1016:
1005:
999:
993:
986:
985:
984:
970:
959:
958:instantaneous
940:
934:
926:
923:(also called
922:
910:
891:
885:
876:
869:
867:
863:
846:
840:
820:
812:
794:
790:
769:
747:
743:
717:
713:
709:
704:
700:
676:
641:
635:
632:
629:
618:
612:
609:
603:
600:
594:
588:
582:
571:
567:
560:
557:
549:
545:
541:
536:
532:
518:
514:
507:
504:
496:
492:
485:
479:
473:
467:
460:
459:
458:
457:
436:
430:
404:
398:
390:
384:
378:
372:
366:
359:
358:
357:
356:
355:
341:
318:
312:
309:
306:
303:
297:
291:
283:
267:
259:
240:
234:
222:
218:
214:
213:
212:
206:
204:
197:
194:
191:
188:
185:
182:
179:
178:
174:
171:
168:
164:
162:Cycle Testing
161:
157:
154:
150:
147:
144:
140:
136:
133:
129:
126:
122:
119:
118:
117:
111:
109:
106:
104:
100:
96:
91:
89:
85:
80:
76:
74:
73:bathtub curve
70:
66:
61:
59:
53:
51:
47:
44:
40:
36:
32:
19:
5431:
5419:
5400:
5393:
5305:Econometrics
5255: /
5238:Chemometrics
5215:Epidemiology
5208: /
5181:Applications
5129:
5023:ARIMA model
4970:Q-statistic
4919:Stationarity
4815:Multivariate
4758: /
4754: /
4752:Multivariate
4750: /
4690: /
4686: /
4460:Bayes factor
4359:Signed rank
4271:
4245:
4237:
4225:
3920:Completeness
3756:Cohort study
3654:Opinion poll
3589:Missing data
3576:Study design
3531:Scatter plot
3453:Scatter plot
3446:Spearman's Ï
3408:Grouped data
3104:
3078:
3046:
3027:
3010:
3006:
2978:
2975:Modarres, M.
2966:
2957:
2953:
2934:
2910:
2891:
2882:
2861:
2841:
2829:
2816:
2797:
2791:
2774:
2770:
2750:
2723:
2717:
2692:
2688:
2682:
2657:
2653:
2609:
2605:
2599:
2572:
2568:
2558:
2531:
2527:
2487:
2483:
2443:
2437:
2426:
2417:
2411:
2401:
2395:
2341:Failure mode
2298:
2198:
2190:
2177:
2173:redundancies
2170:
2162:
2153:
2149:
2147:
2140:
2135:
2133:
2120:
2107:
2104:Applications
2094:
2078:
2069:
2066:
1931:
1809:
1805:"burning in"
1770:
1643:
1533:
1526:
1520:
1398:
1320:
1316:
1310:
1285:
1189:
1152:
1150:
957:
924:
916:
914:
864:
833:. Hence the
665:
226:
220:
210:
201:
155:Life Testing
138:
115:
107:
92:
81:
77:
68:
62:
58:transmission
54:
43:Greek letter
31:Failure rate
30:
29:
5475:Maintenance
5433:WikiProject
5348:Cartography
5310:Jurimetrics
5262:Reliability
4993:Time domain
4972:(LjungâBox)
4894:Time-series
4772:Categorical
4756:Time-series
4748:Categorical
4683:(Bernoulli)
4518:Correlation
4498:Correlation
4294:JarqueâBera
4266:Chi-squared
4028:M-estimator
3981:Asymptotics
3925:Sufficiency
3692:Interaction
3604:Replication
3584:Effect size
3541:Violin plot
3521:Radar chart
3501:Forest plot
3491:Correlogram
3441:Kendall's Ï
3067:|work=
2756:Wierman, A.
2227: hours
1773:memory-less
925:hazard rate
258:probability
84:engineering
5449:Categories
5300:Demography
5018:ARMA model
4823:Regression
4400:(Friedman)
4361:(Wilcoxon)
4299:Normality
4289:Lilliefors
4236:Student's
4112:Resampling
3986:Robustness
3974:divergence
3964:Efficiency
3902:(monotone)
3897:Likelihood
3814:Population
3647:Stratified
3599:Population
3418:Dependence
3374:Count data
3305:Percentile
3282:Dispersion
3215:Arithmetic
3150:Statistics
3037:047147133X
2944:0471511919
2920:0070188521
2901:0135241170
2695:(5): 479.
2388:References
2194:confidence
2167:Additivity
2160:industry.
217:population
148:Prediction
120:Estimation
4681:Logistic
4448:posterior
4374:Rank sum
4122:Jackknife
4117:Bootstrap
3935:Bootstrap
3870:Parameter
3819:Statistic
3614:Statistic
3526:Run chart
3511:Pie chart
3506:Histogram
3496:Fan chart
3471:Bar chart
3353:L-moments
3240:Geometric
3069:ignored (
3059:cite book
2777:: 73â76.
2619:1009.2463
2265:−
2257:×
2237:0.0007998
2009:∫
2005:−
1996:
1990:−
1902:−
1856:−
1793:monotonic
1753:λ
1740:λ
1737:−
1723:λ
1720:−
1712:λ
1620:λ
1617:−
1609:−
1600:τ
1591:τ
1588:λ
1585:−
1577:λ
1563:∫
1445:−
1380:τ
1370:τ
1350:∫
1267:≥
1245:−
1215:≤
1206:
1121:
1096:−
1075:⋅
1069:Δ
1058:Δ
1043:−
1020:→
1014:Δ
968:Δ
710:−
674:Δ
633:⋅
627:Δ
616:Δ
601:−
558:⋅
542:−
505:−
468:λ
367:λ
310:−
235:λ
131:estimate.
103:correlate
35:frequency
5395:Category
5088:Survival
4965:Johansen
4688:Binomial
4643:Isotonic
4230:(normal)
3875:location
3682:Blocking
3637:Sampling
3516:QâQ plot
3481:Box plot
3463:Graphics
3358:Skewness
3348:Kurtosis
3320:Variance
3250:Heronian
3245:Harmonic
3109:Archived
3094:Archived
2850:Archived
2762:(2004).
2636:26570923
2303:See also
2276:failures
2243:failures
1884:′
1313:function
423:, where
159:instead.
99:railways
5421:Commons
5368:Kriging
5253:Process
5210:studies
5069:Wavelet
4902:General
4069:Plug-in
3863:L space
3642:Cluster
3343:Moments
3161:Outline
2856:. 2006.
2836:. 2010.
2697:Bibcode
2674:1266340
2591:2243747
2550:2243910
2506:2243267
2336:Failure
2331:Burn-in
2187:Example
2072:) is a
1783:, or a
1311:density
141:, is a
33:is the
5290:Census
4880:Normal
4828:Manova
4648:Robust
4398:2-way
4390:1-way
4228:-test
3899:
3476:Biplot
3267:Median
3260:Lehmer
3202:Center
3100:, ASQC
3034:
2987:
2941:
2917:
2898:
2804:
2738:
2672:
2634:
2589:
2548:
2504:
2458:
2095:For a
1791:it is
1286:where
221:et al.
5490:Rates
4914:Trend
4443:prior
4385:anova
4274:-test
4248:-test
4240:-test
4147:Power
4092:Pivot
3885:shape
3880:scale
3330:Shape
3310:Range
3255:Heinz
3230:Cubic
3166:Index
3003:(PDF)
2767:(PDF)
2670:JSTOR
2632:S2CID
2614:arXiv
2587:JSTOR
2546:JSTOR
2502:JSTOR
2254:799.8
2136:hours
2130:Units
1521:Many
782:) to
735:from
169:FMEDA
5147:Test
4347:Sign
4199:Wald
3272:Mode
3210:Mean
3071:help
3032:ISBN
2985:ISBN
2939:ISBN
2915:ISBN
2896:ISBN
2802:ISBN
2736:ISBN
2456:ISBN
2279:hour
2246:hour
2223:7502
2148:The
1932:for
1527:see
762:(or
69:only
4327:BIC
4322:AIC
3015:doi
2779:doi
2728:doi
2705:doi
2662:doi
2624:doi
2577:doi
2536:doi
2492:doi
2448:doi
2154:FIT
1993:exp
1323:),
1010:lim
927:),
5451::
3063::
3061:}}
3057:{{
3011:51
3009:.
3005:.
2956:.
2775:32
2773:.
2769:.
2734:.
2703:.
2693:17
2691:.
2668:.
2656:.
2644:^
2630:.
2622:.
2610:48
2608:.
2585:.
2571:.
2567:.
2544:.
2532:16
2530:.
2526:.
2514:^
2500:.
2486:.
2482:.
2470:^
2454:.
2261:10
2087:.
2076:.
1779:,
1536:,
1315:,
1270:0.
1192:,
1155:,
1118:ln
689:=
354:.
52:.
4272:G
4246:F
4238:t
4226:Z
3945:V
3940:U
3142:e
3135:t
3128:v
3073:)
3053:.
3040:.
3021:.
3017::
2993:.
2958:2
2947:.
2923:.
2904:.
2868:.
2810:.
2785:.
2781::
2744:.
2730::
2711:.
2707::
2699::
2676:.
2664::
2658:5
2638:.
2626::
2616::
2593:.
2579::
2573:9
2552:.
2538::
2508:.
2494::
2488:8
2464:.
2450::
2284:,
2268:6
2251:=
2234:=
2213:6
2152:(
2070:t
2047:.
2042:)
2038:t
2035:d
2032:)
2029:t
2026:(
2023:h
2018:t
2013:0
2001:(
1987:1
1984:=
1981:)
1978:t
1975:(
1972:F
1949:)
1946:t
1943:(
1940:F
1914:)
1911:t
1908:(
1905:F
1899:1
1894:)
1891:t
1888:(
1881:F
1874:=
1868:)
1865:t
1862:(
1859:F
1853:1
1848:)
1845:t
1842:(
1839:f
1833:=
1830:)
1827:t
1824:(
1821:h
1756:.
1750:=
1743:t
1733:e
1726:t
1716:e
1706:=
1700:)
1697:t
1694:(
1691:R
1686:)
1683:t
1680:(
1677:f
1671:=
1668:)
1665:t
1662:(
1659:h
1628:,
1623:t
1613:e
1606:1
1603:=
1597:d
1581:e
1572:t
1567:0
1559:=
1556:)
1553:t
1550:(
1547:F
1498:.
1492:)
1489:t
1486:(
1483:R
1478:)
1475:t
1472:(
1469:f
1463:=
1457:)
1454:t
1451:(
1448:F
1442:1
1437:)
1434:t
1431:(
1428:f
1422:=
1419:)
1416:t
1413:(
1410:h
1383:.
1377:d
1373:)
1367:(
1364:f
1359:t
1354:0
1346:=
1343:)
1340:t
1337:(
1334:F
1321:t
1319:(
1317:f
1295:T
1264:t
1260:,
1257:)
1254:t
1251:(
1248:R
1242:1
1239:=
1236:)
1233:t
1230:(
1227:F
1224:=
1221:)
1218:t
1212:T
1209:(
1203:P
1190:t
1172:)
1169:t
1166:(
1163:F
1136:.
1133:)
1130:t
1127:(
1124:R
1112:t
1108:d
1102:d
1093:=
1087:)
1084:t
1081:(
1078:R
1072:t
1064:)
1061:t
1055:+
1052:t
1049:(
1046:R
1040:)
1037:t
1034:(
1031:R
1023:0
1017:t
1006:=
1003:)
1000:t
997:(
994:h
971:t
944:)
941:t
938:(
935:h
911:.
895:)
892:t
889:(
886:h
850:)
847:t
844:(
841:R
821:t
795:2
791:t
770:t
748:1
744:t
723:)
718:1
714:t
705:2
701:t
697:(
677:t
645:)
642:t
639:(
636:R
630:t
622:)
619:t
613:+
610:t
607:(
604:R
598:)
595:t
592:(
589:R
583:=
577:)
572:1
568:t
564:(
561:R
555:)
550:1
546:t
537:2
533:t
529:(
524:)
519:2
515:t
511:(
508:R
502:)
497:1
493:t
489:(
486:R
480:=
477:)
474:t
471:(
440:)
437:t
434:(
431:f
408:)
405:t
402:(
399:R
394:)
391:t
388:(
385:f
379:=
376:)
373:t
370:(
342:t
322:)
319:t
316:(
313:F
307:1
304:=
301:)
298:t
295:(
292:R
268:t
244:)
241:t
238:(
223:)
46:λ
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
