Gap Analysis - Revision history

Lisa Hacker at 21:15, 18 September 2023

2023-09-18T21:15:51Z

← Older revision		Revision as of 21:15, 18 September 2023
Line 1:		Line 1:
	<noinclude>{{Banner RGA Articles}}		<noinclude>{{Banner RGA Articles}}
	''This article also appears in ~~ReliaSoft's~~ Reliability ~~Growth and Repairable System Analysis Reference~~.'' </noinclude>		''This article also appears in the [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference].'' </noinclude>

	Most of the reliability growth models used for estimating and tracking reliability growth based on test data assume that the data set represents all actual system failure times consistent with a uniform definition of failure (complete data). In practice, this may not always be the case and may result in too few or too many failures being reported over some interval of test time. This may result in distorted estimates of the growth rate and current system reliability. This section discusses a practical reliability growth estimation and analysis procedure based on the assumption that anomalies may exist within the data over some interval of the test period but the remaining failure data follows the Crow-AMSAA reliability growth model. In particular, it is assumed that the beginning and ending points in which the anomalies lie are generated independently of the underlying reliability growth process. The approach for estimating the parameters of the growth model with problem data over some interval of time is basically to not use this failure information. The analysis retains the contribution of the interval to the total test time, but no assumptions are made regarding the actual number of failures over the interval. This is often referred to as ''gap analysis''.		Most of the reliability growth models used for estimating and tracking reliability growth based on test data assume that the data set represents all actual system failure times consistent with a uniform definition of failure (complete data). In practice, this may not always be the case and may result in too few or too many failures being reported over some interval of test time. This may result in distorted estimates of the growth rate and current system reliability. This section discusses a practical reliability growth estimation and analysis procedure based on the assumption that anomalies may exist within the data over some interval of the test period but the remaining failure data follows the Crow-AMSAA reliability growth model. In particular, it is assumed that the beginning and ending points in which the anomalies lie are generated independently of the underlying reliability growth process. The approach for estimating the parameters of the growth model with problem data over some interval of time is basically to not use this failure information. The analysis retains the contribution of the interval to the total test time, but no assumptions are made regarding the actual number of failures over the interval. This is often referred to as ''gap analysis''.

Lisa Hacker at 22:52, 15 September 2023

2023-09-15T22:52:08Z

← Older revision		Revision as of 22:52, 15 September 2023
Line 1:		Line 1:
	<noinclude>{{Banner RGA Articles}}		<noinclude>{{Banner RGA Articles}}
	''This article also appears in ~~the [[Crow-AMSAA (NHPP)\|~~Reliability Growth and Repairable System Analysis Reference]].'' </noinclude>		''This article also appears in ReliaSoft's Reliability Growth and Repairable System Analysis Reference.'' </noinclude>

	Most of the reliability growth models used for estimating and tracking reliability growth based on test data assume that the data set represents all actual system failure times consistent with a uniform definition of failure (complete data). In practice, this may not always be the case and may result in too few or too many failures being reported over some interval of test time. This may result in distorted estimates of the growth rate and current system reliability. This section discusses a practical reliability growth estimation and analysis procedure based on the assumption that anomalies may exist within the data over some interval of the test period but the remaining failure data follows the Crow-AMSAA reliability growth model. In particular, it is assumed that the beginning and ending points in which the anomalies lie are generated independently of the underlying reliability growth process. The approach for estimating the parameters of the growth model with problem data over some interval of time is basically to not use this failure information. The analysis retains the contribution of the interval to the total test time, but no assumptions are made regarding the actual number of failures over the interval. This is often referred to as ''gap analysis''.		Most of the reliability growth models used for estimating and tracking reliability growth based on test data assume that the data set represents all actual system failure times consistent with a uniform definition of failure (complete data). In practice, this may not always be the case and may result in too few or too many failures being reported over some interval of test time. This may result in distorted estimates of the growth rate and current system reliability. This section discusses a practical reliability growth estimation and analysis procedure based on the assumption that anomalies may exist within the data over some interval of the test period but the remaining failure data follows the Crow-AMSAA reliability growth model. In particular, it is assumed that the beginning and ending points in which the anomalies lie are generated independently of the underlying reliability growth process. The approach for estimating the parameters of the growth model with problem data over some interval of time is basically to not use this failure information. The analysis retains the contribution of the interval to the total test time, but no assumptions are made regarding the actual number of failures over the interval. This is often referred to as ''gap analysis''.

Lisa Hacker at 21:00, 24 June 2015

2015-06-24T21:00:52Z

← Older revision		Revision as of 21:00, 24 June 2015
Line 1:		Line 1:
	<noinclude>{{Banner RGA Articles}}~~[[Category: Shared Articles]]~~		<noinclude>{{Banner RGA Articles}}
	''This article also appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]].'' </noinclude>		''This article also appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]].'' </noinclude>

David J. Groebel at 00:13, 23 April 2014

2014-04-23T00:13:39Z

← Older revision		Revision as of 00:13, 23 April 2014
Line 1:		Line 1:
	<noinclude>{{Banner RGA Articles}}[[Category: Shared Articles]]		<noinclude>{{Banner RGA Articles}}[[Category: Shared Articles]]
	''This article also appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]] ~~book~~.'' </noinclude>		''This article also appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]].'' </noinclude>


	Most of the reliability growth models used for estimating and tracking reliability growth based on test data assume that the data set represents all actual system failure times consistent with a uniform definition of failure (complete data). In practice, this may not always be the case and may result in too few or too many failures being reported over some interval of test time. This may result in distorted estimates of the growth rate and current system reliability. This section discusses a practical reliability growth estimation and analysis procedure based on the assumption that anomalies may exist within the data over some interval of the test period but the remaining failure data follows the Crow-AMSAA reliability growth model. In particular, it is assumed that the beginning and ending points in which the anomalies lie are generated independently of the underlying reliability growth process. The approach for estimating the parameters of the growth model with problem data over some interval of time is basically to not use this failure information. The analysis retains the contribution of the interval to the total test time, but no assumptions are made regarding the actual number of failures over the interval. This is often referred to as ''gap analysis''.		Most of the reliability growth models used for estimating and tracking reliability growth based on test data assume that the data set represents all actual system failure times consistent with a uniform definition of failure (complete data). In practice, this may not always be the case and may result in too few or too many failures being reported over some interval of test time. This may result in distorted estimates of the growth rate and current system reliability. This section discusses a practical reliability growth estimation and analysis procedure based on the assumption that anomalies may exist within the data over some interval of the test period but the remaining failure data follows the Crow-AMSAA reliability growth model. In particular, it is assumed that the beginning and ending points in which the anomalies lie are generated independently of the underlying reliability growth process. The approach for estimating the parameters of the growth model with problem data over some interval of time is basically to not use this failure information. The analysis retains the contribution of the interval to the total test time, but no assumptions are made regarding the actual number of failures over the interval. This is often referred to as ''gap analysis''.

David J. Groebel at 15:51, 4 February 2014

2014-02-04T15:51:16Z

← Older revision		Revision as of 15:51, 4 February 2014
Line 15:		Line 15:
	In general, these equations cannot be solved explicitly for <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math>, but must be solved by an iterative procedure.		In general, these equations cannot be solved explicitly for <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math>, but must be solved by an iterative procedure.

	===~~Missing Data~~ Example===		===Example - Gap Analysis===
	{{:Gap Analysis Example}}		{{:Gap Analysis Example}}

David J. Groebel at 18:44, 27 January 2014

2014-01-27T18:44:35Z

← Older revision		Revision as of 18:44, 27 January 2014
Line 9:		Line 9:
	Let <math>{{S}_{1}}\,\!</math>, <math>{{S}_{2}}\,\!</math> denote the end points of the gap interval, <math>{{S}_{1}}<{{S}_{2}}.\,\!</math> Let <math>0<{{X}_{1}}<{{X}_{2}}<\ldots <{{X}_{{{N}_{1}}}}\le {{S}_{1}}\,\!</math> be the failure times over <math>(0,\,{{S}_{1}})\,\!</math> and let <math>{{S}_{2}}<{{Y}_{1}}<{{Y}_{2}}<\ldots <{{Y}_{{{N}_{1}}}}\le T\,\!</math> be the failure times over <math>({{S}_{2}},\,T)\,\!</math>. The maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math> are values <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math> satisfying the following equations.		Let <math>{{S}_{1}}\,\!</math>, <math>{{S}_{2}}\,\!</math> denote the end points of the gap interval, <math>{{S}_{1}}<{{S}_{2}}.\,\!</math> Let <math>0<{{X}_{1}}<{{X}_{2}}<\ldots <{{X}_{{{N}_{1}}}}\le {{S}_{1}}\,\!</math> be the failure times over <math>(0,\,{{S}_{1}})\,\!</math> and let <math>{{S}_{2}}<{{Y}_{1}}<{{Y}_{2}}<\ldots <{{Y}_{{{N}_{1}}}}\le T\,\!</math> be the failure times over <math>({{S}_{2}},\,T)\,\!</math>. The maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math> are values <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math> satisfying the following equations.

	::<math>\widehat{\lambda }=\frac{{{N}_{1}}+{{N}_{2}}}{S\widehat{_{1}^{\beta }}+{{T}^{\widehat{\beta }}}-S_{2}^{\widehat{\beta }}~~}\,\!</math>~~		:<math>\widehat{\lambda }=\frac{{{N}_{1}}+{{N}_{2}}}{S\widehat{_{1}^{\beta }}+{{T}^{\widehat{\beta }}}-S_{2}^{\widehat{\beta }}}\,\!</math>

	::<math>\widehat{\beta }=\frac{{{N}_{1}}+{{N}_{2}}}{\widehat{\lambda }\left[ S\widehat{_{1}^{\beta }}\ln {{S}_{1}}+{{T}^{\widehat{\beta }}}\ln T-S_{2}^{\widehat{\beta }}\ln {{S}_{2}} \right]-\left[ \underset{i=1}{\overset{{{N}_{1}}}{\mathop{\sum }}}\,\ln {{X}_{i}}+\underset{i=1}{\overset{{{N}_{2}}}{\mathop{\sum }}}\,\ln {{Y}_{i}} \right]}\,\!</math>

			:<math>\widehat{\beta }=\frac{{{N}_{1}}+{{N}_{2}}}{\widehat{\lambda }\left[ S\widehat{_{1}^{\beta }}\ln {{S}_{1}}+{{T}^{\widehat{\beta }}}\ln T-S_{2}^{\widehat{\beta }}\ln {{S}_{2}} \right]-\left[ \underset{i=1}{\overset{{{N}_{1}}}{\mathop{\sum }}}\,\ln {{X}_{i}}+\underset{i=1}{\overset{{{N}_{2}}}{\mathop{\sum }}}\,\ln {{Y}_{i}} \right]}\,\!</math>

	In general, these equations cannot be solved explicitly for <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math>, but must be solved by an iterative procedure.		In general, these equations cannot be solved explicitly for <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math>, but must be solved by an iterative procedure.

Chris Kahn at 21:56, 22 January 2014

2014-01-22T21:56:55Z

← Older revision		Revision as of 21:56, 22 January 2014
Line 1:		Line 1:
	<noinclude>{{Banner RGA Articles}}[[Category: Shared Articles]]		<noinclude>{{Banner RGA Articles}}[[Category: Shared Articles]]
	''This article also appears in the [[Crow-~~AMSAA_-_NHPP~~\|Reliability Growth and Repairable System Analysis Reference]] book.'' </noinclude>		''This article also appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]] book.'' </noinclude>

Kate Racaza: /* Missing Data Example */

2013-11-21T23:17:48Z

Missing Data Example

← Older revision		Revision as of 23:17, 21 November 2013
Line 17:		Line 17:

	===Missing Data Example===		===Missing Data Example===
	Consider a system under development that was subjected to a reliability growth test for <math>T=1,000\,\!</math> hours. Each month, the successive failure times, on a cumulative test time basis, were reported. According to the test plan, 125 hours of test time were accumulated on each prototype system each month. The total reliability growth test program lasted for 7 months. One prototype was tested for each of the months 1, 3, 4, 5, 6 and 7 with 125 hours of test time. During the second month, two prototypes were tested for a total of 250 hours of test time. The next table shows the successive <math>N=86\,\!</math> failure times that were reported for <math>T=1000\,\!</math> hours of testing.		{{:Gap Analysis Example}}


	~~<center><math>~~{{~~X}_{i}},\,\!</math> <math>i=1,2,\ldots ,86\,\!</math>, <math>N = 86, T = 1000\,\!</math></center>~~
	~~{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"~~
	\|-
	~~\|.5 \|\| .6 \|\|10.7\|\| 16.6\|\| 18.3\|\| 19.2\|\| 19.5\|\| 25.3~~
	\|-
	~~\|39.2\|\| 39.4 \|\|43.2\|\| 44.8\|\| 47.4\|\| 65.7\|\| 88.1\|\| 97.2~~
	\|-
	~~\|104.9\|\|105.1\|\| 120.8\|\| 195.7 \|\|217.1 \|\|219\|\| 257.5\|\| 260.4~~
	\|-
	~~\|281.3 \|\| 283.7\|\|289.8\|\| 306.6\|\| 328.6\|\| 357.0\|\| 371.7\|\| 374.7~~
	\|-
	~~\|393.2\|\| 403.2\|\| 466.5\|\| 500.9\|\| 501.5\|\| 518.4\|\| 520.7\|\| 522.7~~
	\|-
	~~\|524.6\|\| 526.9\|\| 527.8\|\| 533.6\|\| 536.5\|\| 542.6\|\| 543.2\|\| 545.0~~
	\|-
	~~\|547.4\|\| 554.0\|\| 554.1\|\| 554.2\|\| 554.8\|\| 556.5\|\| 570.6\|\| 571.4~~
	\|-
	~~\|574.9\|\| 576.8\|\| 578.8\|\| 583.4\|\| 584.9\|\| 590.6\|\| 596.1\|\| 599.1~~
	\|-
	~~\|600.1\|\| 602.5\|\| 613.9\|\| 616.0\|\| 616.2\|\| 617.1\|\| 621.4\|\| 622.6~~
	\|-
	~~\|624.7\|\| 628.8\|\| 642.4\|\| 684.8\|\| 731.9\|\| 735.1\|\| 753.6\|\| 792.5~~
	\|-
	~~\|803.7\|\| 805.4\|\| 832.5\|\| 836.2\|\| 873.2\|\| 975.1~~
	\|}

	~~The observed and cumulative number of failures for each month are:~~

	~~{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"~~
	~~!Month~~
	~~!Time Period~~
	~~!Observed Failure Times~~
	~~!Cumulative Failure Times~~
	\|-
	~~\|1\|\| 0-125\|\| 19\|\| 19~~
	\|-
	~~\|2\|\| 125-375\|\|13\|\| 32~~
	\|-
	~~\|3\|\| 375-500 \|\|3\|\| 35~~
	\|-
	~~\|4\|\| 500-625 \|\|38\|\| 73~~
	\|-
	~~\|5\|\| 625-750 \|\|5\|\| 78~~
	\|-
	~~\|6\|\| 750-875 \|\|7\|\| 85~~
	\|-
	~~\|7\|\| 875-1000\|\|1\|\| 86~~
	\|}


	~~:1) Determine the maximum likelihood estimators for the Crow-AMSAA model.~~
	~~:2) Evaluate the goodness-of-fit for the model.~~
	~~:3) Consider <math>(500,\ 625)\,\!</math> as the gap interval and determine the maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math>.~~

	~~'''Solution'''~~

	~~:1) For the time terminated test:~~

	~~::<math>\begin{align}~~
	~~& \widehat{\beta }= & 0.7597 \\~~
	~~& \widehat{\lambda }= & 0.4521~~
	~~\end{align}\,\!</math>~~

	~~:2) The Cramér-von Mises goodness-of-fit test for this data set yields:~~

	~~::<math>C_{M}^{2}=\tfrac{1}{12M}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{\left[ (\tfrac{{{T}_{i}}}{T})\widehat{^{\beta }}-\tfrac{2i-1}{2M} \right]}^{2}}= 0.6989\,\!</math>~~

	:The critical value at the 10% significance level is 0.173. Therefore, the test indicated that the analyst should reject the hypothesis that the data set follows the Crow-AMSAA reliability growth model. The following plot shows <math>\ln N(t)\,\!</math> versus <math>\ln t\,\!</math> with the fitted line <math>\ln \hat{\lambda }+\hat{\beta }\ln t\,\!</math>, where <math>\widehat{\lambda }=0.4521\,\!</math> and <math>\widehat{\beta }=0.7597\,\!</math> are the maximum likelihood estimates.


	~~[[Image~~:~~rga5.14.png\|center\|400px\|Observed and estimated number of failures for the full data set.]]~~

	:Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months. A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.

	:3) Considering the problem interval <math>(500,625)\,\!</math> as the gap interval, we will use the data over the interval <math>(0,500)\,\!</math> and over the interval <math>(625,1000).\,\!</math> The [[Crow-AMSAA_-_NHPP#Missing_Data\|equations for analyzing missing data]] are the appropriate equations to estimate <math>\lambda \,\!</math> and <math>\beta \,\!</math> because the failure times are known. In this case <math>{{S}~~_{1~~}~~}=500,\,{{S}_{2}}=625\,\!</math> and <math>T=1000,\ {{N}_{1}}=35,\,{{N}_{2}}=13\,\!</math>. The maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math> are:~~

	~~::<math>\begin{align}~~
	~~& \widehat{\beta }= & 0.5596 \\~~
	~~& \widehat{\lambda }= & 1.1052~~
	~~\end{align}\,\!</math>~~


	~~:The next figure is a plot of the cumulative number of failures versus time. This plot is approximately linear, which also indicates a good fit of the model.~~

	~~[[Image:rga5.15.png\|center\|400px\|Observed and estimated number of failures for the gap data set.]]~~

Kate Racaza: Created page with '{{Banner RGA Articles}}Category: Shared Articles ''This article also appears in the [[Crow-AMSAA_-_NHPP|Reliability Growth and Repairable System Analysis Reference…'

2013-02-07T21:04:00Z

Created page with '<noinclude>{{Banner RGA Articles}}Category: Shared Articles ''This article also appears in the [[Crow-AMSAA_-_NHPP|Reliability Growth and Repairable System Analysis Reference…'

New page

<noinclude>{{Banner RGA Articles}}[[Category: Shared Articles]]
''This article also appears in the [[Crow-AMSAA_-_NHPP|Reliability Growth and Repairable System Analysis Reference]] book.'' </noinclude>

Most of the reliability growth models used for estimating and tracking reliability growth based on test data assume that the data set represents all actual system failure times consistent with a uniform definition of failure (complete data). In practice, this may not always be the case and may result in too few or too many failures being reported over some interval of test time. This may result in distorted estimates of the growth rate and current system reliability. This section discusses a practical reliability growth estimation and analysis procedure based on the assumption that anomalies may exist within the data over some interval of the test period but the remaining failure data follows the Crow-AMSAA reliability growth model. In particular, it is assumed that the beginning and ending points in which the anomalies lie are generated independently of the underlying reliability growth process. The approach for estimating the parameters of the growth model with problem data over some interval of time is basically to not use this failure information. The analysis retains the contribution of the interval to the total test time, but no assumptions are made regarding the actual number of failures over the interval. This is often referred to as ''gap analysis''.

Consider the case where a system is tested for time <math>T\,\!</math> and the actual failure times are recorded. The time <math>T\,\!</math> may possibly be an observed failure time. Also, the end points of the gap interval may or may not correspond to a recorded failure time. The underlying assumption is that the data used in the maximum likelihood estimation follows the Crow-AMSAA model with a Weibull intensity function <math>\lambda \beta {{t}^{\beta -1}}\,\!</math>. It is not assumed that zero failures occurred during the gap interval, rather, it is assumed that the actual number of failures is unknown, and hence no information at all regarding these failure is used to estimate <math>\lambda \,\!</math> and <math>\beta \,\!</math>.

Let <math>{{S}_{1}}\,\!</math>, <math>{{S}_{2}}\,\!</math> denote the end points of the gap interval, <math>{{S}_{1}}<{{S}_{2}}.\,\!</math> Let <math>0<{{X}_{1}}<{{X}_{2}}<\ldots <{{X}_{{{N}_{1}}}}\le {{S}_{1}}\,\!</math> be the failure times over <math>(0,\,{{S}_{1}})\,\!</math> and let <math>{{S}_{2}}<{{Y}_{1}}<{{Y}_{2}}<\ldots <{{Y}_{{{N}_{1}}}}\le T\,\!</math> be the failure times over <math>({{S}_{2}},\,T)\,\!</math>. The maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math> are values <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math> satisfying the following equations.

::<math>\widehat{\lambda }=\frac{{{N}_{1}}+{{N}_{2}}}{S\widehat{_{1}^{\beta }}+{{T}^{\widehat{\beta }}}-S_{2}^{\widehat{\beta }}}\,\!</math>

::<math>\widehat{\beta }=\frac{{{N}_{1}}+{{N}_{2}}}{\widehat{\lambda }\left[ S\widehat{_{1}^{\beta }}\ln {{S}_{1}}+{{T}^{\widehat{\beta }}}\ln T-S_{2}^{\widehat{\beta }}\ln {{S}_{2}} \right]-\left[ \underset{i=1}{\overset{{{N}_{1}}}{\mathop{\sum }}}\,\ln {{X}_{i}}+\underset{i=1}{\overset{{{N}_{2}}}{\mathop{\sum }}}\,\ln {{Y}_{i}} \right]}\,\!</math>

In general, these equations cannot be solved explicitly for <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math>, but must be solved by an iterative procedure.

===Missing Data Example===
Consider a system under development that was subjected to a reliability growth test for <math>T=1,000\,\!</math> hours. Each month, the successive failure times, on a cumulative test time basis, were reported. According to the test plan, 125 hours of test time were accumulated on each prototype system each month. The total reliability growth test program lasted for 7 months. One prototype was tested for each of the months 1, 3, 4, 5, 6 and 7 with 125 hours of test time. During the second month, two prototypes were tested for a total of 250 hours of test time. The next table shows the successive <math>N=86\,\!</math> failure times that were reported for <math>T=1000\,\!</math> hours of testing.

<center><math>{{X}_{i}},\,\!</math> <math>i=1,2,\ldots ,86\,\!</math>, <math>N = 86, T = 1000\,\!</math></center>
{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|-
|.5 || .6 ||10.7|| 16.6|| 18.3|| 19.2|| 19.5|| 25.3
|-
|39.2|| 39.4 ||43.2|| 44.8|| 47.4|| 65.7|| 88.1|| 97.2
|-
|104.9||105.1|| 120.8|| 195.7 ||217.1 ||219|| 257.5|| 260.4
|-
|281.3 || 283.7||289.8|| 306.6|| 328.6|| 357.0|| 371.7|| 374.7
|-
|393.2|| 403.2|| 466.5|| 500.9|| 501.5|| 518.4|| 520.7|| 522.7
|-
|524.6|| 526.9|| 527.8|| 533.6|| 536.5|| 542.6|| 543.2|| 545.0
|-
|547.4|| 554.0|| 554.1|| 554.2|| 554.8|| 556.5|| 570.6|| 571.4
|-
|574.9|| 576.8|| 578.8|| 583.4|| 584.9|| 590.6|| 596.1|| 599.1
|-
|600.1|| 602.5|| 613.9|| 616.0|| 616.2|| 617.1|| 621.4|| 622.6
|-
|624.7|| 628.8|| 642.4|| 684.8|| 731.9|| 735.1|| 753.6|| 792.5
|-
|803.7|| 805.4|| 832.5|| 836.2|| 873.2|| 975.1
|}

The observed and cumulative number of failures for each month are:

{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
!Month
!Time Period
!Observed Failure Times
!Cumulative Failure Times
|-
|1|| 0-125|| 19|| 19
|-
|2|| 125-375||13|| 32
|-
|3|| 375-500 ||3|| 35
|-
|4|| 500-625 ||38|| 73
|-
|5|| 625-750 ||5|| 78
|-
|6|| 750-875 ||7|| 85
|-
|7|| 875-1000||1|| 86
|}

:1) Determine the maximum likelihood estimators for the Crow-AMSAA model.
:2) Evaluate the goodness-of-fit for the model.
:3) Consider <math>(500,\ 625)\,\!</math> as the gap interval and determine the maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math>.

'''Solution'''

:1) For the time terminated test:

::<math>\begin{align}
& \widehat{\beta }= & 0.7597 \\
& \widehat{\lambda }= & 0.4521
\end{align}\,\!</math>

:2) The Cramér-von Mises goodness-of-fit test for this data set yields:

::<math>C_{M}^{2}=\tfrac{1}{12M}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{\left[ (\tfrac{{{T}_{i}}}{T})\widehat{^{\beta }}-\tfrac{2i-1}{2M} \right]}^{2}}= 0.6989\,\!</math>

:The critical value at the 10% significance level is 0.173. Therefore, the test indicated that the analyst should reject the hypothesis that the data set follows the Crow-AMSAA reliability growth model. The following plot shows <math>\ln N(t)\,\!</math> versus <math>\ln t\,\!</math> with the fitted line <math>\ln \hat{\lambda }+\hat{\beta }\ln t\,\!</math>, where <math>\widehat{\lambda }=0.4521\,\!</math> and <math>\widehat{\beta }=0.7597\,\!</math> are the maximum likelihood estimates.

[[Image:rga5.14.png|center|400px|Observed and estimated number of failures for the full data set.]]

:Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months. A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.

:3) Considering the problem interval <math>(500,625)\,\!</math> as the gap interval, we will use the data over the interval <math>(0,500)\,\!</math> and over the interval <math>(625,1000).\,\!</math> The [[Crow-AMSAA_-_NHPP#Missing_Data|equations for analyzing missing data]] are the appropriate equations to estimate <math>\lambda \,\!</math> and <math>\beta \,\!</math> because the failure times are known. In this case <math>{{S}_{1}}=500,\,{{S}_{2}}=625\,\!</math> and <math>T=1000,\ {{N}_{1}}=35,\,{{N}_{2}}=13\,\!</math>. The maximum likelihood estimates of <math>\lambda \,\!</math> and <math>\beta \,\!</math> are:

::<math>\begin{align}
& \widehat{\beta }= & 0.5596 \\
& \widehat{\lambda }= & 1.1052
\end{align}\,\!</math>

:The next figure is a plot of the cumulative number of failures versus time. This plot is approximately linear, which also indicates a good fit of the model.

[[Image:rga5.15.png|center|400px|Observed and estimated number of failures for the gap data set.]]