Template:Hypothesis tests rsa - Revision history

Richard House: Redirected page to RGA Appendix B

2012-08-23T23:06:52Z

← Older revision		Revision as of 23:06, 23 August 2012
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	~~=Hypothesis Tests=~~		#REDIRECT [[RGA_Appendix_B]]
	~~{{common beta hypothesis test rsa}}~~

	~~{{laplace trend test rsa}}~~

	~~{{critical values for cramer-von mises test}}~~

Nicolette Young: /* Critical Values for Cramér-von Mises Test */

2012-01-10T23:11:04Z

Critical Values for Cramér-von Mises Test

← Older revision		Revision as of 23:11, 10 January 2012
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	{{laplace trend test rsa}}		{{laplace trend test rsa}}

	~~==Critical Values for Cramér-von Mises Test==~~		{{critical values for cramer-von mises test}}
	~~<br>~~
	~~Table B.2 displays the~~ critical values for ~~the Cramér~~-von ~~Mises goodness-of-fit~~ test ~~given the sample size, <math>M</math> , and the significance level, <math>\alpha </math> .~~

	~~<br>~~
	~~<br>~~

	~~{\|style= align="center" border="1"~~
	\|-
	~~\|colspan="6" style="text-align:center"\|Table B.2 - Critical values for Cramér-von Mises test~~
	\|-
	~~\| \|\|colspan="5" style="text-align:center;"\|<math>\alpha </math>~~
	\|-
	~~\|<math>M</math>\|\| 0.20\|\| 0.15\|\| 0.10\|\| 0.05\|\| 0.01~~
	\|-
	~~\|2\|\| 0.138\|\| 0.149\|\| 0.162\|\| 0.175\|\| 0.186~~
	\|-
	~~\|3\|\| 0.121\|\| 0.135\|\| 0.154\|\| 0.184\|\|0.23~~
	\|-
	~~\|4\|\| 0.121\|\| 0.134\|\| 0.155\|\| 0.191\|\|0.28~~
	\|-
	~~\|5\|\| 0.121\|\| 0.137\|\| 0.160\|\| 0.199\|\|0.30~~
	\|-
	~~\|6\|\| 0.123\|\| 0.139\|\| 0.162\|\| 0.204\|\|0.31~~
	\|-
	~~\|7\|\| 0.124\|\| 0.140\|\| 0.165\|\| 0.208\|\|0.32~~
	\|-
	~~\|8\|\| 0.124\|\| 0.141\|\| 0.165\|\| 0.210\|\|0.32~~
	\|-
	~~\|9\|\| 0.125\|\| 0.142\|\| 0.167\|\| 0.212\|\|0.32~~
	\|-
	~~\|10\|\| 0.125\|\| 0.142\|\| 0.167\|\| 0.212\|\|0.32~~
	\|-
	~~\|11\|\| 0.126\|\| 0.143\|\| 0.169\|\| 0.214\|\|0.32~~
	\|-
	~~\|12\|\| 0.126\|\| 0.144\|\| 0.169\|\| 0.214\|\|0.32~~
	\|-
	~~\|13\|\| 0.126\|\| 0.144\|\| 0.169\|\| 0.214\|\|0.33~~
	\|-
	~~\|14\|\| 0.126\|\| 0.144\|\| 0.169\|\| 0.214\|\|0.33~~
	\|-
	~~\|15\|\| 0.126\|\| 0.144\|\| 0.169\|\| 0.215\|\|0.33~~
	\|-
	~~\|16\|\| 0.127\|\| 0.145\|\| 0.171\|\| 0.216\|\| 0.33~~
	\|-
	~~\|17\|\| 0.127\|\| 0.145\|\| 0.171\|\| 0.217\|\| 0.33~~
	\|-
	~~\|18\|\| 0.127\|\| 0.146\|\| 0.171\|\| 0.217\|\| 0.33~~
	\|-
	~~\|19\|\| 0.127\|\| 0.146\|\| 0.171\|\| 0.217\|\| 0.33~~
	\|-
	~~\|20\|\| 0.128\|\| 0.146\|\| 0.172\|\| 0.217\|\| 0.33~~
	\|-
	~~\|30\|\| 0.128\|\| 0.146\|\| 0.172\|\| 0.218\|\| 0.33~~
	\|-
	~~\|60\|\| 0.128\|\| 0.147\|\| 0.173\|\| 0.220\|\| 0.33~~
	\|-
	~~\|100\|\| 0.129\|\| 0.147\|\| 0.173\|\| 0.220\|\| 0.34~~
	\|}

	~~For application of the Cramér-von Mises critical values, refer to Sections 5.5.1 and 10.1.6.1.~~

Nicolette Young: /* Laplace Trend Test */

2012-01-10T23:10:19Z

Laplace Trend Test

@@ Line 2: / Line 2: @@
 {{common beta hypothesis test rsa}}
-==Laplace Trend Test==
+{{laplace trend test rsa}}
 ==Critical Values for Cramér-von Mises Test==

Nicolette Young: /* Common Beta Hypothesis Test */

2012-01-10T23:09:56Z

Common Beta Hypothesis Test

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Nicolette Young: Created page with '=Hypothesis Tests= ==Common Beta Hypothesis Test== The Common Beta Hypothesis (CBH) Test is applicable to the following data types: Multiple Systems-Concurrent Operating Times, R…'

2012-01-10T23:09:39Z

Created page with '=Hypothesis Tests= ==Common Beta Hypothesis Test== The Common Beta Hypothesis (CBH) Test is applicable to the following data types: Multiple Systems-Concurrent Operating Times, R…'

New page

=Hypothesis Tests=
==Common Beta Hypothesis Test==
The Common Beta Hypothesis (CBH) Test is applicable to the following data types: Multiple Systems-Concurrent Operating Times, Repairable and Fleet. As shown by Crow [17], suppose that <math>K</math> number of systems are under test. Each system has an intensity function given by:

::<math>{{u}_{q}}(t)={{\lambda }_{q}}{{\beta }_{q}}{{t}^{{{\beta }_{q}}-1}}</math>

where <math>q=1,\ldots ,K</math> . You can compare the intensity functions of each of the systems by comparing the <math>{{\beta }_{q}}</math> of each system. When conducting an analysis of data consisting of multiple systems, you expect that each of the systems performed in a similar manner. In particular, you would expect the interarrival rate of the failures across the systems to be fairly consistent. Therefore, the CBH Test tests the hypothesis, <math>{{H}_{o}}</math> , such that <math>{{\beta }_{1}}={{\beta }_{2}}=\ldots ={{\beta }_{K}}</math> . Let <math>{{\tilde{\beta }}_{q}}</math> denote the conditional maximum likelihood estimate of <math>{{\beta }_{q}}</math> , which is given by:

::<math>{{\tilde{\beta }}_{q}}=\frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{M}_{q}}}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{{{M}_{q}}}{\mathop{\sum }}}\,\ln \left( \tfrac{{{T}_{q}}}{{{X}_{iq}}} \right)}</math>

where:
 
:• <math>K=1.</math>
:• <math>{{M}_{q}}={{N}_{q}}</math> if data on the <math>{{q}^{th}}</math> system is time terminated or <math>{{M}_{q}}=({{N}_{q}}-1)</math> if data on the <math>{{q}^{th}}</math> system is failure terminated ( <math>{{N}_{q}}</math> is the number of failures on the <math>{{q}^{th}}</math> system).
:• <math>{{X}_{iq}}</math> is the <math>{{i}^{th}}</math> time-to-failure on the <math>{{q}^{th}}</math> system.
 
Then for each system, assume that:

::<math>\chi _{q}^{2}=\frac{2{{M}_{q}}{{\beta }_{q}}}{{{{\tilde{\beta }}}_{q}}}</math>

are conditionally distributed as independent Chi-Squared random variables with <math>2{{M}_{q}}</math> degrees of freedom. When <math>K=2</math> , you can test the null hypothesis, <math>{{H}_{o}}</math> , using the following statistic:

::<math>F=\frac{\tfrac{\chi _{1}^{2}}{2{{M}_{1}}}}{\tfrac{\chi _{2}^{2}}{2{{M}_{2}}}}</math>

If <math>{{H}_{o}}</math> is true, then <math>F</math> equals <math>\tfrac{{{{\tilde{\beta }}}_{2}}}{{{{\tilde{\beta }}}_{1}}}</math> and conditionally has an F-distribution with <math>(2{{M}_{1}},2{{M}_{2}})</math> degrees of freedom. The critical value, <math>F</math> , can then be determined by referring to the Chi-Squared tables. Now, if <math>K\ge 2</math> , then the likelihood ratio procedure [17] can be used to test the hypothesis <math>{{\beta }_{1}}={{\beta }_{2}}=\ldots ={{\beta }_{K}}</math> . Consider the following statistic:

::<math>L=\underset{q=1}{\overset{K}{\mathop \sum }}\,{{M}_{q}}\ln ({{\tilde{\beta }}_{q}})-M\ln ({{\beta }^{*}})</math>

where:
 
:• <math>M=\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{M}_{q}}</math>
:• <math>{{\beta }^{*}}=\tfrac{M}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\tfrac{{{M}_{q}}}{{{{\tilde{\beta }}}_{q}}}}</math>
 
Also, let:

::<math>a=1+\frac{1}{6(K-1)}\left[ \underset{q=1}{\overset{K}{\mathop \sum }}\,\frac{1}{{{M}_{q}}}-\frac{1}{M} \right]</math>

Calculate the statistic <math>D</math> , such that:

::<math>D=\frac{2L}{a}</math>

The statistic <math>D</math> is approximately distributed as a Chi-Squared random variable with <math>(K-1)</math> degrees of freedom. Then after calculating <math>D</math> , refer to the Chi-Squared tables with <math>(K-1)</math> degrees of freedom to determine the critical points. <math>{{H}_{o}}</math> is true if the statistic <math>D</math> falls between the critical points.
 
 
'''Example'''
 
Consider the data in Table B.1.

 
{|align="center" border="1"
|-
|colspan="4" style="text-align:center"|Table B.1 - Repairable system data
|-
| ||System 1||System 2||System 3
|-
|Start||0||0||0
|-
|End||2000||2000||2000
|-
|Failures||1.2||1.4||0.3
|-
| ||55.6 ||35 ||32.6
|-
| ||72.7||46.8||33.4
|-
| ||111.9||65.9||241.7
|-
| ||121.9||181.1||396.2
|-
| ||303.6||712.6||444.4
|-
| ||326.9||1005.7||480.8
|-
| ||1568.4||1029.9||588.9
|-
| ||1913.5||1675.7||1043.9
|-
| || ||1787.5||1136.1
|-
| || ||1867||1288.1
|-
| || || ||1408.1
|-
| || || ||1439.4
|-
| || || ||1604.8
|}

Given that the intensity function for the <math>{{q}^{th}}</math> system is <math>{{u}_{q}}(t)={{\lambda }_{q}}{{\beta }_{q}}{{t}^{{{\beta }_{q}}-1}}</math> , test the hypothesis that <math>{{\beta }_{1}}={{\beta }_{2}}</math> while assuming a significance level equal to 0.05. Calculate <math>{{\tilde{\beta }}_{1}}</math> and <math>{{\tilde{\beta }}_{2}}</math> using Eqn. (CondBeta). Therefore:

::<math>\begin{align}
& {{{\tilde{\beta }}}_{1}}= & 0.3753 \\
& {{{\tilde{\beta }}}_{2}}= & 0.4657
\end{align}</math>

Then <math>\tfrac{{{{\tilde{\beta }}}_{2}}}{{{{\tilde{\beta }}}_{1}}}=1.2408</math> . Using Eqn. (ftatistic) calculate the statistic <math>F</math> with a significance level of 0.05.

::<math>F=2.0980</math>

Since <math>1.2408<2.0980</math> we fail to reject the null hypothesis that <math>{{\beta }_{1}}={{\beta }_{2}}</math> at the 5% significance level.
Now suppose instead it is desired to test the hypothesis that <math>{{\beta }_{1}}={{\beta }_{2}}={{\beta }_{3}}</math> . Calculate the statistic <math>D</math> using Eqn. (Dtatistic).

::<math>D=0.5260</math>

Using the Chi-Square tables with <math>K-1=2</math> degrees of freedom, the critical values at the 2.5 and 97.5 percentiles are 0.1026 and 5.9915, respectively. Since <math>0.1026<D<5.9915</math> , we fail to reject the null hypothesis that <math>{{\beta }_{1}}={{\beta }_{2}}={{\beta }_{3}}</math> at the 5% significance level.
 
 

==Laplace Trend Test==
 
The Laplace Trend Test tests the hypothesis that a trend does not exist within the data. The Laplace Trend test is applicable to the following data types: Multiple Systems-Concurrent Operating Times, Repairable and Fleet. The Laplace Trend Test can determine whether the system is deteriorating, improving, or if there is no trend at all. Calculate the test statistic, <math>U</math> , using the following equation:

::<math>U=\frac{\tfrac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{X}_{i}}}{N}-\tfrac{T}{2}}{T\sqrt{\tfrac{1}{12N}}}</math>

where:
 
:• <math>T</math> = total operating time (termination time)
:• <math>{{X}_{i}}</math> = age of the system at the <math>{{i}^{th}}</math> successive failure
:• <math>N</math> = total number of failures
 
The test statistic <math>U</math> is approximately a standard normal random variable. The critical value is read from the Standard Normal tables with a given significance level, <math>\alpha </math> .
 
 
'''Example'''
 
Consider once again the data in Table B.1. Check for a trend within System 1 assuming a significance level of 0.10. Calculate the test statistic <math>U</math> for System 1 using Eqn. (Utatistic).

::<math>U=-2.6121</math>

From the Standard Normal tables with a significance level of 0.10, the critical value is equal to 1.645. If <math>-1.645<U<1.645</math> then we would fail to reject the hypothesis of no trend. However, since <math>U<-1.645</math> then an improving trend exists within System 1. 
If <math>U>1.645</math> then a deteriorating trend would exist.

 

==Critical Values for Cramér-von Mises Test==
 
Table B.2 displays the critical values for the Cramér-von Mises goodness-of-fit test given the sample size, <math>M</math> , and the significance level, <math>\alpha </math> .

 
 

{|style= align="center" border="1"
|-
|colspan="6" style="text-align:center"|Table B.2 - Critical values for Cramér-von Mises test
|-
| ||colspan="5" style="text-align:center;"|<math>\alpha </math>
|-
|<math>M</math>|| 0.20|| 0.15|| 0.10|| 0.05|| 0.01
|-
|2|| 0.138|| 0.149|| 0.162|| 0.175|| 0.186
|-
|3|| 0.121|| 0.135|| 0.154|| 0.184||0.23
|-
|4|| 0.121|| 0.134|| 0.155|| 0.191||0.28
|-
|5|| 0.121|| 0.137|| 0.160|| 0.199||0.30
|-
|6|| 0.123|| 0.139|| 0.162|| 0.204||0.31
|-
|7|| 0.124|| 0.140|| 0.165|| 0.208||0.32
|-
|8|| 0.124|| 0.141|| 0.165|| 0.210||0.32
|-
|9|| 0.125|| 0.142|| 0.167|| 0.212||0.32
|-
|10|| 0.125|| 0.142|| 0.167|| 0.212||0.32
|-
|11|| 0.126|| 0.143|| 0.169|| 0.214||0.32
|-
|12|| 0.126|| 0.144|| 0.169|| 0.214||0.32
|-
|13|| 0.126|| 0.144|| 0.169|| 0.214||0.33
|-
|14|| 0.126|| 0.144|| 0.169|| 0.214||0.33
|-
|15|| 0.126|| 0.144|| 0.169|| 0.215||0.33
|-
|16|| 0.127|| 0.145|| 0.171|| 0.216|| 0.33
|-
|17|| 0.127|| 0.145|| 0.171|| 0.217|| 0.33
|-
|18|| 0.127|| 0.146|| 0.171|| 0.217|| 0.33
|-
|19|| 0.127|| 0.146|| 0.171|| 0.217|| 0.33
|-
|20|| 0.128|| 0.146|| 0.172|| 0.217|| 0.33
|-
|30|| 0.128|| 0.146|| 0.172|| 0.218|| 0.33
|-
|60|| 0.128|| 0.147|| 0.173|| 0.220|| 0.33
|-
|100|| 0.129|| 0.147|| 0.173|| 0.220|| 0.34
|}

For application of the Cramér-von Mises critical values, refer to Sections 5.5.1 and 10.1.6.1.