1P-Weibull with Zero Failure Data: Difference between revisions

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{{Reference_Example_Heading1}}
{{Reference_Example_Heading1}}
The data from Table 8.2 on page 196 of the book ''Statistical Methods for Reliability Data'' by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998 is used.
The data set from Table 8.2 on page 196 of the book ''Statistical Methods for Reliability Data'' by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998 is used.




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The formulas for calculating the <math>\eta \,\!</math> at a given confidence level of <math>1 - \alpha\,\!</math> is on page 195.  
The formulas for calculating the <math>\eta \,\!</math> at a given confidence level of <math>1 - \alpha\,\!</math> is on page 195.  


::<math>\hat{\eta}_{L} = \left (\frac{2\sum_{i=1}^{n} t^{\beta}_{i}}{x^{2}_{(1-\alpha ;2)}}\right ) ^{\beta}</math>
::<math>\hat{\eta}_{L} = \left (\frac{2\sum_{i=1}^{n} t^{\beta}_{i}}{X^{2}_{(1-\alpha ;2)}}\right ) ^{\beta}</math>




The 95% lower confidence bound on <math>\eta \,\!</math> when <math>\beta = 2\,\!</math> is:
The 95% lower confidence bound on <math>\eta \,\!</math> when <math>\beta = 2\,\!</math> is:


::<math>\hat{\eta}_{L} = \left (\frac{2\sum_{i=1}^{n} t^{\beta}_{i}}{x^{2}_{(1-\alpha ;2)}} \right )^{\beta} = 10250\,\!</math>
::<math>\hat{\eta}_{L} = \left (\frac{2\sum_{i=1}^{n} t^{\beta}_{i}}{X^{2}_{(1-\alpha ;2)}} \right )^{\beta} = 10250\,\!</math>




{{Reference_Example_Heading4}}
{{Reference_Example_Heading4}}


The following picture shows the results in Weibull++:


[[Image:1PW_no_failures.png|center]]
[[Image:1PW_no_failures.png|center]]

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1P-Weibull with Zero Failure Data

This example compares the calculation for a 1-parameter Weibull with zero failure data.


Reference Case

The data set from Table 8.2 on page 196 of the book Statistical Methods for Reliability Data by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998 is used.


Data

Number in State State F or S Time to Failure
10 S 500
12 S 1000
8 S 1500
9 S 2000
7 S 2500
9 S 3000
6 S 3500
3 S 4000


Result

The formulas for calculating the [math]\displaystyle{ \eta \,\! }[/math] at a given confidence level of [math]\displaystyle{ 1 - \alpha\,\! }[/math] is on page 195.

[math]\displaystyle{ \hat{\eta}_{L} = \left (\frac{2\sum_{i=1}^{n} t^{\beta}_{i}}{X^{2}_{(1-\alpha ;2)}}\right ) ^{\beta} }[/math]


The 95% lower confidence bound on [math]\displaystyle{ \eta \,\! }[/math] when [math]\displaystyle{ \beta = 2\,\! }[/math] is:

[math]\displaystyle{ \hat{\eta}_{L} = \left (\frac{2\sum_{i=1}^{n} t^{\beta}_{i}}{X^{2}_{(1-\alpha ;2)}} \right )^{\beta} = 10250\,\! }[/math]


Results in Weibull++

The following picture shows the results in Weibull++:

1PW no failures.png
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