Template:Grouped data rga

Grouped Data

Parameter estimation for grouped data using the Crow-AMSAA model is the same as described in Chapter 5. The equations used to estimate the parameters of the Crow Extended model are presented next. For test-find-test data, the maximum likelihood estimates of [math]\displaystyle{ {{\lambda }_{BD}} }[/math] and [math]\displaystyle{ {{\beta }_{BD}} }[/math] are calculated using the first occurrences of the BD modes such that:

[math]\displaystyle{ \underset{i=1}{\overset{k}{\mathop \sum }}\,{{n}_{i}}\left[ \frac{T_{i}^{\widehat{\beta }}\ln {{T}_{i}}-T_{i-1}^{\widehat{\beta }}\ln {{T}_{i-1}}}{T_{i}^{\widehat{\beta }}-T_{i-1}^{\widehat{\beta }}}-\ln {{T}_{k}} \right]=0 }[/math]

[math]\displaystyle{ \widehat{\lambda }=\frac{n}{T_{k}^{\widehat{\beta }}} }[/math]

where [math]\displaystyle{ {{n}_{i}} }[/math] is the number of distinct BD modes within the [math]\displaystyle{ {{i}^{th}} }[/math] interval. For test-fix-find-test data, the maximum likelihood estimates of [math]\displaystyle{ {{\lambda }_{BC}} }[/math] and [math]\displaystyle{ {{\beta }_{BC}} }[/math] are estimated in the same manner using the first occurrences of the BC modes.

Confidence Bounds

The confidence bounds on the parameters for the Crow Extended model for grouped data are calculated using the same as the procedure presented in Chapter 5. If there are no BC modes, the confidence bounds on the demonstrated failure intensity and MTBF, projected failure intensity and MTBF and growth potential failure intensity and MTBF are the same as the procedure presented in Section 9.5. If there are BC modes, then the confidence bounds on the demonstrated failure intensity and MTBF are the same as the procedure presented in Chapter 5 and the confidence bounds on the projected failure intensity and MTBF and growth potential failure intensity and MTBF are the same as in Section 9.5. The confidence bounds on time are the same as the procedure presented in Chapter 5.