Template:Chi-squared test rsa

Chi-Squared Test

The parametric Cramér-von Mises test described above requires that the starting time, [math]\displaystyle{ {{S}_{q}} }[/math] , be equal to 0 for each of the [math]\displaystyle{ K }[/math] systems. Although not as powerful as the Cramér-von Mises test, the Chi-Squared test can be applied regardless of the starting times. The expected number of failures for a system over its age [math]\displaystyle{ (a,b) }[/math] for the Chi-Squared test is estimated by [math]\displaystyle{ \widehat{\lambda }{{b}^{\widehat{\beta }}}-\widehat{\lambda }{{a}^{\widehat{\beta }}}=\widehat{\theta } }[/math] , where [math]\displaystyle{ \widehat{\lambda } }[/math] and [math]\displaystyle{ \widehat{\beta } }[/math] are the maximum likelihood estimates. The computed [math]\displaystyle{ {{\chi }^{2}} }[/math] statistic is:

[math]\displaystyle{ {{\chi }^{2}}=\underset{j=1}{\overset{d}{\mathop \sum }}\,{{\frac{\left[ N(j)-\theta (j) \right]}{\widehat{\theta }(j)}}^{2}} }[/math]

where [math]\displaystyle{ d }[/math] is the total number of intervals. The random variable [math]\displaystyle{ {{\chi }^{2}} }[/math] is approximately Chi-Square distributed with [math]\displaystyle{ df=d-2 }[/math] degrees of freedom. There must be at least three intervals and the length of the intervals do not have to be equal. It is common practice to require that the expected number of failures for each interval, [math]\displaystyle{ \theta (j) }[/math] , be at least five. If [math]\displaystyle{ \chi _{0}^{2}\gt \chi _{\alpha /2,d-2}^{2} }[/math] or if [math]\displaystyle{ \chi _{0}^{2}\lt \chi _{1-(\alpha /2),d-2}^{2} }[/math] , reject the null hypothesis.