Table 1 Proof that \(\varvec{\varTheta }_{\mathcal {EW}}\) is identifiable
Cases | Hypothesis: \(\varvec{\theta }_{i} \ne \varvec{\theta }_{j}\) | Implication for the thesis |
|---|---|---|
1 | \(b_{i} \ne b_{j}\), \(c_{i}=c_{j}\) and \(\beta _{i}=\beta _{j}\) | \(F_{EW}\left( t; \varvec{\theta }_{i}\right) \ne F_{EW}\left( t; \varvec{\theta }_{j}\right)\) |
2 | \(b_{i}=b_{j}\), \(c_{i}=c_{j}=c\) and \(\beta _{i} \ne \beta _{j}\) | \(\beta _{i} \ne \beta _{j} \Rightarrow \left( c t\right) ^{\beta _{i}} \ne \left( c t\right) ^{\beta _{j}}\) |
3 | \(b_{i}=b_{j}\), \(c_{i} \ne c_{j}\) and \(\beta _{i}=\beta _{j}=\beta\) | \(c_{i} \ne c_{j} \Rightarrow \left( c_{i} t\right) ^{\beta } \ne \left( c_{j} t\right) ^{\beta }\) |
4 | \(b_{i}=b_{j}\), \(c_{i} \ne c_{j}\) and \(\beta _{i} \ne \beta _{j}\) | \(c_{i} \ne c_{j} \Rightarrow \left( c_{i} t\right) ^{\beta _{i}} \ne \left( c_{j} t\right) ^{\beta _{j}}\) |
5 | \(b_{i} \ne b_{j}\), \(c_{i} \ne c_{j}\) and \(\beta _{i}=\beta _{j}=\beta\) | \(c_{i} \ne c_{j} \Rightarrow \left( c_{i} t\right) ^{\beta } \ne \left( c_{j} t\right) ^{\beta }\) |
6 | \(b_{i} \ne b_{j}\), \(c_{i}=c_{j}=c\) and \(\beta _{i} \ne \beta _{j}\) | \(c_{i}= c_{j} \Rightarrow \left( c t\right) ^{\beta _{i}} \ne \left( c t\right) ^{\beta _{j}}\) |
7 | \(b_{i} \ne b_{j}\), \(c_{i} \ne c_{j}\) and \(\beta _{i} \ne \beta _{j}\) | \(c_{i} \ne c_{j} \Rightarrow \left( c_{i} t\right) ^{\beta _{i}} \ne \left( c_{j} t\right) ^{\beta _{j}}\) |