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Table 1 Proof that \(\varvec{\varTheta }_{\mathcal {EW}}\) is identifiable

From: Analyzing and solving the identifiability problem in the exponentiated generalized Weibull distribution

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 \(\beta _{i} \ne \beta _{j}\) \(c_{i} \ne c_{j} \Rightarrow \left( c_{i} t\right) ^{\beta _{i}} \ne \left( c_{j} t\right) ^{\beta _{j}}\)