Equation below refers to the likelihood for my statistical model. I need to show the model has identifiability issues. The hint that I have been given is that I should attempt to interchange the parameters in the likelihood function.
$$ L(\alpha , \beta ; (x_{i}, .., x_{n})) = \left(\frac{\alpha \beta}{\beta-\alpha}\right)^{n}\prod_{i=1}^{n}\left((1-\alpha)^{r_i-1}-(1-\beta)^{r_{i}-1}\right)\ $$
Edit: explanation of model likelihood.
Graphical representation of the Markov chain:
The model is a discrete-time markov chain. Assume that we always start in state 1. $j$ is the time at which we observe being in state 2, and $r$ is the time at which we observe being in state 3.
The joint probability for one observation from this markov chain would be $$P(Y_{1}=y_{1},\ldots,Y_{n}=y_{n}) = (1-\alpha)^{i-1} \alpha (1-\beta)^{r-i-1}\beta$$
Now the issue is, that we can only observe $r$, the time at which we are in state 3. To marginalise out $i$, I think we would sum over all possible values of $i$.
\begin{align} P(Y_{1}=y_{1},\ldots,Y_{n}=y_{n}) &= \alpha\beta\sum_{i=1}^{r-1} (1-\alpha)^{i-1}(1-\beta)^{r-i-1} \\&= \left(\frac{\alpha \beta}{\beta-\alpha}\right)\left((1-\alpha)^{r-1}-(1-\beta)^{r-1}\right) \end{align}
Now we multiply each of the joint probability distributions $X_{1}, X_{2}, ..,X_{n}$ to form the likelihood that we have above.
Note: $0\leq\alpha, \beta \leq1$
Any help would be greatly appreciated. Thank you in advance.