this my second question, so I'm still new... thanks in advance for any help! Basically, I'm looking for some references and tools to study the following problem.
Consider the following function $f(\bullet;\alpha,\beta): [0,+ \infty) \rightarrow R$ that depends on two real parameters $\alpha$ and $\beta$. So, for each combination of the two parameters, we have a real valued function. We assume that $f$ is a smooth function of $t$, and also with respect to the parameters. For example, we may have: $f(t,\alpha,\beta)=(\alpha + \beta)t$.
Thus, we can define a relation such that for each $(\alpha,\beta) \in R^2$ we associate the function $f(t;\alpha,\beta)$. The general question is: which condition we can impose on the function $f$ such that the previous relation is ''injective''. That is, under which conditions we can guarantee that $f(\bullet;\alpha,\beta) \neq f(\bullet;\hat \alpha, \hat \beta)$, for all $\alpha,\beta, \hat \alpha, \hat \beta$ such that $\alpha \neq \hat \alpha$ and $\beta \neq \hat \beta$.
Some remarks.
1) If $f(\bullet;\alpha,\beta)$ is a probability distribution function, then the problem falls in the identification theory. But, note that in our case the function does not need to be a probability distribution function.
2) A result that I think is clearly true is the following. If $f$ can be written as $f(t;\alpha,\beta)=f(t,g(\alpha,\beta))$, then the relation is not injective.
3) A straightforward example of a function such that the relation is injective is: $f(t;\alpha,\beta)=(\alpha+\beta) + (\alpha-\beta)t$.
As I wrote, I'm interested in conditions that guarantees that the relation is ''injective". Additionally, I'm wondering what happens if $f$ is not a real function but a sequence, that is: $t = 0, 1, 2, \ldots$.
Thanks a lot! and I hope you will find the problem interesting.
