I am trying to make sense of the bootstrap method.
I am studying on Rice, "mathematical statistics and data analysis"
Here it is its explanation of the bootstrap method:
Imagine for the moment that we knew the true values $\lambda_0$ and $\alpha_0$.We could generate many, many samples of size $n$ from the gamma distribution with these parameter values, and from each of these samples we could calculate estimates of$\lambda$ and $\alpha$. A histogram of the values of the estimates of $\lambda$, for example, should then give us a good idea of the sampling distribution of $\hat \lambda$.
This makes sense. But now:
The only problem with this idea is that it requires knowing the true parameter values. So we substitute our estimates of $\lambda$ and $\alpha$ for the true values; that is we draw many, many samples of size $n$ from a gamma distribution with parameters $\alpha = \hat \alpha(\omega)$ and $\lambda = \hat \lambda(\omega)$
(Since $\hat \alpha$ and $\hat \lambda$ are random variables and not numbers, with $\hat \alpha(\omega) $ I mean the actual value that we observe with our data set (ie $\omega$) )
Now this does not make much sense; We want to study the probability distributions of our variables $\hat \alpha$ and $\hat \lambda$, but if in doing do we basically start with the assumption that $\alpha = \hat \alpha(\omega)$ and $\lambda = \hat \lambda(\omega)$!
Is there something I am missing? It does not seem very rigorous to me