# Maximum-Likelihood estimator

Imagene we have a sequence of i.i.d random variables $(Y(t))_{1\le t\le s}$. It is possible to derive the density of $Y(t)$ and it is a function of parameters of interest $f(p,\rho, y(t))$. To have a Maximum-Likelihood estimator all i need to do is \begin{align*} \widehat{(p,\rho)}^{MLE}=\arg\max\limits_{p,\rho} \prod\limits_{t=1}^sf(p,\rho,y(t)) \end{align*} Sadly $Y(t)$ isn't directly observable. With a given dataset it is possible to construct MLE for realisation of $Y(t)$.

My question is: If $y(t)$ in above equation is replaced by $\widehat{y(t)}^{MLE}$, can the estimator above still be considered as a maximum-likelihood estimator?

I tend more to a "no" answer, but have no proof.

If it isn't MLE anymore, what about asymptotic normality? Hope at least this still usable.

Situation 1: You have $Y(t)$ assumed to be latent variables. In this case, integrate out (possibly using numerical method) to obtain the marginal likelihood of the observable variables, and you are essentially in the same setting as classical likelihood.