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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.

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I am not entirely sure if I understand your settings, however, these are usually two relevant situation.

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.

Situation 2: What you propose is to estimate certain parameter and then plug-in such estimators to derive estimators for other parameters. This is usually referred to as a 2-step estimator. There are some results available for the limiting properties (for one, see section 5.4 of van der Vaart, Asymptotic Statistics). Unfortunately, deriving such results can be often rather cumbersome. An commonly accepted alternative is to use bootstrap.

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