extremum Estimator (possibly tobit?) This question is related to extremum estimators. I am looking to propose an estimator that I think may be a Tobit estimator, but I am unsure. The question is as follows: 

Let $z$ be a random variable with a strictly increasing CDF $F( \dot ) $ everywhere on its support. The median of $Z$, $\gamma$ is defined as: 

$ \int _{-\infty, \gamma} dF(z) &gt; \frac{1}{2}$ ; $ \int _{\gamma, \infty} dF(z) &gt; \frac{1}{2}$

It is known that $\gamma$ minimizes an expected absolute deviatin criterion:
 $\gamma = argmin_e E(|Z-e| - |Z|)$. 

We consider a model $Y=max\{c, f(X; \theta)+\epsilon\}$ where c is a known constant and the conditional median of $\epsilon$ given  $X$ is 0. Suppose we have an i.i.d sample $\{X_i, Y_i\}$, that is generated by this model. 

1) Propose an extremum estimator for $\theta$. 

Here, I was thinking that a Tobit model would be appropriate since the model is truncated by the constant $c$. 

2) Show the estimator you proposed is consistent.
First of all, this is not a Tobit-type estimator. A Tobit estimator is a maximum-likelihood estimator, which is based on a full specification of the error distribution. You have only the median specified, so it can't be an MLE.

What can we do? To begin with, note that, for a random variable $Z$, the median of $\max(c,Z)$ is $\max(c, \mathrm{med}(Z))$, which can be proved by using the definition of the median.

Thus, for your problem, you have
$$\mathrm{med}(y|X) = \max(c, f(X,\theta))$$
under the assumption that $\mathrm{med}(\epsilon|X)=0$. A natural extremum estimator would be the minimizer of
$$\frac{1}{n} \sum_{i=1}^n |y_i - \max(c, f(X_i,\theta)) |.$$
You defined the median as the minimizer of $E(|Z-e|-|Z|)$, but this is the same as the minimizer of $E|Z-e|$ without $-|Z|$, so I dropped the "$-|y_i|$" from the objective function.

This is a generalization of Powell (1984), Least absolute deviations estimator for the censored regression model, Journal of Econometrics, 25, 303-325. See the paper for more details including consistency. https://web.stanford.edu/~doubleh/eco273/powell1984.pdf