If one records whether the observations (taken from $N(\mu,1)$) are less than zero or not, how can I find the MLE of $\mu$? Let $X_1,\cdots,X_n \sim N(\mu,1)$, where $\mu$ is unknown. However, one only records whether the observations $X_1,\cdots,X_n$ are less than $0$. Suppose we know that there are $m$ out of the $n$ observations that are less than $0$. How can I find the MLE of $\mu$?
My idea:
Let $Y_i=1_{\{-\infty,0\}}(X_i)$. Then, $\sum^n_{i=1}Y_i=m$.
$P(Y_i=1)=P(X_i < 0)$ and $P(Y_i=0)=P(X_i \ge 0)$
I don't really understand how I can derive a likelihood distribution for $Y_1,\cdots,Y_n$ that depends on $\mu$.
How would any of you even approach this problem?
 A: There are two steps to solving this problem.  The first is based on the realization that what you actually observe are observations $y_i = 1_{(x_i < 0)}$ which are drawn from a Binomial distribution with probability parameter $p(\mu) = \text{P}(x<0 |\, \mu, 1)$, with $\text{P}$ of course being the cumulative Normal distribution.
We form the maximum likelihood estimate of $p(\mu)$ in the usual way: $\hat{p}(\mu) = m/n$, where $m$ is the number of $x_i < 0$ and $n$ is the total sample size.
The second step is based on the fact that the maximum likelihood estimator of the value of a function of a parameter $p$, label it $f(p)$, for which we have an MLE $\hat{p}$ is just the plug-in estimator $f(\hat{p})$.  In this case, $p(\mu)$ is a continuous, strictly monotonically decreasing function of $\mu$, so it is invertible; we can write the inverse function as $\mu(p)$.  The maximum likelihood estimator of $\mu$ is $\hat{\mu} = \mu(\hat{p})$.
Operationally speaking, we just find the (guaranteed unique) value of $\mu$ such that $\text{P}(X < 0 | \,\mu, 1) = m/n$, keeping in mind the problems that will arise if $m = 0$ or $m = n$.
