Identification from minimum value of truncated distribution Suppose that a given population is endowed with a pair of characteristics $T$ and $K$. Let's think of these characteristics as random variables   $$(T,K) \sim \operatorname{BiNormal}((\mu_T, \mu_K), (\sigma_T,\sigma_K), \rho)$$
Suppose now that I observe the realisations of $T$ for a sample consisting of those individuals with $K<a$, where the selection threshold $a$ is unknown. 
I have a theoretical model that predicts the value of $a$ for any set of the underlying parameters $(\mu_T, \mu_K)$ and $ (\sigma_T,\sigma_K), \rho$. 
Assuming I know all the parameters but one (say, $\rho$), I could use the theoretical model to predict $a$ (say $\hat{a}$), and then estimate  $\rho$ using minimum distance (i.e. find the value of $\rho$ that minimises difference between $E[T|K<\hat{a}]$ and the observed mean of the selected sample).
Now assume that I know all the parameters but two (say $\rho$ and $\sigma_T$). I could similarly estimate $\rho$ and $\sigma_T$ by simultaneously minimising the distance for $Var[T|K<\hat{a}]$ and $E[T|K<\hat{a}]$.
Now I've run out of usual suspects to target... Is it possible to identify another unknown parameter (say $\sigma_K$), for example, from the observed minimum value of $T$ in the selected sample?
 A: This is close to a Type II Tobit, or Heckit model.
In such models, we posit latent variables
$$
(T, K) \sim \mbox{BiNormal}(\mu, \Sigma)
$$
and observe
$$
Y = \begin{cases}
T, K>a \\
0, K \leq a
\end{cases}
$$
You have it at least as difficult as the Heckit model (since you don't even get to know that $K$ was below a threshold).  In fact, you have only the second term of the likelihood in equation 11 of the linked reference
$$
P(T=t|K>a) = \frac{1}{\sigma_T} \phi \bigl( (t-\mu_T)/\sigma_T \bigr) \frac{1}{\sqrt{1-\rho^2}} \Phi \left(\frac{a-\mu_K - \rho\sigma_T/\sigma_K(t-\mu_T) }{\sqrt{1-\rho^2}} \right),
$$
where $\phi(z)$ and $\Phi(z)$ are the standard normal pdf and cdf, respectively, evaluated at $z$.
(You can show this by writing $P(T,K>a)=P(K>a|T) P(T)$ and observing that $P(K|T)$ is Normal.)
Identifiability
I can easily see the following (which are true in the Heckit model as well)


*

*Only $\mu_K-a$ is estimible.

*$\sigma_K$ and $a-\mu_K$ are not simultaneously identifiable.


In general, it's not simple to show that identifiability holds, since it boils down to finding the roots of $P_\theta(t|K>a) - P_{\theta'}(t|K>a') = 0$, which is a transcendental equation!  
Showing that identifiability fails to hold is easier.  If the variance-covariance matrix of the MLE is singular then you have a recipe for generating new $\theta$ that will return the same likelihood, so you have good evidence that your model is singular.
