Suppose the following equation holds
\begin{align*} p_2=\int\limits_{-\infty}^{\Phi^{-1}(p)}\int\limits_{-\infty}^{\Phi^{-1}(p)} \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\bigg({-\frac{1}{2}\frac{x^2-\rho xy+y^2}{1-\rho^2}}\bigg)dxdy \end{align*}
$\Phi^{-1}$ is the inverse of cumulative distribution function and I am able to estimate $p$ and $p_2$ from the data. I could use R for example to give me an unique solution for $\rho$.
- Can I make any statement about expectation or variance of my estimator for $\rho$?
If you can name the needed R commands and sketch the way to do it, this will be enough. Also analytical formulas will be enough, will just code them in R by myself.
(The problem is related to Asset-Value-Model and Vasicek distribution.)
This could be usefull information aswell: We are going to use default data, where $d(t)$ stands for amount of defaults in period $t$ and $t \in \{1,\ldots , s\}$. $n(t)$ is total number of credits. One can use intuitive estimators (will be unbiased) \begin{align*} \frac{\sum\limits_{t=1}^s \frac{d(t)}{n(t)}}{s} \end{align*} for $p$. And
\begin{align*} \frac{\sum\limits_{t=1}^s \frac{d(t)(d(t)-1)}{n(t)(n(t)-1)}}{s} \end{align*} for $p_2$. There are another possible estimators, for example via Bayes approach.