Suppose the random variable $Y$ has a multivariate normal (MVN) distribution, and consider truncating $Y$ in some way to create $T$. Given $T$'s mean and covariance matrix, I'd like to obtain $Y$'s mean and covariance matrix.
Below I state the above problem more precisely, ask three specific questions, give an example, and close with several remarks.
Suppose $Y$ is a random variable whose $d$-variate normal ($d$VN) distribution has mean $\mu = \mathrm{E}(Y)$ and covariance matrix $\Sigma = \mathrm{Cov}(Y)$, so $Y \sim \mathcal{N}_d(\mu, \Sigma)$. Let $T = Y | (Y \in \mathcal{A})$ be a truncated version of $Y$ that takes only values of $Y$ in the set $\mathcal{A}$ and has mean $\eta = \mathrm{E}(T)$ and covariance matrix $\Omega = \mathrm{Cov}(T)$. For instance, in the example below $\mathcal{A}$ consists of vectors that form valid correlation matrices. Collect $\mu$'s and $\Sigma$'s $d(d+3) / 2$ distinct parameters in the vector $\theta_Y$, and collect their counterparts from $\eta$ and $\Omega$ in $\theta_T$. Here are three questions about obtaining $\theta_Y$ for a specified $\theta_T$:
Under what conditions does a unique $\theta_Y$ exist for a specified $\theta_T$? This may be a question of invertibility: If $f$ is a function such that $\theta_T = f(\theta_Y)$, under what conditions is $f$ invertible?
In a condition from #1 above, how can we obtain $\theta_Y$ for a specified $\theta_T$, either exactly or by numerical approximation or simulation? For instance, can this be framed as a problem of optimization or solving a nonlinear system of equations?
If we approximate or simulate $\theta_Y$ in #2 above, how can we control the approximation's or estimate's accuracy? For instance, we might want an estimate of $\theta_Y$ to be near $\theta_Y$ with high probability.
EXAMPLE: Consider a fairly simple version of the above problem where $d = 3$, so $Y \sim \mathcal{N}_3(\mu, \Sigma)$, and $\mathcal{A}$ used for truncation is the set of all 3-vectors containing Fisher-z transformations of a valid Pearson-r correlation matrix's distinct elements (e.g., below the diagonal in row-major order). For instance, $\mathcal{A}$ contains [1.20 1.20 0.50] but not [1.20 1.20 0.40]. Let's specify $T$'s desired mean and covariance matrix -- with commas separating rows -- as follows, along with $\theta_T$ (e.g., $\eta$ followed by $\Omega$'s lower triangle in row-major order):
$\eta$ = [0.40 0.60 0.80]
$\Omega$ = [0.25 0.12 0.06, 0.12 0.16 0.06, 0.06 0.06 0.09]
$\theta_T$ = [0.40 0.60 0.80 0.25 0.12 0.16 0.06 0.06 0.09]
An ad hoc method -- applying an iterative truncate-and-transform technique each of several large 3VN samples -- yields the following estimates of $Y$'s mean and covariance matrix and, hence, $\theta_Y$:
$\hat \mu$ = [0.329 0.631 0.832]
$\hat \Sigma$ = [0.292 0.101 0.043, 0.101 0.163 0.064, 0.043 0.064 0.095]
$\hat \theta_Y$ = [0.329 0.631 0.832 0.292 0.101 0.163 0.043 0.064 0.095]
These estimates are meant to be within about 0.1% of their estimands (in $\theta_Y$) with probability at least .95.
REMARKS: Answering the above questions would probably be easier if $f$ in #1 above were available in closed form, but that's difficult for some truncation rules of interest (i.e., some sets $\mathcal{A}$). For now I'd mainly like an accurate solution to these questions, but I'd eventually like to create a fast implementation. I'm also interested in extensions of this problem, such as working with a mixture of $d$VNs for $Y$ or finding $\mu$ and $\Sigma$ for specified $\mathrm{E}(G)$ and $\mathrm{Cov}(G)$, where $G = g(T)$ is a nonlinear function of $T$. These questions arose while developing meta-analytic methods for functions of multivariate effect sizes, where between-studies heterogeneity in an effect-size parameter (e.g., correlation matrix) induces heterogeneity in a function of that parameter (e.g., path-model coefficients).