I have an iid sample, $X_1,\dots,X_N \in R^d$, from a multivariate normal density with mean $\mu$ and covariance matrix $\Sigma$.
I am estimating the density $p(y) = N(y| \mu, \Sigma)$, using $\hat{p}(y) = N(y| \hat{\mu}, \hat{\Sigma})$, where $\hat{\mu}$ and $\hat{\Sigma}$ are sample mean and covariance matrix and $y$ is a fixed vector.
I would like to work out the bias of $\hat{p}(y)$, that is $E\{\hat{p}(y) - p(y)\}$. Using the answer to this question and this article, it should be possible to derive the bias of $\log \hat{p}(y)$, but this is not quite what I need.
Given the Gaussian setting, I expected such a result to be "out there", but I was unable to find it. Maybe someone is aware of relevant results?
