0

I have a question that I cannot answer myself. I am close and seem to know the answer, but I do not know how to get there. My question is closely related to this question. That is, let's call the sampled values from a multivariate normal distribution $X_i$. What is an estimator of the population variance of the sampled values $X_i$? I assume that all the means $\mu$ are the same, there is one common variance $\sigma_i^2$, and all covariances $\sigma_{ij}$ are the same.

The linked question states that you can take the expectation of the sampling variance, which seems to make sense to me. That is, we need to solve

$$\begin{align*} \mathbb{E} \left( \frac{1}{k} \sum_{i=1}^k \left( X_i - \frac{1}{k} \sum_{j=1}^k X_j \right)^2 \right) &= \frac{1}{k} \sum_{i=1}^k \mathbb{E}(X_i^2) - \frac{2}{k^2} \sum_{i=1}^k \sum_{j=1}^k \mathbb{E}(X_i X_j) + \frac{1}{k^2} \sum_{j=1}^k \sum_{\ell=1}^k\mathbb{E}(X_j X_{\ell}). \\ &= \frac{1}{k} \sum_{i=1}^k \mathbb{E}(X_i^2) - \frac{1}{k^2} \sum_{i=1}^k \sum_{j=1}^{k} \mathbb{E}(X_i X_j). \end{align*}$$ where we sum over $i=1, ..., k$.

However, I do not know how to take the expectation in case of a multivariate normal distribution. I know how to do this for a (univariate) normal distribution but not for a multivariate. I think the answer is $\sigma_i^2 - \sigma_{ij}$, but I am not totally sure.

Thank you in advance for your help!

User33
  • 101
  • what is your definition of variance and covariance? – Exodd May 26 '20 at 11:46
  • I do not fully understand your question, so hopefully I am clarifying it. What I want to obtain is an estimator of the population variance of the sampled values from multivariate normal distribution. That is, I sample $k$ values from a multivariate normal distribution and I want to get an estimator of the mean of these sampled values. – User33 May 26 '20 at 11:56
  • I asked you, how do you define $\sigma_i$ and $\sigma_{i,j}$ in relation to the $X_i$. It is fundamental to conclude your computation – Exodd May 26 '20 at 11:59
  • OK, let's say that $X \sim MVN(\mu,\Sigma)$ where the variance-covariance matrix is: $\Sigma$ = \begin{pmatrix} \sigma^2_1 & \sigma_{12} & \sigma_{13} & \cdots & \sigma_{1k} \ \sigma_{21} & \sigma^2_2 & \sigma_{23} & \cdots & \sigma_{2k} \ \vdots & \vdots& \vdots & \ddots & \vdots \ \sigma_{k1} & \sigma_{k2} & \sigma_{k3} & \cdots & \sigma^2_k \end{pmatrix} where $\sigma^2_i$ and $\sigma_{ij}$ can be any positive number. – User33 May 26 '20 at 14:08

0 Answers0