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Is there an analytical expression for the distribution of variances of MVN RVs? I mean if $$X=[x_1, \dots, x_D]\sim \mathcal{N}(0, \Sigma)$$ where $$\Sigma$$ is a $$D$$-dimensional covariance matrix, is there an analytical expression for the distribution of $$V(X) = \frac{1}{D}\sum_d(x_d-\bar{X})^2$$ where $$\bar{X}=\frac{1}{D}\sum_d x_d$$?

Even better would be an expression for the distribution of $$\frac{V(X)}{V'(X)}$$ where $$V'(X)$$ is the variance of a subset of the $$D$$ components of $$X$$. That is $$V'(X) = \frac{1}{D}\sum_{d \in \mathcal{D}}(x_d-\bar{X}')^2$$, where $$\mathcal{D} \subseteq \{1, \dots, D\}$$ and $$\bar{X}'$$ is the mean over these same dimensions. I'm guessing there is no such expression but it never hurts to ask.

There is a related question http://stats.stackexchange.com/questions/17231/second-moment-of-draws-from-a-multivariate-normal-covariance-matrixSecond moment of draws from a multivariate normal covariance matrix but there they just ask for the moments not the full distribution.

Is there an analytical expression for the distribution of variances of MVN RVs? I mean if $$X=[x_1, \dots, x_D]\sim \mathcal{N}(0, \Sigma)$$ where $$\Sigma$$ is a $$D$$-dimensional covariance matrix, is there an analytical expression for the distribution of $$V(X) = \frac{1}{D}\sum_d(x_d-\bar{X})^2$$ where $$\bar{X}=\frac{1}{D}\sum_d x_d$$?

Even better would be an expression for the distribution of $$\frac{V(X)}{V'(X)}$$ where $$V'(X)$$ is the variance of a subset of the $$D$$ components of $$X$$. That is $$V'(X) = \frac{1}{D}\sum_{d \in \mathcal{D}}(x_d-\bar{X}')^2$$, where $$\mathcal{D} \subseteq \{1, \dots, D\}$$ and $$\bar{X}'$$ is the mean over these same dimensions. I'm guessing there is no such expression but it never hurts to ask.

There is a related question http://stats.stackexchange.com/questions/17231/second-moment-of-draws-from-a-multivariate-normal-covariance-matrix but there they just ask for the moments not the full distribution.

Is there an analytical expression for the distribution of variances of MVN RVs? I mean if $$X=[x_1, \dots, x_D]\sim \mathcal{N}(0, \Sigma)$$ where $$\Sigma$$ is a $$D$$-dimensional covariance matrix, is there an analytical expression for the distribution of $$V(X) = \frac{1}{D}\sum_d(x_d-\bar{X})^2$$ where $$\bar{X}=\frac{1}{D}\sum_d x_d$$?

Even better would be an expression for the distribution of $$\frac{V(X)}{V'(X)}$$ where $$V'(X)$$ is the variance of a subset of the $$D$$ components of $$X$$. That is $$V'(X) = \frac{1}{D}\sum_{d \in \mathcal{D}}(x_d-\bar{X}')^2$$, where $$\mathcal{D} \subseteq \{1, \dots, D\}$$ and $$\bar{X}'$$ is the mean over these same dimensions. I'm guessing there is no such expression but it never hurts to ask.

There is a related question Second moment of draws from a multivariate normal covariance matrix but there they just ask for the moments not the full distribution.

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# Sampling distribution of variances of multivariate normal RVs

Is there an analytical expression for the distribution of variances of MVN RVs? I mean if $$X=[x_1, \dots, x_D]\sim \mathcal{N}(0, \Sigma)$$ where $$\Sigma$$ is a $$D$$-dimensional covariance matrix, is there an analytical expression for the distribution of $$V(X) = \frac{1}{D}\sum_d(x_d-\bar{X})^2$$ where $$\bar{X}=\frac{1}{D}\sum_d x_d$$?

Even better would be an expression for the distribution of $$\frac{V(X)}{V'(X)}$$ where $$V'(X)$$ is the variance of a subset of the $$D$$ components of $$X$$. That is $$V'(X) = \frac{1}{D}\sum_{d \in \mathcal{D}}(x_d-\bar{X}')^2$$, where $$\mathcal{D} \subseteq \{1, \dots, D\}$$ and $$\bar{X}'$$ is the mean over these same dimensions. I'm guessing there is no such expression but it never hurts to ask.

There is a related question http://stats.stackexchange.com/questions/17231/second-moment-of-draws-from-a-multivariate-normal-covariance-matrix but there they just ask for the moments not the full distribution.