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1
vote
1answer
105 views

Deriving the sampling distribution of MLE for Normal distribution

Let $X_1,\ldots,X_n$ be an observed random sample from $N_p(\mu, \Sigma)$. I know that the MLE of $\Sigma$ is $\frac{1}{n} \sum_i^n(X_i -\bar X)(X_i -\bar X)^T$, which is biased. We define $S = \...
0
votes
0answers
125 views

Property of Wishart distribution

I am solving the next exercise about a property of a Wishart Distribution: $$M_1\sim W_p(\Sigma,n_1)$$ $$M_2\sim W_p(\Sigma,n_2)$$ are independent, then $M_1+M_2\sim W_p(\Sigma,n_1+n_2)$ I have ...
2
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0answers
88 views

expectation associated with Wishart distribution

Suppose that $W$ follows the Wishart distribution with parameter $\Sigma$ and d.f. $n$. Then, I would like to know the result of the following expectations. $E[tr(WAWB)tr(WC)]$ $E[tr(WAWB)tr(WCWD)]$...
3
votes
1answer
161 views

Can Wishart be close to Normal distribution?

I'm trying to figure out if Wishart can be close to Normal for a number of degrees of freedom enough large. About chi-squared distribution, Wikipedia states: ...
1
vote
0answers
35 views

Eigenvalue order of magnitude for Wishart random matrix

If we have a $P\times N$ matrix $\mathbf{A}$ whose elements $A_i$ are samples (in this case, P samples) from a multivariate gaussian distribution in an $N$ dimensional space, we can define the Wishart ...
2
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0answers
59 views

Show that distribution of $\small(n-1)\overline{X}'(S^{-1}-\frac{S^{-1}\mu_0\mu_0'S^{-1}}{\mu_0'S^{-1}\mu_0})\overline{X}$ is $\small T^2(p-1,n-1)$

Show that the distribution of $(n-1)\overline{X}'\left(S^{-1}-\dfrac{S^{-1}\mu_0\mu_0'S^{-1}}{\mu_0'S^{-1}\mu_0}\right)\overline{X}$ is $T^2(p-1,n-1)$ where $X_i\sim N_p(\mu,\Sigma)$, where $\Sigma$ ...
1
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0answers
78 views

Deriving expectation involving Wishart distributions $E[\bf{A(A'WA)^-A'W}]=\bf{A(A'\Sigma A)^-A'\Sigma}$

I have a problem deriving two expectation involving Wishart distributions with mean zero. Let $\bf{W} \sim {W_p}({\bf{\Sigma }},$$n\bf{)}$ and $\bf{A}$$: p\times q$. Prove that $E[\bf{A(A'WA)^-A'W}]...
5
votes
0answers
184 views

Joint distribution of two distances

Suppose there are three points in 3D space, each with coordinates $A_i=(X_i,Y_i,Z_i)\leadsto \mathcal{N}(\mu_i,\tau^2\mathbb{I}_3)$. We compute the distance between the three points, e.g. $D_{ij} = \|...
3
votes
0answers
372 views

Distribution of a normalized inverse Wishart times Gaussian

Suppose $z\sim\mathcal{N}\left(\lambda^2 e_1,I_n\right)$ where $e_1$ is the first column of the $n$-dimensional identity matrix, denoted here as $I_n$. Suppose $S\sim\mathcal{W}\left(m,I_n\right)$ is ...
16
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0answers
525 views

Distribution of inverse Wishart to a power?

In a related question, I had asked about the norm induced by an inverse Wishart matrix. I am interested in generalizing that result somewhat. Let $A\sim\mathcal{W}_p\left(I,n\right)$, a Wishart matrix ...