All Questions
8 questions
2
votes
0
answers
97
views
Robust Covariance in Multivariate / Multi-response OLS
Assume we are in the OLS setting with $y = X\beta + \epsilon$. When $y$ is a response vector, and $X$ are covariates, we can get two types of covariance estimates:
The homoskedastic covariance
$cov(\...
- 1,662
1
vote
1
answer
661
views
What is the conditional covariance matrix of $(X_2,X_3)^T$ given $X_1$?
$X=(X_1,X_2,X_3)^T\sim N_3(\mu,\Sigma).$ Suppose $X_1,...,X_{20}$ are i.i.d. observations from $X$. The sample mean vector and the covariance matrix are then defined by
$$ \bar{x} = (1,0,2)^T,\quad S=...
- 314
1
vote
0
answers
996
views
two-way MANCOVA with two covariates in SPSS
I am running a two-way MANCOVA which needs to be adjusted by two covariates. Problem is, I am not entirely sure whether I clarified all assumptions correctly and how to finally deal with two ...
- 11
3
votes
1
answer
74
views
Does $\text{cov}(a_1' X, a_2' X) = 0$ imply $a_1 \cdot a_2 = 0$?
Let $X$ be a $p$-dimensional random vector with $p$ principal components $y_1, y_2, \dots, y_p$. By definition, a restriction put on the second principal component $y_2 = a_2'X$ is
$$
\text{cov}(y_1, ...
- 1,817
2
votes
1
answer
1k
views
How to show sample correlation is sample covariance for standardized values?
Given a matrix $X$ and the resulting sample correlation matrix $R$, consider the standardized observations:
$$\frac{(x_{jk} - \bar x)} {\sqrt{S_{kk}}} \quad k=1,2,...,p \quad j=1,2,...,n$$
Show that ...
- 163
6
votes
1
answer
400
views
Are all symmetric matrices with diagonal elements 1 and other values between -1 and 1 correlation matrices?
A question for the statisticians and other math lovers: Are all symmetric matrices with diagonal elements 1 and other values between $-1$ and 1 correlation matrices?
- 61
6
votes
1
answer
4k
views
sequential/recursive/online calculation of sample covariance matrix
I am solving the next exercise, but I have spent a lot of time and I can´t.
For random vectors, $X_1,X_2,...\in\mathbb{R}^p$ The sample covariance with $\Sigma_1=0$ is given by:
$$\hat\Sigma_n=\frac{1}...
- 1,003
2
votes
1
answer
337
views
Methods to prove that a guess for the covariance matrix is correct
Suppose we are interested in the covariance matrix $\Sigma$ of a few MLE estimators $\hat \theta_1,\hat \theta_2,\cdots,\hat \theta_n$. For each $j$, $\hat \theta_j$ is normally distributed and ...
- 831