Questions tagged [multivariate-statistical-analysis]
87 questions
3
votes
1 answer
What is the probability that a bivariate Gaussian generates a point beyond a certain line?
Context: Consider the problem of identifying the error probability associated with a given classifier. Assume that the points produced by the two classes, namely A and B, are characterized by bivariate Gaussians. In this context, the classifier may…
Daniel S.
- 863
2
votes
1 answer
Ratio of maxima of equicorrelated normal variables increasing property
Let $X_i \sim N(0,1)$ for $i = 1, \dots, n$ with $Cov(X_i, X_j) = \rho \in (0,1)$ for $i \ne j$.
For $k = 0, \dots, n-1$, let $r_k : \mathbb{R} \rightarrow \mathbb{R}$, be such that
$$ r_k(t) = \frac{\Pr\{X_1 \leq t, \dots, X_{k+1} \leq…
2
votes
0 answers
What is the most efficient algorithm for computing $E[x_1 x_2 \cdots x_n]$ in a multivariate normal distribution?
I am working with a multivariate normal distribution $\mathbf{x} = [x_1, x_2, \ldots, x_n] \sim \mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$, and I need to compute the expectation $E[x_1 x_2 \cdots x_n]$ efficiently for arbitrary $n$.
What I…
cloudmath
- 21
2
votes
0 answers
Expectation of the norm of a random vector multiplied by a matrix
Lets say we are given random (row) vector $x \in \mathbb{R}^n$, and a non-random matrix $M \in \mathbb{R}_{nxn}$
I came across a claim (which does not impose any assumptions on the distribution of $x$ a priori), that the following expectation…
giorgio
- 693
2
votes
0 answers
Find $S$ such that $P(X+Y\in S)\geq 0.9$ only using marginal distributions of $X$ and $Y$
Let $X, Y$ be continuous random variables with their distributions $F_X, F_Y$, finite second moments and correlation $\rho$.
I would like to find a smallest possible set $S$ (set such that its Lebesgue measure is as small as possible) satisfying…
Albert Paradek
- 897
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- 19
2
votes
1 answer
Show that every set of positive measure in $R^n$ must contain a basis for $R^n$
The intuition comes from the fact that every open set in $R^n$ must contain a basis for $R^n$. Now, I want to extend this to a set of positive measure. But I don't know how to start it.
Lin Xuelei
- 561
2
votes
1 answer
one issue of matrix determinant in Reduced Rank Regression
The equation is Eq 11.20 in ECONOMETRICS by BRUCE E. H ANSEN (2021).
I have no idea on the following equation. Which property of determinant is applied in this…
Hagan Ross
- 45
2
votes
0 answers
Lower Tail Bounds for Euclidean Norm of (Sub)gaussian Vectors
While working on a research problem I got stuck at the following question. Suppose I have a mean zero multivariate Gaussian random vector $\mathbf{X}\sim N\left(\mathbf{0}_{n\times 1}, \Sigma_{n\times n}\right)$, with subgaussian parameter…
youngtableaux
- 431
2
votes
0 answers
How to solve the matrix equation $\sum_iA_i^TA_iXB_iB_i^T=\sum_iA_i^TC_iB_i^T$ efficiently?
How would I go about solving the matrix equation $\sum_iA_i^TA_iXB_iB_i^T=\sum_iA_i^TC_iB_i^T$ for $X$?
The simplest thing to do would be to, of course, consider $Y=\sum_iA_i^TC_iB_i^T$, vectorise and use Krönecker…
2
votes
1 answer
Why is Hotelling's $T^2 \sim \chi^2_p$ for large $n$?
I'm interested in some proof (simple if possible) as to why Hotelling's $T^2$ is chi-squared distributed for large n. I understand and can show that the Mahalanobis Distance is in fact chi-squared distributed (as bellow), but I have a little bit of…
2
votes
0 answers
Minima of a cdf of multivariate normal distribution with respect to a parameter
Let $\mathrm{X}\sim\mathcal{N}_{3}(\boldsymbol{\mu},\mathrm{\Sigma})$ where
\begin{equation}
\boldsymbol{\mu} = n[(\mu_1-\mu_2)\sqrt{\xi_1\xi_2/(\xi_1+\xi_2)},…
SP SINGH
- 83
2
votes
0 answers
CDF of the distance from origin to the hyperplane passing through $d$ i.i.d. points in $\mathbb{R}^d$
I am stuck with a problem in multivariable statistics. The problem can be stated as follows:
For a spherically symmetric distribution in $\mathbb{R}^d$, it can be specified completely by the function $F(r)=\Pr(\|X\|>r)$. For example, 2d standard…
zhaofeng-shu33
- 448
- 3
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2
votes
2 answers
How to calculate median, variance and correlation on two-dimensional random variable.
Two random variables, X and Y, have the joint density function:
$$f(x, y) = \begin{cases} 2 & 0 < x \le y < 1 \\ 0 & ioc\end{cases}$$
Calculate the correlation coefficient between X and Y.
I am pretty much stuck because y is an upper limit for x,…
AWDn0n
- 23
- 4
2
votes
2 answers
Finding the covariance matrix of $Y=AX$ where $X$ is a known multivariate gaussian random variable
For $X=(X_1,X_2,X_3)$ with Gaussian distribution, covariance matrix $2I_3$, and mean vector $\mu$ = $(3,3,3)^T$, I want to find the covariance matrix of $Y=(Y_1,Y_2,Y_3)^T$ where
$$Y=\begin{pmatrix}1/\sqrt 2 & 0 & -1/\sqrt 2 \\1/\sqrt 3 & 1/\sqrt 3…
FD_bfa
- 4,757
2
votes
0 answers
Given $c$ and $a_i$, how many solutions does $c=\sum\nolimits_{i}{{{n}_{i}}{{a}_{i}}}$ have?
Consider the following equation:
$$
c=\sum\limits_{i=1}^{N}{{{n}_{i}}}{{a}_{i}}
$$
where $c$ and $a_i$ are positive real numbers, and $n_i$ is an integer equal or larger than zero.
Given $c$ and $a_i$, how many solutions (i.e., ensembles $\{n_i\}$)…
Barb20
- 31