Mathematics

Questions tagged [multivariate-statistical-analysis]

81 questions
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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.
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Quantile Regression variance attribution

For multi OLS regression. I normally use variance attribution to explain how much contribution an independent/predictor variable has to the dependent variable. For example, a standard two factor regression (independent variables $X_1$ and $X_2$ )…
theQDude
  • 21
2
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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…
Chava Candelas
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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 Know:…
cloudmath
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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
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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
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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
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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 derivation? \begin{equation}…
Hagan Ross
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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
  • 441
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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 products. \begin{align*}…
Rahul Bordoloi
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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…
Willian Leite
  • 21
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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
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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
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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
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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…
FD_bfa
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