Questions tagged [multivariate-statistical-analysis]

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CDF and PDF of dependent variables originating from a real life shock models

Suppose there are two sources of shocks and a system with two components. Shock A can affect the first component and Shock B can affect both the components. I am trying to find the joint distribution ...
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Conditions on formula of expectation of sum of infinitely random variables

My book states that it's not always true that: $$E(\sum_{i=1}^{\infty} X_i) = \sum_{i=1}^{\infty} E(X_i)$$ and what makes it not true in general is this equality: $$E(\sum_{i=1}^{\infty} X_i) = E(\...
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Latent Change Score Modelling

I am currently undertaking a research project that aims to assess the effectiveness of an intervention program. However, I am encountering difficulties in locating suitable resources for my study. ...
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Orthant probabilities for equicorrelated multivariate normal

Calculate $\Phi_n((-\infty)_n, 0_n; 0_n, I_n + \rho e_n e_n')$, where $\Phi_n((a)_n, (b)_n; \mu, \Sigma) = P(X_i \in (a_i, b_i) \forall i=1,2\dots n)$ for $X$ n-variate normal with mean $\mu$ and ...
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Expected value of $X$ given $a^TX$ if X has a multivariate normal distribution.

Let $X$ be a normally distributed random vector with mean vector $\mu$ and covariance matrix $\Sigma$. Also, let Y be a random vector such that $Y = a^TX$ where $a$ is a constant vector. How would I ...
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Linear system with multivatiate bernoulli elements

I have the following system $$ \mathbf M \left(\alpha_1,\alpha_2,...,\alpha_n \right) \vec v = \vec b $$ $$ v_{out} = \vec c^T \vec v $$ where $\mathbf M \left(\alpha_1,\alpha_2,...,\alpha_n \right)$ ...
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Change of Variables: Multivariate Integrals

I was asked to show that $\mu$ and $\Sigma$ are respectively the Mean and Covariance of the Multivariate Gaussian Distribution. $X$ is a Random Vector, and $x \in \Re ^n$ $$P(X = x) = \frac{1}{(2\pi)^\...
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Vine Copulas: Conditional Copula cdf in higher dimensions

Hello :) I am currently working on a project where I am stuck calculating the following term: $u_{i(e)|D(e)} = C_{i(e)|D(e)}(u_{i(e)}|D(e))$ $u_{j(e)|D(e)} = C_{i(e)|D(e)}(j_{i(e)}|D(e))$ The ...
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Statistical test for null hypothesis $\|p-q\|\le \epsilon$ for $k$-dimensional categorical data

Is there any known (asymptotic) statistical test for the null hypothesis $$\|p-q\|\le \epsilon$$ for $k$-dimensional categorical data independently taken from two societies for some given norm $\|\...
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What is the probability that a bivariate Gaussian generates a point beyond a certain line? [closed]

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 ...
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Tsiatis Semiparametric Theory Chapter 2 Exercise 3(b) Use Change of Variable to Get a NEW Probability Density

I have been reading the book Semiparametric Theory and Missing Data by Tsiatis. The question is about exercise 3(b) in chapter 2 Hilbert Space for Random Vectors. Here is the problem Let $Z=\left(Z_1, ...
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Obtaining multivariate linear regression coefficients from residual values of simple linear regressions

We are interested in a multivariate linear regression of $Y$ against $X_1$ and $X_2$. More specifically we want to know the regression coefficients of $X_1$. There is a sample drawn of size $n$, but ...
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Kullback–Leibler divergence of two multivariate normal distribution

I have a problem: I want to calculate Kullback-Leibler (KL) divergence of two dataset where $X_{1}$ has $M$ features with its multivariate normal distribution $\mathcal{N}(\mu_1, \sigma_1)$ and $X_{2}$...
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Work someone check my work using Bayes Classifier?

Background: Suppose we have two classes for p=2 variate observations, where the probability for class 1 follows MVN($\mu_1$, $\Sigma$) and the population for class 2 follows MVN($\mu_2$, $\Sigma$) ...
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Transform Wishart distribution to Chi-square distribution

This's actually what I'm trying to prove: $$ \frac {a^{'}\Sigma^{-1}a}{a^{'}W^{-1}a} \sim \chi^{2}_{n-p+1} $$ $a$ is any P-dimensional nonzero constant vector, and $W \sim W_{p}(n,\Sigma)$, $\Sigma$ ...
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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 ...
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Does Independence of Conditional Expectation still hold without multivariate normal distribution

Suppose that X,$Z_1 ... Z_n$ have a multivariate normal distribution, and X has zero mean. Furthermore, suppose that $Z_1 ... Z_n$ are independent. Show that $$\mathbb{E}[X|Z_1 ... Z_n] = \sum_{i=1}^...
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Chi-square and Wishart distribution with non-symmetric quadratic forms

I am studying the textbook An Introduction to Mutivariate Statistical analysis by T.W Anderson and was stuck by the following exercise $\textbf{7.13}$ Let $Z_1,\ \dots,\ Z_n$ be independently ...
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Justify that $\operatorname{Cov}(y, y^\top\!Ay) = 2\Sigma A\mu$

If $A$ is a $p \times p$ symmetric matrix of constants, and $y \sim N_p(\mu, \Sigma)$, justify the following result: $$ \newcommand{\Cov}{\operatorname{Cov}} \Cov(y, y^\top\!Ay) = 2\Sigma A\mu $$ I ...
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Would someone check my work finding the joint distribution for two RVs X and Y?

Question U, V, follow independent univariate normal with mean zero and variance one. Let X = U and Y = pU+$\sqrt{1-p^2}$V where $|p|$ < 1. $\Sigma_{XY} = \pmatrix{1 & p \\ p & 1}$ Find the ...
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Matrix integral involving a Kronecker product

I need to evaluate the following matrix integral: $\int_{P>0}\mathrm{det}(P)^{-1/2}\mathrm{det}(I+M\otimes P)^{-1/2}~_0F_1\left(\frac{d}{2}, -\frac{1}{4}SS^TP\right)~dP$ where $M$ and $P$ are $d\...
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Calculation on Quadrant Probability for Bivariate data using Copula

I have a question regarding the computation on bivariate probability when using copula function. Let $u=F_X(x)$ and $v=F_T(t)$ be the CDF for marginals of $X$ and $T$ respectively, a joint ...
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Eigenvalues of multivariate t-wishart and wishart matrix

If $W$ is a wishart matrix with Identity Covariance and $n$ degrees of freedom. And another matrix $X= v*W*S$ where $S$ is a diagonal matrix with diagonal elements as iid inverse-$\chi^2$ distributed ...
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Simplify $\text{Tr}[(A-I)\Sigma_X(A-I)^\top+\Sigma_{\xi}]$

I encounter a problem as below: $$\mathcal{L}(A,\Sigma_{\xi})=\text{Tr}[(A-I)\Sigma_X(A-I)^\top+\Sigma_{\xi}].$$ One approach I'm trying is to replace $(A, \Sigma_{\xi})$ with $(\tilde A, I)$ with $\...
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Using goodness of fit statistics to cluster data

We are working on a wrapper that clusters observations, wrapping around a multivariate linear regression model within each cluster of observations. The idea is to use some goodness of fit statistic ...
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Can I use assume a Normal Distribution to calculate a Confidence Interval for multiple random variables with different means and std each?

Let's say, I've a table with a list of trips I need to make this week and I want to calculate a confidence interval for the average trip duration time for all the trips I'll make in the week. Var ...
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Suppose that the variance-covariance matrix of a $p$-dimensional random vector $X$ is

Suppose that the variance-covariance matrix of a $p$ -dimensional random vector $X$ is $\Sigma=\sigma_{ij}$ for all $i,j=1, 2, ...,p$. Show that the coefficients of the first principal component have ...
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Gaussian Markov Random Fields - Conditional distribution from jointly normal random vector with known joint precision matrix

Suppose I have jointly normal random vectors $[\bf{v_1}, \ldots, \bf{v_K}]$' with mean $ \bf{M}$ and joint block tridiagonal precision matrix $ \bf{P}$: $$ \bf{M}= \begin{bmatrix}\bf{\mu_1} \\\ldots \\...
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Find conditional distribution from jointly normal with vec operator

I have two random matrices one on the top of the other: $ \begin{bmatrix}\boldsymbol{B_1} \\ \boldsymbol{B_2} \end{bmatrix}$. and they are both of dimension $k \times N$. I have that: $ vec\begin{...
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How to find the distribution of a multivariate normal with mean 0?

I am relatively new to higher-level statistics and I just cannot seem to wrap my head around this. I am trying to derive ELBO for a Variational Autoencoder. Here is the formula for the PDF of a ...
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Algorithm implementing multivariate normal distribution from Sobol QRNG

My goal is to implement a mathematically correct multivariate normal distribution using Sobol QRNG sequence as a source of randomness. The implementation should NOT produce the whole set of a given ...
Dmitry Mikushin's user avatar
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Is it impossible that $(X,Y)$ follows bivariate normal if they always have the same sign? [closed]

Suppose $X$ and $Y$ follows normal distribution respectively. I heard that if $X$ and $Y$ always have the same sign, then $(X,Y)$ cannot follow the bivariate normal distribution. However, I can't see ...
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Sampling from a preference table

Suppose we have a table of a group of friend's preferences for food. Each person will receive a unique meal. What is the approach to sample from this preference table so that each time this group ...
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Expectation of a random quadratic form [closed]

Let ${\bf X}_{1},\cdots,{\bf X}_{p}$ be $p$ independent Gaussian random variables, in particular, ${\bf X}_{j}\sim N_{n}({\bf 0},\lambda_{j} {\bf I})$, where $\lambda_{j}$s are not all equal, $j=1,\...
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Modeling angles and magnitudes using a bi-variate gaussian.

I have a bunch of points in n-d space who's coordinates follow a Normal distribution $(X=x_1,x_2,...,x_n\sim N(0,1) )$. The coordinates form an angle $\theta$ (with respect to some arbitrary vector $V$...
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Moments of uniform distribution on Stiefel manifold

Suppose I have a $p \times n$ (with $p \geq n$) matrix $\bf U$ such that ${\bf U}' {\bf U} = {\bf I}_{n}$ and that $\bf U$ is uniformly distributed on the Stiefel manifold $V_{n,p}$. I would like to ...
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Linear transformation of random vector has bounded moments?

Suppose that the random $p$-vector $\mathbf{y}=(Y_1, Y_2,\ldots, Y_p)'$ with $p\to\infty$ satisfies: $\mathrm{E}Y_i = 0$, $\mathrm{E}Y_i^2=1$ for any $1\leqslant i \leqslant p$; $\mathrm{Cov}(Y_i,Y_j)...
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Probability density function of truncated multivariate Gaussian

I'm trying to understand what is the correct way to define the PDF of a multivariate truncated Gaussian that is supported on a closed ball, as I find the literature in this subject very limited. ...
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Can the joint distribution of N normally distributed random variables not be the multivariant normal distribution?

Given a random vector of size $n$, where the marginal distribution of each component $x_i$ is a normal distribution: $x_i \sim N(\mu_i,\sigma_i)$, is it possible that the joint distribution is not the ...
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Why is $\operatorname{Cov}\left[v^TX,v_1^TX\right] = v^T\operatorname{Var}\left(X\right)v_1$? [closed]

I came across the following result in one of the proofs exposed in my multivariate analysis course and I haven't been able to prove it : Let $X=(X_1,\dots,X_p)$ be a random vector in $\mathbb{R}^p$. ...
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Why do normalized gaussian points, with the same magnitudes, create a uniformly random spherical shell?

I’m creating random x,y,z points on a sphere for a background I’m making. I first tried rectangular coordinates created from spherical coordinates by making angles phi (0,2pi) and theta (0,pi) ...
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How to get to this result $ (\mu_2-\mu_1)^{´}V^{-1}\bigl( \frac{\mu_2+\mu_1 }{2}\bigr)=0$ using matrix operations?

In the book, $\mu_1$ and $\mu_{2}$ are a vector with dimension , $p\times 1$ y $V^{-1}$ inverse covariance matrix with dimension $p\times p$. I have the following identity $$ \frac{1}{2}\mu_{1}^{´}V^{-...
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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 ...
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Inferenc on factor model with GMM

I'm considering inference of the factor loading in the classical factor analysis. There is the model of classical factor analysis: suppose there are $T$ i.i.d samples from the data generating process $...
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Confusion in multivariate Gaussian distribution and circular symmetric complex Gaussian distribution

Let $W=[X_1,Y_1,\cdots,X_N,Y_N]^T$, and $Z=[X_1+iY_1,\cdots,X_N+iY_N]^T$, where $W\sim~N(0,\Sigma)$, and $W\sim~CN(0,M)$. According to Wiki, the pdf of $2N$ multivariate of zero mean Gaussian ...
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$X\sim N(\mu,\sigma^2)$, $X'PX$ follows?

I posted a question that went unanswered, but managed to figure that what I need is something to do with the following. If $X\sim N(\mu,\sigma^2)$ and $P$ is an idempotent matrix. Then what can be ...
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Expectation of $||P_VY||^2$

Given that $Y\sim N(\mu,\sigma^2I)$ where $\mu\in V$ ($V$ is a vector space with $\dim(V)=p$. If $P_V$ is the projection matrix of $Y$ on $V$ then, we define $$\hat{\mu}=P_VY.$$ Then, $$E(||\hat{\mu}||...
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Representation of 3d Gaussian as vector of functions of 3iid gaussians.

If I have a multivariate guassian. $X \sim \mathcal{N}(0^3,\Sigma)$,where $\Sigma= \begin{bmatrix} 1 &r_{12}&r_{13}\\ r_{12}&1&r_{23}\\r_{13}&r_{23}&1\end{bmatrix}$ I want to ...
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Updating the covariance of bivariate normal after a signal

Let $X$ and $Y$ be two bivariate normally distributed random variables with means $\mu_X$ and $\mu_Y$ and variances $\sigma_X^2$ and $\sigma_Y^2$. The covariance is $Cov(X,Y) = \rho\sigma_X\sigma_Y$, ...
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Inclusion-exclusion in a set of multivariables

I faced a challenge leading me to ask this question: If we have a set of n random variables, can we apply inclusion-exclusion for this set? Clearly, if we have a portfolio containing $X_{1}$, $X_{2}$, ...
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