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Questions tagged [multivariate-normal-distribution]

The multivariate normal distribution, is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. (Also called, multivariate Gaussian)

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Why does the sufficient statistic for the bivariate normal not imply a sufficient statistic for the correlation under bivariate normality?

This question links to a document by Jon Wellner that defines the sufficient statistic for the multivariate normal (p. 7, Example 2.7). The result follows from the factorization theorem and is proven ...
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Normal Distribution on a Line Segment [closed]

Let $h_{old}^{1}$ and $h_{old}^{2}$ be our initial line segment. and we have points $r_{old}$ that is uniformly distributed on this line segment. We change one of the vertices of the old line ...
Rust32627's user avatar
5 votes
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How to write a function for the normal copula in R?

How can I write the following function for the normal copula in R? $$ C_\theta(u, v)=\Phi_\theta\left(\Phi^{-1}(u), \Phi^{-1}(v)\right), $$ where $\Phi$ is the $N(0,1)$ cdf, $\Phi^{-1}$ is the ...
Aria's user avatar
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Derivative of the multivariate normal cumulative distribution function (CDF) with reparameterisation [duplicate]

I would like to learn how to calculate the derivatives of a multivariate normal cumulative distribution function (MVN CDF) w.r.t. certain elements by using the derivatives of the same MVN CDF w.r.t. ...
Kirin G.'s user avatar
3 votes
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Probability that normally distributed variables will have a specific ranking

There are $k$ players playing a game, each gives a performance $X_k \sim N(\mu_k, 1)$ and we observe their ranking from best to worst (a permutation of the player indexes). How to calculate the ...
fhucho's user avatar
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Does Partial Correlation Affect Likelihood of Multivariate Normal? [duplicate]

Suppose I have a 3-dimensional multivariate normal distribution characterized by the following variance-covariance matrix $$ \begin{bmatrix} V_{X} & C_{XY} & C_{XZ} \\ C_{XY} & V_{Y} & ...
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Constructing Joint Multivariate Normal Distribution from Marginal Distribution (detrended data)

I have two n-length data vectors $X_1 \sim N(\mu_{1},\Sigma_{1})$ and $X_{2} \sim N(\mu_{2}, \Sigma_{2})$ which may or may not have a covariance. To see whether they do or not, I detrend them by ...
A Friendly Fish's user avatar
2 votes
2 answers
87 views

How to test for equal spread in bivariate normal samples with equal means?

I'm working with samples taken from a bivariate normal distribution, where the differences in means is not relevant since all samples are scaled to mean (0,0) anyway, and I'm trying to remember how to ...
epistemetrica's user avatar
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18 views

Distribution of between-group sum of squares (SSB)?

Consider a standard multivariate random effects model $$\mathbf{X}_{ij} = \boldsymbol{\mu} + \boldsymbol{\alpha}_i +\boldsymbol{\varepsilon}_{ij}$$ for $i = 1,\ldots,m$ and $j = 1,\ldots,n_i$, and ...
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Initialising Multivariate linear regression with the maximum likelihood method

I am attempting to use maximum likelihood estimation to fit a multivariate linear regression problem. I have 5 predictor variables and a similar number of response variables. I am using a correlation-...
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Characterize conditions in which Taylor moment approximation is good

I am working with the Projected Gaussian, or Angular Gaussian distribution, which is given by $z = \frac{x}{||x||}$, where $x \sim \mathcal{N}(\mu, \Sigma)$. This is a distribution on the sphere in $\...
dherrera's user avatar
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Test for independence of multivariate normality in R

I have 3 normal data matrices, i.e., that in each matrix, the rows are iid Multivariate Normal, but the rows of different matrices need not be identically distributed, but they have the same dimension....
Shaikh Ammar's user avatar
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Posterior distribution for multivariate Gamma-Normal model

Let $\theta \in \mathbb{R}_{>0}^n$ be a random variable with prior distribution $p(\theta)$: \begin{equation} p(\theta) = \prod_{i=1}^n \text{Ga}(\alpha_i, \beta_i)(\theta_i), \end{equation} where $...
Mathieu le provost's user avatar
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Show convergence in distribution to a bivariate normal vector

Let $Y_1,...,Y_n$ be iid exponential random variables with mean $\theta>0$. Let $$ \hat\alpha_n:= \frac{1}{n} \sum_{i=1}^n Y_i \quad \text{and} \quad \hat\beta_n:=\sqrt{\frac{1}{n} \sum_{i=1}^n (...
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Projection of i.i.d. Gaussians onto correlated Gaussian is bounded from below by chi-squared

Let $\varepsilon \sim \mathcal{N}(0, \mathrm{Id}_k)$ and $\varepsilon_0 \in \mathbb{R}$ with $\varepsilon_0 \neq 0$. For some matrix or vector $A \in \mathbb{R}^{k \times p}$, let $P_A := A (A^T A)^{-...
M. Londschien's user avatar
1 vote
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32 views

Closed Form Solution for MLE parameter defining Linear Combination of two multivariate normal distributions

I have one set of $n$ observations which can be described as a single vector sampled from a multivariate normal distribution of the following form: $$ (1-\lambda)\mathbb{I}_n + \lambda \Sigma_{n} $$ ...
A Friendly Fish's user avatar
1 vote
0 answers
14 views

Correlation between variable and variable conditioned on the sign agreement [closed]

Suppose I have two variables X1 and X2, following a bivariate standard normal distribution with a correlation coefficient of 0.2, if I create a new variable X3 that is equal to X2 if X1 and X2 have ...
user99's user avatar
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1 answer
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How to perform a z-test for a 2d-gaussian?

I have N 2-dimensional points (X,Y) where X and Y are gaussian distributed and have a strong correlation, i.e., X and Y are not independent. What is the simplest way to perform a z-test? In other ...
Roshan Satapathy's user avatar
1 vote
1 answer
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SEM: Robust estimation in Onyx

Does anyone know if there is a possibility for robust estimation (e. g. Yuan-Bentler or Satorra-Bentler) in Onyx, because the normal distribution assumption is not given? I only see the options ...
anonymoususer's user avatar
2 votes
1 answer
95 views

SEM: Multivariate normality of the residuals?

I am currently reviewing the assumptions related to structural equation modeling and I read the following in an article on the subject: "Second, the assumption of normality applies to the ...
anonymoususer's user avatar
2 votes
2 answers
79 views

Large sample size and multivariate normality assumption

I am working on a student research project using structural equation models for longitudinal analyses and am currently checking the assumptions. My sample comprises around 300-400 people. If I have ...
anonymoususer's user avatar
1 vote
1 answer
94 views

Multivariate normality assumption of SEMs

I am calculating structural equation models as part of my thesis and am currently in the process of checking the assumptions. One assumption is multivariate normal distribution. This applies to all ...
anonymoususer's user avatar
1 vote
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Is the assumption of a diagonal covariance matrix on the latent space in a variational autoencoder in any way restrictive?

The covariance matrix in an autoencoder is assumed to be diagonal. And, I see it mentioned in good places that this is a fairly restrictive assumption. To quote However, in order to simplify the ...
figs_and_nuts's user avatar
1 vote
0 answers
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Truncated Multivariate Normal expected value approximation

I have $\vec{x} \sim N(\vec{\mu}, \Sigma)$. I would like to calculate $$E[x_i | \vec{x} \geq 0]$$ There are libraries like tmvtnorm (in R) that calculates this for me. However, it seems to be very ...
JEK's user avatar
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1 vote
3 answers
101 views

E(X1 | X2 > X3) for (X1,X2,X3) multivariate normal

I'd like a closed form solution for $E(X1 | X2 > X3)$ where $(X1, X2, X3)$ is multivariate normal with possibly arbitrary mean vector and covariance matrix. The conditional distribution $f(X1 | X2, ...
frelk's user avatar
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48 views

Normal or Linear relationship?

I'm generating simulated data from a multivariate normal distribution with a variance-covariance matrix and then fitting it by either A) finding the maximum likelihood parameter estimates for the ...
A Friendly Fish's user avatar
2 votes
0 answers
49 views

Notation for distributions on subspaces

Question What notation should I use for a normal distribution on a subspace $U \subset \mathbb{R}^n$? Motivation I have a geometric intuition for Bessel's correction which goes something like this: A ...
Steven Gubkin's user avatar
3 votes
1 answer
145 views

How to "see" the covariance matrix and mean vector?

I am working with following model specifications (Regression, Modelle, Methoden und Anwendungen, Springer-Verlag Berlin Heidelberg (2009), p. 147): $$Y \sim MVN(X\beta, \sigma^2I)$$ $$\beta|\sigma^2 \...
BlankerHans's user avatar
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How to derive conditional destribution of MVN variable

I am working with following model specifications (Regression_ Modelle, Methoden und Anwendungen-Springer-Verlag Berlin Heidelberg (2009), p. 147): $$Y \sim MVN(X\beta, \sigma^2I)$$ $$\beta|\sigma^2 \...
BlankerHans's user avatar
0 votes
1 answer
27 views

Understanding the multivariate normal density proportional

I don't understand the second line of the following equation I get: $$f(x) \propto exp(-\frac{1}{2}(x-\mu)^T \Sigma^{-1}(x-\mu))$$ $$=exp(-\frac{1}{2}x^T\Sigma^{-1}x+\frac{1}{2}x^T\Sigma^{-1}\mu+\...
BlankerHans's user avatar
3 votes
1 answer
93 views

Distribution of $\max_i \bar{X}-X_i$

Let $X_1, \ldots, X_n$ be i.i.d. random variables from the standard normal distribution and let $\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i$ be their sample mean. I'm interested in the distribution of the $...
Theo Mary's user avatar
2 votes
0 answers
19 views

Probability of N-dim Gaussian marginalized over a plane, and a likelihood of a point on a plane

Context: I'd like to fit a series of coordinates, each with their uncertainties, to a (hyper)plane. I am trying to derive the likelihood for a point on a plane observed at a point. I need a ...
Hojin Cho's user avatar
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6 votes
1 answer
154 views

Why do the quantiles of a multivariate normal not work properly?

Let's say we have a multivariate normal distribution with two components. The two means $\mu_1$ and $\mu_2$ are both equal to 0 and the covariance matrix is a simple 2x2 square matrix with ...
Denzo's user avatar
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1 answer
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Hotelling's $T^2$ test on one-sample population

I'm trying to get a grasp with Hotelling's $T^2$ test in the case of a one-sample population. This exercise is a part of a training set; it's not an homework, and I'm not looking for a complete ...
Thiagals's user avatar
1 vote
2 answers
111 views

How to calculate Mardia's multivariate normality in Mplus?

I would like to assess the multivariate normality of the structural equation model (SEM) in Mplus. I have a few questions regarding this matter: How can I examine multivariate normality, specifically ...
İpek Gülsün's user avatar
4 votes
0 answers
51 views

Resulting univariate marginal distributions are not $t$ distributed - why? (S Wood Core Statistics)

In the Core Statistics by Simon Wood it says: "If we replace the random variables $Z_i\sim_\text{i.i.d.} N(0,1)$ with random variables $T_i \sim_\text{i.i.d.} t_k$ in the definition of a ...
Alex S.'s user avatar
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0 answers
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Distribution of covariance parameter resulting from sum of known- and unknown-covariance noise processes

Let's say I have a set $X$ of $N$ random $p$-dimensional vectors generated by $\mathbf{x}_i = \boldsymbol{\mu} + \Psi_i^{1/2} \boldsymbol{\xi}_i + \Sigma^{1/2} \boldsymbol{\zeta}_i$, where $\...
hank's user avatar
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1 vote
0 answers
35 views

Derivatives of matrices

Consider multiple response data from a matrix normal distribution $Y_{n\times m}\sim\text{Normal}(\mathbf{M},\mathbf{U},\mathbf{V})$, where $\mathbf{U}=\mathbf{I}_n$ is the variance among rows and $\...
Chewysplace's user avatar
1 vote
0 answers
24 views

Multiple IVs and DVs with non-normally distributed data

I am performing a research study aiming at answering the research question: How does job design satisfaction differ between employees working in hybrid and remote work arrangements? For that I ...
JoaoFilipeClementeMartins's user avatar
6 votes
1 answer
31 views

How can I quantify uncertainty for a least squares estimator in a multivariate linear regression with covariance structure?

Suppose that we have $$\mathbf{y}\sim\text{N}(\mathbf{X}\boldsymbol{\beta},\sigma^2\mathbf{M}\mathbf{M}'),$$ and let $\boldsymbol{\hat{\beta}}$ be the least squares estimator for $\boldsymbol{\beta}$. ...
Ron Snow's user avatar
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1 vote
1 answer
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Using linear combination of multivariate normal $X$

I have a question and its answer but I don't understand it. would you please explain it to me? QUESTION: Let $X$ follow a $N(0, 1)$ and $Z$ a random variable independent of $X$ following a $U(-1, 1)$. ...
mormey's user avatar
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0 answers
73 views

MLE of multivariate normal distribution when the VCV matrix is full of equations [duplicate]

Short Version: Given a variance covariance matrix for my multivariate normal distribution where the entries are equations of other parameters, how do I find which of those parameter values maximizes ...
A Friendly Fish's user avatar
1 vote
0 answers
22 views

How to Handle Non-Multinormality in the Context of Exploratory Factor Analysis for Logistic Regression

I'm trying to follow the book A Step-by-Step Guide to Exploratory Factor Analysis with R and Rstudio, by Marley W. Watkins, and apply the principles in the book to a real-world data set. Ultimately, ...
Adrian Keister's user avatar
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0 answers
42 views

meaning of empirical argument in mvrnorm

I'm using the mvrnorm function from MASS library, and I have a question regarding the ...
locus's user avatar
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Is this the formula for the conditional covariance of normally distributed random variables? [duplicate]

Assuming $X$, $Y$, and $Z$ are normally distributed random variables, is it true that: $Cov[X, Y | Z] = Cov[X, Y] - Cov[X, Z]Cov[Y, Z] / Var[Z]$ Could you provide a simple derivation?
anonymous 's user avatar
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0 answers
24 views

How to compute a posterior for the parameter of a model?

I have quite the specific question over a model for which I am asked to compute a posterior. Here are all the details : Bayesian model . For clarity purposes, let $X=[x_1^{\top}, \ldots, x_T^{\top}]^{\...
user20920567's user avatar
11 votes
2 answers
375 views

Formulas, approximations, or bounds for $\mathbb{E}\left( \frac{X}{\lVert X \rVert} \right)$, $X\sim N(\mu, \Sigma)$?

In another question, I asked for $\mathbb{E}\left( \frac{X}{\lVert X \rVert} \right)$, in the case where $X \in \mathbb{R}^d \sim N(\mu, I_{d})$. Somebody posted an exact formula based on the symmetry ...
dherrera's user avatar
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4 votes
1 answer
206 views

Bivariate normal covering circles and ellipses

I am looking at covering circles for cartesian coordinates given by independent bivariate random variables $X, Y \sim N(0, \sigma)$. The radius of a circle that will cover proportion p of these ...
feetwet's user avatar
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0 votes
0 answers
13 views

How do I adequately calculate the distribution of function of the multivariate normal distribution using the R mvtnorm package?

I am currently working on an analysis project involving simultaneous testing of several hypotheses. I intend to use the multcomp R package, which calculates the family-wise error rate for multiple ...
Sebastian Gerdes's user avatar
0 votes
0 answers
47 views

How to find $\mathbb{P}(\textbf{Y}^T\mathbf{A}\textbf{Y}>0)$ where $\mathbf{Y}$ is a vector of independent normal distributions? [duplicate]

Say $\mathbf{Y} \sim \mathcal{N}(\mathbf{\mu}, \mathbf{D})$ where $\mathbf{D}$ is diagonal and $A$ is any real matrix, how could you calculate $\mathbb{P}(\mathbf{Y}^T\mathbf{A}\mathbf{Y}>0)$? In ...
mrepic1123's user avatar

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