# 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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### How to properly "subtract" a known covariance component from a sample covariance? regression

I have a situation where observed random variables $X_i$ are the sum of two independent (but unobserved) variables, $$X_i = S_i + N_i,$$ (e.g. what you observe is a random signal plus random noise). I ...
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### Convergence in distribution of a sequence of contrasts applied to a random vector which is converging to a multivariate normal

Suppose that we know that $$\Sigma_n^{-1/2}(\hat{\theta}_n - \theta^*_n) \overset{\text{d}}{\longrightarrow} N_{k}(0, I_{k})$$ where $\hat{\theta}_n$ is a random vector of length $k$, $\theta_n^*$ is ...
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### Closed Form Solution for Gaussian Matrix which is Convex Combination?

I already asked a pretty similar question here, but the answer was inconclusive and now this related problem has come up again here. My problem is as follows, I have a $2n$-dimensional multivariate ...
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### Hypothesis Test Finite Sample Spatial Gaussian Mixture Model

I have $n$ observations of pairs $(x, y)$ and three different models I would like to compare. Model0 is nested within Model1. Model0 is also nested within Model2. I would like to do hypothesis ...
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### Relating covariances for (θ, Χ) and (cos(θ), Χ)

From basic error propagation rules, we have σ(cos(θ)) = |sin(θ)| σ(θ). Question: does something similar hold for the covariance cov(cos(θ),X) and cov(θ,Χ)?
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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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### 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 ...
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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. ...
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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 ...
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### 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 ...
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### 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 ...
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### 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+\...
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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 $... 2 votes 0 answers 20 views ### Probability of N-dim Gaussian marginalized over a plane, and a likelihood of a point on a plane Context: I'd like to fit a series of coordinates, each with their uncertainties, to a (hyper)plane. I am trying to derive the likelihood for a point on a plane observed at a point. I need a ... • 131 6 votes 1 answer 176 views ### Why do the quantiles of a multivariate normal not work properly? Let's say we have a multivariate normal distribution with two components. The two means$\mu_1$and$\mu_2$are both equal to 0 and the covariance matrix is a simple 2x2 square matrix with ... • 632 0 votes 1 answer 53 views ### Hotelling's$T^2$test on one-sample population I'm trying to get a grasp with Hotelling's$T^2$test in the case of a one-sample population. This exercise is a part of a training set; it's not an homework, and I'm not looking for a complete ... • 11 1 vote 2 answers 168 views ### How to calculate Mardia's multivariate normality in Mplus? I would like to assess the multivariate normality of the structural equation model (SEM) in Mplus. I have a few questions regarding this matter: How can I examine multivariate normality, specifically ... • 141 4 votes 0 answers 55 views ### Resulting univariate marginal distributions are not$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$in the definition of a ... • 153 0 votes 0 answers 21 views ### Distribution of covariance parameter resulting from sum of known- and unknown-covariance noise processes Let's say I have a set$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$, where$\...
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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 \$\...