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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How to Derive the Conditional Variance in a Bivariate Normal Distribution After Bayesian Updating?
I'm working with a bivariate normal distribution of two variables, $\theta_1$ and $\theta_2$ in a Bayesian framework, with an intial joint prior distribution defined as:
$$\begin{pmatrix}
\theta_1 \\
\...
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How to calculate $P(f_1(X) = \text{max}(f_1(X), \dots, f_K(X))$ when $X$ is multivariate Normal?
Let's say I have a multivariate distribution $\mathbf{X} \sim \text{MVN}(\mathbf{\mu}, \mathbf{\Sigma})$ and a set of $K$ scalar functions of $\mathbf{X}$, $f_1(\mathbf{X}), \dots, f_K(\mathbf{X})$. ...
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Distribution of the correlation coefficient based on quadratic forms
Let $x,y$ be two independent random correlated vectors following the same multivariate (real or complex) centred normal distribution, and let $A$ be a non-negative linear operator.
We can read here, ...
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Equivalence of inverse transformations under distributional equivalence
Consider continuous, invertible transformations $g,h : \Bbb{R}^d \rightarrow \Bbb{R}^d$ and suppose $g(Y) \overset{d}{=} h(Y)$, where $Y$ is a $N(0, I)$ random variable. Then what can we infer about ...
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Is it possible for some p-values to be impossible? (because statistic generated by parametric bootstrap is mostly the same value.)
I am using a parametric bootstrap/monte carlo hypothesis testing method to generate the null distribution of the log likelihood ratio statistic. However, I am worried I might be doing it wrong ...
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How to evaluate this conditional expectation for the E-step in expectation-maximisation?
I'm trying to devise an expectation-maximisation algorithm for a certain problem but I'm unable to derive the conditional expectation in the E-step. For the purpose of this question I'll simplify the ...
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Estimating correlation parameter from known value of bivariate normal distribution
I want to estimate the correlation parameter $\rho$ using the following expression taken from this paper (equation 10 on page 17):
$$ \hat{s}^2+\hat{\mu}^2=N_2(N^{-1}(\hat{\mu}),N^{-1}(\hat{\mu}), \...
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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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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
$$
\begin{bmatrix}
V_{X} & C_{XY} & C_{XZ} \\
C_{XY} & V_{Y} & ...
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Constructing Joint Multivariate Normal Distribution from Marginal Distribution (detrended data)
I have two n-length data vectors $X_1 \sim N(\mu_{1},\Sigma_{1})$ and $X_{2} \sim N(\mu_{2}, \Sigma_{2})$ which may or may not have a covariance. To see whether they do or not, I detrend them by ...
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How to test for equal spread in bivariate normal samples with equal means?
I'm working with samples taken from a bivariate normal distribution, where the differences in means is not relevant since all samples are scaled to mean (0,0) anyway, and I'm trying to remember how to ...
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Distribution of between-group sum of squares (SSB)?
Consider a standard multivariate random effects model
$$\mathbf{X}_{ij} = \boldsymbol{\mu} + \boldsymbol{\alpha}_i +\boldsymbol{\varepsilon}_{ij}$$
for $i = 1,\ldots,m$ and $j = 1,\ldots,n_i$, and ...
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Initialising Multivariate linear regression with the maximum likelihood method
I am attempting to use maximum likelihood estimation to fit a multivariate linear regression problem. I have 5 predictor variables and a similar number of response variables. I am using a correlation-...
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Characterize conditions in which Taylor moment approximation is good
I am working with the Projected Gaussian, or Angular Gaussian distribution, which is given by $z = \frac{x}{||x||}$, where $x \sim \mathcal{N}(\mu, \Sigma)$. This is a distribution on the sphere in $\...
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Test for independence of multivariate normality in R
I have 3 normal data matrices, i.e., that in each matrix, the rows are iid Multivariate Normal, but the rows of different matrices need not be identically distributed, but they have the same dimension....
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Posterior distribution for multivariate Gamma-Normal model
Let $\theta \in \mathbb{R}_{>0}^n$ be a random variable with prior distribution $p(\theta)$:
\begin{equation}
p(\theta) = \prod_{i=1}^n \text{Ga}(\alpha_i, \beta_i)(\theta_i),
\end{equation}
where $...
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Show convergence in distribution to a bivariate normal vector
Let $Y_1,...,Y_n$ be iid exponential random variables with mean $\theta>0$. Let
$$ \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)^{-...
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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:
$$
(1-\lambda)\mathbb{I}_n + \lambda \Sigma_{n}
$$
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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E(X1 | X2 > X3) for (X1,X2,X3) multivariate normal
I'd like a closed form solution for $E(X_1 \mid X_2 > X_3)$ where $(X_1, X_2, X_3)$ is multivariate normal with possibly arbitrary mean vector and covariance matrix.
The conditional distribution $f(...
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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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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):
$$Y \sim MVN(X\beta, \sigma^2I)$$
$$\beta|\sigma^2 \...
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How to derive conditional destribution of MVN variable
I am working with following model specifications (Regression_ Modelle, Methoden und Anwendungen-Springer-Verlag Berlin Heidelberg (2009), p. 147):
$$Y \sim MVN(X\beta, \sigma^2I)$$
$$\beta|\sigma^2 \...
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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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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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 $\...
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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 ...
