Questions tagged [multivariate-normal-distribution]

Question 1

I have a multivariate normal likelihood of the form $$p(y| \mu_\alpha, \Sigma_y + \Sigma_\alpha) = N(\mu_\alpha, \Sigma_y + \Sigma_\alpha) $$ where $y$ is observed data, $\Sigma_y$ is known, and $(\...

Question 2

In my research involving statistical inference for the bivariate normal distribution under a special sampling scheme, I encountered the following integral: \begin{eqnarray} I=\int_{-\sqrt{s_{11}}}^{\...

Question 3

I am studying about Discriminant Functions For The Normal Density (2.6) from the book Pattern Classification (Second Edition) by Duda,Hart and Stork.The book suggest following function for ...

Question 4

I have encountered a problem when trying to simulate from a multivariate normal distribution with a covariance matrix built to encode specific correlations. Most of my parameters are estimated from ...

Question 5

Consider the problem of classifying $x \in \mathbb{R}^2$ into one of two classes, $c1$ and $c2$, with known distributions \begin{align} & p(x\mid c1) \sim \mathcal{N}\left(\begin{bmatrix} 0 \\ 0 \...

Question 6

Assume that I have four groups of samples, each from a $d$-dimensional normal population $N(\mu_i,\Sigma_i)$ ($1\le i\le 4$). All the $\mu_i, \Sigma_i$ are unknown parameters. I need to test the ...

Question 7

As the title says, I would like to know if there any method already implemented in R that efficiently samples from the truncated multivariate normal distribution. More precisely, given $\textbf{X}\sim\...

Question 8

Is this true: Let $$Z \sim \mathcal N (0, I).$$ Then if $$X = \Sigma^{1/2}Z + \mu$$ then $$X \sim \mathcal N(\mu, \Sigma).$$ That is, we can nonstandardize a multivariate normal in the exact way we do ...

Question 9

As the title says, I would like to know if the following relation is true: \begin{align*} \mathbb{E}[\textbf{X}\textbf{X}' \mid \textbf{X} \leq \textbf{C}\textbf{b}] = \boldsymbol{\Sigma} + \mathbb{E}[...

Question 10

Suppose $(X_1,Y_1),\ldots,(X_n,Y_n)$ is a random sample from a bivariate normal $N_2(\boldsymbol \mu, I_2)$ distribution where $\lVert \boldsymbol \mu \rVert=1$. Let $\widehat{\boldsymbol\mu}$ be the ...

Question 11

In linear regression, I'm assuming that when $y$ belongs to a normal distribution it's because there's only one variable, as in it's a simple linear regression, for example $y = \beta_{0} + \beta_1 ...

Question 12

I'm interested in knowing whether the following statement is true, and if it's false, what condition should be satisfied on $f$: If $x \in \mathbb R^m$ and $y = f(x) \in \mathbb R^n$ follow two (...

Question 13

Background Often we have multiple composite endpoints reported from the same clinical trial. Let's say we have the hazard rate per 100 patient-years reported (in a publication, we don't have the raw ...

Question 14

Okay, here is the scenario. We have a bunch of data, say x1,...,x300 that we have collected for a bunch of samples, y1,...,y1000. The data x1,...,x300 is dependent (value of x1 is correlated to the ...

Question 15

Let $X$ be a random vector with multivariate normal distribution: $X\sim \mathcal{N}(\mu,\Sigma)$. Let $L$ be the Cholesky factorization of $\Sigma$: $\Sigma = LL'$, so that we can write $$X=L X_0 +\...

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Questions tagged [multivariate-normal-distribution]

Conjugate prior for normal likelihood when the likelihood covariance is shifted

Change of Variables in a Double Integral Involving Bivariate Normal Sample and Spectral Decomposition

Why don't we ignore dividing by variance in discriminant functions for the normal density in case where covariance matrix for all classes is sigma*I?

Covariance matrix construction problem for multivariate normal sampling

Asymmetric Bayes error $\mathcal{N}\left(0,\begin{bmatrix} \sigma_1^2 & 0 \\ 0 & \sigma_2^2 \end{bmatrix}\right)$ vs $\mathcal{N}(0,I)$ classification

Test the hypothesis $\langle \mu_1,\mu_2\rangle= \langle \mu_3,\mu_4\rangle$ for four $d$-dimensional groups of normal population

How can we efficiently sample from the truncated multivariate normal distribution? [closed]

How can we generate a nonstandard multivariate normal from a standard multivariate normal? [duplicate]

Is the expression for $\mathbb{E}[\textbf{X}\textbf{X}' \mid \textbf{X} \leq \textbf{C}\textbf{b}]$ correct?

Asymptotic variance of MLE of $\boldsymbol \mu$ when $\lVert \boldsymbol \mu \rVert=1$

Difference between normal and multivariate normal distribution for Y in linear regression?

Non-monotonic function preserving Gaussian distribution

Correlation of estimated rates for overlapping composite endpoints from the same study

Independence of Principal Components

Cholesky factorization of a conditional multivariate normal distribution

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