Questions tagged [multivariate-distribution]

Probability distribution over vectors (as opposed to univariate distributions that are over numbers).

Filter by
Sorted by
Tagged with
0
votes
0answers
8 views

Calculating the credible interval of each variable in a multivariate distribution

I have a k-dimensional Dirichlet posterior with parameters $\alpha_1,...,\alpha_k$ and variables $\theta_1,...,\theta_k$. This posterior comes from a Dirichlet prior and a multinomial likelihood. ...
4
votes
1answer
160 views

Multivariate normal distribution transformation

Suppose that $X $ has a multivariate normal distribution $X\sim MVN (\mu, \Sigma) $, How can I transform $X$ into $Z$ so that $Z\sim MVN(\mu, I) $ where $I$ is the identity matrix? For instance, ...
0
votes
1answer
23 views

DeGroot P.155 integration problem for multivariate distributions

I am stuck with the integral for equation 3.7.4 and do not see how it was done. Could someone provide me with some hints or resources to read around?
0
votes
1answer
28 views

Confusion over multinomial and multivariate- hypergeometric distributions

You, your parents, your sister, go to visit grandma for her birthday. Grandma made a cake for the party. If she puts $20$ raisins in the cake at random in the cake, and she divides the cake into $5$ ...
1
vote
0answers
29 views

Simulations using correlated random numbers from Multivariate Normal and fat-tailed distributions

This question makes use of the LaplacesDemon package in R, but it is not a coding question, so I believe this is the most appropriate forum. First, the unsurprising results. I generate k correlated ...
0
votes
0answers
9 views

How to check if set of 50-d vectors comes from the multivariable normal distribution in Python?

I have a data set that consists of 18000 rows and 50 columns. Each row represents an observation and each observation is a vector with 50 components. Is there any way in Python for me to check if ...
0
votes
0answers
14 views

Predict vector of random variables from historical data?

There's historical prices for gold, sp500, silver, iron. ...
3
votes
1answer
57 views

Generating pairs of random variables with given covariance and gamma marginals

I have shape parameters $k_X, k_Y$ and scale parameters $\theta_X, \theta_Y$, as well as a covariance $\sigma_{XY}$. How do I generate random variables $(X,Y)$ such that the marginals are gamma ...
1
vote
0answers
34 views

What information in general is necessary to fully specify a multivariate distribution?

Given some multivariate probability distribution, we can fully describe it with its density or mass function -- we can associate each point in the space with either a probability density or mass, ...
0
votes
0answers
10 views

Tail Dependency of Multivariate T-distribution

In my time series class, my prof said if $\mathbf{Y}$ has a multivariate t-distribution, then $Y_i$ and $Y_j$ are dependent because of the tail dependence. Can someone give an intuitive and/or ...
2
votes
0answers
77 views

Sum of Log Chi-Squared Asymptotic Distribution

I'd like to find the asymptotic distribution of $$\sqrt{n}\left(\log|\mathbf{S}| - \log|\boldsymbol{\Sigma}|\right), ~~~~~n \rightarrow \infty$$ where $\mathbf{S} \sim W_j\left(n, \frac{\...
1
vote
1answer
78 views

How to perform joint inference on multivariate normal variables?

Suppose I have the following model: $$\begin{aligned} \text C &\sim \mathcal N \left(\mu, \delta^2\right) \\ \forall i: \text L_i | \text C = c &\sim \mathcal N \left(c, \lambda_i^2 \right) \\...
3
votes
1answer
62 views

Ornstein-Uhlenbeck process

Given a multivariate Ornstein-Uhlenbeck process that is a stochastic process, is it correct that each component of this process is a univariate Ornstein-Uhlenbeck process?
1
vote
1answer
52 views

Something like Mahalanobis distance when the copula is not Gaussian

Mahalanobis distance accounts for different variances of the marginal variables and correlations between the marginal variables. However, there is an implicit (maybe explicit) assumption that ...
2
votes
1answer
133 views

From beta distribution to Dirichlet: Estimation of the concentrantion parameters

Searching at least 3 hours about the connection between beta distribution and dirichlet. My problem is: I have a collection of random variables $X_i \sim Beta(a_i, b_i)$. The parameters $a_i$ and $...
0
votes
0answers
43 views

How to generate multiple, non-independent samples from a multivariate normal distribution?

Suppose I have a multivariate normal (MVN) distribution: $$\textbf{X} \sim MVN({\mu},\Sigma)$$ where $\Sigma \neq \sigma^2\textbf{I}$ i.e. the RVs within $\textbf{X}$ have some correlation structure....
2
votes
1answer
46 views

Decomposing a random variable into marginals and copula

I’m having trouble getting understanding how to actual construct a copula, from my understanding it captures the purely joint features of a joint distribution. I’ve been working with the following ...
0
votes
0answers
24 views

Quantifying information loss (KL divergence?) between a multivariate and a univariate discrete distribution

Let's say I have n discrete variables, n1, n2, ... n_n, each with a different scale, and another discrete variable ...
2
votes
0answers
85 views

Estimation of common variance in elliptical distribution

Suppose you observe a random $p$-vector $X$ which follows an elliptical distribution with mean zero, covariance $\sigma^2 I$ and (unknown) distribution function $g$. Given a single observation of $X$...
0
votes
1answer
54 views

How to prove that Normal Squared Distances follow a Chi-Square distribution?

Given a multivariate normal distribution $f(x) = \frac{1}{\sqrt{(2 \pi)^n|\Sigma|}} \times \exp\left( -\frac{1}{2} (x-\mu)' \Sigma^{-1} (x-\mu)\right)$ how can I prove that $ (x-\mu)' \Sigma^{-1} (x-...
0
votes
1answer
75 views

Distribution of transformed multivariate log-normal

Let $\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)$ and $\mathbf{Y} = \text{exp}(\mathbf{X})$. If $Y_i$ is one of the components of $\mathbf{Y}$, what is the distribution of $\frac{\mathbf{Y}}{...
1
vote
1answer
376 views

Marginalizing a high-dimensional multivariate Gaussian distribution

I have an 11-dimensional multivariate Gaussian, with a covariance matrix with non-zero values in every element. My goal is to marginalize this down to 4 dimensions, but I'm having some computational ...
1
vote
0answers
33 views

Diffusion tensor as a covariance matrix

TLDR: In nuclear magnetic resonance (NMR), to study molecular diffusion we assume that molecules displace in 3D space according to a trivariate gaussian distribution. The variables are then the ...
4
votes
2answers
192 views

Derive multivariate from bivariate normal distribution

Could anyone help me on the following. Let $K$ and $M$ be integers so that $K\geq3$ and $2\leq M < K$. Let $\boldsymbol{X}=(X_1, ..., X_K)^T$ be a random vector, $\boldsymbol{\mu}$ be a $K\times 1$...
0
votes
0answers
41 views

How Do I interpret this Q-Q plot?

I am new to statistics and I am having trouble interpreting the Q-Q plot above. Given the number of outliers in the plot, can I assume that the dataset being tested has a lot of bad data? Should I ...
0
votes
1answer
360 views

When does Bayesian classifier act as linear classifier?

I am reviewing my lectures in Machine Learning and my current topic is Bayesian Classifier. The context is the classification of two classes C1 and C2. My book (neural networks and learning machines ...
0
votes
1answer
25 views

Correlation between two multivariate measures

I'm reading a paper, but I'm with a problem. The authors say: Let $\boldsymbol{X} = (X_1, \ldots, X_p)^T$ be a vector $m \times 1$ whose the estimative of variance is proportional to $\boldsymbol{\hat{...
0
votes
0answers
17 views

Frank copula with no dependence

From Wikipedia, the Frank copula is the function $C(u, v)$ such that: $$C(u, v) = \frac{1}{\theta} \log\!\left[ 1+\frac{(\exp(-\theta u)-1)(\exp(-\theta v)-1)}{\exp(-\theta)-1} \right]$$ for $\theta\...
1
vote
0answers
38 views

Sufficient conditions for multivariate MGF to be finite in neighborhood of zero

In the one-dimensional case, we have the following fact: (see Existence of the moment generating function and variance for proof) Proposition: The mgf $m(t)$ is finite in an open interval $(t_n,t_p)$ ...
1
vote
0answers
34 views

Estimating Pearson correlation for multivariate t distribution using Kendall rank correlation

I had a task to generate a certain amount of samples for $(X_{1},X_{2})$ with a bivariate $t$ distribution $t_{2}(\nu,\mu,\Sigma)$ and then estimate Pearson's correlation coefficient $\rho$ by using ...
0
votes
0answers
38 views

Projection of multivariate distribution to lower dimensional subspace

Say that $X \in \mathbb{R}^n$ is a vector of $n$ r.v.'s with pdf $p(x_1,\ldots,x_n)$. Let's consider now the linear map $Y = A X$ where $Y \in \mathbb{R}^m$ with $m < n$. I am seeking $p(y_1,\ldots,...
1
vote
1answer
306 views

Multivariate conditional entropy

I would like to take data columns and compute the multivariate conditional entropy. For instance, suppose I have columns $A, B, C D, E$ and I want to compute the conditional entropy $H(E | A,B,C,D)$. ...
0
votes
1answer
49 views

Multivariate distribution: calculate P(Y > b/2)

The joint probability function looks like this: The first step for calculating $P(Y > 2/b)$ is calculating $f_Y(y)$. Which I did like this: The problem here is that my x is still in my indicator,...
1
vote
1answer
496 views

Understanding this expression of the multivariate t-distribution

I found this SO post which expresses the PDF of a multivariate t-distribution in terms of the gamma and normal distribution in python as follows $$ G = \Gamma (k = \nu /2 ; \theta = 2 / \nu)\\ Z = N (...
1
vote
0answers
145 views

Multivariate $t$ ('mvt') - adjustment in emmeans in R

I am doing post-hoc comparisons of contrasts based on linear mixed models I built in R. I am using the emmeans package for the comparisons. One of the default ...
0
votes
0answers
248 views

multivariate normal distribution with mean vector 0 and covariance matrix Σ

I am newby in statistics and I have huge data with "p" variables and "n" samples. My data is a two dimensional matrix with "n" columns (each column is a sample) and "p" rows (each row is a variable). ...
0
votes
1answer
55 views

Unbiased estimator for Theta of a Normal Distribution

If $X_1,\ldots,X_n\sim \operatorname{iid} \operatorname N(\theta, \sigma^2)$, then verify that $\bar{X}_n$ is unbiased estimator for $\theta$ and that Cramer Rao bound is met? I am facing difficulty ...
0
votes
1answer
382 views

Finding joint probability distributions from marginal distributions

Question: I was solving test papers where I found this one. My doubt: I know to work with conditional probabilities and Jaccobian Transformation and part A and B can be done applying the above..But ...
0
votes
1answer
92 views

Mean and variance of probability density with multidimensional indicator function

I encountered the following question while studying machine learning: We are asked to calculate mean and covariance of a given probability density function $$p(x) = \frac{1}{16} \cdot 1_{0 \leq x_1 ...
1
vote
0answers
52 views

How to compare two multivariate datasets

I have two datasets consisting of 20 features, one set contains 50k records and the other 70k. I want to check if they are from the same population. Datasets contain discrete features and the ...
5
votes
2answers
61 views

How to compare a new measurement to an existing multivariate distribution?

I have a dataset that describes the position and rotation of an object at different points in time using four dimensions. I want to use this sample of observations to get a sense of what positions and ...
3
votes
0answers
52 views

Equality of two multivariate normal CDF's

Let $\pmb{X} \sim N_d(\pmb{\mu}, \pmb{\Sigma})$ and $\pmb{Y} \sim N_d(\pmb{\nu}, \pmb{\Omega})$; $\pmb{\mu} \neq \pmb{\nu}, \pmb{\mu} \neq \pmb{0}, \pmb{\nu} \neq \pmb{0}$, and $\pmb{\Sigma}\neq\pmb{\...
5
votes
1answer
152 views

multivariate Student's t distribution: intuition for non-independence?

Consider a multivariate Student's t distribution, with parameters $\nu$ (d.f.), $\mu$ (location) and $\Sigma$ (shape). Does anyone have a good intuition for the individual components not being ...
0
votes
1answer
179 views

Variance of sum of random vectors - a proof

For nonrandom matrices $A(rXk)$,$ B(rXm)$, and $c(rX1)$, how does one show that $$\newcommand{\Var}{{\rm Var}}\newcommand{\Cov}{{\rm Cov}}\newcommand{\*}{{\times}} \Var(AX+BY+c)=A\Var(X)A′+ B \Var(...
1
vote
0answers
37 views

Multivariate Test for Mean Equivalence

I am looking for an equivalence test for multivariate means (arbitrary or normally distributed). Any suggestions or hints in the right direction are appreciated.
1
vote
0answers
40 views

Expectation of expressions involving sample covariance matrix and inverse of covariance matrix

Let $y_{ij}$, $i=1,2,\cdots,n_j$ be a random sample from $N_p(\mu_j,\Sigma_j)$, $j=1,2$. Let $$\overline{y}_j=\frac{1}{n_j}\sum_{i=1}^{n_j}y_{ij} \hspace{2mm} \mathrm{and}$$ $$S_j=\frac{1}{n_j-1}\sum_{...
2
votes
0answers
59 views

Check whether a random sample comes from an elliptical distribution?

How can I check whether it is a reasonable assumption to say that a multivariate sample $x_1,...,x_n$ comes from an elliptical distribution, such as a normal distribution or a t-distribution? In the ...
4
votes
1answer
85 views

A Multivariate Distribution for Linear Combinations of Independent Exponential Random Variables

Consider a random vector $\mathbf{X} \in \mathbb{R}^r$ whose components $X_j$ are independent exponential variables with different scale parameters $\beta_j$, $j=1,\dots,r$. Suppose I have a general $...
1
vote
0answers
55 views

Mutual information between multivariate random variables? [duplicate]

I have read that mutual information only works with two random variables, and that for 3 or more RVs there seems to be a variety of different measures (synergy, partial information decomposition, and ...
1
vote
1answer
136 views

Conditional distribution of multivariate Rayleigh distribution

The correlated Rayleigh envelopes using a set of zero-mean complex Gaussian RVs (Random Variables) is given by $$G_{k}=\sigma_{k}(\sqrt{1-\lambda_k^2}X_k+\lambda_kX_0)+i\sigma_{k}(\sqrt{1-\lambda_k^2}...