Questions tagged [multivariate-distribution]

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

Filter by
Sorted by
Tagged with
1
vote
0answers
11 views

Multivariate Normal distribution Cramer-Rao bound

How to compute the Cramer-Rao lower bound for $\mu$ of a multivariate normal distribution $\mathcal{N}_{p}(2\mu,\Sigma_0)$. Also I would like to know the minimum variance unbiased estimators for $\mu$....
1
vote
0answers
10 views

Bivariate random effect problems in selection models (Mixture Cure model)

I am currently working on a mixed effects selection model. The selection model is a logistic model with a Gaussian random effect. The principal model is a survival model with a Gaussian random effect (...
2
votes
1answer
44 views

Find expectation of conditional normal distribution

I am struggling with some finding expectation value question . the question is to find $E[Y|X]$ from the result $P(Y|X)$ with given mean and covariance $$\mu=[\mu_x, \mu_y]^T$$ $$\Sigma=\begin{bmatrix}...
1
vote
0answers
20 views

What is the entropy of multivariate data multiplied by a vector?

It is a general rule that for multivariate data $\boldsymbol{X}$ and a matrix $\boldsymbol{A}$, their entropy is $$h(\boldsymbol{A} \boldsymbol{X}) = h(\boldsymbol{X}) + \ln |\det \boldsymbol{A}|$$ (...
1
vote
0answers
41 views

What is the distribution of the CDF of a sample drawn from a multivariate normal?

Introduction: Lets say we have a random variable $X$ that follows a normal distribution, $X \sim N(\mu, \sigma^2)$ , with a CDF function $F_X(x) = P(X \leq x)$. Then we draw some random samples $S_1$, ...
2
votes
1answer
30 views

What is the fisher information matrix of the multivariate t distribution?

$\newcommand{\bx}{\mathbf{x}}$ $\newcommand{\bSigma}{\boldsymbol{\Sigma}}$ $\newcommand{\bE}{\mathbf{E}}$ $\newcommand{\bD}{\mathbf{D}}$ Consider the multivariate central t distribution with p.d.f. \...
14
votes
4answers
2k views

Is this causation?

Consider the following joint distribution for the random variables $A$ and $B$: $$ \begin{array} {|r|r|}\hline & B=1 & B=2 \\ \hline A=1 & 49\% & 1\% \\ \hline A=2 & 49\% & 1\...
0
votes
0answers
7 views

Is the mvt (multivariate t) adjustment for multiplicity appropriate for repeated measures and longitudinal trials?

As in the title. If I have a longitudinal experiment with t0...tn timepoints, and want to verify the Dunnett contrast (all versus baseline), can I use the mvt adjustment for multiplicity? I think it ...
3
votes
1answer
61 views

Correlation Coefficient for Hypergeometric-type Distribution

Problem Statement: A box contains $N_1$ white balls, $N_2$ black balls, and $N_3$ red balls $(N_1+N_2+N_3=N).$ A random sample of $n$ balls is selected from the box (without replacement). Let $Y_1,Y_2,...
2
votes
1answer
79 views

Probability distribution of the product of two dependent random variables

It is well known that being $X$ and $Y$ two independent random variables with distributions $f_X(x)$ and $f_Y(y)$, respectively, then the probability distribution of the multiplicative function $z = ...
2
votes
1answer
42 views

What does i.i.d. mean for multivariate case?

When we say a random variable is i.i.d., it's often used to describe the dependency between the observations of that random variable, which I call the row dimension, indexed by time if it's a time ...
0
votes
0answers
17 views

How does the maximum entropy principle affect joint entropy, mutual information, and other info measures?

The maximum entropy principle says that we should use the probability distribution of a univariate dataset that has the highest level of entropy because it offers the lowest information. How does the ...
0
votes
0answers
46 views

Rescaling multivariable normal pdf and normalizing constant

I am trying to understand change of variables for a random variable and how it changes the pdf and the normalizing constant. Let $\mathbf{x}$ be $N$-dimensional normal variable and let $\mathbf{\...
1
vote
0answers
26 views

What are the best known techniques to verify that a GAN samples correctly from a given distribution?

I would like to know what are the best known techniques to check that a generative adversarial network (GAN) samples from the correct distribution. Naively, I would say it all boils down to a ...
0
votes
0answers
17 views

Multivariate rare events

My data is something like this: I have U urns, and I have taken a bag of $n$ objects from each urn. Each urn has $N$ objects, and I have sampled $n$ with replacement. $n$ is comprised of coloured ...
2
votes
0answers
31 views

Algorithm to generate non normal (discrete and continuous) correlated variables using a correlation matrix

What I am trying to do here is, given a dataset (let's say n observations of N variables), and thus a correlation matrix M that result from said dataset, I would like to write a Python algorithm that ...
0
votes
0answers
26 views

Empirical conditional density of continuous variables

I have a dataframe, with data of several continuous variables. The variables are not independent. My goal is to sample from the distribution that generated this data. What's a relatively easy and ...
0
votes
1answer
51 views

Multivariate distributions

Let $x$ and $y$ be random variables with the following joint density function: $f(x,y) = e^{-x}$ for $0<x< \infty$, and $0<y<1$ If $z= x+2y$, what is the joint density function of $x$ and ...
1
vote
1answer
11 views

How to choose what feature vector to plot in multivariate regression analysis?

I'm new to the field of machine learning and I have been having this doubt for a long time now. If we want to plot a scatter plot, we plot it as, x as a function of y where x is a 1-D array. But in ...
1
vote
1answer
61 views

Does the covariance of i.i.d. random vectors/multivariate random variables have any zero terms?

If we have i.i.d. random variables, $X$ and $Y$, then $\text{Cov}(X,Y)=0$. But let's say we have i.i.d. random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$, where $\boldsymbol{X}=(X_{1},...,X_{p})$ ...
1
vote
0answers
41 views

How to draw huge high-dimensional normally distributed vectors in a memory-efficient way

I would like to simulate a few hundred multivariate normally distributed vectors $H$ of dimension $10^6$ with covariance matrix $\Sigma$, preferrably in R. The individual entries of $\Sigma$ are ...
0
votes
1answer
67 views

What's the best function to test multivariate normality when sample size more than 3000?

Before MANOVA, I need to test multivariate normality. Then I tried MVN::mvn() function in R, output as below: ...
1
vote
0answers
14 views

bivariate conditional joint sensor model

I am struggling to find $P\left( V_t | z \right)$ from $P\left( V_t | z , V_p \right)$. Here $z$ and $V_p$ are independent variables. ...
1
vote
0answers
30 views

Calculate fifth and sixth polynomials for Headrick (2002) method for non-normal multivariate distribution

I am trying to perform a 3-variable correlated multivariate Monte Carlo simulation. As the asset class returns are non-normal, I found the following function ...
0
votes
0answers
22 views

Likelihood of three dimensional data

I'm having a lot of trouble finding the likelihood and log-likelihood of $\tau$ that corresponds to the following equation: $a_i = y(t_i, \tau) + \epsilon_i$ where $\epsilon_i \sim \mathcal{N}(0, \...
1
vote
1answer
19 views

Normal random matrix as linear transformation of standard normal random matrix?

Wikipedia has the following definition of a normal random vector: A real random vector $\mathbf{X} = (X_1,\ldots,X_k)^{\mathrm T}$ is called a normal random vector if there exists a random $\ell$-...
0
votes
0answers
31 views

How to learn dependency of variables from data?

I have a data set $X$ that consist of $m$ vectors $\vec{x}$ of $n$ real valued components. Each vector component lies within a corresponding predefined interval of valid values, which is the same for ...
0
votes
0answers
79 views

Probability density of conditional multivariate distribution [duplicate]

We have a multivariate normal vector ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Consider partitioning ${\boldsymbol Y}$ into $${\boldsymbol Y}=\begin{bmatrix}{\boldsymbol y}_1 \\ {...
2
votes
2answers
43 views

How to prove a multivariate r.v. does not follow the nonparanormal distribution?

Background You may find the definition of the non-paranormal distribution at the 2nd paragraph in p.2296 of this paper. In short, $(X_1, \ldots, X_p)$ is non-paranormal if there exists a set of ...
0
votes
0answers
9 views

Modelling probability distribution of multivariate dataset

my background is not in stats so apologies if this is an obvious question. I have a dataset with 60,000 observations, and 128 features. Each feature follows a hurdle distribution with either a gamma, ...
0
votes
0answers
36 views

Distribution of quadratic forms in mixed model

I have a question related to the distribution or asymptotic distribution of quadratic forms that arise in the linear mixed model. Suppose, $$Y=X\beta + H\delta + \epsilon$$ where $\epsilon \sim N_n(0,...
0
votes
1answer
41 views

Conditional Distribution Multivariate Normal Distribution [closed]

Let $X_1, X_2, X_3$, be jointly distributed according to a multivariate normal distribution. $[X_1, X_2, X_3]^T\sim N(\mu=[0,0,0]^T , \Sigma = [[5,0,0],[0,2,1], 0,1,3]])$ $U = X_1 + 2X_2$ and $V = ...
0
votes
0answers
12 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
222 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
69 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
40 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
15 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
18 views

Predict vector of random variables from historical data?

There's historical prices for gold, sp500, silver, iron. ...
3
votes
1answer
89 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
41 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
95 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{\boldsymbol{\...
1
vote
2answers
515 views

What is the conditional expectation of the exponential functional?

Consider the function $g(W)=-e^{-W}$, where $W$ is some random variable s.t.$W=X+YZ$. Furthermore, it holds that all the random variables $X,Y,Z$ follow the normal distribution with the following ...
1
vote
1answer
90 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
87 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
87 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 ...
3
votes
1answer
230 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 $...
1
vote
0answers
60 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
54 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 ...