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 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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