Tagged Questions

0
votes
0answers
28 views

pdf of multivariate normal distribution

I have a question concerning some sentences in the book Structural Equations with Latent Variables (Bollen) at page 132 (bottom) and page 133 (top) regarding the pdf of the multivariate normal ...
1
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1answer
219 views

Hypothesis testing of normal distribution, known mean unknown variance

I've been working on review problems, and this one has me completely stumped. Let $X_1 ... X_{10}$ be a random sample from a $N(3,\sigma^2)$ distribution, where $\sigma^2$ is unknown. Using the ...
0
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1answer
87 views

Spherical Gaussian Sigma dimension

I think I am confused with this thing. If we have a 3 dimension Gaussian then the MLE estimate for $\mu$ is a vector with 3 element $$\mu(1)' = \frac{1}{n}\sum_{j = 1} ^ n x_j\text{ and so on ...
0
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0answers
58 views

Constraints on ML for mixture of Gaussians

I have some data sampled from a mixture of two Gaussians where one of them is known, and the density function is as follows: $f(x, \mu, \sigma) = \frac{1}{2}\frac{1}{\sqrt{2\pi}\sigma} ...
3
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1answer
278 views

Maximum likelihood of function of the mean on a restricted parameter space

I've been trying to teach myself some of the fundamentals of statistics by trying to work through old qualifying exams. Here's a problem: Suppose $X_1, \ldots, X_n$ are a random sample from a normal ...
2
votes
1answer
112 views

Confusion related to derivation of the log likelihood

I was reading the Hastie, Friedman, Tibshirani paper "Sparse Inverse Covariance Estimation with the Graphical Lasso" and it had the following I couldn't get how the following expression was derived ...
9
votes
1answer
242 views

Hypothesis testing on the inverse covariance matrix

Suppose I observe i.i.d. $x_i \sim \mathcal{N}\left(\mu,\Sigma\right)$, and wish to test $H_0: A\ $vech$\left(\Sigma^{-1}\right) = a$ for a conformable matrix $A$ and vector $a$. Is there known work ...
0
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1answer
100 views

How can I replace this condition by a probability?

I want to see if a datapoint x should (or not) be assigned to a nearest component y using the following condition: if ($d > T$) then {do not assign x to y}. With $d = distance(x,y)$ and $T = ...
1
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0answers
186 views

Is there a covariance MLE which takes into account independence relationships?

In the extreme case where all of the components of an $M$-variate observation are pairwise independent from each other, a multivariate normal distribution can be decomposed into the product of $M$ ...
1
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2answers
403 views

How difficult is it to train a gaussian mixture model compared to other models?

I have finally been able to wrap my head around the mechanics of how to initialize and train a multivariate Gaussian mixture model using expectation maximization algorithm. So I wonder how difficult ...
1
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1answer
509 views

Likelihood at MLE and transformations, the multivariate normal case

Given a univariate sample $\vec X = X_1, ..., X_n$ with standard deviation 1 and a strictly monotone transformation $t: R \to R$ with the property that the standard deviation of $t(\vec X)$ is also 1 ...
1
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1answer
716 views

What is the maximum likelihood estimator of the mean of two normally-distributed variables?

Let $Z=(X+Y)/2$, where $X$ and $Y$ are independent normally-distributed random variables with known variances $\sigma^2_X$ and $\sigma^2_Y$ and unknown (and possibly different) means. Given a sample ...