# Tagged Questions

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### Calculating Partitioned Matrices from subs

Say you have a matrix $A$ which is of size $P\times P$ and a number $Q < P$ can be used to take a partition of said matrix, where: $A_1$ is the upper-left sub matrix, with dimension $Q\times Q$, ...
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### Why is $(x'x)^{-1}x' = x(x'x)^{-1}$

If $(AB)'=B'A'$ then $(x'x)^{-1}x'$ should be equal to $x((x'x)^{-1})'$ . However most econometrics textbooks say that this is equal to $x(x'x)^{-1}$ . What happened to the transpose of $(x'x)^{-1}$? ...
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### Finding estimates of a Linear Regression Equation - R

I'm new to Statistics and R. I'm currently looking through a book called "Discovering Statistics using R". Although the book implies you don't need any statistical background, some of the content ...
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### how can I write a regression model in matrix form?

The question is $$y_{ij} = \mu + \nu d_{2j} + \beta x'_{ij} +\varepsilon_{ij} \text{ where }i= 1 ,\ldots, n,\ j= 1,2$$ So I am trying to write this model in matrix form. I know how to put ...
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### How to calculate the covariance matrix

I tried searching a lot on the net and got the following sources: Source One Source Two The first source seems to be incorrect cause when I calculate it using matlab it comes to be different from ...
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### How can I show rank $( AB)$ = rank $( A)$?

Can you help me how to show rank $( AB)$ = rank $( A)$ iff null$(A)$ $\cap$ range$(B) = \{0\}$? I can understand that rank $(AB)$ would be no greater than rank A. But not sure how to show this ...
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### matrix spectral representation

can you give me any idea how to show this? Show that $A \ge I$ implies $A^{-1} \le I$, and use this result to deduce that $A\ge B\ge0$ implies $0\le A^{-1} \le B^{-1}$. I think this question is ...
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### Redundancies in covariance matrix

We know that covariance matrix is symmetrical. I have a vague intuition that there may be some other redundancies beyond that. For example, if A is correlated to B and B is correlated to C then A and ...
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### Covariance matrix always positive semidefinite?

I actually was perusing here right now to see if anything could explain a result I have been getting - of a covariance matrix which has a negative eigenvalue, yet the correlation matrix does not - but ...
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### standard deviation calculation using covariance?

i require a formula to calculate the standard deviation using variances of three or more variables (lets call them a,b,c) and the covariances between them. To complicate matters more i only need a ...
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### Mysterious failure to generate independent set of random variables

After getting my answer in this: Making a well conditioned orthonormal basis I am running into a problem which I do not understand. I have n dependent gaussian random variables that are related by a ...
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### Making a well conditioned orthonormal basis

Ok, so I have n dependent gaussian random variables that are related by a known n x n covariance matrix. What I would like to do, is produce a linear transformation to turn them into n independent ...
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### Is every symmetric positive semi-definite matrix a covariance of some multivariate distribution?

One can easily prove that every covariance matrix is positive semi-definite. I come across many claims that the converse is also true; that is, Every symmetric positive semi-definite matrix is a ...
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### Unbiased estimate of cross-product for unbiased vector

Let $g$ be an unbiased estimate of a vector $G$. Can $g$ be used to find an unbiased estimate of the cross product $GG'$? I'm stuck because naively using $gg'$ is a biased estimator, with the ...
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### Bound for Arithmetic Harmonic mean inequality for matrices?

This comes from my research in econometrics, but it boils down to a pure matrix-algebra question. The framework is as follows: We have a cross-sectional i.i.d. sample $\{\mathbf y, \mathbf X\}$, where ...
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### How to find the best fit when you have a set of ideal ratios, but some of those are below a minimum?

Say you have a set of ideal ratios, whose sum = 1. For example, input = [0.2, 0.2, 0.3, 0.3] But suppose that there is a rule stating that every ratio should be ...
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### sum of squares of dependent gaussian random variables

Ok, so the Chi-Squared distribution with n degrees of freedom is the sum of the squares of n independent gaussian random variables. The trouble is, my gaussian random variables are not independent. ...
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### Can the diagonal elements of a precision matrix be 0

I have this confusion that why the diagonal elements of the precision matrix cannot be 0? Any suggestions will be much appreciated
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### Matrix Algebra in Statistics [duplicate]

We have $y = X\alpha + \xi$, where $\xi\sim N(0, 1)$ and $\xi = (\xi_1\,,\xi_2\,,\xi_3\,,\xi_4)^T$. Moreover $X$ is a 4x4 design matrix such that $x_i$ is a column 4x1 column vector made up of 1's, ...
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### How to combine covariance matrices?

I have a data set of points in three dimensions. I'm calculating the barycenter (mean) and $3\times3$ covariance matrix from this data set. I store the average, the $3\times3$ matrix (where really ...
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### What we can do using fisher information matrix?

I learned in statistic inference course that fisher information is related to the cramer-rao lower bound. I wonder, if fisher information matrix has any other applications ? I am interested in ...
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### How to compute future's (tomorrow's) distribution given today's?

Marital status can be defined as single, married, separated, or divorced. Today's distribution is: Single = 49% Married = 25% Separated = 15% Divorced = 11% I ...
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### Is this subset of positive definite symmetric matrices closed?

Consider the collection of all matrices of the form $$\Lambda \Lambda^T + \Psi$$ where $\Psi$ is $n \times n$,positive definite, and diagonal and $\Lambda$ is $n \times m$ with $m < n$. In ...
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### Simplifying covariance matrices in distributions

In the multivariate Gaussian distribution, it is required that the covariance matrix be positive semidefinite. I have read that a positive semidefinite matrix $\Sigma$ can be written as $LL^{T}$. I ...