Covariance is a quantity used to measure the strength and direction of the linear relationship between two variables. The covariance is unscaled, & thus often difficult to interpret; when scaled by the variables' SDs, it becomes Pearson's correlation coefficient.

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Covariance matrix of estimated parameters of SVAR

How to get covariance matrix of estimated parameters of structural VAR? Any reference, if possible. Thank you in advance.
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8 views

How does unbiased cov.wt compute its value?

What is exactly the calculation used when calling cov.wt((faithful,c(1,2,3,4))$cov)? (Say the data faithful has 4 rows and 2 columns, coordinates x,y) I.e.: What ...
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1answer
41 views

A sufficient statistic for Laplace distribution

Suppose we have p dimensional vector of $X =[X_1 \dots X_n]$ where X is Laplace distributed. What will be a sufficient statistics for estimating covariance of $X$? Would it be the sample ...
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1answer
17 views

Covariance between a variable and an interaction variable

If I have two independent variables $X$ and $Y$, then $Cov(X,Y)=0$. Now let $Z = X*Y$. Then I would assume $Cov(X,Z)\ne 0$, but given the expecations, variances and covariances of $X$ and $Y$ is there ...
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11 views

Variance of sum of datasets with different sampling intervals

I have two randomly distributed datasets which are to be added together element-wise, and I am looking to calculate the uncertainty of the mean of the result. One dataset had a sampling rate of 1 ...
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19 views

How to equal weight and value weight a portfolio of mutual fund returns?

I have the historical monthly returns of around 3000 mutual funds. How would I equal weight them? And does 'value weighting' them include multiplying each set of fund returns by a factor ...
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8 views

Gaussian Mixture Model: bandwidth parameter versus variogram fitting?

I'm estimating a stationary, spatially random variable over a 2-dimensional domain. I have ground-truth measurements in several locations, over time. I need some way of spatially-interpolating ...
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1answer
140 views

Scaling a Covariance Matrix by a factor?

I have a $n\times n$ covariance matrix representing the covariance between n random processes. I want to increase all of the covariances by a certain factor, say $1.5x$. Multiplying each covariance ...
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21 views

In logistic regression, what is the expected correlation between prediction and the dependent variable?

In multiple logistic regression: what is the expected covariance between the dependant variable $Y_i$ and prediction $expit(X_i\hat{\beta})$? what is the expected covariance between the dependant ...
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+50

What is the meaning of expression $\sum E[y_{t-1}y_{t-1}^T]$ & how to implement?

Eq(1) represents an Autoregressive model driven by a zero mean input $x_t$ where $h = [h1,h2,..,hp]^T$; $y[t-1] = [y[t-1],y[t-2],...,y[t-p]^T$. Output of these observations are corrupted by zero ...
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38 views

Maximum likelihood estimation for Cauchy noise

What is the maximum likelihood estimator of the covariance matrix for a given vector in the presence of Cauchy noise? How can we calculate it given that the Cauchy distribution has infinite variance? ...
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27 views

high correlated variables -> ill-conditioned covariance matrix?

I was wondering why the covariance matrix of high correlated variables is ill-conditioned. Is there any (logical) explanation for this? Thanks!
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18 views

Covariance Matrix vs. Pairwise Covariance Matrix?

I found this equation here to calculate a covariance matrix of any number of variables using matrix algebra. $1/N(X - 1\bar{x})^T(X - 1\bar{x}^T)$ For a given matrix $X$ with $N$ samples. The ...
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2answers
114 views

What is the expected value of $\frac{X}{X+Y}$?

I am trying to find the expected value of $\displaystyle E\Bigg[\frac{X}{X+Y}\Bigg]$. I started with writing $\displaystyle E\Bigg[\frac{X}{X+Y}\Bigg] = ...
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8 views

Picking the Right Co-Variance

I am conducting a repeated measures regression, in a linear mixed model. I am also analyzing the data with a general estimating equation. The dependent variable is continuous (a measure of arousal) ...
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1answer
36 views

Alternative parameterization for the covariance matrix via Euler angles

Using spectral decomposition, we can write any symmetric matrix as $$\Sigma = Q \Lambda Q^{\top}$$ where $Q$ is orthonormal, and $$\Lambda = \text{diag}(\lambda_1, ..., \lambda_p)$$ with ...
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15 views

General definition for higher order co-moment matrix

Is there a general equation or procedure for computing higher-order co-moment matrices (i.e., coskewness matrix, cokurtosis matrix, etc) for a vector of random variables? For example, the covariance ...
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10 views

Correct formula for unbiased estimate of population covariance [duplicate]

Is this formula a correct way to estimate population covariance from a sample? $$s_{xy} = \frac{1}{n-1}\sum\limits_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})$$
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42 views

How can I extract a residual variance-covariance matrix in lme?

I have been using MCMCglmm in R to fit bivariate (two response traits) mixed models in R, but now I need to move to lme to account for temporal autocorrelation of the residuals. In MCMCglmm I can fit ...
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16 views

Covariance of ARMA (2,1)

I am preparing for an exam and need help. Consider the following estimated ARMA(2,1) model, y(t) = 0,05 + 0,83y(t-1) + 0,13y(t-2)- 0,15e(t-1) + e(t): Given the unconditional variance and 1st order ...
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1answer
19 views

How do you define the structure of a Covariance Matrix

I am reading a paper by Ledoit and Wolf,2001 on Improved Estimation of the Covariance Matrix of Stock Returns. I am a little confused by some points, and will appreciate a broader explanation. The ...
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23 views

Fit multiple regression model with pairwise deletion (or on a correlation/covariance matrix) in R

I'm trying to fit a multiple regression model with pairwise deletion in the context of missing data. lm() uses listwise deletion, which I'd prefer not to use in my ...
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27 views

slope comparisons given by independant measurements

With the data used in a previous question matplot/ggplot2 ...
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2answers
88 views

Is $\text{Cov}(|a|,|b|)\geq \text{Cov}(a,b)$?

The above seems intuitively true, (where $|a|$ refers to the absolute value of $a$), but I'm struggling to prove it - would be very grateful for either a proof or a reference.
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Computing covariance between normal and uniform distributions

I think that this question is related: Covariance of a compound distribution, but it's too abstract for me (I believe that what I have to do must be simpler). Here's the given: machines $A$ and $B$ ...
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22 views

Testing for structural break in the covariance

I estimated a bivariate VAR(p) model and assume that there exist two covariance regime $\Sigma_1$ for the period 1 to $T_B$ and $\Sigma_2$ for the period $T_B+1$ to $T$. I am now interest in testing ...
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Does a “pruned” i.i.d. multivariate sample behave as the i.i.d. sample?

Let $z_1,\cdots,z_n$ be $n$ points drawn i.i.d. from $\mathbb{C}N_p(0,\Sigma_n)$. The distribution of the covariance $S_n=\frac1n\sum_{i=1}^n z_i z_i^*$ is well known in the limit as $n,p(n)\to\infty$ ...
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1answer
32 views

How do i show multiplication of covariances?

i am trying to show the following $$Cov(a,b) = \frac {Cov(a,x) * Cov(b,x)} {Var(x)}$$ I am a bit lost on how to expand the numerator term, so i wrote the following $$Cov(a,b) = \frac ...
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22 views

Multifactor Covariance Matrix

hanks for taking a look. I am struggling to understand a rather simple concept. I ran a simple linear regression of the form $$A= \alpha+ \beta X + E$$ $$C = \alpha +\beta X + E$$ Then i ...
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12 views

On the correlation function of a stationary time series (spectral analysis)

I am following a proof of the following fact of which I do not understand only the last step. I will post it entirely for the sake of completeness but do not hesitate to just look at my question at ...
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7 views

Distribution of the maximum vector (over i.i.d. set) and its dot product with the eigenvectors spanning the rest.

Let $z_1,\dots,z_n$ be i.i.d. draws from $N(0,\Sigma)$, where $\Sigma$ is a $p\times p$ matrix. Assume that $p>n$. Suppose (up to re-labeling) that $z_n=\max_i \|z_i\|_2$. Consider the eigenvectors ...
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Distribution of correlation coefficient in compound symmetric covariance model

Suppose $X_i \overset{iid}{\sim} N_m(0, \Sigma)$ for $i=1, \dots, n$ for the case that $$\Sigma = \sigma^2((1-\rho)I_m + \rho \mathcal{1}\mathcal{1}^\prime),$$ where $\mathcal{1} \in \mathbb{R}^m$ is ...
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46 views

Covariance of Categorical variables

I know that the following is true for one categorical variable $X \in \{1,...,k\}$ $cov(X=i, X=j) = -p_i p_j$ However, I don't have intuition behind this and cannot find a proof for this. Is there a ...
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1answer
33 views

estimators with singular covariance matrix

Suppose I have 2 vectors of random variables $\boldsymbol\theta_1 \in \mathbb{R^n}$ and $\boldsymbol\theta_2 \in \mathbb{R^m}$ with asymptotic covariance $\Sigma_1$ and $\Sigma_2$ respectively. I want ...
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47 views

Sample covariance matrix and its inverse

Assume we have the sample covariance matrix $S_1 = XX'/k$ which is not positive definite (in fact it is positive semi-definite) and not well conditioned in very large dimension (large $p$, small $k$). ...
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12 views

How to find covariance matrix from correlation if mean is not given?

I'm given autocorrelation function of gaussian random process: $$ R_x(\tau) = 3e^{|-\tau/3|} $$ Now I should find covariance matrix. I know the formula and solutions, where $$ C_{xx} = R - E[X]^2 $$ ...
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19 views

Interpretation of (diagonalized) inverse covariance matrix

There are several threads here about covariance matrix and inverse covariance matrix interpretation (here, here or here). However, I was wondering how to interpret the inverse covariance matrix (or ...
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1answer
73 views

When the sample covariance matrix becomes singular

Assume a data set $X$ which contains $k$ iid random vectors of size $p$. Denote by $S$ the sample covariance matrix. Really I have some questions and I need your very appreciated opinions: 1) Ledoit ...
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59 views

How to generate a sparse inverse covariance matrix for sampling multivariate Gaussian vectors?

I need to generate a sparse 100x100 precision matrix to sample multivariate Gaussian random vectors using the inverse of it as the covariance matrix. To be a valid precision matrix, the matrix I ...
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2answers
42 views

Covariance estimation of overlapping time series

I have two series $x_t,z_t$, and compute the differences like $\Delta_h x_t=x_t-x_{t-h}$. What is a good estimator of the covariance of changes? $$Cov[\Delta_h x_t,\Delta_h z_t]$$ The intervals are ...
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42 views

Regularization parameter to generate inverse covariance matrix

My data consists of approx. 5 Million binary strings (n) and every string is 2788 characters long. My goal is to find out if position i is correlated with position j. I approximated a covariance ...
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1answer
40 views

What is the duality relationship between eigensystems of X X' vs X' X?

I want to know the relationship between the eigensystems of two non-negative-definite (covariance-like) matrices. Both are derived from X which is a T-by-K real matrix (wlog say K > T). I avoid ...
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37 views

Calculating the covariance of effect sizes for multiple-treatment multiple-enpoint studies

I'm interested in finding the covariance of effect sizes for multiple-treatment multiple-outcome design given the scenario below. Gleser and Olkin (2009) provided the formula for the estimating the ...
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In an analysis of covariance table why do the sums of squares for the treatment, covariate and residual not add up to the total sum of squares?

I am trying to understand why this is the case, since in an analysis of variance table the sums of squares in the rows prior to the total row add up to the total sum of squares?
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1answer
30 views

GARCH and (non) covariance stationarity

I am trying to model two series of financial returns using a bivariate BEKK GARCH estimation. From the output of the statistical software, I see that covariance stationarity is not ensured in my case ...
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1answer
63 views

Shrinkage estimation of Efron and Morris (1972)

I read this article: Article1 and which was refined by the second article Article2 that was considered as a generalization of the James-Stein estimator. In article 1 for example, they considered the ...
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35 views

Where is the dominated convergence theorem being used?

I am trying to fully understand the proof of a theorem, I only have a problem with the application of the dominated convergence theorem. For the sake of completeness I will upload the whole statement ...
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1answer
29 views

Covariance of 2 sequences of random variables

We have a random number generator that generates random integers independently uniformly distributed from 1 to 5, inclusive. This random number generator is used to generate a sequence on $n$ ...
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1answer
40 views

Incremental Gaussian Process Regression

I want to implement an incremental gaussian process regression using a sliding window over the data points which arrives one by one through a stream. Let $d$ denote the dimensionality of the input ...
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1answer
61 views

Property of the autocovariance function in time series

In the framework of time series analysis Why does $\lim_{n \rightarrow \infty} n^{-1} \sum_{|h| <n} |\gamma(h)| = \lim_{n \rightarrow \infty} 2|\gamma(n)| $? The LHS (left hand side) sequence of ...