# Questions tagged [symmetry]

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### The Wilcoxon signed-rank test without symmetry caused by one outlier

I am comparing two algorithms on the same input data. Now I want to see whether the difference in output is significant. For this I need to use the Wilcoxon signed-rank test, since my data is paired ...
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### In estimating $X + Y$, is it helpful if I know random variables $X$ and $Y$ are identically and independently distributed?

Suppose I have $$X \sim Dist_1$$ $$Y \sim Dist_2$$ and I want to estimate $X + Y$. I can sample from $Dist_1$ and $Dist_2$ and generate samples for $X + Y$. So far so good. Now suppose I discover that ...
1 vote
34 views

### Is there a statistical test for matrix symmetry? [closed]

I have collected some data and done some processing until the question I'm faced with is - "is the data matrix $X$ a symmetric matrix?". Note that elements of $X$ represents event counts. ...
42 views

### Is a bivariate copula relevant in this physics setting manifesting uniform univariate marginals--and, if so, how can it be constructed?

To quickly place our probabilistic (copula) question in its subject matter setting, we note that a fundamental concept in quantum theory is that of entanglement QuantumEntanglement. The states of ...
1 vote
52 views

### What are examples of symmetric copulas $f(x,y)=f(y,x)$ having relative minima for $f(x,x)$?

In a previous posting on this site RepulsiveBehavior I attempted to detail a quantum-information-theoretic separability/entanglement problem I am pursuing. Detailed issues of sampling sizes for a data ...
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### About unique determination of symmetric point (or center) of a distribution based on pdf or cdf

Suppose we have a distribution that is known to be continuous and symmetric, and is otherwise unknown. We want to decide whether it is actually centered at zero using an equation involving pdf or cdf. ...
730 views

### Is the definition of symmetric distribution using cdf correct?

Based on wikipedia (https://en.wikipedia.org/wiki/Symmetric_probability_distribution), a distribution is symmetric about $x_0$ if and only if it is a distribution whose pdf(or pmf) $f(\cdot)$ ...
1 vote
29 views

### Efficiently sampling a symmetric posterior with MCMC

I am using MCMC (via emcee) to sample a posterior distribution $p(\vec\theta|Y)$ where $\vec\theta = (\theta_1, \theta_2, \ldots)$ are parameters for a physical model of the process generating an ...
50 views

### Symmetry of distribution function is defined as $f(x-a)=f(-(x-a))$, then expectation is $'a'$. i.e $E(X)=a$ [duplicate]

I came across this statement in a book. While I know, how to prove $E(X) = a$ is using $f(x+a)=f(x-a)$. I cannot seem to prove it using $f(x-a)=f(-(x-a))$. I keep going on in a loop, no matter what I ...
1 vote
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### Is 0 the unique center for the mixture density of $N(-a,\sigma^2)$ and $N(a,\sigma^2)$, each with weight 0.5? [duplicate]

Suppose $f_{-a}(x)$ is the pdf for $N(-a,\sigma^2)$ and $f_{a}(x)$ is the pdf for $N(a,\sigma^2)$. Let $f(x)=0.5f_{-a}(x)+0.5f_{a}(x)$ be the mixture density. Is $c=0$ the unique center for $f(x)$ in ...
1 vote
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### How to analyse association between two (paired) sets of measurements, when arbitrary to which set each member of a pair belongs

I have pairs of measurements and need advice to select a measure of association between the measurements. The special aspect that has me confused is the symmetry: there is no reason to allocate a ...
418 views

### Correlation is a symmetric measure, but scatter plot matrix shows asymmetric dependence

The correlation matrix demonstrates that correlation is a symmetric measure: $\rho(X,Y) = \rho(Y,X)$ since the lower off-diagonals are mirror images of the upper off-diagonals. The scatterplot matrix ...
1 vote
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### Error distributions and consistent and unbiased OLS

If OLS estimator is unbiased and consistent, what does it imply about the distribution of error terms? In linear regression model: $y_i = \boldsymbol{x_i' \beta} + \epsilon_i$ if the OLS estimator ...
1 vote
67 views

### Is there a signed (ie anti-symmetric) version of SMAPE?

The symmetric mean absolute percent error (SMAPE) is a symmetrized version of percent error with the formula: $$\frac{200\%}{n}\sum_i\frac{|x_i - y_i|}{|x_i| + |y_i|}$$ SMAPE is symmetric: ...
4k views

### What is a symmetric distribution symmetric about if it has zero skewness? [duplicate]

We know that a distribution with zero Skewness are symmetric. A quick Google search or looking up in textbooks says that Symmetric distributions are distributions where the left side mirrors the ...
446 views

### What would be the output distribution of ReLu(X) activation (In case that the distribution of X is unknown)?

Suppose E[X]=0, var(X)=1 and we know X has a symmetric distribution, What would be the distribution of 𝑌=ReLU(𝑋)=max{0,𝑋}? I have seen this question What would be the output distribution of ReLu ...
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226 views

### Symmetry group in posterior distribution/inference

Here's a scenario: Suppose I collect a dataset $\{x_i\}_{i=1}^k\subseteq\mathbb R$ of data points $x_i$, and I wish to explain it using a mixture of two Gaussians; assume the unknown parameters are ...
173 views

### What can we say about distributions of random variables $X$ such that $X$ and its inverse $1/X$ have the same distribution?

What can we say about random variables such that it and its inverse have the same distribution? One example is Cauchy distributed random variables, easily proved via the fact that if $X, Y$ are IID ...
I've two related propositions which seem correct intuitively, but I struggle to prove them properly. Question 1 Prove or disprove: If $X$ and $Y$ are independent and have identical marginal ...
### When $(X_1-X_0, X_1-X_2)\sim (X_2-X_0, X_2-X_1)\sim(X_0-X_1, X_0-X_2)$?
Consider a bivariate distribution function $P: \mathbb{R}^2\rightarrow [0,1]$. I have the following question: Are there necessary and sufficient conditions on $P$ (or on its marginals) ensuring that ...