# Questions tagged [symmetry]

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### Should point estimates for a parameter always be exactly in the middle of their 95% CI or does it depend on the distribution? [duplicate]

I'm modelling some count data using negative binomial regression with glm.nb in R. I've noticed that my point estimates are quite consistently not at the midpoint of the 95% CIs and wondering if this ...
674 views

### Why must a product of symmetric random variables be symmetric?

I was reading about weight initialization in neural networks (He et. al, 2015) when I came across this statement: "If we let $w_{l-1}$ have a symmetric distribution around zero and $b_{l} = 0$, ...
84 views

### Is multinomial logistic regression symmetric?

Simple linear regression is symmetric in the sense that, if I regress $Y$ on $X$ or $X$ on $Y$, I get the same $R^2$ and result from the overall $F$-test. ...
1 vote
31 views

### How to design a Gaussian Process which respects input order symmetry

I am trying to design a Gaussian Process model for optimizing experimental parameters. In particular, an experiment requires n parameterizations of points on the 2D Cartesian plane. For a particular ...
153 views

### Symmetry assumption in the Wilcoxon-Mann-Whitney test

The Wilcoxon-Mann-Whitney test requires that two distributions are symmetrical How can I check this assumption by using a hypothesis test and how apply it in R? If this assumption is not met what test ...
972 views

### Does no correlation but dependence imply a symmetry in the joint variable space?

I was looking through the answers to this question, and all of them seem to have some form of symmetry between the variables. I'll walk through the examples in that question so you can see what I mean....
1 vote
167 views

### Symmetry of standardized regression coefficient (=Pearson correlation) in linear regression

Suppose 2 continuous variables X and Y. Their Pearson correlation equals 0.8. This correlation is symmetric (it does not assume a dependent or independent variable). We proceed to a linear regression, ...
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### Verification of proof that a point of symmetry is mode (Casella-Berger 2.27)

I was solving the following problem 2.27 from Statistical Inference by Casella-Berger. 2.27 Let $f(x)$ be a pdf, and let $a$ be a number such that if $a\geq x\geq y$, then $f(a)\geq f(x) \geq f(y)$, ...
1 vote
55 views

### Standard deviation of symmetric data

Within my field a recent study suggested to use the symmetric properties of certain image datasets to improve signal to noise ratio (SNR). I will spare you the details, but in the end one can get a ... 624 views

### Difference between Symmetrically normalized Laplacian matrix versus graph laplacian matrix

I am trying to understand the graph laplacian matrix in Graph Convolution networks. To get a basic understanding of graph laplacian matrix I am referring to this https://mbernste.github.io/posts/...
1 vote
214 views

### 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 ...
39 views

### 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
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### 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. ...
44 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
85 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. ...
1k 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
77 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 ...
51 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
47 views

### 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
131 views

### 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 ...
794 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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1 vote
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Suppose $k(\cdot)$ is a univariate kernel function of order 2 in the sense that $\int x k(x)dx=0$, and $\int x^2 k(x)dx\neq0$. $k(\cdot)$ equals 0 outside a bounded interval, and $k(-x)=k(x)$ for any $... 2 votes 0 answers 31 views ### If the joint density$f_{X_1,...,X_n}(x_1,...,x_n)$is symmetric about the origin, does this imply that each marginal cdf$F_{X_i}(0)=1/2$? If the joint density$f_{X_1,...,X_n}(x_1,...,x_n)$is symmetric about the origin in the sense that for any$(x_1,...,x_n)$, it holds that$f_{X_1,...,X_n}(x_1,...,x_n)=f_{X_1,...,X_n}(-x_1,...,-x_n)$... 0 votes 0 answers 462 views ### Symmetrization in Proof of Hoeffding's Lemma This alternative proof of a slightly weaker version of Hoeffding's Lemma features in Stanford's CS229 course notes. What's notable about this proof is its use of symmetrization. However, I find this ... 1 vote 0 answers 284 views ### What is the expected cost of using LDA? Suppose that you observe$(X_1,Y_1),...,(X_{100}Y_{100})$, which you assume to be i.i.d. copies of a random pair$(X,Y)$taking values in$\mathbb{R}^2 \times \{1,2\}$. I have that the cost of ... 2 votes 1 answer 49 views ### In this case, no problem for initializing weights in deep learning networks to 0 Deep learning textbooks say that initializing all weights of neural networks to 0 will be problematic as it breaks symmetry. I tried with a simple 1-layer neural network but found such is not the ... 2 votes 1 answer 1k views ### Which test(s) are alternative to Mann-Whitney test for non-parametric continues data when symmetry assumption is violated From here and here I see that we cannot use Mann-Whitney test if symmetry assumption is violated. Which test(s) can we use instead of Mann-Whitney test for non-parametric continues data if symmetry ... 1 vote 0 answers 29 views ### Asymmetric robust regression What are the methods for robust regression with asymmetric distribution of outliers? I am specifically interested in equivalents of Huber and Tukey M-estimators. However, asymmetric heavy-tailed ... 1 vote 1 answer 71 views ### Help - Expectations and Ratios I would need your help for a problem I have and I don't know how to solve. I would like to know whether I could prove that : $$E[0.5X/(0.5X+0.25)] = E[0.5(1-X)/(0.5(1-X)+0.25)]$$ knowing that$E[X] =...
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 ...
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: ...