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Questions tagged [cauchy]

Cauchy distribution is a symmetric density which equals the t distribution with one degree of freedom. The expectation and variance of the cauchy distribution do not exist. See https://en.wikipedia.org/wiki/Cauchy_distribution

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26 views

Number of samples to estimate Cauchy probability distribution?

I wonder how many samples (approximately) are needed to fit the parameters of a Cauchy probability distribution. I'm guessing probably more than with a normal.
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Central Tendency of Cauchy distribution

How do you measure the central tendency of a cauchy distribution? I'm aware that the mean is not a good measure of the central location for cauchy. Can I use median? I've been searching with the key ...
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Empirical versus Theoretical Convergence of Ratio of Normal Distributions

I have observed that if you take the ratio of two normal variables distributed as $N(1, \sigma) / N(1, \sigma)$, empirically this ratio distribution approaches normality as sigma approaches 0 (shown ...
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Why is the Cauchy Distribution so useful?

Could anyone give me some practical examples of the Cauchy Distribution? What makes it so popular?
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Probability integral transforms - Cauchy distribution of 1/x and X

When revising for exams, I recently came across the following question: Suppose that $X$ is Cauchy distributed, ie has a density function $$f_X(x) = \frac{1}{\pi(1+x^2)}$$ Show that $1/X$ is ...
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How to calculate coverage probability in R for cauchy distribution?

I know that the coverage probability is $P(|\hat{\theta}-\theta|\leq\epsilon)$. My task is to compare estimators of the Cauchy distribution for the location $\theta$. I've plotted the estimators and ...
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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 ...
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Cauchy distribution: R code [closed]

Generate 1000 sets of numbers from the Cauchy distribution. Do this for set size 2, 5, 10 and 20. Compute the median of each set. Find the distribution of the medians, for each set size. How do I ...
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Is Cauchy distribution somehow an “unpredictable” distribution?

Is Cauchy distribution somehow an "unpredictable" distribution? I tried doing cs <- function(n) { return(rcauchy(n,0,1)) } in R for a multitude of n values ...
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Sum of powers of standard normal random variables

Context: While trying to teach the Central Limit Theorem I thought it would be a good idea to show a case where it breaks down. Question: Consider the sum of increasing powers of standard normal ...
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599 views

Consistent unbiased estimator for the location parameter of Cauchy (theta, 1)

Given Cauchy distribution with pdf $p(x) = \frac{1}{\pi ((x - \theta)^2 + 1)}$ how can I find a consistent unbiased estimator for $\theta$? My reasoning so far Tried MLE, but there seems to be no ...
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Are there any distributions other than Cauchy for which the arithmetic mean of a sample follows the same distribution?

If $X$ follows a Cauchy distribution then $Y = \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$ also follows exactly the same distribution as $X$; see this thread. Does this property have a name? Are there ...
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How is the family of distributions with PDF proportional to $(1+ax^2)^{-1/a}$ called?

Consider a family of distributions with PDF (up to a proportionality constant) given by $$p(x)\sim \frac{1}{(1+\alpha x^2)^{1/\alpha}}.$$ How is it called? If it does not have a name, how would you ...
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Formutaion of Least Squares Problem

In general, to use the method of least squares, a linear stochastic system is modeled as: \begin{equation} y = ax + \eta \end{equation} where, $y$, is an observed variable, $x$ is an input while $\...
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Find the Maximum Likelihood Estimator given two pdfs

From the book Introduction to Mathematical Statistics by Hogg, McKean and Craig (# 6.1.12): Let $X_1,X_2,\cdots,X_n$ be a random sample from a distribution with one of two pdfs. If $\theta=1$,...
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Is multivariate Cauchy stable?

I am trying to prove (if possible), for given $A_{n\times n}$ and $B_{n\times n}$, there exists a $C_{n\times n}$ satisfying $$A\pmb{X}_1 + B\pmb{X}_2 \stackrel{D}{=} C\pmb{X},$$ where $X_1, ~X_2$, ...
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How to determine if a distribution is Cauchy?

Sorry if this is a dumb question, but I am making a Cauchy random number generator and I want to make some tests to determine if my code is correct. What are some simple tests I can do to show that ...
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311 views

Standard Deviation of Cauchy distribution on a given interval

In general Cauchy distribution doesn't have standard deviation defined, though it should be possible to calculate it for a given interval. This is the formula that I'm trying to use: PDF for Cauchy ...
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Simulating KL Divergence between Cauchy RV and the MLE estimate of the RV - Multimodality seems wrong

I'm working on a (what I think is a fairly simple, straightforward) explanation of how it's really hard to approximate distributions with fat tails accurately in the tails. I started looking at an ...
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Distribution of the ratio of two i.i.d standard normals

I am reading an article in which the author states that given two i.i.d random variables $X,Y\sim\mathcal{N}(0,1)$, we have $\mathbb{P}(\frac{X}{Y}\le t)=\mathbb{P}(\frac{X}{|Y|}\le t)$ since the ...
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1answer
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Difference between a Student-T vs Cauchy distribution

What are the real and practical differences between the student t distribution and the Cauchy distribution? The results I get when I use them as priors in Bayesian linear models appear to be ...
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Correlation matrix for multivariate Cauchy distribution

I have found an equation for the entropy of a $p$-variate Cauchy distribution here [page 70]: $H(X,R) = \frac{1}{2}\log(\det(R))+f(p)\,,$ where $X=(X_1,X_2,\dots,X_p)$ is vector of random variables ...
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What distributions don't follow the central limit theorem?

The regularity restrictions for the CLT aren't that strong, it seems. As a result, most sums of iid RVs converge toward a normal distribution. I know there are examples that fail. The Cauchy ...
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Interpretation of Constraint in Maximum Entropy Derivation of Cauchy distribution

As per Wikipedia: The Cauchy distribution is the maximum entropy probability distribution for a random variate $X$ for which $$ {\displaystyle \operatorname {E} [\log(1+(X-x_{0})^{2}/\gamma ^{...
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Frequentist Predictive Distribution for a Cauchy variable

I have not been able to find this in the literature, but that probably means I am looking in the wrong spot. I am looking to find the Frequentist predictive distribution, assuming it exists, for a ...
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How can I find the distribution of sample mean of Cauchy distribution? [duplicate]

Let $Y_1,...,Y_n \sim Cauchy(0,1), i,i,d.$ How can I find the distribution of $\bar{Y}$? I know that $Y_1+Y_2\sim Cauchy(0,2)$ by some integration, but I don't know how to proceed.
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Median of the half-Cauchy distribution

The probability density function $f(x)$ of a Cauchy distributed random variable $x$ is given by: $$f(x; x_0,\gamma) = { 1 \over \pi \gamma } \left[ { \gamma^2 \over (x - x_0)^2 + \gamma^2 } \right]$$...
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Mixture probability from mixtures of Cauchy distributions

Suppose that $Z \sim \mathrm{Bern}(p)$ and $Y_1,Y_2 \sim \mathcal{N}(0,1)$ be Gaussian random variables and let all of them be independent. Suppose I am given samples from the following random ...
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A variant of polar ratio of uniforms method for Cauchy variables

The Handbook of Monte Carlo Methods (page 107., Algorithm 4.27) presents a variant of the polar ratio of uniforms method for generating standard Cauchy distributed variables: The better known version ...
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How can you determine if a function is heavy tailed from its characteristic function?

The question is given as follow: Let $N$ have a Poisson distribution with mean $\lambda$. $X_i$ is Cauchy distribution with mode 0 and and scaling parameter $1$. Find the characteristic function ...
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How does the maximum distance between adjacent values vary for increasing $n$

That is, when is the $\underset{n \to \infty}{\lim} \max (X_i-X_{i-1})\rightarrow 0$, where $1<i\leq n$, and $X_i\geq X_{i-1}$ and when is the limit $\neq 0$? The question supposes that the ...
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Does this box plot indicate that an extreme value exists?

This is the boxplot of the data. Did Extreme Value occur in this data?
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Simulation Using Cauchy Distribution as Error Disturbance

I had run artificial data using y=a+ax1+ax2+e. x1 is generated using Normal Distribution and e generated using Cauchy and Normal Distribution. The model i want to compare is ANN and SVM. When using ...
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1answer
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Cauchy distribution and Bartlett test

I want to proof that Bartlett test is highly nonrobust in the case of nonnormal data. Is it correct to use Cauchy distribution which has undefined variance to estimate the significance level of this ...
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Is the ratio between two correlated time series significant?

I have the following two correlated seasonal time series for the same time period: ...
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Cauchy regression estimator

I have a decent understand of OLS regression. Now, what if my observation isn't normally distributed anymore, how can I estimate my parameter of the regression model? I was trying to estimate the ...
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What is the distribution of sample means of a Cauchy distribution?

Typically when one takes random sample averages of a distribution (with sample size greater than 30) one obtains a normal distribution centering around the mean value. However, I heard that the Cauchy ...
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1answer
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What are the properties of a half Cauchy distribution?

I am currently working on a problem, where I need to develop a Markov chain Monte Carlo (MCMC) algorithm for a state space model. To be able to solve the problem, I have been given the following ...
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3answers
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Tail probability for heavy tailed distributions

For some data (where I have the mean and standard deviation) I currently estimate the probability of getting samples greater than some x by using the Q function; i....
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Posterior distribution under Cauchy prior?

I have a (I hope) simple question! If I had a linear regression, $Y_t = \alpha + \beta X_t + \epsilon_t$ with $\epsilon_t \sim N(0,\sigma^2)$ and I assume a Cauchy prior for $\sigma$, is it ...
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Is it possible to calculate a pseudo-R squared for a binomial GLMM with a cauchit link?

I'm modeling some repeated-measures presence-absence data using a binomial GLMM in lme4. I've been using the method suggested by Nakagawa and Schielzeth (2013) to calculate a marginal and conditional ...
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1answer
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Cauchy Distribution used in error term in simulation [closed]

What is the reason used Cauchy distribution in error term for simulation of data. I see a lot of researcher used the distribution but does not stated the reason why used Cauchy Distribution.
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Is the sum of a large number of independent Cauchy random variables Normal?

By Central Limit Theorem, the probability density function of the the sum of a large independent random variables tends to a Normal. Therefore can we say that the sum of a large number of independent ...
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1answer
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Standard notation to indicate certain distributions

On a figure that I wish to annotate to indicate what prior distributions were used in an analysis, I need a shorter way of indicating a $\text{Cauchy}(0, \sigma)$ distribution and saving just 3 or 4 ...
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1answer
689 views

What is the distribution of the ratio of two normals?

I need to use the ratio of two variables as the dependent variable in a regression. Both variables are normally distributed but with positive values. I can either center them or use as it is. If I ...
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Can the Cauchy distribution work well for modelling up-vote/down-vote ratio?

Say I run an A/B test on a Facebook post. I could monitor up-vote/down-vote ratios of the post with A and with B. Now I could model two Cauchy distributions, which are good for modelling ratios. Then ...
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What are the recent real life use or applications of the Cauchy Random Variable?

We have a short assignment on the described question and I already have gone through a lot of trash results from Google. I can't seem to find any. I don't know where else to post this question. ...
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How to check if a distribution has undefined variance?

How can I determine if experimental data comes from a distribution where the variance is undefined (e.g. the Cauchy distribution)? I honestly have no idea how to attack this problem in a sensible way,...
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1answer
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convergence of Cauchy distribution

It is known that the Large Number Theorem does not apply to Cauchy distribution since it does not have an expectation value. That said, $S_n / n$ does not converge in any sense (almost sure, in ...
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Fitting an analytic distribution

I am performing a Bayesian model comparison and attempting to ensure that the priors for the two models are "fair". Unfortunately there is no direct one-to-one mapping between the model parameters for ...