# 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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429 views

### Does there exist a Frequentist or Non-Bayesian solution to Gull's Lighthouse Problem?

Does there exist a Frequentist or ODE or Non-Bayesian solution to Gull's Lighthouse Problem which is correctly modeled with cauchy distribution? See The Lighthouse Problem and Dave Harris' answer to ...
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### Bias-Variance Trade Off with Cauchy Estimator

I'm having a look at the bias and standard error of a set of estimators. I expected to see the trade off when varying the parameter of the estimator, but I see that both the bias and the variance ...
91 views

### Ratio of two independent normal : cumulative sum [duplicate]

Given two independent normal variables, $X\sim N(0, \sigma^2_X)$ and $Y\sim N (0, \sigma_Y^2)$, find the probability that $\frac{X}{Y} < 1$. I seem to prove: $Z = \frac X Y \sim$ Cauchy ...
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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 $\theta=1$,...
107 views

### 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?

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 the distribution of the generated values ...
485 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 ...
752 views

### 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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### 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 ...
361 views

### 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 ...
549 views

### 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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### 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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### 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: ...
355 views

### 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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### 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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### 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 ...
270 views

### 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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### 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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### 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 ...