# Questions tagged [heavy-tailed]

The tag has no usage guidance.

66 questions
0answers
6 views

### Estimating tail share of apparently subexponential distributions drawn from finite population, given a finite sample

Suppose I have data on a large sample of some units of observation, where the observed quantity has meaningful differences and ratios. The sample is much smaller than the population, but both are ...
0answers
8 views

### statistics of (possibly) non-independent interacting systems with fat tails

I apologize if this question has already been asked. I don't know the proper terminology to find my answer - so if it has, please direct me to it! Problem setup: I have many time series of a ...
1answer
65 views

### Distribution of the inverse square of a non-standard normal random variable multiplied by a constant

It's a somewhat complicated situation and sorry about my phrasing, but it's my first time here. Suppose I have random normal variable $X$ ~ $N( \mu, \sigma^2)$, which represents some true effect(s). ...
2answers
311 views

### Quantifying dependence of Cauchy random variables

Given two Cauchy random variables $\theta_1 \sim \mathrm{Cauchy}(x_0^{(1)}, \gamma^{(1)})$ and $\theta_2 \sim \mathrm{Cauchy}(x_0^{(2)}, \gamma^{(2)})$. That are not independent. The dependence ...
3answers
109 views

0answers
159 views

### definition of Black swan random variables

What's the precise mathematical definition of a black swan random variable? Taleb describes it approximately here(http://highlands.vmhost.psu.edu/_reading/docs/blackswan.pdf), where he describes ...
0answers
2k views

### Difference between light tail and heavy tail distribution [duplicate]

I read numerous websites including wikipedia on heavy and light tail distributions. However, I am not quite understanding the distinction between the two. In some sources, it notes that light tail ...
1answer
151 views

### Besides the Pareto and Zipfian distributions, which distributions obey the power-law?

I need a list of distributions that obey the power-law, beside the commonly used Pareto and Zipfian distributions. A comprehensive list or a reference to a comprehensive list will be particularly ...
2answers
323 views

### A Tail Bound For Poisson Binomial Distribution?

Consider the Poisson-Binomial Distribution with two components. Let $Y_0\sim bin(n,p_0)$, $Y_1\sim bin(n,p_1)$, and let $Y=Y_0+Y_1$. For any $k>n(p_0+p_1)$, Can we upper bound the tail probability ...
0answers
217 views

### which percentiles can be used to better describe the tail distribution

I understand that the extreme value distribution is concerned about the density of the tail which describes the losses if I am not wrong. In such a case which part of the density(expressed as ...
1answer
1k views

### How is the tail of a distribution defined (about heavy-tailed distributions)?

Some distributions are said to be heavy-tailed. It seems that one definition of a heavy-tailed distribution is that its tails are heavier than the tails of an exponential distribution. However, how ...
1answer
122 views

### Tail equivalence for heavy-tailed data

Is it possible for a distribution $F(x)$ that has not a Pareto ($G(x)$) right tail equivalence to fit well heavy-tailed data? That is, to have ${\lim_{x\rightarrow\infty}}\frac{1-F(x)}{1-G(x)}=0$ ...
2answers
2k views

### Linear Regression with heavy tailed noise

The model is linear $y_i = a\cdot x_i + b + e_i,~ i = 1,2,\ldots,N$. It is given that the noise is heavy tailed. However the distribution of noise conditional on $x$ is the same for all data points. ...
1answer
89 views

### Fractional moments of the Laplacian distribution larger than of the normal

How can I show that the fractional moments of the (unit variance) Laplacian distribution are higher than of the standard normal distribution, for moments higher than 2? Formally, if \$l \sim Laplace(...
1answer
2k views

### Best transformation for a heavy tailed distribution

I am anaylsing a data set, which displays a heavy-tailed distribution when examined on a Quantile-Quantile plot. What is (or are) the best transformation(s) to use to correct a dataset with a heavy-...
1answer
488 views

### How are errors terms calculated in GARCH model by rugarch package?

I am fitting a GARCH(1,1) model to the data and want to look at the innovation distribution. ...
0answers
109 views

### Predictive modeling of an complex panel of heavy-tailed data

I am struggling to develop a sensible strategy or protocol for the predictive modeling of a complex set of data. Apologies in advance for the indeterminate nature of some of this description but it’s ...
2answers
580 views

### Data transformation for heavy tailed data for mixed model use

I am trying to transform my data to meet assumptions of a mixed model (lme4). This is the qqplot for the data. I have tried the traditional transformations: log & square root.
3answers
1k views

### Are two asymptotic values enough to fail the test of normality?

Looking at this post I started to wonder about the gestalt interpretation of the QQ plots generated by qqnorm in R. Here's the plot to avoid having to go to the ...
1answer
713 views

### Bernstein's inequality for heavy-tailed random variables

It is known that for independent sub-exponential random variables, the following Bernstein-type inequality holds: \begin{align} \mathbb{P}\biggl(\biggl| \sum_{i=1}^N a_i X_i\biggr| >t \biggr) \...