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Questions tagged [heavy-tailed]

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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 ...
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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 ...
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1answer
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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). ...
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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 ...
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109 views

What does it mean to say that $X_1, X_2$ have a “common” Normal distribution?

An exercise question asks Let $X_1, X_2$ be rvs having a common Normal distribution $N(0,1)$ with $\operatorname{Corr}(X_1, X_2) = \rho$. Calculate the coefficient of upper tail-dependence for all $...
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1answer
45 views

need explanation about the exponent parameter s in zipf distribution

I need to model the popularity of some requested files from a library with Zipf distribution and I want to simulate it in MATLAB. I don't know what's the effect of parameter s on my result. for ...
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153 views

Is the truncated power law a heavy-tailed distribution?

A heavy-tailed distribution is often defined as a distribution with a tail that is not exponentially bounded. A truncated power law (or power law with exponential cut-off) is a distribution that ...
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1answer
79 views

Transform Heavy right tailed data

I am clustering (K-MEANS) a data 1.7million observations, which displays a heavy-tailed distribution when examined by plot. What is the best transformation to correct it. does log can handle this?
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1answer
148 views

A question about qqplot

This is my qq plot : Its concave-convex curve so it indicates light tails. But my mean excess plot : is increases which means the tail of the distribution of my data is heavy-tailed. I don't ...
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595 views

In a left skewed distribution, how can the range where 95% data lies?

In a simulation that I ran, I have the following graph as a result. How can I find the 95% confidence interval (i.e. the range where 95% of data lies for me). Since I am not expert in stats, please do ...
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1answer
50 views

Decomposition of the probability of the sum

I cannot understand how is gotten the following decomposition. Supposing that $X_1,...,X_n$ random variables i.i.d with heavy tailed distribution $S_n=\sum_{i=1}^nX_i$ In the article that I m ...
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Determining the distribution of data

Hi, I am a student learning financial modelling. I would like some help in determining the distribution of the data given the plots above. I am reluctant to assume normal distribution of the data due ...
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Determining the number of observations within the tail of a given distribution

I am wondering how to determine the number of observations that fall within the tail of a distribution. I am reading a paper and the authors use the assumption that 50 observations need to fall into ...
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51 views

Left “tail” of one-tailed distributions

I think of the "tail" of a probability distribution as the behavior of its PDF $f(x)$ as $x\rightarrow +\infty$. For some PDFs with complicated expressions, it is sometimes easy to study their ...
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3answers
59 views

If the best-fitting distribution has infinite variance, should low observed variance be troubling?

Suppose you have observations which, over the observed range of outcomes, are well-fitted by some distribution like the Pareto that, for certain parameter values, has a an infinite variance. For ...
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1answer
96 views

Does the normality assumption hold? Is this an outlier?

I am trying to fit a multiple linear regression (OLS) model with IPO underpricing as dependent variable. As part of my master thesis I would like to analyze the effect of venture capital ...
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1answer
42 views

A random variable $X$ on $(0,\infty)$ which behaves like Exp for small $x$ and Pareto for large $x$

Are there any examples of distributions which behave like Exponential for small values and like Pareto for large values. $$\ln \mathbb{P}[X>x] \sim -\lambda x, \qquad \text{ for } x \text{ small}, ...
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Regression: zeros in heavy-tailed independent variable from quantization

This question is about handling zeros in an independent variable for a regression. In particular, the zeros are not missing data or true zeros, but occur because of quantization. As a concrete ...
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Rigorous threshold determination for heavy-tailed data?

Overview: I'm trying to design a change-point detection system for univariate, non-normal, skewed, heavy-tailed data that I believe is generated by a stable random process (i.e. stable-distributed ...
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1answer
333 views

Visual fitting of tails of density plots on log scale (R)

Density plots are useful in confirming the fit of a distribution or assessing which distribution to try in order to give the best fit. However, when looking at the tails on log scales, especially for ...
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387 views

tail dependence calculation

If I have the tail dependence value calculated using Joe (1997) in fact $R$ gives the result for any family copula. Using Caillault and Guegan method called "naive" what are the main differences? Why ...
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2answers
245 views

inferring heavy-tail distribution from finite sample of histogram data

I have some data in the form of bins and counts. Here is one complete non-truncated example: ...
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68 views

Parametric modelling of survival data: when there is no event in a long tail, is there information?

I have a survival analysis question. Let's take a look at the below curve: For the red curve, there are no more events after 48 months. For the black curve, there are no more events after 60-ish ...
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1answer
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On the existence of a heavy tailed c.d.f. given a condition based on the CLT

My question is about the existance of a heavy tailed distribution $F$ such that: given two i.i.d. samples $\{X_1,\ldots,X_n\}$ and $\{Y_1,\ldots,Y_n\}$, from $F$, we have $$\sqrt{n}\frac{\hat{\mu}_X -...
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167 views

Relationship between the length of the whiskers and the tails of a population in boxplot

Why in the boxplots long whiskers are associated with a heavy tailed distribution and short whiskers are associated with light tailed distributions populations?
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1answer
2k views

Difference between long tail and short tail distribution?

I want to understand the difference between these distribution types. What is the difference between a long and short-tailed distribution?
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1answer
94 views

how to choose a transformation for left heavy tailed data due to arbitrary upper limit imposed by experimenter?

I have two vectors that look like this: I would like to make a significance test on the paired differences between the two vectors, as well as measure association. The process of data generation is "...
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1answer
27 views

Are the family of stable distributions differentiable everywhere on the real line?

Are stable distributions smooth enough for each index of stability $\alpha$ between 0 and 2, and skewness parameter $\beta$ between 0 and 1? Where there any papers that mention this?
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1answer
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Is there an analytical way to find the area under the probability density function of a stable distribution over a particular interval I=[a,b]

I am trying to find the area over the interval I=[a,b] for a stable distribution. As you know, in general, the densities of stable laws do not have explicit expressions via elementary functions. ...
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1answer
229 views

Distributions that being to domain of attraction of a stable law that are not unimodal?

I was wondering whether there are any distribution that belongs to the domain of attraction of a stable law that is not unimodal. It is known that distribution in that law converge to a stable ...
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789 views

What options are there to 'normalize' values from a heavy-tailed distribution?

We have some data collected acquired from a complex setup, that is expected to come from a "fairly normal" underlying distribution. However when I investigate the data it appears that it's fairly ...
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1answer
269 views

What is the heaviest tail possible for a continuous normalizable distribution?

The heaviest tailed smooth normalizable continuous distributions that I am familiar with are those with fat power-law tails $\frac{1}{x^{1+\alpha}}$, e.g. a Pareto with $\alpha\rightarrow 0^+$ or a ...
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What is the necessity of log concave/ convex for tail distributions?

What is the impact of log concave density for light tailed distribution? How it could be substantiated with real time example? Most of the books and research papers highlights that a distribution ...
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656 views

Fat tail? Short tail? Long tail? Where do I go from here?

I am running a linear mixed model with 4 fixed factors and 1 random factor. The response variable is %growth and it has negative values (some of my animals shrunk). The problem I'm having is the ...
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1answer
283 views

Bounding tail probabilities of the Poisson Binomial Distribution?

In a problem I'm working on, I have Bernoulli random variables $X_1,X_2,\dots,X_k$ ($k$ is odd) and I am interested in their sum $Y = \sum_{i=1}^k X_i$. In this problem, $P(X_i=1) = p_i$ and $P(X_i=0) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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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 ${\lim_{x\rightarrow\infty}}\frac{1-F(x)}{1-G(x)}=0$ ...
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2answers
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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. ...
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1answer
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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(...
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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-...
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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. ...
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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 ...
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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.
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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 ...
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1answer
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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) \...