Questions tagged [heavy-tailed]

Heavy-tailed distributions have tails that are not exponentially bounded (eg, log-normal & Pareto [heavy right tail], & t [both]). For general questions about fat tails, use the [kurtosis] tag.

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How to interpret Hill estimate of tail index

I'm seeking a non-technical explanation of how to interpret the Hill estimate of the tail index for fat-tailed data, and, if possible, some explanation of seemingly contradictory results that ...
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28 views

Question based on the @whuber detailed answer on fat tailed term [duplicate]

There is one accepted answer, but I don't get it quite. In fact that answer raised the few additional questions I liked to share. 1. How would you define the tails of the distribution? Seams that ...
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Estimation of the mean of the autocorrelation of the time series

I have asked a similar question before, and I was encouraged to repost the question with more details. I am not all that familiar with the jargon, so feel free to edit the post or add the tags. I ...
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69 views

Linear model with heavy tails

I am just approaching statistics and I find myself trying to fit different linear mixed models for my experimental data. My previous experience was mostly on lmer with binomial distributions, which ...
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1answer
24 views

H2O GLM for heavy-tailed data [closed]

I am trying to run H2O GLM (OLS, lasso, ridge, EN) for stock returns, which have very heavy tails (i.e. potentially infinite variance). Is there a robust loss function modification for this, say Huber ...
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52 views

Estimation of the mean of a long tailed distribution

I want to calculate the average of a data set in which elements are distributed according to a PDF that seems to have a quite long tail. This means that when I bin elements of this set I get a PDF ...
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14 views

Goodness of fit for heavyLme

I was analyzing repeated measures data with linear mixed models with the lmer function, but due to heavy tailed residuals (image below) I am now using the heavyLme function. I wanted to compare the ...
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44 views

How to prove a distribution is or is not heavy tailed?

I've seen the definition for a distribution that has a heavy right tail but I can't seem to prove to myself that a distribution has a heavy right tail or not. How would you prove that for the normal ...
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30 views

Heavy right tail distributions [duplicate]

Does the standard normal distribution have a heavy right tail? How to prove it? Does the standard log-normal distribution have a heavy right tail? How to prove it? Thanks!
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53 views

Fat tails vs heavy tails

There is a lot of confusion about these two terms also in previous asked questions. I am interested in this part of the Wikipedia article (I am only considering right-tailed distributions): https://en....
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164 views

Family=scat in mgcv not behaving as expected

I am analysing the eye-tracking data of an experiment in psycholinguistics I ran some time ago, and after fitting a model that captures the data pretty well, I ran a number of model checks and found ...
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1answer
20 views

Which is the right measure of dispersion to be used as a proxy for risk for a fat tail distribution?

Which is the right measure of dispersion to be used as a proxy for risk for a fat tail distribution ? Standard Deviation, Mean deviation, Value at risk, what else?
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How to measure the level of tail dependence in copula?

In the Clayton copula below, we see that there is stronger lower tail dependence (bottom left corner of Clayton) than upper tail dependence (upper right corner) because the pseudo-observation pairs in ...
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1answer
47 views

A tail bound for an unknown distribution via sampling

I know that many results exist for making an argument about the tail of a distribution, i.e., for a random variable $X$, one can find a bound $\epsilon$ such that $\Pr[X \geq a]<\epsilon$. Some ...
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667 views

In comparison with a standard gaussian random variable, does a distribution with heavy tails have higher kurtosis?

Under a standard gaussian distribution (mean 0 and variance 1), the kurtosis is $3$. Compared to a heavy tail distribution, is the kurtosis normally larger or smaller?
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3answers
558 views

Sampling from heavy vs light tailed distribution

I am having some issue understanding the behavior of such distributions when generating random numbers. I was under the impression that heavy tailed distributions have "heavier" tails, so ...
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1answer
109 views

Short tailed/Long tailed distributions and their effects on p-value interpretation when assuming normality

Can anyone offer better insight into the comparison of how p-values for hypothesis tests are affected when your distribution is short/long tailed but we assume it is normally distributed? I'm ...
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60 views

Heteroskedasticity tests: heavy-tailedness of squared estimated errors

I have a time series model and obtain the following distribution of estimated errors: I suspect that the errors are heteroscedastic in the sense that their variance depends on the level of one or ...
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86 views

How do we call a more extreme case of fat tails than a power law?

According to Wikipedia the most extreme case of a fat tail follows a power law: The most extreme case of a fat tail is given by a distribution whose tail decays like a power law. That is, if the ...
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35 views

Can someone please explain “L1 Regularization is inefficient when the errors in data have heavy tail distribution”

I was reading into L0, L1, and L2 regularization and I found this research paper. On page 2, paragraph 3 it mentions the line L1 Regularization is inefficient when the errors in data have heavy ...
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198 views

linear mixed effect model with symmetrical heavy tailed errors distribution

I am using the Lmer function from the lmerTest in R to test the significant of fixed effects ...
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0answers
60 views

GAM scaled t family for heavy tailed distributions

I have some heavy tailed data I wish to model using the mgcv package in R with a t-distribution. Reproducible example: ...
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20 views

Appropriate Data Analysis when Criterion has Heavy-Tailed Distribution

I have a data set where my independent variables (i.e., personality assessment scores) are continuous and follow a normal distribution. The criterion, sales performance, is heavy-tailed and follows a ...
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2answers
2k views

Regression with heavy-tailed response variable

I have a response variable that is unbounded and continuous, but has heavier tails and violates some of the assumptions of normality (see plots below). This variable represents selection coefficients ...
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1answer
96 views

Appropriate to fit lognormal model to data with heavy tail?

I am attempting to standardize recreational fishery CPUE data. I am using a delta approach, with a binomial model fit to the presence/absence data and a lognormal model fit to the positive ...
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1answer
173 views

Connection between subgaussian/subexponential and exponential family

I am wondering if there is any relationship between subgaussian/subexponential with (one parameter) exponential family. In particular, is there any sub-family density that belongs to both ...
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1answer
135 views

Want to know the differences between exponential and heavy-tailed distributions in terms of first and third quartiles

What are the differences between exponential and heavy-tailed distributions, please illustrate this difference by explaining how well the first and 3rd quartiles describe them. I know what are ...
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1answer
197 views

Correlation of heavy tailed variables

I have two heavy-tailed random variables and want to know if they are correlated. While I only have estimates for the tail exponent, it is in a ballpark so that variance would normally not exist. The ...
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2answers
500 views

Bootstrap confidence interval on heavy tailed distribution

I read from Wikipedia: ... if one performs a naive bootstrap on the sample mean when the underlying population lacks a finite variance (for example, a power law distribution), then the ...
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2answers
199 views

Identifying the tail of a heavy tailed distribution [closed]

I have several distributions with a heavy right hand tail as shown below. I am not interested in analyzing the tail of this distribution. Is there any official definition on where a tail begins on a ...
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1answer
160 views

Are the skew-normal distribution and the skew-Cauchy distribution heavy-tailed?

I think the title is self-explanatory. I understand that the skewness and the tail behavior of some distribution are completely unrelated as any symmetric distribution will have a skewness of zero ...
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184 views

When I fit my data with GEV, I got positive parameter, but when I fit it with GPD, I got negative parameter?

My data is the total annual precipitation in Australia. My purpose is to observe the extreme precipitation on the right end tail. When I fit my data with Generalized Extreme Value, I got positive ...
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32 views

How can Pareto(alpha = 5, x_min = 2) be heavy-tailed where alpha is the shape parameter or the tail index?

Point 1 : It's known that, usually, when the tail-index (alpha) is between 0 and 2, of a certain data set, the distribution is considered as heavy-tailed. Point 2 : It's know that Pareto ...
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1answer
334 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). ...
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1answer
117 views

Alternative to Chow test in the case of heavy tailed residual distribution

I would like to check if two subpopulations of my data have the same parameters in a model. Model 1 is based on subpopulation 1 and Model 2 is based on subpopulation 2. Model 1: $y=x^\alpha + \...
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2answers
463 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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3answers
149 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
112 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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1answer
93 views

How can I derive the function curve from a histogram of observed data

I'm analysing some datasets that produce heavy tailed data when plotted as a histogram. My initial goal was to attempt to fit a known distribution to my dataset. Thereafter I use to the properties of ...
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2answers
4k views

Formal definition of the qqline used in a Q-Q plot

I'm doing some distribution fitting work and I'm looking at Q-Q plots and how they can be used visually to interpret goodness of fit. My data is heavy-tailed so I am looking at Weibull, log-normal, ...
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1answer
333 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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455 views

Choosing bins for chi-square testing distributional fits for distribution similar to normal/heavy-tailed

Chi-squared test can be used to check the hypothesis, that given sample is from the given theoretical distribution. If the distribution is continuous then we should properly choose bins. Question 1 : ...
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1answer
355 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
221 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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2k 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
55 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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140 views

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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42 views

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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0answers
93 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
91 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 ...