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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fitting GAM to longitudinal heavy-tailed count data

I am trying to fit a generalized additive model to the sum of events occurring over a fixed interval (count data >= 1). I would like to model these data as a function of day-of-year and include ...
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Quantile regression for a sample of city sizes

I would like to estimate the following regression: $$ \ln(Rank) = \alpha + \beta \ln(Population) $$ For an ordered sample of city sizes (from biggest to smallest), where $Rank$ is 1 for the biggest 2 ...
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A follow-up to 'The meaning of an analyt. result concerning the… mean of the square of a reciprocal of a norm. distrib. rand. variable'

This question concerns the same subject matter as this previous question of mine. However, a moderator felt that the questions I posed there are significantly different from the question I am about ...
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63 views

The meaning of an analytical result concerning the (formally nonexistent) mean of the square of a reciprocal of a normally distributed random variable

This question arose as I was writing this answer to this question. Let $X$ be normally distributed with mean $\mu$ and standard deviation $\sigma$, and let $Y=1/X$. First, note that the integral ...
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How do I interpret parameters from a generalized Pareto distribution?

I fit a generalized Pareto distribution using the function "fitdistcens" from the package "fitdistrplus": ...
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129 views

Definition of heavy-tailed distribution

I'm reading about heavy-tailed distributions, the definition states that: The distribution of a real-valued random variable $X$ is said to have a heavy right tail if the probabilities $\mathbb{P}(X &...
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1answer
98 views

Student's t as a power law distribution

I'm currently reading about power laws and I have came across an answer stating: The density function of a Student's t-distribution with $n$ degrees of freedom is: $$f(x) \sim (1 + x^2 / n)^{-(n+1)/2}...
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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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34 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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39 views

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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89 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
31 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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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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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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70 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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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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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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220 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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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
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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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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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855 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
199 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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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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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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267 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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85 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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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
116 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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2answers
394 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
219 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
266 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
614 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
302 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
233 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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210 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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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
391 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
138 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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509 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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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
146 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
117 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
5k 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
380 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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551 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
463 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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228 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 ...