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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Monte Carlo Integration Results in Heavy Tailed Distribution

I am running a Monte Carlo simulation that results in an heavy-tailed distribution. The image below shows the distribution of 1,200 runs of the Monte Carlo simulation, where each run consists of ...
2 votes
1 answer
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Bootstrap in cluster experiments

I am planning an AB-test for something like a call-center. We are testing a new interface for the call-center operators. It has to be tested within a small group of roughly 20 operators. The main ...
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outliers for right heavy tails distribtuions

There is plenty of information on how to detect outliers in a sample when assuming that this sample was derived from a normal distribution. Sometimes it seems to me as if when we talk about outliers ...
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2 answers
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Heavy vs light tail distributions when modelling with outliers

I am reading this lecture notes on using the MLEs from other distributions (as Laplace) rather than a Gaussian when dealing with outliers. The lecture notes came from Oxford University: https://www.cs....
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Generalized Gamma for survival analysis , with no defined mean, fat tailed

I have a unique problem in fitting generalized gamma to a survival analysis problem. I used flexsurv package source with following parameterization: If $g \sim \...
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Partial first moments of stable distributions?

Given a stable distribution with parameters $(\alpha, \beta), \alpha>1$ is it true that its all first partial moments (i.e. the integrals of the form $\int_a^b x f(x) dx$, where $a$ and $b$ could ...
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Linear Mixed Model - Heavy Tailed QQ Plot

I'm running a study in which participants rate the politeness and superiority of two different types of smiles (two levels: rewarding and affiliative) presented in three different situational contexts ...
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2 votes
1 answer
100 views

Log Cauchy distribution as lifetime distribution

Though I know that the moments of Log-Cauchy distribution do not exist, is it possible to use the Log-Cauchy distribution as a lifetime distribution under Type-II censoring? Because the MLE of the ...
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2 votes
1 answer
128 views

Heavy tailed residuals in linear mixed model

I'm a new user of linear mixed models and I'm experiencing some troubles with that. I have a dataset with 680000 measures of milk production from 2017 to 2020, from a population of almost 37000 cows ...
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2 votes
1 answer
610 views

Best way to plot a heavy tailed distribution?

Log-log seems more conventional to plot a probability distribution to look for evidence of a heavy tail. Why is this the case? For data with a heavier tail than an exponential distribution, wouldn't ...
2 votes
0 answers
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Test the difference of one side tails from two ditributions?

wilcox.test tests the median difference between two distributions. ks.test tests for any difference between two distributions. ...
1 vote
0 answers
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Weibull and Lognormal Fits

I have simulated failure time data for some components. I tested whether a Weibull or lognormal distribution best fit the data. Please see Figures below. The figures provide median and $95\%$ ...
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2 votes
1 answer
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What does "tail" mean in probability distribution? And what do we mean by heavy or light tail?

I have seen this word so many times while studying the probability distributions, till I should probably want to know exactly what it means, to boost my understanding of probability concepts.
3 votes
1 answer
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How to show that regularly varying distributions are heavy tailed?

Let distribution $F$ be regularly varying with index $\alpha \geq 0$ (denote $F \in R_{\alpha}$), i.e. its tail $\bar{F} = 1 - F$ satisfies $\lim_{x\rightarrow\infty} \frac{\bar{F}(xy)}{\bar{F}(x)} = ...
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1 answer
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Interpreting QQ plot (Normal vs Heavy-tailed)

I'm having some trouble interpreting the shape of this distribution. It is a distribution of price differences between an estimate and actual price. There are 219 points. I'm not sure if I can call it ...
1 vote
0 answers
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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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2 votes
1 answer
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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 ...
3 votes
1 answer
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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 ...
8 votes
1 answer
437 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 &...
4 votes
1 answer
279 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}...
2 votes
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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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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 ...
1 vote
0 answers
241 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 ...
2 votes
1 answer
138 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 ...
1 vote
0 answers
362 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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269 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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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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1 answer
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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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1 vote
0 answers
219 views

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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2 votes
1 answer
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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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15 votes
4 answers
2k 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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1 vote
3 answers
2k 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 ...
2 votes
1 answer
440 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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1 vote
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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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7 votes
0 answers
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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 ...
3 votes
0 answers
632 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 ...
0 votes
0 answers
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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: ...
1 vote
0 answers
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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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8 votes
2 answers
6k 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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0 votes
1 answer
312 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 ...
2 votes
2 answers
3k 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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0 votes
1 answer
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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 ...
1 vote
1 answer
569 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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6 votes
3 answers
1k 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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0 votes
2 answers
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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 ...
3 votes
1 answer
548 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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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 ...
0 votes
0 answers
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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 ...
1 vote
1 answer
616 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). ...