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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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Correction for heavy-tailed distribution of residuals?

I'm interested in studying the effect of $x$ on $y$ using a fixed effects method. The residuals follow a heavy tail distribution, as the normal Q-Q plot suggests. For inference, I need a normal ...
TFT's user avatar
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2 votes
0 answers
62 views

Decision theory when distributions don't have a first moment? [closed]

Lets say that we are presented with two gambling opportunities and would like to decide between them in a decision-theoretic framework. For gamble 1, the cost is $1$ and the payoff is $X_1$ where $X_1 ...
QMath's user avatar
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1 vote
1 answer
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Are there any heavy tailed distributions available for GAMM?

I have a gamm that looks to be heavy tailed according to the qqplot so I'd like to account for this. According to this page things like scaled t distributions for heavy tailed data are only available ...
adkane's user avatar
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1 vote
1 answer
67 views

How to estimate how heavy a tail is?

Suppose I have data coming from a single variate distribution. I want to estimate how heavy the tail of the distribution is. For example, if the data comes from the Zipf distribution, I would want the ...
user2316602's user avatar
14 votes
1 answer
922 views

Intuitive explanation for the fat tails of the t-distribution

Given some standard assumptions, the test statistic $$ \frac{\Delta\bar{X}}{\sigma/\sqrt{N}} $$ is normally distributed if $\sigma$ is known and t-distributed if $\sigma$ has to be estimated from the ...
monade's user avatar
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0 votes
0 answers
78 views

How Anderson-Darling test with Braun's method works in R?

I generally work with skewed and heavy-tailed lifetime distributions, so I need to check the goodness-of-fit of certain data to such distributions. After some exploration, I came to know that the AD ...
DevD's user avatar
  • 115
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0 answers
26 views

Characterizing/Estimating heaviness of tails using ratio of moments

For any probability distribution function (PDF), $p(x)$, which has finite moments $\left<X^k\right>$ defined upto $k=N$, is it possible to say something about the heavyness of the tails by ...
user35952's user avatar
  • 101
2 votes
1 answer
204 views

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 ...
hipHopMetropolisHastings's user avatar
2 votes
1 answer
92 views

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 ...
Tim's user avatar
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0 votes
0 answers
173 views

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 ...
Alex Il's user avatar
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1 vote
2 answers
552 views

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....
cgo's user avatar
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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 ...
user363270's user avatar
2 votes
1 answer
144 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 ...
DevD's user avatar
  • 115
2 votes
1 answer
527 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 ...
RoBeDo's user avatar
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3 votes
1 answer
2k 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 ...
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2 votes
0 answers
58 views

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. ...
user1424739's user avatar
1 vote
0 answers
377 views

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 $95\%$ ...
JLee's user avatar
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2 votes
1 answer
530 views

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.
Melchizedek McAaron's user avatar
3 votes
1 answer
671 views

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)} = ...
Igalala's user avatar
  • 119
1 vote
1 answer
11k views

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 ...
Som Naik's user avatar
1 vote
0 answers
239 views

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 ...
SGE's user avatar
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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 ...
linguisticturn's user avatar
3 votes
1 answer
94 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 ...
linguisticturn's user avatar
9 votes
1 answer
630 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 &...
Blg Khalil's user avatar
4 votes
1 answer
444 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}...
Blg Khalil's user avatar
2 votes
0 answers
2k views

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 ...
jason's user avatar
  • 21
0 votes
0 answers
46 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 ...
Easy Points's user avatar
1 vote
0 answers
389 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 ...
Locoric Polska's user avatar
2 votes
1 answer
192 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 ...
HeavyTailedH2O's user avatar
1 vote
0 answers
552 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 ...
Djole's user avatar
  • 11
0 votes
0 answers
371 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 ...
Noam_I's user avatar
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0 votes
0 answers
37 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!
Mer's user avatar
  • 1
0 votes
0 answers
613 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 ...
DJL's user avatar
  • 1
0 votes
1 answer
125 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?
user avatar
2 votes
0 answers
420 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 ...
develarist's user avatar
  • 3,987
3 votes
1 answer
277 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 ...
Bashir's user avatar
  • 31
15 votes
4 answers
3k 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?
user321627's user avatar
  • 4,428
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 ...
Marco De Virgilis's user avatar
2 votes
1 answer
560 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 ...
OGV's user avatar
  • 133
1 vote
0 answers
74 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 ...
nluckn's user avatar
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8 votes
0 answers
323 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 ...
Sextus Empiricus's user avatar
3 votes
0 answers
807 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 ...
Emilie Durosoir's user avatar
0 votes
0 answers
263 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: ...
Berthrand Eros's user avatar
1 vote
0 answers
46 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 ...
syehtre's user avatar
  • 71
9 votes
2 answers
10k 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 ...
mmill's user avatar
  • 93
0 votes
1 answer
448 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 ...
Dr.FishGirl's user avatar
2 votes
2 answers
5k 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 ...
shong's user avatar
  • 103
1 vote
1 answer
747 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 ...
user11534659's user avatar
1 vote
1 answer
737 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 ...
0range's user avatar
  • 131
7 votes
3 answers
2k 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 ...
Eric Kim's user avatar
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