# 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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49 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 ...
23 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. ...
1 vote
39 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 $95\%$ ...
224 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.
17 views

### Exponentially distributed bins for viewing and analysing the tails of a histogram

Is there a rule for visualising and analysing density estimates of heavy tailed observations on a log-log scale? For example, equispaced bins and exponentially distributed bins: ...
81 views

175 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}...
379 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 ...
35 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 ...
1 vote
145 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 ...
87 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
173 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 ...
147 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 ...
32 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!
307 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 ...
26 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? 1 vote
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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 ...
126 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 ...
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?
1 vote
1k 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 ...
323 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 ...
1 vote
68 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 ...
122 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 ...
424 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 ...
138 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: ...
1 vote
31 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 ...
4k 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 ...
192 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 ...
1 vote
1k 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 ...
400 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 ...
1 vote
387 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 ...
853 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 ...
487 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 ...
390 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 ...
274 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 ...
48 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 ...
1 vote
507 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). ...
164 views