Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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.

Heavy-tailed distributions are probability distributions whose tails are not exponentially bounded; that is, they have heavier tails than the exponential distribution. Examples are the log-normal and Pareto (heavy right tail) and t distributions (both tails heavy). These are distributions for which the moment generating function do not exist for positive argument t.

There are more details and discussion at Differences between heavy tail and fat tail distributions, and at wikipedia. See also Chapter XIII in Applied Probability and Queues.

126 questions
51
votes
3 answers

Which has the heavier tail, lognormal or gamma?

(This is based on a question that just came to me via email; I've added some context from a previous brief conversation with the same person.) Last year I was told that the gamma distribution is heavier tailed than the lognormal, and I've since been…
Glen_b
  • 257,508
  • 32
  • 553
  • 939
28
votes
3 answers

Differences between heavy tail and fat tail distributions

I thought heavy tail = fat tail, but some articles I read gave me a sense that they aren't. One of them says: heavy tail means the distribution have infinite jth moment for some integer j. Additionally all the dfs in the pot-domain of attraction of…
Melon
  • 391
  • 2
  • 4
  • 6
19
votes
2 answers

Heavy-tailed errors in mixed-effects model

I'm relatively new to statistical modelling and `R', so please let me know If I should provide any further information/plots. I did originally post this question here, but unfortunately have not received any responses yet. I am using the lme()…
Dom
  • 191
  • 4
15
votes
4 answers

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
  • 2,511
  • 3
  • 13
  • 49
14
votes
1 answer

Example of heavy-tailed distribution that is not long-tailed

From readings about heavy-, and long-tailed distributions, I understood that all long-tailed distributions are heavy-tailed, but not all heavy-tailed distributions are long-tailed. Could somebody please provide an example of: a continuous,…
toliveira
  • 287
  • 1
  • 9
11
votes
3 answers

t-distribution having heavier tail than normal distribution

In my lecture notes it says, t-distribution looks like normal, though with slightly heavier tails. I understand why it would look normal (because of the Central Limit Theorem). But I am having hard time understanding how to mathematically prove…
hmi2015
  • 223
  • 2
  • 7
9
votes
1 answer

Asymptotic normality of order statistic of heavy tailed distributions

Background: I have a sample which I want to model with a heavy tailed distribution. I have some extreme values, such that the spread of the observations are relatively large. My idea was to model this with a generalized Pareto distribution, and so I…
Erosennin
  • 1,384
  • 17
  • 31
8
votes
1 answer

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 $\mathbb{P}(X > x)$ decay more slowly than those of any…
Blg Khalil
  • 608
  • 5
  • 12
8
votes
2 answers

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 structure of random variables can often be quantified…
Jonas
  • 716
  • 4
  • 12
8
votes
3 answers

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 $\rho \in [-1, 1]$. What does it mean with it says…
FoetDen
  • 81
  • 1
8
votes
2 answers

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, Pareto and log-logistic distributions…
Jonathan Dunne
  • 452
  • 4
  • 15
7
votes
3 answers

Nonhomogeneous Poisson and Heavy tail inter arrival time distribution

What is the relationship between a Nonhomogeneous Poisson process and a process that has heavy tail distribution for its inter arrival times? Any pointer to a resource that can shed some light on this question would be hugely appreciated
MarkSAlen
  • 2,559
  • 5
  • 24
  • 25
7
votes
0 answers

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 complementary cumulative distribution of a random…
Sextus Empiricus
  • 43,080
  • 1
  • 72
  • 161
7
votes
1 answer

Bernstein's inequality for heavy-tailed random variables

It is known that for independent sub-exponential random variables, the following Bernstein-type inequality holds: \begin{align} \mathbb{P}\biggl(\biggl| \sum_{i=1}^N a_i X_i\biggr| >t \biggr) \leq 2 \exp\left[ -c\min \left(\frac{t^2}{K^2 \|…
Steve
  • 287
  • 1
  • 6
6
votes
2 answers

Regression: zeros in heavy-tailed independent variable from quantization

This question is about handling zeros in an independent variable for a regression. In particular, the zeros are not missing data or true zeros, but occur because of quantization. As a concrete example, lets say the observations are cities, and the…
elplatt
  • 190
  • 1
  • 5
1
2 3
…
8 9