Questions tagged [heavy-tailed]

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41
votes
3answers
14k views

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 ...
14
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1answer
2k views

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 ...
10
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3answers
7k views

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 ...
8
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3answers
114 views

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 $...
7
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2answers
382 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 ...
6
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3answers
6k views

what is the meaning of 'tail' of kurtosis?

There are two kurtosis types : positive(leptokurtic) and negative(platykurtic). leptokurtic is heavy tailed, and platykurtic is thin tailed. But leptokurtic is more thinner and pointy than platykurtic ...
6
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1answer
858 views

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) \...
6
votes
1answer
2k views

How is the tail of a distribution defined (about heavy-tailed distributions)?

Some distributions are said to be heavy-tailed. It seems that one definition of a heavy-tailed distribution is that its tails are heavier than the tails of an exponential distribution. However, how ...
6
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2answers
155 views

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 ...
5
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3answers
866 views

Fat tail? Short tail? Long tail? Where do I go from here?

I am running a linear mixed model with 4 fixed factors and 1 random factor. The response variable is %growth and it has negative values (some of my animals shrunk). The problem I'm having is the ...
4
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3answers
1k views

Are two asymptotic values enough to fail the test of normality?

Looking at this post I started to wonder about the gestalt interpretation of the QQ plots generated by qqnorm in R. Here's the plot to avoid having to go to the ...
4
votes
1answer
167 views

Besides the Pareto and Zipfian distributions, which distributions obey the power-law?

I need a list of distributions that obey the power-law, beside the commonly used Pareto and Zipfian distributions. A comprehensive list or a reference to a comprehensive list will be particularly ...
3
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2answers
3k views

How to best plot a heavy vs light tailed distribution in R

I want to create a plot that overlays a heavy vs light tailed distribution as an example and am trying to figure out the best way to do this. I can plot gamma distribution which is light tailed and a ...
3
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3answers
65 views

If the best-fitting distribution has infinite variance, should low observed variance be troubling?

Suppose you have observations which, over the observed range of outcomes, are well-fitted by some distribution like the Pareto that, for certain parameter values, has a an infinite variance. For ...
3
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2answers
327 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 ...
3
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2answers
370 views

What is the heaviest tail possible for a continuous normalizable distribution?

The heaviest tailed smooth normalizable continuous distributions that I am familiar with are those with fat power-law tails $\frac{1}{x^{1+\alpha}}$, e.g. a Pareto with $\alpha\rightarrow 0^+$ or a ...
3
votes
1answer
83 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 ...
3
votes
1answer
253 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 ...
2
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2answers
180 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 ...
2
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2answers
425 views

inferring heavy-tail distribution from finite sample of histogram data

I have some data in the form of bins and counts. Here is one complete non-truncated example: ...
2
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1answer
64 views

On the existence of a heavy tailed c.d.f. given a condition based on the CLT

My question is about the existance of a heavy tailed distribution $F$ such that: given two i.i.d. samples $\{X_1,\ldots,X_n\}$ and $\{Y_1,\ldots,Y_n\}$, from $F$, we have $$\sqrt{n}\frac{\hat{\mu}_X -...
2
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1answer
93 views

Fractional moments of the Laplacian distribution larger than of the normal

How can I show that the fractional moments of the (unit variance) Laplacian distribution are higher than of the standard normal distribution, for moments higher than 2? Formally, if $l \sim Laplace(...
2
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2answers
513 views

Modeling web traffic distribution

I have to test some functionalities of a software. I need to simulate the web traffic to a server in terms of the numbers of access. What it is the most suitable function distribution for this task?
2
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1answer
781 views

Is there a binomial regression model that captures data with fat tails?

Specifically, are there any binomial regression models that use a kernel with heavier tails and higher kurtosis than the standard kernels (logistic/probit/cloglog)? As a function of the linear ...
2
votes
1answer
43 views

A random variable $X$ on $(0,\infty)$ which behaves like Exp for small $x$ and Pareto for large $x$

Are there any examples of distributions which behave like Exponential for small values and like Pareto for large values. $$\ln \mathbb{P}[X>x] \sim -\lambda x, \qquad \text{ for } x \text{ small}, ...
2
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1answer
566 views

How are errors terms calculated in GARCH model by rugarch package?

I am fitting a GARCH(1,1) model to the data and want to look at the innovation distribution. ...
2
votes
1answer
1k views

Sampling from a long tailed distribution

I have about 500K records from a dataset that consists of counts (so every record is a count of something, like the number of attempts by an IP address to connect to a website). I know, a priori, ...
2
votes
1answer
3k views

Difference between long tail and short tail distribution?

I want to understand the difference between these distribution types. What is the difference between a long and short-tailed distribution?
2
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1answer
342 views

Bounding tail probabilities of the Poisson Binomial Distribution?

In a problem I'm working on, I have Bernoulli random variables $X_1,X_2,\dots,X_k$ ($k$ is odd) and I am interested in their sum $Y = \sum_{i=1}^k X_i$. In this problem, $P(X_i=1) = p_i$ and $P(X_i=0) ...
2
votes
1answer
49 views

Fitting upper and lower student-t separately

I'm doing some research on fitting marginal distributions to data, in particular distributions with a focus on heavy tails. In one report, albeit several years old, the authors suggested fitting a ...
2
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0answers
89 views

Parametric modelling of survival data: when there is no event in a long tail, is there information?

I have a survival analysis question. Let's take a look at the below curve: For the red curve, there are no more events after 48 months. For the black curve, there are no more events after 60-ish ...
2
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0answers
123 views

What is the necessity of log concave/ convex for tail distributions?

What is the impact of log concave density for light tailed distribution? How it could be substantiated with real time example? Most of the books and research papers highlights that a distribution ...
2
votes
0answers
120 views

Predictive modeling of an complex panel of heavy-tailed data

I am struggling to develop a sensible strategy or protocol for the predictive modeling of a complex set of data. Apologies in advance for the indeterminate nature of some of this description but it’s ...
2
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0answers
151 views

Exponent value in long tail phenomenon of online store

I've read (pdf) that there is a long-tail phenomenon (or generalized Zipf law) in popularity of retail items: If I sort items for sale by their popularity, I will see a long tail of not very popular ...
2
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2answers
2k views

Linear Regression with heavy tailed noise

The model is linear $y_i = a\cdot x_i + b + e_i,~ i = 1,2,\ldots,N $. It is given that the noise is heavy tailed. However the distribution of noise conditional on $x$ is the same for all data points. ...
1
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1answer
55 views

Decomposition of the probability of the sum

I cannot understand how is gotten the following decomposition. Supposing that $X_1,...,X_n$ random variables i.i.d with heavy tailed distribution $S_n=\sum_{i=1}^nX_i$ In the article that I m ...
1
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2answers
418 views

Required number of random numbers for using Central Limit Theorem

I wanted to know how many i.i.d random variables have to be summed in order to be able to use Central Limit Theorem. I know it varies depending on the distribution, but does there exist any number $N$,...
1
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1answer
165 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). ...
1
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1answer
139 views

Tail equivalence for heavy-tailed data

Is it possible for a distribution $F(x)$ that has not a Pareto ($G(x)$) right tail equivalence to fit well heavy-tailed data? That is, to have ${\lim_{x\rightarrow\infty}}\frac{1-F(x)}{1-G(x)}=0$ ...
1
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1answer
443 views

Extreme Value Theory and heavy (long) tailed distributions

I'm analyzing data about which I have a strong suspicion that it is self-similar (Hurst parameter ranging from 0.60 to 0.78 depending on estimation method and sample sequence). I also observe high ...
1
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1answer
348 views

Heavy-tailed distribution with closed-form ML fit from data

Which (if any) heavy-tailed distributions can we compute the maximum likelihood parameters of, given some data to fit the distribution to?
1
vote
1answer
34 views

Are the family of stable distributions differentiable everywhere on the real line?

Are stable distributions smooth enough for each index of stability $\alpha$ between 0 and 2, and skewness parameter $\beta$ between 0 and 1? Where there any papers that mention this?
1
vote
1answer
287 views

Distributions that being to domain of attraction of a stable law that are not unimodal?

I was wondering whether there are any distribution that belongs to the domain of attraction of a stable law that is not unimodal. It is known that distribution in that law converge to a stable ...
1
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1answer
617 views

Regression plot and function for: heavy-tailed probability distribution

I've got data points from a simulation as coordinates in a text-files like so: ...
1
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0answers
14 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 ...
1
vote
1answer
61 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 ...
1
vote
1answer
179 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 ...
1
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0answers
59 views

Determining the distribution of data

Hi, I am a student learning financial modelling. I would like some help in determining the distribution of the data given the plots above. I am reluctant to assume normal distribution of the data due ...
1
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1answer
137 views

Does the normality assumption hold? Is this an outlier?

I am trying to fit a multiple linear regression (OLS) model with IPO underpricing as dependent variable. As part of my master thesis I would like to analyze the effect of venture capital ...
1
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0answers
52 views

Rigorous threshold determination for heavy-tailed data?

Overview: I'm trying to design a change-point detection system for univariate, non-normal, skewed, heavy-tailed data that I believe is generated by a stable random process (i.e. stable-distributed ...