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.
145 questions
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Maximum Likelihood estimation for heavy tailed and binned data
I have binned loss data where each bin is defined by:
A minimum loss and maximum loss (the bin boundaries)
A probability of occurrence for that bin
The probabilities across all bins sum to 1. ...
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Why GPD MLE (EVT) fails to estimate Tail Index for Student T?
Why Generalized Pareto Distribution (GPD) MLE estimation of Tail Index fails?
On the chart multiple simulation of the same $\text{StudentT}(\nu=4)$ with tail estimated with GPD estimator (blue lines).
...
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Moment based tail bound for concentration of softmax transform of i.i.d. gaussians
This question is in the same context as this one.
Consider $n$ i.i.d. standard Gaussian random variables, denoted by $X_1, \ldots, X_n$, and I am trying to characterise the concentration of
$$T_n = \...
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Concentration of softmax transform of i.i.d. gaussians
Consider $n$ i.i.d. standard Gaussian random variables, denoted by $X_1, \ldots, X_n$. I am looking to characterise the concentration of functions like $\sum_{i=1}^n X_i e^{-X_i/\tau}$ and $\sum_{i=1}^...
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Where's heavy tail in the distribution of the stock price changes?
There's hypothesis that stock price changes have hybrid distribution - a combination of Normal for Head and Power law for Tail.
The Power law $f(x) = C x^{-\alpha}$ could be identified on log-log plot ...
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Conditional Remaining Time to Event Paradox
The conditional expected time remaining for an event to occur seems to grow with waiting time. This seems either wrong or like some sort of paradox.
Let's take the Lomax distribution as an example. ...
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How can I measure Monte Carlo convergence in distribution with heavy tails?
I'm performing a Monte Carlo study on a simple agent based simulation, and I'm trying to formulate a heuristic for the number of MC samples to use. I'm able to measure convergence of statistics like ...
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GLMM for not so gaussian data
I am having an issue with GLMM and hope you could advice me.
So basically I have data from microscopy experiment of three independent groups (variable: subfolder) nested within 4 experimental ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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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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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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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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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 ...
user142054
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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.
...
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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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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.
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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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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 ...
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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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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 ...
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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 ...
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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 &...
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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}...
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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 ...
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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 ...
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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 ...
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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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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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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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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?
user14351387
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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 ...
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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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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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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 ...
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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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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 more ...
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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 ...
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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 ...
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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:
...
