Questions tagged [kurtosis]

a normalized fourth moment of a distribution or dataset, or other aspects of fat tails

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20 views

How the kurtosis value can determine the unhealthy event

I read a paper regarding air quality. The author mentioned that "The PM10 pollutant also shows the highest of kurtosis value, which indicates that it appears most frequently in unhealthy ...
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7 views

Calculating the moments of ordered timeseries

I'm interested in calculating the moments of ordered data (monthly rainfall) so that calculation takes into account the order in which the values accumulate. For the mean and variance order shouldn't ...
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1answer
37 views

Is my understanding of the Gini plot to detect fat tails correct?

I'm trying to reproduce the following plot: which was generated on the Danish dataset of fire insurance claims using the ineq() function (a wrapper for functions ...
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1answer
75 views

Fat tails equal higher probability of non-extreme values according to Nassim Taleb?

I just came across the following passage written by Nassim Taleb Link: The fattest tail distribution has just one very large extreme deviation, rather than many departures form the norm. [...] if we ...
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1answer
49 views

Why is a Fréchet distribution slowly varying, and what is the intuition behind it?

The Fréchet distribution: $$\Phi_\alpha(x)=\begin{cases}0 & & x\leq 0,\\[6pt]e^{-x^{-\alpha}} & & x>0,\end{cases}$$ is regularly varying as stated here (page 19): It is not ...
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1answer
37 views

Why does the tail of a Fréchet distribution decay as a power law?

The Fréchet distribution: $$\Phi_\alpha(x)=\begin{cases}0,\; x\leq 0\\e^{-x^{-\alpha}},\; x>0\end{cases}$$ shows a power law decay at the tail (survival): $$1- \Phi_\alpha(x) = 1 -e^{-x^{-\alpha}}\...
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2answers
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What is an intuitive explanation for Q90 (X+Y) > Q90(X) + Q90(Y) in fat-tailed variables. Non Subadditivity

In a business situation, management keeps a reserve of money for a 'rainy day' just in case costs are more than expected. The 90th percentile ($Q_{90}$ in the following) might be an indicator of how ...
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3answers
50 views

Excess kurtosis in the error term

I am modelling the residual errors terms in my time-series model using Student's $t$-distribution. This distribution allows for more kurtosis (‘heavy tailedness’) than the Gaussian distribution. ...
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How to interpret cokurtosis results?

I calculated a cokurtosis matrix of the returns of multiple stocks. I obtained the following results but not sure how to interpret them. I understand that when two random variables exhibit a high ...
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31 views

Low kurtosis (platykurtic) distribution transform to normal distribution

I use QQ-plot on residuals in time series analysis and find the distribution is not normally distributed, but somehow thin-tail distributed: I know that if the residuals are not a normal distribution,...
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71 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 ...
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1answer
37 views

Kurtosis comparison with different range

I simulated an agent based modeling that individuals can end up with a different range of attitudes. Individuals in Model 1 end up with the attitude range -3 +3 Individuals Model 2 end up with the ...
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1answer
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Replicate distribution image kurtosis

I am trying to replicate one of the images from this post. In particular I want to plot something similar to this image in matlab: I.e. I want these three differently peaked distributions, where the ...
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1answer
87 views

Prove that Kurtosis is at least one more than the square of the skewness

Wikipedia claims it, and on reading the paper that it linked I found that the proof that was written there was quite difficult. Is there a simple proof possible for this identity? The proof given in ...
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1answer
70 views

How to efficiently calculate Skewness and Kurtosis of data having value with repetitions?

I am doing some research on stock data and somewhat new to advanced statistics. The data is for example Price --> Volume 100 ---> 1234 101 ---> 123456 102 ---> 6678 103 ---> 3456 104 ---...
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Fat tails vs heavy tails

There is a lot of confusion about these two terms also in previous asked questions. I am interested in this part of the Wikipedia article (I am only considering right-tailed distributions): https://en....
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Dealing with heavy-tailed residuals and clustered-like relationship

I have two financial time-series (daily log returns) and used the OLS (ordinary least squares) to fit a linear regression model. We can see that the scatterplot of two time-series shows points in a ...
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1answer
98 views

Geometric Brownian motion with target skewness and kurtosis

The Cholesky inversion method can be adopted to set a target correlation matrix when artificially generating a multivariate geometric Brownian motion dataset Can the moments of a univariate GBM be ...
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1answer
20 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?
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1answer
172 views

How to set shape parameters for Johnson SU distribution in python scipy? [closed]

The Johnson SU distribution has 4 parameters ($\delta,\gamma,\lambda,\xi$), but scipy.stats.johnsonsu only has 2 parameters ($a,b$). Why the difference. how can I ...
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2answers
233 views

Long Tail Distribution and importance in classification/prediction problems?

I came across this interview question online: Explain what a long tailed distribution is and provide three examples of relevant phenomena that have long tails. Why are they important in classification ...
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85 views

Derivation of skewness and kurtosis algebra of random variables

In algebra of random variables, the symbolic rule for computing variance of random variable $X\in\mathbb{R}^{n\times p}$ multiplied by a coefficent vector, $a\in\mathbb{R}^p$, is $$\text{Var}(X\cdot a)...
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1answer
205 views

Why aren't the coskewness and cokurtosis matrices square like the covariance matrix?

The variance-covariance matrix is shaped $p\times p$, whereas the co-skewness matrix is shaped $p\times p^2$ and the co-kurtosis matrix is $p\times p^3$. Why is this, given that skewness and kurtosis ...
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How to find eigenvalues and eigenvectors of the cokurtosis matrix?

Kurtosis is the fourth statistical moment of a random variable's distribution. Unlike the variance-covariance matrix $\Sigma$, which had a shape of $p\times p$, the kurtosis-cokurtosis matrix is ...
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1answer
50 views

Connection between copula tail dependence and kurtosis?

Tail dependence in the pseudo-observations of bivariate copula imply that extreme upper or lower samples move together in some way suggesting correlation between the two marginals' (variables') ...
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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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How is the kurtosis on the Cullen and Frey graph in the R package fitsidtrplus calculated?

I am using the package fitdistrplus in R to fit distributions to my data. My first step was to check my data against the Cullen and Frey graph that is produced using the descdist function. This is ...
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1answer
405 views

Generating distributions with with given variance, skewness, and kurtosis

I would like to generate distributions that are as close to normal as possible, except for the deviations shows below. The options I've located have properties that are related to the families of ...
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766 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?
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2answers
86 views

Why is there a kurtosis condition for joint distributions to be elliptical?

I read that if x1, x2 are 2 random variables with different excess kurtosis, their joint distribution cant be elliptical. Is there an intuition or proof of that? It is not very clear to me. Edit- in ...
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56 views

Use of Kurtosis statistic for understanding lognormality

To help clarify my understanding of this statistic, I'd appreciate feedback on the rationale presented here. Assume we have a distribution that seems potentially lognormal. Checking the median against ...
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1answer
62 views

Relationship between the number of moments and Tail of the distribution?

While studying about kurtosis and extreme value theory, I came across the concept of tails of the distribution. So I wanted to ask that why is it such that distribution with higher number of moments ...
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2answers
281 views

Is power law distribution for extreme event special like normal distribution?

The power-law distribution is defined as below in Wikipedia article: The most extreme case of a fat tail is given by a distribution whose tail decays like a power law. $$ \mathrm{Pr}[X>x] \sim x^{-...
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1answer
116 views

What are the basic affine transformations on a distribution for the various moments?

To change a distribution's mean, we do the translation affine transformation. E.g. add a constant to every data point. To change a distribution's variance, we do the scale affine transformation. E.g....
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42 views

What is the maximum entropy distribution with a mean of 0, stdev of 1, skew of 1 (instead of zero), and kurtosis of 1 (instead of 3)?

As we know, there are an infinite number of distributions with mean of zero and variance of one. One of the special things about the normal distribution is that is has the maximum entropy. (https://...
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87 views

Standardize first four moments: match sample moments with population moments

Let $X$ be a sample from $N(0,1)$ and $m$, $v$, $s$, $k$ denote sample mean, variance, skewness and kurtosis of $X$. I want to transform the sample $X$ such that the sample moments equal the true ...
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42 views

Standard skew / kurtosis distribution

What is the common choice of distribution to fit skewed data? I am in the process of studying the effect of skew on qqplot (and varies other). So I want to know the standard version of skewed ...
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31 views

Is the leptokurtic distribution always time varying and what will the impact be on a weak stationary time series

I hope this alteration will make more sense. A normal distribution is defined by its first and second moment. When these are finite, as in the case of weak stationarity, then the weak stationarity ...
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1answer
723 views

Sufficient Statistic for Normal Distribution | Mean, Variance & Kurtosis

I have seen multiple times that a normal distribution is fully specified by mean and variance. It is obvious that the third moment is not necessary for a perfect normal distribution as it is 0. I ...
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39 views

Calculate fifth and sixth polynomials for Headrick (2002) method for non-normal multivariate distribution

I am trying to perform a 3-variable correlated multivariate Monte Carlo simulation. As the asset class returns are non-normal, I found the following function ...
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0answers
41 views

Using quantile regression results to select and weight variables for models

Linear regression is commonly used to identify predictor(s) (e.g., scores on cognitive ability or personality assessments) of job performance. Typically, predictors that exhibit a significant ...
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2answers
748 views

PDF Formula for distribution with mean, standard deviation, skew, and kurtosis

What would the probability density function be for a graph with input variables: mean, standard deviation, skewness, and kurtosis? For example, if the inputs were confined only to mean and standard ...
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1answer
100 views

What does it mean if data follows a t-distribution?

If I make some experiment, which I repeat $N$ times, and look at the distribution of the experiment. I would usually expect (and hope) for a Gaussian or normal distribution. If however my data is T-...
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3answers
387 views

Test which distribution has a “longer tail”

I have measured two non-negative random variables, A and B. Their true underlying probabilities are unknown, however, it may be assumed that the probabilities are largest at zero and monotonically ...
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1answer
1k views

How to reduce kurtosis of data

I'm trying to reduce the kurtosis of my dataset and make it approximately Gaussian, with a common-sense uni-modal shape. The raw data looks like this: I first tried ...
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0answers
134 views

Skewness and kurtosis using quantiles and mean/variance

I would like to ask if there is a way to get skewness and kurtosis measures if we only know the distribution's mean, variance, and certain quantiles. Basically, the problem that I am facing is I have ...
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1answer
154 views

Why Pareto distribution has pointy head? How its symmetrical version looks like?

I want to use Pareto distribution for the stock price probability distribution. They say (B. Mandelbrot and N. Taleb Mild vs Wild Randomness) it represents the price changes, especially the tail ...
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What is the connection between fat tailedness of student t distribution and sparsity inducement in lower dimensions in context of t-sne

I have read that the t distribution is a heavy tailed distribution in comparison to Normal distribution. Also some say that heavy tailed distributions help in creating more sparsity. My question is ...
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1answer
405 views

Notation for skewness and kurtosis

Traditionally (see Johnson et al., 2005, for instance), the (population) standardised skewness and kurtosis can be denoted as $\sqrt{\beta_1}= \frac{\mu_3}{\sigma^3}\;\;$ and $\;\;\beta_2= \frac{\...
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72 views

Better measure of tail extremity than kurtosis

According to Wikipedia, the only correct interpretation of kurtosis is "tail extremity," the logic being that datapoints within one standard deviation of the mean are raised to the fourth power and ...

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