# Questions tagged [kurtosis]

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

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1answer
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 ...
0answers
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 ...
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 ...
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 ...
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 ...
1answer
37 views

1answer
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### 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 ...
0answers
37 views

### 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 ...
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') ...
0answers
28 views

### 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 ...
0answers
33 views

### 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 ...
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 ...
4answers
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?
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 ...
0answers
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 ...
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 ...
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^{-...
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....
0answers
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://...
0answers
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 ...
0answers
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 ...
0answers
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 ...
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 ...
0answers
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 ...
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 ...
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 ...
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-...
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 ...
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 ...
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 ...
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 ...
0answers
61 views

### 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 ...
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{\...
1answer
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 ...