# Questions tagged [kurtosis]

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

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### How should I best to use reported stats on the Tippy-top?

Suppose I have a large population, in the millions, drawn from some underlying distribution, which we will take as a member of a known distributional family with unknown parameters. Assume the ...
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### How can we efficiently find the fourth moment of a Poisson distribution?

Suppose we have $X\sim \textrm{Poisson}(\lambda)$ and we know that moment generating function $M(t)=\mathbb{E}(e^{tX})$. How do we use the moment generating function property $M^k(0)=\mathbb{E}(X^k)$ ...
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### Finding a distribution where skewness and kurtosis do not depend on each other. Does it even make sense?

I am simulating non-normal data to investigate how this affects some diagnostical methods that assume normality. In particular I'm interested in seeing how skewness and kurtosis affects the results. I'...
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### Formal testing for differences in kurtosis between two samples when bootstrapping suggests a difference

My question is similar to Testing difference in kurtosis between two samples where a comment suggested Unless you are looking for an enormous difference in kurtosis, it's unlikely any physically ...
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### Variance of Fourth Sample Central Moment [closed]

I am trying to derive a formula for the variance of the fourth sample central moment $m_4=\frac{1}{n}\sum_{i=1}^n (X_i-\bar{X})^4$ (where $X_i$ is the $i$th realization of a random variable, $\bar{X}$ ...
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### Pooled Kurtosis Estimator Using Pooled Cumulant Estimators

I am trying to come up with a statistically sensible pooled kurtosis estimator that is based on pooled cumulant estimators. Specifically, I have unbiased estimators of the second and fourth cumulant ...
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### Can you do a log transformation for excess kurtosis, or is that mainly used for skewness?

I am planning on doing a regression analysis on STATA on the financial performance of private equity funds. On my descriptive statistics, I saw higher levels of kurtosis and skewness. I decreased ...
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### Probability that a sample drawn from one distribution is lower than a sample drawn from another distribution?

Context: we don't know the exact distribution parameters, however in practice we can obtain many samples from each distribution. Case 1: let's say that I have a sample of size N from each distribution....
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### Spliced Distributions Framework for python

There is an article Fat-Tailed Regression Modeling with Spliced Distributions that describes fat-tailed regression modeling by fitting the distribution consisting of N components (different ...
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### Timeseries Anomaly Detection using Rolling Kurtosis?

I'm working on anomaly detection for multiple streaming time series datasets. Due to the vast number of datasets, I'm seeking a scalable, generalized method without resorting to adaptive thresholds ...
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### GAM: mgcv model with kurtosis: Does this need to be solved and how?

I am trying to model CO2 fluxes (fco2) using a number of environmental parameters using a GAM in mgcv. Specifically, I have leaf temperature (tl), vapour pressure deficit (vpd), and transpiration (tr)....
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### What does it mean when dots on a residual vs fitted graph are clumped like a shotgun result? How do I fix it if it needs to be fixed?

Here's the code for these graphs ...
1 vote
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### Non-negative fat-tailed "almost stable" family of distribution with finite mean?

I am looking for a finite-dimensional family of distributions $F_X(x)$ with all the following properties: Supported on $[0, +\infty)$, Fat tailed, i.e. $(1-F_X(x)) \sim x^{-\alpha}$ for $x\to +\infty$...
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### Calculating Kurtosis for Groups Containing Fewer Than 4 Observations

Based on some preliminary exploration, here are some interesting observations about kurtosis for when you're calculating kurtosis for groups that have fewer than 4 observations. First, here's the ...
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### Can I do skewness on multiple standard deviation?

I have a 1000 sample of an electrical test at each 4 different time, so I do simple descriptive statistic to obtain standard deviation at each 4 different time. Can I then use skewness on standard ...
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### Meaning of Skewness and Kurtosis values of Residual Errors in Time Series Forecasting Problem using LSTM

I have developed different kinds of RNNs (such as LSTM,GRU etc.)to predict future values of thermocouple measurements. The residual errors look like they do not follow normal distribution, so I wanted ...
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### Cut off value of +/- 1.5 for Skewness and Kurtosis (Tabachnick & Fidell)

I've read multiple posts/papers citing Tabachnick and Fidell's cut off of +/- 1.5 as the acceptable range for skewness and kurtosis to determine normality; however, I cannot find it in their book. Can ...
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1 vote
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### Inaccuracies due to initial values in GARCH(1, 1) simulation

I'm experimenting with non-normal innovations standard GARCH(1, 1) model $$\epsilon_t = \sqrt h_t z_t$$ $$h_t = \omega + \alpha \epsilon_{t-1} + \beta h_{t-1}$$ Where $E[z_t] = 0$, $E[z_t^2] = 1$, but ...
1 vote
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### Does Bayesian modeling result in fat-tail distributions?

Let's say we have an univariate dataset x that follow a gaussian with parameters (m, s). Under a frequentist methodology, m and s are estimated using MLE and x is modeled as N(m_hat, s_hat). Using the ...
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1 vote
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### Power analysis to detect non-zero skew/kurtosis

Tests exist to determine whether a distribution is normal. For example the Shapiro-Wilk’s method. I'm wondering how to determine whether I'm powered to detect that my distribution is non-normal (e.g., ...
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### A question about the variance of sample variance? [duplicate]

if I have negative kurtosis in my distribution (meaning negative exceeded kurtosis) - which indicates that the tails are lighter than normal distribution tails. Does it also mean that I have more ...
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1 vote
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### Is there a relationship between the alpha of fat tails and kurtosis

From Wikipedia I have the compliment of the CDF parameterized for fat-tails distributions. $$\Pr[X>x] \sim x^{- \alpha}\text{ as }x \to \infty,\qquad \alpha > 0.\$$ In forecasting literature a ...
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1 vote
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### Transform a sample to have target values for the first 3 moments (Equating moments)

I want to transform the first three moments of a sample X of size N to get a new sample Y with moments ($\mu_Y, \sigma_Y, \nu_Y$). The first two moments can be mapped to the target values using the ...
1 vote
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### Can we improve variance estimate by using sample kurtosis?

The sample variance is the minimum variance unbiased estimator and its variance is related to kurtosis $$\operatorname{Var}(S^2) = \frac1n\left(\mu_4 -\frac{n-3}{n-1}\sigma^4\right)$$ Is there some ...
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### Is my data fat tailed in terms of alpha

From Wikipedia I have the compliment of the CDF parameterized for fat-tails distributions. $$\Pr[X>x] \sim x^{- \alpha}\text{ as }x \to \infty,\qquad \alpha > 0.\,$$ Here $\alpha$ is the ...
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### Rejection Sampling Kurtosis and Number of Iterations

I tried to implement a rejection sampling method in python based on something explained during class. The target distribution is the normal distribution and the proposal is the exponential ...
1 vote
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### Normality tests for latent variable in probit regression

I am performing a probit regression where the latent variable y* is conceptually important. I already have the model defined with regressors: categorical variables, quadratic terms, continuous ...
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### Kurtosis is greater or equal to square of skewness plus one [duplicate]

Given is a random variable $X$ with finite fourth moment. Let $\gamma_3$ and $\gamma_4$ denote its skewness and kurtosis respectively. I want to prove that $$\gamma_4\geq 1+ \gamma_3^2$$ I have seen a ...
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### How to deal with high skewness and kurtosis

I have two dependent variables (soccer dataset) that I'm interested in. They have the following skewness and kurtosis: Variable A: % of minutes played --> Skewness: 0.145 | Kurtosis: -1.03 ...
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### How can I standardize a sample to fixed skewness and kurtosis? [closed]

Standardizing a sample of a random variable to mean 0 and standard deviation 1 are common practice. However, I would like to also standardize its skewness to 0 and its kurtosis to 3, but preserve ...
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### Does skewness multiplied by kurtosis have some intuitive meaning?

I understand the individual measures of skewness (how asymmetric a distribution is) and kurtosis (how "large" the tails are in a distribution). For this specific case, I am referring to ...
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### Transformation of discrete data to reduce Skew

In a dataset I have running through on kaggle's tabular playground, some of the measurement columns that are missing values follow a gaussian distribution. I want to fix these missing values by ...
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### Probability distribution with independent expectation, variance, skewness and kurtosis

There are 4 measures for the characterization of the shape of a probability distribution: expectation (1st order raw moment), variance (2nd order central moment), skewness (expression in 3rd and 2nd ...
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### Are Poisson distributions with low mean heavy-tailed?

It is very apparent to me how using the normal distribution to estimate the probability of large, Poisson-distributed events may lead to significant underestimates of the probability of these events, ...
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### F-test and violated assumptions

I am researching the effect of parameters on energy consumption. To determine the effect of parameters, I want to use the R Studio's F-test. In this way I want to investigate if the model with the ...
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### Standard deviation and long-tail data

Sorry, I don't know the math words. I have a data set that looks like the following. I am counting things, and the graph shows how many of each thing there is. for example, I have a LOT of thing #0, ...
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### Is there a way to define significance to say something has "fatter" tails?

I have two distributions of data. The means are about the same, but one data set has longer, fatter tails. Is there a way I can say one distribution has significantly "fatter" or "...
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### Best way to find confidence interval for a mean of a symmetric leptokurtic distribution?

I have a very symmetric distribution with kurtosis of 10 and sample size of more than 100. Here is the Histogram https://ibb.co/ws7vBjd This histogram was obtained by asking participants in the ...
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### Are there distributions for skewness and kurtosis? Similarly to mean (normal) and variance (chi-squared)

My question is really straightforward. The distribution of the sample means approaches a normal distribution (CLT). The distribution of the sample variance approaches a chi-square distribution (...
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### Which method is best?

I have an independent variable which is BMI and catagorized into 1- under weight 2- normal weight 3- over weight 4- obese Sample size is 100 The dependent variable for group 1,2 and 3 is not normally ...
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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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### How can I see Kurtosis in a box plot?

How do I know whether a distribution is leptokurtic or platykurtic by only having the box plot?