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Questions tagged [kurtosis]

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

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Kurtosis of b(n,p) - binomial distribution

So I have this problem that I’m trying to do. I been at this for hours. It’s to find the kurtosis of a binomial distribution. So far, I have that M’’’’(0) = $n[(n-1)(n-2)(n-3)p^4 + 6(n-1)(n-2)p^3 +7(n-...
maria guallpa's user avatar
8 votes
4 answers
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Estimate Box-Cox Transformation Lambda Using Skewness and Kurtosis

I would be interested in a method to find an appropriate Lambda parameter for the Box-Cox transformation based on only the skewness and the kurtosis of a given sample. I.e, if the skewness and ...
Hiro's user avatar
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Derivation of a dynamical Generalized Pareto distribution

I'm currently reading a paper for my master thesis on the tail index estimation for asset returns using the peak over threshold method. In this paper the authors introduce the cumulative distribution ...
data_science_101's user avatar
2 votes
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Why do the skewness and kurtosis formulae have powers of the variance in the denominator?

We calculate the variance as the centered 2nd moment $E[(X-\mu)^2]$. So when it comes to the skewness and kurtosis, why are the 3rd and 4th moments divided by the 3rd and 4th powers of $\sigma$? Why ...
ahron's user avatar
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occurence of a n-sigma event in symmetric distribution

Is it possible to approximate the frequency of occurence of a n-sigma event in a symmetrical (skew=0) unimodal distribution with mean/mode/median=0, but with fat tails, with given kurtosis =k. I was ...
dayum's user avatar
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Inequality regarding measure of skewness & kurtosis [duplicate]

The measures of skewness and kurtosis respectively are $b_1=\frac{m_3^2}{m_2^3}$(skewness) and $b_2=\frac{m_4}{m_2^2}$(Kurtosis) where $m_r$ is the central moment of $rth$ order. That is $m_r = \frac{\...
Loves Mathematics's user avatar
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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 ...
andrewH's user avatar
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8 votes
5 answers
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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 $M^k(0)=\mathbb{E}(X^k)$ ...
Kai's user avatar
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6 votes
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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'...
Vilman's user avatar
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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 ...
StrongBad's user avatar
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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, $\bar{X}$ ...
Hiro's user avatar
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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 ...
Hiro's user avatar
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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 ...
Lucy's user avatar
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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....
daruma's user avatar
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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 ...
franz-german's user avatar
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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 ...
The One's user avatar
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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 ...
Rachelf's user avatar
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 $x\to +\infty$...
AndreA's user avatar
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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 ...
David Moore's user avatar
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1 answer
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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 ...
Ali Aizat's user avatar
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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 ...
Angel's user avatar
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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 ...
user62348's user avatar
1 vote
0 answers
71 views

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 ...
denizen of the north's user avatar
1 vote
0 answers
36 views

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., ...
David B's user avatar
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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 ...
Alex Il's user avatar
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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 ...
Alex's user avatar
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1 vote
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25 views

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 ...
KhanSamnida's user avatar
1 vote
0 answers
34 views

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 ...
jf328's user avatar
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2 answers
223 views

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 ...
Alex's user avatar
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0 answers
64 views

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 ...
STU273's user avatar
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1 vote
0 answers
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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 ...
Neo Avi's user avatar
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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 ...
stack_math's user avatar
-1 votes
1 answer
974 views

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 ...
Lasnik23's user avatar
1 vote
0 answers
257 views

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 ...
danijar's user avatar
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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 ...
PyRsquared's user avatar
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0 answers
51 views

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 ...
Arif Meighan's user avatar
2 votes
0 answers
172 views

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 ...
BeBlunt's user avatar
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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, ...
deppep's user avatar
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0 votes
1 answer
192 views

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 ...
Siebe's user avatar
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0 votes
0 answers
233 views

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, ...
Tony Ennis's user avatar
0 votes
2 answers
90 views

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 "...
Phil_T's user avatar
  • 133
2 votes
2 answers
153 views

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 ...
peter56's user avatar
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4 votes
0 answers
139 views

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 (...
José Carlo's user avatar
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38 views

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 ...
Stats34's user avatar
  • 57
2 votes
0 answers
59 views

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. ...
user1424739's user avatar
2 votes
2 answers
2k views

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?
StatisticsNoobie's user avatar
2 votes
1 answer
162 views

Mean substitution - skewness and kurtosis

I am doing the review questions from Thompson: foundations of behavioral statistics, chapter 4, question 2. I cannot seem to conceptualize the correct answer. Would appreciate any help as I need to ...
Alison Field's user avatar
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0 answers
21 views

Best way to code ratio of two skewed, heavy-tailed continuous variables with zero values present in denominator?

I'm modeling the ratio of total medical cost in one 12 month period and the subsequent 12 month period (total_medical_cost_ratio = post_total_medical_cost/pre_total_medical_cost). The data is skewed, ...
RobertF's user avatar
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