Generate random normal distribution with kurtosis greater than 3

Question

A normal distribution has a kurtosis of 3. With an increase in outliers in the distribution, the tails become "fat" and the kurtosis increases beyond 3.

How do I generate a random distribution between two numbers with kurtosis greater than 3 (preferably around 5-7)?

Imports

import numpy as np
import scipy.stats import kurtosis

Random Uniform between 0.01-0.10

# Random Uniform Distribution
runif = np.random.uniform(0.01, 0.10, 10000)

kurtosis(runif, fisher=False)

1.8124891901330156

Random Normal between 0.01-0.10

lower = 0.01
upper = 0.10
mu = (upper)/2
sigma = 0.01
N = 10000
retstats = scipy.stats.truncnorm.rvs((lower-mu)/sigma,(upper-mu)/sigma,loc=mu,scale=sigma,size=N)

mean = .05
stdev = .01  # 99.73% chance the sample will fall in your desired range

values = [gauss(mean, stdev) for _ in range(10000)]

kurtosis(values, fisher=False)

3.015004351756201

Random Normal with fat-tails between 0.01-0.10

???

Answer 1

Since kurtosis is not one of the parameters of the normal distribution function, you have to use another method that generates a function that approximates the normal distribution. It gets complicated. Have a look at this: https://stats.stackexchange.com/questions/43482/transformation-to-increase-kurtosis-and-skewness-of-normal-rv

The link above gives examples using R (sigh) code, but I think it is straightforward enough to allow you to code the equivalent in Python. This is one of several expansions (ie layering of functions) that I am aware of that allow you to achieve this.

Unfortunately, no easy solution that I am aware of.

Answer 2

A normal distribution always has a kurtosis of 3. A uniform distribution has a kurtosis of 9/5. Long-tailed distributions have a kurtosis higher than 3. Laplace, for instance, has a kurtosis of 6. [Note that typically these distributions are defined in terms of excess kurtosis, which equals actual kurtosis minus 3.] See the table here: http://mathworld.wolfram.com/KurtosisExcess.html

By cutting off the tails, however, you only reduce the kurtosis. By cutting tails, it is impossible to generate a normal distribution with kurtosis higher than 3. In order to generate a distribution with limited range and high kurtosis, you will need to ensure that the cut has a minimal effect on the tails and start with a long-tailed (not normal) distribution. Colloquially, you'll need to have a very spiky distribution. I produce one below using Laplace with a small exponential decay parameter.

import numpy as np                                                                       
from scipy.stats import kurtosis                                                         

min_range = 0.01                                                                         
max_range = 0.10                                                                         
midpoint = (max_range + min_range)/2                                                     
samples = 10000                                                                          

def filter_tails(x):                                                                     
    return x[(x >= min_range) & (x <= max_range)]                                        

runif = np.random.uniform(min_range, max_range, samples)                                 
value = kurtosis(filter_tails(runif), fisher=False)                                      
print(f"uniform kurtosis = {value}")                                                     

sigma = 0.01                                                                             
runif = np.random.normal(midpoint, sigma, samples)                                       
value = kurtosis(filter_tails(runif), fisher=False)                                      
print(f"gaussian kurtosis = {value}")                                                    

exponential_decay = 0.001                                                                
runif = np.random.laplace(midpoint, exponential_decay, samples)                          
value = kurtosis(filter_tails(runif), fisher=False)                                      
print(f"laplace kurtosis = {value}")

Running the script, I get:

uniform kurtosis = 1.8011863970680828
gaussian kurtosis = 3.0335178694177785
laplace kurtosis = 5.76290423111418