[Python] Kumaraswamy-Laplace Distribution using scipy rv_continuous

I am trying to create a script to be able to fit Kumaraswamy-Laplace distribution,

• As defined in this paper, 1
• Inspiring from these posts, 2 3 4
• To replicate market returns fitting like in this study 5

The fitting always end up with NaN error somewhere.

With the following attempt:

import numpy as np
import pandas as pd
import scipy.stats as st
from scipy.special import beta

class kl(st.rv_continuous):

def _pdf(self, x, mu, sigma, p, q):

v = ( x - mu ) / sigma
#I tried two ways of defining of the function
#y = .5 + .5 * np.sign( v ) * ( 1 - np.exp( - np.abs( v ) ) )
#fx = p * q *.5 / sigma * np.exp( - np.abs( v ) ) * y ** ( p - 1 ) * ( 1 - y ** p ) ** ( q - 1 )
fx = np.where( v < 0 ,
p * q * 2. **(-p) * np.exp( p * v ) * (1. - 2. ** ( - p ) * np.exp( p * v)) **(q - 1.),
.5 * p * q * np.exp (-v) * (1.-.5*np.exp(-v))**(p-1.) * (1.-(1.-.5*np.exp(-v))**p)**(q-1.)
)


return fx

def _argcheck(self, mu, sigma, p, q):

s = sigma > 0
p_bool = p > 0
q_bool = q > 0

all_bool = s & p_bool & q_bool

return all_bool


It seems that I always end up with the error ValueError: The function value at x=nan is NaN; solver cannot continue. when attempting to fit the distribution.

Even though, plotting it on given parameters looks correct. I suppose it comes from an overflow, but I am not sure, and I don't know how to solve it!

Could someone please provide some understanding of the fitting and numerical analysis requirements I am missing?

Note: Fitting a Laplace works for the same dataset since it is already defined in the library of scipy.

The code to fit:

initial_guess = [np.mean(sample_data), np.std(sample_data), 1.0, 1.0] # [mu, sigma, p, q]
estimated_params = my_sgt_distribution.fit(sample_data, *initial_guess)


And to plot:

import matplotlib.pyplot as plt
import numpy as np

# Explicitly given parameters (replace with your desired values)
mu = 0
sigma = 1
p = 0.6
q = 0.7

# Create an instance of custom distribution
my_kl_distribution = kl()

# Generate x values for plotting
x_values = np.linspace(-5, 5, 1000)

# Calculate PDF and CDF values
pdf_values = my_kl_distribution.pdf(x_values, mu, sigma, p, q)
cdf_values = my_kl_distribution.cdf(x_values, mu, sigma, p, q)

# Plot the PDF
plt.figure(figsize=(10, 5))
plt.plot(x_values, pdf_values, label='PDF')
plt.title('Probability Density Function (PDF) of Kumaraswamy Laplace Distribution')
plt.xlabel('x')
plt.ylabel('Density')
plt.legend()
plt.grid(True)
plt.show()

# Plot the CDF
plt.figure(figsize=(10, 5))
plt.plot(x_values, cdf_values, label='CDF', color='orange')
plt.title('Cumulative Distribution Function (CDF) of Kumaraswamy Laplace Distribution')
plt.xlabel('x')
plt.ylabel('Probability')
plt.legend()
plt.grid(True)
plt.show()

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