# Negative binomial with continuous parameters in python

So scipy doesn’t support a negative binomial for a continuous r parameter. The expression for its pdf is $P(k)=\frac{\Gamma(k+r)}{k!\,\Gamma(r)} (1-p)^rp^k$. I coded a small class which computes the pdf and is also able to find MLE estimates for p and k given some data. It relies on the mpmath arbitrary precision library since the gamma function values can get quite large and overflow a double. It might be useful to someone so here’s the code below.

import scipy.special as special
import scipy.optimize as optimize
import numpy as np
import mpmath

class negBin(object):
def __init__(self, p = 0.1, r = 10):
nbin_mpmath = lambda k, p, r: mpmath.gamma(k + r)/(mpmath.gamma(k+1)*mpmath.gamma(r))*np.power(1-p, r)*np.power(p, k)
self.nbin = np.frompyfunc(nbin_mpmath, 3, 1)
self.p = p
self.r = r

def mleFun(self, par, data, sm):
'''
Objective function for MLE estimate according to
https://en.wikipedia.org/wiki/Negative_binomial_distribution#Maximum_likelihood_estimation

Keywords:
data -- the points to be fit
sm -- \sum data / len(data)
'''
p = par
r = par
n = len(data)
f0 = sm/(r+sm)-p
f1 = np.sum(special.psi(data+r)) - n*special.psi(r) + n*np.log(r/(r+sm))
return np.array([f0, f1])

def fit(self, data, p = None, r = None):
if p is None or r is None:
av = np.average(data)
va = np.var(data)
r = (av*av)/(va-av)
p = (va-av)/(va)
sm = np.sum(data)/len(data)
x = optimize.fsolve(self.mleFun, np.array([p, r]), args=(data, sm))
self.p = x
self.r = x

def pdf(self, k):
return self.nbin(k, self.p, self.r).astype('float64')