Fitting negative binomial in python

Question

In scipy there is no support for fitting a negative binomial distribution using data (maybe due to the fact that the negative binomial in scipy is only discrete).

For a

Answer 1

Not only because it is discrete, also because maximum likelihood fit to negative binomial can be quite involving, especially with an additional location parameter. That would be the reason why .fit() method is not provided for it (and other discrete distributions in Scipy), here is an example:

In [163]:

import scipy.stats as ss
import scipy.optimize as so
In [164]:
#define a likelihood function
def likelihood_f(P, x, neg=1):
    n=np.round(P[0]) #by definition, it should be an integer 
    p=P[1]
    loc=np.round(P[2])
    return neg*(np.log(ss.nbinom.pmf(x, n, p, loc))).sum()
In [165]:
#generate a random variable
X=ss.nbinom.rvs(n=100, p=0.4, loc=0, size=1000)
In [166]:
#The likelihood
likelihood_f([100,0.4,0], X)
Out[166]:
-4400.3696690513316
In [167]:
#A simple fit, the fit is not good and the parameter estimate is way off
result=so.fmin(likelihood_f, [50, 1, 1], args=(X,-1), full_output=True, disp=False)
P1=result[0]
(result[1], result[0])
Out[167]:
(4418.599495886474, array([ 59.61196161,   0.28650831,   1.15141838]))
In [168]:
#Try a different set of start paramters, the fit is still not good and the parameter estimate is still way off
result=so.fmin(likelihood_f, [50, 0.5, 0], args=(X,-1), full_output=True, disp=False)
P1=result[0]
(result[1], result[0])
Out[168]:
(4417.1495981801972,
 array([  6.24809397e+01,   2.91877405e-01,   6.63343536e-04]))
In [169]:
#In this case we need a loop to get it right
result=[]
for i in range(40, 120): #in fact (80, 120) should probably be enough
    _=so.fmin(likelihood_f, [i, 0.5, 0], args=(X,-1), full_output=True, disp=False)
    result.append((_[1], _[0]))
In [170]:
#get the MLE
P2=sorted(result, key=lambda x: x[0])[0][1]
sorted(result, key=lambda x: x[0])[0]
Out[170]:
(4399.780263084549,
 array([  9.37289361e+01,   3.84587087e-01,   3.36856705e-04]))
In [171]:
#Which one is visually better?
plt.hist(X, bins=20, normed=True)
plt.plot(range(260), ss.nbinom.pmf(range(260), np.round(P1[0]), P1[1], np.round(P1[2])), 'g-')
plt.plot(range(260), ss.nbinom.pmf(range(260), np.round(P2[0]), P2[1], np.round(P2[2])), 'r-')
Out[171]:
[]