Fit a distribution by maximum likelihood

In this example we are going to estimate the parameters of a parametric by generic numerical optimization of the likelihood.

The likelihood of a sample according to a parametric density function is:

from __future__ import print_function
import openturns as ot
import math as m
ot.Log.Show(ot.Log.NONE)

Create data from a normal PDF with , .

sample = ot.Normal(4.0, 1.5).getSample(200)

Create the search interval of (, ) : the constraint is .

lowerBound = [-1.0, 1.0e-4]
upperBound = [-1.0, -1.0]
finiteLowerBound = [False, True]
finiteUpperBound = [False, False]
bounds = ot.Interval(lowerBound, upperBound, finiteLowerBound, finiteUpperBound)

Create the starting point of the research:

• for : the first point,

• for : a value evaluated from the two first points.

mu0 = sample[0][0]
sigma0 = m.sqrt((sample[1][0] - sample[0][0]) * (sample[1][0] - sample[0][0]))
startingPoint = [mu0, sigma0]

Create the estimator from a parametric PDF.

pdf = ot.Normal()
factory = ot.MaximumLikelihoodFactory(pdf)
factory.setOptimizationBounds(bounds)

Set the starting point via the solver.

solver = factory.getOptimizationAlgorithm()
solver.setStartingPoint(startingPoint)
factory.setOptimizationAlgorithm(solver)

Estimate the parametric model.

distribution = factory.build(sample)
str(distribution)

Out:

'Normal(mu = 3.94775, sigma = 1.49821)'

Retrieve the estimated parameters.

print(distribution.getParameter())

Out:

[3.94775,1.49821]