Get the asymptotic distribution of the estimators

In this example we introduce the buildEstimator method to obtain the asymptotic distribution of the parameters of a fitted distribution obtained from a Factory.

import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt

ot.Log.Show(ot.Log.NONE)

Set the random generator seed

ot.RandomGenerator.SetSeed(0)

The standard Normal

The parameters of the standard Normal distribution are estimated by a method of moments. Thus the asymptotic parameters distribution is Normal and estimated by bootstrap on the initial data.

distribution = ot.Normal(0.0, 1.0)
sample = distribution.getSample(50)
estimated = ot.NormalFactory().build(sample)

We take a look at the estimated parameters :

print(estimated.getParameter())

[0.0353171,0.968336]

The buildEstimator method gives the asymptotic parameters distribution.

fittedRes = ot.NormalFactory().buildEstimator(sample)
paramDist = fittedRes.getParameterDistribution()

We draw the 2d-PDF of the parameters

graph = paramDist.drawPDF()
graph.setXTitle(r"$\mu$")
graph.setYTitle(r"$\sigma$")
graph.setTitle(r"Normal fitting : $(\mu, \sigma)$ iso-PDF")
view = viewer.View(graph)

We draw the mean parameter distribution

graph = paramDist.getMarginal(0).drawPDF()
graph.setTitle(r"Normal fitting : PDF of $\mu$")
graph.setXTitle(r"$\mu$")
graph.setLegends(["PDF"])
view = viewer.View(graph)

We draw the scale parameter distribution

graph = paramDist.getMarginal(1).drawPDF()
graph.setTitle(r"Normal fitting : PDF of $\sigma$")
graph.setXTitle(r"$\sigma$")
graph.setLegends(["PDF"])
view = viewer.View(graph)

We observe on the two previous figures that the distribution is Normal and centered around the estimated value of the parameter.

The Pareto distribution

We consider a Pareto distribution with a scale parameter , a shape parameter and a location parameter . We generate a sample from this distribution and use a ParetoFactory to fit the sample. In that case the asymptotic parameters distribution is estimated by bootstrap on the initial data and kernel fitting (see KernelSmoothing).

distribution = ot.Pareto(1.0, 1.0, 0.0)
sample = distribution.getSample(50)
estimated = ot.ParetoFactory().build(sample)

We take a look at the estimated parameters :

print(estimated.getParameter())

[0.39768,0.696349,0.690277]

The buildEstimator method gives the asymptotic parameters distribution.

fittedRes = ot.ParetoFactory().buildEstimator(sample)
paramDist = fittedRes.getParameterDistribution()

We draw scale parameter distribution

graph = paramDist.getMarginal(0).drawPDF()
graph.setTitle(r"Pareto fitting : PDF of $\beta$")
graph.setXTitle(r"$\beta$")
graph.setLegends(["PDF"])
view = viewer.View(graph)

We draw the shape parameter distribution

graph = paramDist.getMarginal(1).drawPDF()
graph.setTitle(r"Pareto fitting : PDF of $\alpha$")
graph.setXTitle(r"$\alpha$")
graph.setLegends(["PDF"])
view = viewer.View(graph)

We draw the location parameter distribution

graph = paramDist.getMarginal(2).drawPDF()
graph.setTitle(r"Pareto fitting : PDF of $\gamma$")
graph.setXTitle(r"$\gamma$")
graph.setLegends(["PDF"])
view = viewer.View(graph)

plt.show()