# Wilks and empirical quantile estimators¶

In this example we want to evaluate a particular quantile, with the empirical estimator or the Wilks one, from a sample of a random variable.

Let us suppose we want to estimate the quantile of order of the variable : , from the sample of size , with a confidence level equal to .

We note the sample where the values are sorted in ascending order. The empirical estimator, noted , and its confidence interval, is defined by the expressions:

The Wilks estimator, noted , and its confidence interval, is defined by the expressions:

Once the order has been chosen, the Wilks number is evaluated, thanks to the static method of the Wilks object.

In the example, we want to evaluate a quantile , with a confidence level of thanks to the th maximum of the ordered sample (associated to the order ).

Be careful: means that the Wilks estimator is the maximum of the sample: it corresponds to the first maximum of the sample.

[30]:

from __future__ import print_function
import openturns as ot
import math as m

[31]:

model = ot.SymbolicFunction(['x1', 'x2'], ['x1^2+x2'])
R = ot.CorrelationMatrix(2)
R[0,1] = -0.6
inputDist = ot.Normal([0.,0.], R)
inputDist.setDescription(['X1','X2'])
inputVector = ot.RandomVector(inputDist)

# Create the output random vector Y=model(X)
output = ot.CompositeRandomVector(model, inputVector)

[32]:

# Quantile level
alpha = 0.95

# Confidence level of the estimation
beta = 0.90

[33]:

# Get a sample of the variable
N = 10**4
sample = output.getSample(N)
ot.UserDefined(sample).drawCDF()

[33]:

[34]:

# Empirical Quantile Estimator
empiricalQuantile = sample.computeQuantile(alpha)

# Get the indices of the confidence interval bounds
aAlpha = ot.Normal(1).computeQuantile((1.0+beta)/2.0)[0]
min_i = int(N*alpha - aAlpha*m.sqrt(N*alpha*(1.0-alpha)))
max_i = int(N*alpha + aAlpha*m.sqrt(N*alpha*(1.0-alpha)))
#print(min_i, max_i)

# Get the sorted sample
sortedSample = sample.sort()

# Get the Confidence interval of the Empirical Quantile Estimator [infQuantile, supQuantile]
infQuantile = sortedSample[min_i-1]
supQuantile = sortedSample[max_i-1]
print(infQuantile, empiricalQuantile, supQuantile)

[4.28423] [4.38014] [4.51415]

[52]:

# Wilks number
i = N - (min_i+max_i)//2 # compute wilks with the same sample size
wilksNumber = ot.Wilks.ComputeSampleSize(alpha, beta, i)
print('wilksNumber =', wilksNumber)

wilksNumber = 10604

[51]:

# Wilks Quantile Estimator
algo = ot.Wilks(output)
wilksQuantile = algo.computeQuantileBound(alpha, beta, i)
print('wilks Quantile 0.95 =', wilksQuantile)

wilks Quantile 0.95 = [4.52594]