Estimate quantile confidence intervals from data

In this example, we introduce use two methods to estimate confidence intervals of the level quantile (). This example is adapted from [meeker2017] pages 76 to 81.

import openturns as ot
import openturns.experimental as otexp

The data represents the ordered measured amount of a chemical compound in parts per million (ppm) taken from randomly selected batches from the process (see 5.1 data p. 76).

data = [
    1.49,
    1.66,
    2.05,
    2.24,
    2.29,
    2.69,
    2.77,
    2.77,
    3.10,
    3.23,
    3.28,
    3.29,
    3.31,
    3.36,
    3.84,
    4.04,
    4.09,
    4.13,
    4.14,
    4.16,
    4.57,
    4.63,
    4.83,
    5.06,
    5.17,
    5.19,
    5.89,
    5.97,
    6.28,
    6.38,
    6.51,
    6.53,
    6.54,
    6.55,
    6.83,
    7.08,
    7.28,
    7.53,
    7.54,
    7.62,
    7.81,
    7.87,
    7.94,
    8.43,
    8.70,
    8.97,
    8.98,
    9.13,
    9.14,
    9.22,
    9.24,
    9.30,
    9.44,
    9.69,
    9.86,
    9.99,
    11.28,
    11.37,
    12.03,
    12.32,
    12.93,
    13.03,
    13.09,
    13.43,
    13.58,
    13.70,
    14.17,
    14.36,
    14.96,
    15.89,
    16.57,
    16.60,
    16.85,
    17.16,
    17.17,
    17.74,
    18.04,
    18.78,
    19.84,
    20.45,
    20.89,
    22.28,
    22.48,
    23.66,
    24.33,
    24.72,
    25.46,
    25.67,
    25.77,
    26.64,
    28.28,
    28.28,
    29.07,
    29.16,
    31.14,
    31.83,
    33.24,
    37.32,
    53.43,
    58.11,
]
sample = ot.Sample.BuildFromPoint(data)

In the example, we consider the quantile of level , with a confidence level of (see example 5.7 p. 85).

alpha = 0.1
beta = 0.95
algo = otexp.QuantileConfidence(alpha, beta)

Estimate bilateral rank. Indices are given in the integer interval whereas the book gives them in .

n = len(sample)
l, u = algo.computeBilateralRank(n)
print(f"l={l} u={u}")

l=3 u=15

Estimate the bilateral confidence interval of the 0.1 quantile from the order statistics:

..math:

\Prob{X_{(k_1)} \leq x_{\alpha} \leq X_{(k_2)}} \geq \beta

ci = algo.computeBilateralConfidenceInterval(sample)
print(f"ci={ci}")

ci=[2.24, 4.04]

In this example, we consider the quantile of level , with a confidence level of (see example 5.1 p. 81).

algo.setAlpha(0.9)

Estimate the one-sided rank for the upper side (in ).

k = algo.computeUnilateralRank(n, True)
print(f"k={k}")

k=84

Estimate the one-sided upper quantile confidence interval of the 0.9 quantile

ci = algo.computeUnilateralConfidenceInterval(sample, True)
print(f"ci={ci}")

ci=[24.33, (1.79769e+308) +inf[