FDistribution.inverseCumulativeProbability returns incorrect quantile (orders of magnitude error) for small degrees of freedom

[yin-mao]

Description

The method FDistribution.inverseCumulativeProbability(p) returns a value that does not satisfy the expected definition of a quantile.

Specifically, the returned value is not the smallest x such that CDF(x) >= p, and is in fact several orders of magnitude larger than the correct solution.

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Reproducible Example

double numeratorDf = 0.10006;
double denominatorDf = 1.51904;

FDistribution dist = new FDistribution(numeratorDf, denominatorDf);
double p = 0.16038;

double x = dist.inverseCumulativeProbability(p);

System.out.println("x = " + x);
System.out.println("CDF(x) = " + dist.cumulativeProbability(x));

double x2 = x - 1e-9;
System.out.println("CDF(x - 1e-9) = " + dist.cumulativeProbability(x2));

// Scan for smaller valid x
int steps = 1000000;
double max = 1e-8;

for (int i = 0; i < steps; i++) {
double testX = i * (max / steps);
double cdf = dist.cumulativeProbability(testX);
if (cdf > p) {
System.out.println("First x where CDF > p: " + testX + ", CDF=" + cdf);
break;
}
}

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Observed Behavior

• inverseCumulativeProbability(p) returns:
  x ≈ 1.35e-9

• CDF(x) ≈ 0.3069, which is significantly larger than p = 0.16038

• A much smaller value exists:
  x ≈ 3.13e-15
  CDF(x) ≈ 0.16038

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Expected Behavior

The method should return a value x such that:

• CDF(x) ≈ p
• or at least the smallest x such that CDF(x) >= p

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Severity of the issue

This is not a small numerical error.

The returned value is several orders of magnitude larger than the correct solution region:

• Returned x ≈ 1.35e-9
• However, values as small as 1e-14 already satisfy CDF(x) > p

This demonstrates that the correct solution lies far below the returned value.

Additionally:

• CDF(returned x) ≈ 0.3069
• target p = 0.16038

So the returned value corresponds to a probability almost twice as large as requested.

This indicates that the root-finding algorithm fails to locate the correct region,
rather than suffering from minor floating-point inaccuracies.

Analysis

This behavior suggests a failure in the root-finding process used in inverseCumulativeProbability.

Possible causes include:

• incorrect bracketing of the root
• premature convergence
• numerical instability when the numerator degrees of freedom is very small

The issue appears when the CDF changes very rapidly near zero.

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Additional Notes

This issue was originally observed in Apache Commons Math 3.6.1 and appears to persist in Hipparchus.