Home / Tolerance Interval / Nonparametric Method

Nonparametric Method¶

The nonparametric (distribution-free) method makes no assumption about the shape of the population distribution — it only assumes the distribution is continuous. The bounds are order statistics: actual values from the sorted sample, \(X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}\). Which order statistics to use is determined from the binomial distribution, and is independent of the population's shape.

Because the bounds can only move in whole steps from one sorted data value to the next, the interval usually cannot hit the requested confidence \(C\) exactly. Quantum XL therefore also reports the Attained Confidence — the confidence the returned interval actually achieves.

See Tolerance Interval for the notation used on this page.

One-sided bounds¶

Let \(Y \sim \mathrm{Bin}(n,\ 1-P)\). The bound uses the largest rank \(k \in \{1, \dots, n\}\) that still attains the requested confidence:

\[ \Pr\left(Y \geq k\right) \;=\; 1 - F_{\mathrm{Bin}(n,\,1-P)}(k-1) \;\geq\; C \]

Interval type	Bound
One-sided lower bound	\(L = X_{(k)}\)
One-sided upper bound	\(U = X_{(n-k+1)}\)

The attained confidence of the returned bound is:

\[ C_{\text{attained}} \;=\; 1 - F_{\mathrm{Bin}(n,\,1-P)}(k-1) \]

The fraction of the population actually captured by the interval is itself random; it follows a Beta distribution,

\[ \text{coverage} \;\sim\; \mathrm{Beta}\left(n-k+1,\ k\right) \]

so the coverage that can be guaranteed at confidence \(C\) is the \(\alpha\)-quantile of that Beta distribution:

\[ P_{\text{guaranteed}} \;=\; F^{-1}_{\mathrm{Beta}(n-k+1,\ k)}(\alpha) \]

When the sample is too small

If no rank \(k\) satisfies the confidence requirement, the widest possible interval is returned (\(k = 1\): the bound is the sample minimum or maximum), and the Attained Confidence shows the confidence that interval actually provides — which will be less than requested. The remedy is a larger sample.

Two-sided bounds¶

Let \(V \sim \mathrm{Bin}(n,\ P)\) — note this uses \(P\), where the one-sided rule uses \(1-P\). The method finds the smallest span \(k \in \{1, \dots, n+1\}\) satisfying:

\[ F_{\mathrm{Bin}(n,\,P)}(k-1) \;\geq\; C \]

The interval returned is \(\left(X_{(r)},\ X_{(s)}\right)\), placed so that the \(n - 1 - k\) excluded order statistics are split as evenly as possible between the two tails, with the extra one (when the count is odd) removed from the upper tail:

\[ r = \left\lfloor \frac{n-1-k}{2} \right\rfloor + 1 \qquad\qquad s = n - \left\lceil \frac{n-1-k}{2} \right\rceil \]

so that \(s - r = k\) (whenever \(k \leq n-1\); smaller samples fall back to the full range as described below). This returns the tightest interval attaining the requested confidence, rather than a symmetric one.

The attained confidence is computed from the width of the interval actually returned:

\[ C_{\text{attained}} \;=\; F_{\mathrm{Bin}(n,\,P)}\left((s - r) - 1\right) \]

The coverage of \(\left(X_{(r)}, X_{(s)}\right)\) follows \(\mathrm{Beta}\left(s-r,\ n-s+r+1\right)\), so the guaranteed coverage at confidence \(C\) is:

\[ P_{\text{guaranteed}} \;=\; F^{-1}_{\mathrm{Beta}(s-r,\ n-s+r+1)}(\alpha) \]

When the sample is too small

If even the full sample range cannot attain the requested confidence, the full range \(\left(X_{(1)},\ X_{(n)}\right)\) is returned and the Attained Confidence reports what that range truly achieves. For example, with \(n = 25\) at 95% coverage the full sample range attains only about 35.8% confidence — far short of a 95% request, signaling that a much larger sample is needed.

References¶

Wilks, S. S. (1941). "Determination of Sample Sizes for Setting Tolerance Limits." The Annals of Mathematical Statistics, 12(1), 91–96.
Krishnamoorthy, K. and Mathew, T. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. Wiley. https://onlinelibrary.wiley.com/doi/book/10.1002/9780470473900
NIST/SEMATECH e-Handbook of Statistical Methods, "Tolerance intervals based on the largest and smallest observations." https://www.itl.nist.gov/div898/handbook/prc/section2/prc264.htm
Hahn, G. J., Meeker, W. Q., and Escobar, L. A. (2017). Statistical Intervals: A Guide for Practitioners and Researchers, 2nd ed. Wiley.